Category: Music

Philip Ball’s The Music Instinct explores the multifaceted nature of music, examining its scientific underpinnings and its profound impact on human experience. The book investigates how our brains process sound, perceive melody and harmony, and respond emotionally to music across diverse cultures and historical periods. Ball considers the universality of music, the evolution of musical scales and structures, and the ongoing debate about music’s meaning and purpose. Through explorations of acoustics, psychology, neuroscience, and cultural studies, the book seeks to understand why music is so integral to humanity.

The Science and Art of Music

Music is not simply a kind of mathematics but rather a remarkable blend of art and science, logic and emotion, physics and psychology. The study of how music works involves both scientific investigation and an appreciation for its artistic qualities.

Here are some aspects of the relationship between music and science discussed in the sources:

• The Physics of Sound and Music: Musical notes can be understood in terms of their acoustic frequencies. The relationship between pitch and frequency seems simple, with higher frequencies generally corresponding to higher pitches. However, the selection of discrete notes used in music across cultures is not solely determined by nature. The interaction of nature and culture shapes the diverse palettes of notes found in different musical traditions. Helmholtz combined his knowledge of the ear’s workings with the mathematics of vibration to understand how we hear tones, producing a significant early scientific exposition on music cognition in his 1863 book “On the Sensations of Tone as a Physiological Basis for the Theory of Music”. He also explored the ancient question of consonance, noting the historical preference for intervals with simple frequency ratios.
• The Neuroscience of Music: When we listen to music, our brains perform complex feats of filtering, ordering, and prediction automatically and unconsciously. Neuroscience seeks to identify which brain regions are used for different musical tasks, providing insights into how the brain classifies and interprets music. For example, pitch perception appears to be mostly localized in the right hemisphere. Pitch intervals and melody are processed in areas like Heschl’s gyrus and the planum temporale. The brain also engages in sophisticated streaming and binding of sound to distinguish different musical elements and create a coherent perception. Musical training can alter the brain, leading to more analytical processing in musicians and changes in the corpus callosum and auditory cortex. However, the precise link between the rich experience of music and brain activity remains a significant challenge for neuroscience. The “Mozart Effect,” which suggested a positive effect of listening to Mozart on general intellect, has been qualified by findings showing that children might respond best to their favorite kind of music, leading to the idea of a “Blur Effect” as well.
• Music Cognition and Psychology: The science of music cognition is increasingly exploring the universal aspects of music by breaking it down into basic structural elements like pitch, tone, and rhythm. However, emotional, social, and cultural factors also significantly influence music perception. For instance, the perception of melodic pitch steps shows probability distributions that are fairly universal across Western and many other musical traditions. Music psychologists study how we process melodies, which involves learning expectations about pitch steps. They also investigate how we decode sound, including the streaming and binding of different musical voices. The field of music and emotion has become central to music cognition, moving away from purely atomistic dissections of music to examine responses to actual music. Theories like Meyer’s and Narmour’s attempt to explain emotional responses in terms of expectation, tension, and release.
• Music as Organized Sound: Avant-garde composer Edgar Varèse defined his music as “organized sound,” distinguishing his experimental sonic explorations from conventional music. This definition highlights the role of organization in what we perceive as music, although the listener also actively participates in this organization.
• Music and Language: Some researchers propose an evolutionary link between music and language, suggesting a common ancestral “musilanguage”. This theory posits that musilanguage might have contained features like lexical tone, combinatorial phrases, and expressive phrasing. Even today, non-vocal music seems to share speech-like patterns, such as pitch contours (prosody). Studies suggest that the rhythmic and melodic patterns of language may have shaped the music of composers from the same linguistic background. While there are neurological dissociations between language and music processing (amusia and aphasia), some theories suggest that syntactic processing in both domains might share neural resources.
• The Meaning of Music: The question of whether music has inherent meaning is debated. Some believe music is purely formal and does not “say” anything. Others argue that music can convey and elicit emotions , although the precise relationship is complex. Musical affect might arise from underlying principles that can be analyzed rationally. Composers and musicians intuitively manipulate human characteristics to create musical effects.

In conclusion, the study of music is deeply intertwined with various scientific disciplines. Acoustics provides the foundation for understanding musical sound, neuroscience explores the brain’s engagement with music, and music cognition investigates how we perceive and process musical information. While music is undoubtedly an art form, scientific inquiry continues to shed light on the intricate mechanisms underlying our musical experiences.

The Fundamentals of Musical Scales

Musical scales are fundamental to most musical traditions, serving as the set of pitches from which melodies and harmonies are constructed. They represent a selection of discrete pitches from the continuous spectrum of audible frequencies.

Here are key aspects of musical scales discussed in the sources:

• Definition and Basic Concepts: A musical scale is a set of discrete pitches within the octave that a tradition uses to build its music. Unlike the smoothly varying pitch of a siren, a scale is like a staircase of frequencies. Most musical systems are based on the division of pitch space into octaves, a seemingly fundamental aspect of human pitch perception. Within this octave, different cultures choose a subset of potential notes to form their scales. This selection is not solely determined by nature but arises from an interaction of nature and culture.
• Western Scales and Their Development:
• Pythagorean Scales: One of the earliest theoretical frameworks for Western scales is attributed to Pythagoras, though the knowledge was likely older. Pythagorean scales are derived mathematically from the harmonious interval of a perfect fifth, based on the simple frequency ratio of 3:2. By repeatedly stepping up by a perfect fifth from a tonic and folding the resulting notes back into an octave, the major scale can be generated. This scale has an uneven pattern of whole tones and semitones. The Pythagorean system aimed to place music on a solid mathematical footing, suggesting music was a branch of mathematics embedded in nature. However, the cycle of fifths in Pythagorean tuning does not perfectly close, leading to an infinite number of potential notes, which can be problematic if music modulates between many keys.
• Diatonic Scales: Western music inherited diatonic scales from Greek tradition, characterized by seven tones between each octave. The major and minor scales became the basis of most Western music from the late Renaissance to the early twentieth century. Each note of a diatonic scale has a specific order, with the tonic being the starting and central note.
• Chromatic Scale: In addition to the seven diatonic notes, there are five other notes within an octave (like the black notes on a piano within a C major scale). The scale that includes all twelve semitones is called the chromatic scale, and music that uses notes outside the diatonic scale is considered chromatic.
• Modes: Before diatonic scales became dominant, Western music utilized modes, which can be thought of as scales using the same notes but starting in different places, each with a different sequence of step heights. Medieval modes had anchoring notes called the final and often a reciting tone called the tenor. The Ionian and Aeolian modes introduced later are essentially the major and a modern minor scale, respectively.
• Accidentals, Transposition, and Modulation: Sharps and flats (accidentals) were added to the modal system to preserve pitch steps when transposing melodies to different starting notes (keys). This also enabled modulation, the process of moving smoothly from one key to another, which became central to Western classical music. Transposition and modulation necessitate the introduction of new scales and notes.
• Non-Western Scales: Musical scales vary significantly across cultures.
• Javanese Gamelan: Gamelan music uses non-diatonic scales like pélog and sléndro, which have different interval structures compared to Western scales. The sléndro scale is a rare exception with equal pitch steps.
• Indian Music: The Indian subcontinent has a rich musical tradition with non-diatonic scales that include perfect fifths. North Indian music employs thirty-two different scales (thats) of seven notes per octave, drawn from a palette of twenty-two possible pitches. These scales (ragas) have tunings that can differ significantly from Western scales.
• Arab-Persian Music: This tradition also uses pitch divisions smaller than a semitone, with estimates ranging from fifteen to twenty-four potential notes within an octave. However, some of these might function as embellishments rather than basic scale tones.
• The existence of diverse scale systems demonstrates that the selection of notes is not solely dictated by acoustics or mathematics.
• Number and Distribution of Notes: Most musical systems use melodies constructed from four to twelve distinct notes within an octave. This limitation likely stems from cognitive constraints: too few notes limit melodic complexity, while too many make it difficult for the brain to track and organize the distinctions. The unequal pitch steps found in most scales (with sléndro being an exception) are thought to provide reference points for listeners to perceive the tonal center or key of a piece. Scales with five (pentatonic) or seven (diatonic) notes are particularly widespread, possibly because they allow for simpler interconversion between scales with different tonic notes during modulation.
• Cognitive Processing of Scales: Our brains possess a mental facility for categorizing pitches, allowing us to perceive melodies as coherent even on slightly mistuned instruments. We learn to assign pitches to a small set of categories based on interval sizes, forming mental “boxes”. To comprehend music, we need to discern a hierarchy of status between the notes of a scale, which depends on our ability to intuit the probabilities of different notes occurring.
• Alternative Scales: Some twentieth-century composers explored non-standard scales to create unique sounds, such as Debussy’s whole-tone scale, Messiaen’s octatonic scales, and Scriabin’s “mystic” scales.

In essence, musical scales are carefully chosen sets of pitches that provide the foundational elements for musical expression. Their structure and the specific notes they contain vary greatly across historical periods and cultural traditions, reflecting both acoustic principles and human cognitive and cultural preferences.

The Perception of Melody in Music

Melody perception is a complex cognitive process through which we hear a sequence of musical notes as a unified and meaningful whole, often referred to as a “tune”. However, the sources clarify that “melody” is a more versatile term than “tune,” as not all music has a readily identifiable tune like “Singin’ in the Rain”. For instance, Bach’s fugues feature short, overlapping melodic fragments rather than a continuous, extended tune.

Pitch and Pitch Relationships:

The foundation of melody perception lies in our ability to process pitch, which is processed by pitch-selective neurons in the primary auditory cortex. These neurons have a unique one-to-one mapping for pitch, unlike our perception of other senses. While pitch increases with acoustic frequency, our auditory system creates a cyclical perception where pitches an octave apart sound similar, a phenomenon called octave equivalence. This is a unique perceptual experience in music. However, the sources emphasize that simply having the correct pitch classes in different octaves does not guarantee melody recognition. When listeners were presented with familiar tunes where the octave of each note was randomized, they couldn’t even recognize the melody. This suggests that register or ‘height’ (which octave a note is in) is a crucial dimension of melody perception, alongside chroma (the pitch class).

Our brains possess a remarkable mental facility for categorizing pitches, allowing us to perceive melodies as coherent even if played on slightly mistuned instruments. We learn to assign pitches to mental “boxes” representing intervals like “major second” or “major third,” classifying any pitch close enough to that ideal interval size.

Melodic Contour:

The contour of a melody, or how it rises and falls in pitch, is a vital cue for memory and recognition. Even infants as young as five months respond to changes in melodic contour. Interestingly, both children and untrained adults often think melodies with the same contour but slightly altered intervals are identical, highlighting the primacy of contour in initial recognition. Familiar tunes remain recognizable even when the melodic contour is “compressed”. Composers can create repeating contour patterns to help bind a melody together, even if they are not exact repeats, adapting the contour to fit the specific pitch staircase of a scale. Diana Deutsch refers to these building blocks as “pitch alphabets,” which can be compiled from scales and arpeggios.

Tonal Hierarchy and Expectation:

Our perception of melody is deeply influenced by the tonal hierarchy, which is our subjective evaluation of how well different notes “fit” within a musical context or key. Even listeners without extensive musical training have a mental image of this hierarchy and constantly refer to it to form anticipations and judgments about a tune. This is supported by experiments where listeners consistently rated the “rightness” of notes within a set tonal context. The tonal hierarchy helps us organize and understand music, making it sound like music rather than a random sequence of notes. Music that ignores these hierarchies can be harder to process and may sound bewildering.

Gestalt Principles and Binding:

Underlying melody perception is the brain’s constant search for coherence in the auditory stimuli it receives. We mentally and unconsciously “bind” a string of notes into a unified acoustic entity, a tune. This process aligns with principles of gestalt psychology, where the brain seeks to perceive patterns. For example, large intervals can create a discontinuity, challenging the brain’s ability to perceive the melody as a single “gestalt”. Conversely, repetition of notes or contours can strengthen the perception of a unified melody. The auditory picket-fence effect demonstrates our ability to perceive a continuous tone even when interrupted by noise, highlighting the brain’s tendency to “fill in” gaps to maintain a coherent auditory stream. In sequences with large pitch jumps, listeners may even separate the notes into two distinct melodic streams.

Phrasing and Rhythm:

Phrasing, the way a melody is divided into meaningful segments, is crucial for perception. Click migration experiments show that listeners tend to perceive breaks between notes that delineate musical phrases. Phrasing is closely linked to rhythmic patterns, which provide a natural breathing rhythm to music and help us segment it into manageable chunks. The duration and accentuation of notes contribute to our perception of rhythmic groupings.

Memory and Context:

When we listen to a melody, we hear each note in the context of what we have already heard, including previous notes, the melodic contour, repeated phrases, the established key, and even our memories of other music. This constant referencing and updating of information shapes our perception of the unfolding melody.

Brain Processing:

The brain processes melody through various regions, including the lateral part of Heschl’s gyrus and the planum temporale in the temporal lobe, which are involved in pitch perception and sophisticated auditory attributes. The anterior superior temporal gyrus also handles streams of sound like melodies. Research suggests that the right hemisphere discerns the global pattern of pitch contour, while the left hemisphere processes the detailed aspects of pitch steps.

Atonal Music:

Music that rejects tonal hierarchies can be harder to process because it goes against our learned expectations about note probabilities. While some theories attempt to analyze atonal music through concepts like pitch-class sets, these approaches often don’t explain how such music is actually perceived.

In summary, melody perception is a dynamic process involving the processing of pitch and its relationships, the recognition of melodic contour, the influence of tonal hierarchies and learned expectations, the brain’s ability to bind sequences of notes into coherent units, the segmentation of melodies into phrases guided by rhythmic patterns, and the crucial role of memory and context. These elements work together to allow us to experience a series of discrete musical notes as a meaningful and unified melodic line.

Understanding Harmony and Dissonance in Music

Harmony is about fitting notes together. Conventionally, combinations that fit well are called consonant, and those that fit less well are dissonant. In a reductive formulation, consonance is considered good and pleasing, while dissonance is bad and unsettling. However, these concepts are often misunderstood and misrepresented.

Historical Perspectives on Consonance and Dissonance:

• In tenth-century Europe, a perfect fifth was generally not deemed consonant; only the octave was.
• When harmonizing in fifths became common, fourths were considered equally consonant, which is different from how they are perceived today.
• The major third (C-E), part of the “harmonious” major triad, was rarely used even by the early fourteenth century and was not fully accepted as consonant until the High Renaissance.
• The tritone interval, supposedly dissonant, becomes pleasing and harmonious when part of a dominant seventh chord (e.g., adding a D bass to C-FG).
• The whole polarizing terminology of consonance and dissonance is a rather unfortunate legacy of music theory.

Sensory (or Tonal) Dissonance:

• There is a genuinely physiological aspect of dissonance, distinguished from musical convention, called sensory or tonal dissonance.
• This refers to the rough, rattle-like auditory sensation produced by two tones closely spaced in pitch.
• It is caused by the beating of acoustic waves when two pure tones with slightly different frequencies are played simultaneously. If the beat rate exceeds about 20 Hz, it is heard as roughness.
• The width of the dissonant region depends on the absolute frequencies of the two notes. An interval consonant in a high register may be dissonant in a lower register. Therefore, there is no such thing as a tonally dissonant interval independent of register.
• In the mid-range of the piano, minor thirds generally lie beyond the band of roughness, while even a semitone does not create roughness for high notes. However, in the bass, even a perfect fifth can become dissonant in sensory terms, explaining the “gruffness” of low chords.

Consonance, Dissonance, and Overtones:

• Tones played by musical instruments are complex, containing several harmonics.
• Two simultaneously sounded notes offer many possibilities for overtones to clash and produce sensory dissonance if close enough in frequency.
• Hermann von Helmholtz calculated the total roughness for all overtone combinations, generating a curve of sensory dissonance with dips at various intervals of the chromatic scale. The octave and fifth have particularly deep “consonant” valleys.
• However, the depths of several “consonant” valleys don’t differ much. The modern dissonance curve shows that most intervals between the major second and major seventh lie within a narrow band of dissonance levels, except for the perfect fifth. Even the tritone appears less dissonant than major or minor thirds according to some measurements.
• The greatest sensory dissonance is found close to the unison, particularly the minor second, predicted to sound fairly nasty. However, such intervals can be used for interesting timbral effects.
• The brain is insistent on “binding” overtones into a single perceived pitch. If a harmonic is detuned, the brain tries to find a new fundamental frequency that fits, and only when the detuning is too large does it register the “bad” harmonic as a distinct tone. Percussive instruments often produce inharmonic overtones, resulting in an ambiguous pitch.

Cultural Influences and Learning:

• Whether we experience note combinations as smooth or grating is not solely a matter of convention, but there is a physiological aspect. However, likes and dislikes for certain combinations probably involve very little that is innate and are mostly products of learning.
• What is disliked is probably not the dissonances themselves but how they are combined into music.
• Acculturation can overcome sensory dissonance, as seen in the ganga songs of Bosnia and Herzegovina, where chords of major and minor seconds are considered harmonious.
• People tend to like best what is most familiar. Western listeners, being accustomed to tonal music, will be acclimatized to octaves, fifths, thirds, etc., and hear less common intervals as more odd.
• Studies suggest that cultural tradition exerts a stronger influence than inherent qualities in determining the emotional connotations of music, implying that perceptions of consonance and dissonance can also be culturally influenced.

Harmony in Musical Composition:

• In polyphonic music, harmony fills out the musical landscape. If melody is the path, harmony is the terrain.
• Harmonization is the process of fitting melodic lines to chords. This is often where music comes alive.
• Harmonization is generally more sophisticated in classical music, tending to use voice-leading, where accompanying voices have their own impetus and logic, rather than being monolithic chords.
• Harmonic progressions are sequences of chords. In Western classical music until the mid-nineteenth century, these tended to be formulaic and conservative, involving transitions to closely related chords. Pop and rock music have inherited much of this tradition.
• Modulation is the alteration of the key itself within a harmonic progression.
• Music theorists and psychologists have attempted to create a cartography of chords and keys, trying to map out relationships in harmonic space. Carol Krumhansl’s research suggests that the perceived relatedness of keys aligns with the cycle of fifths.

Harmony, Dissonance, and Musical Style/Emotion:

• Many classical-music traditionalists deny enjoying dissonance, associating it with jarring modern music. However, even composers like Chopin use dissonance extensively.
• The use of dissonance by modernist composers was seen by some as an affront to music itself. However, champions of atonalism argued that aversion to dissonance is culturally learned.
• “Dissonant” intervals like major sixths, sevenths, and ninths can create luxuriant sounds in the hands of composers like Debussy and Ravel.
• Composers may confuse our expectations regarding harmony to introduce tension and emotion.
• Expectations about harmony are crucial for our emotional response to music. Composers manipulate these expectations through devices like cadences, anticipation notes, and suspensions.
• Ambiguity in harmony and tonality can also create a powerful effect, with pleasure arising from the resolution of confusion.
• Different musical genres establish their own harmonic schemas, which they can then use to manipulate tension.

Dissonance in Polyphony:

• In early medieval polyphony, it was considered better to compromise the melody than to incur dissonance. However, composers increasingly prioritized maintaining good melodies in each voice, even if it led to occasional dissonances.
• This led to rules governing permissible dissonances in counterpoint. In Palestrina’s counterpoint, dissonances often occur on “passing tones” leading towards a consonance, and strong consonances are achieved at the beginnings and ends of phrases. The main objective is to maintain horizontal coherence of each voice while enforcing vertical integration through judicious use of consonance and controlled dissonance.
• Streaming of sound can offer a barrier to the perception of dissonance in polyphony. If voices are sufficiently distinct, potentially dissonant intervals may not be registered as jarring. Bach’s fugues, for example, contain striking dissonances that can go unnoticed due to the independence of the voices.
• Harmony can support the mental juggling act of listening to multiple melodies simultaneously, especially when the melodies are in the same key. Harmonic concordance seems to assist cognition.
• The composer doesn’t always want polyphonic voices to be clearly defined. In hymn singing, the focus is on creating a sense of unity through harmonies, resulting in a more homophonic texture where voices combine to carry a single melody, as opposed to the elaborate interweaving of voices in Bach’s contrapuntal music.

In conclusion, harmony and dissonance are fundamental aspects of music that involve both acoustic/physiological phenomena and cultural learning and conventions. Their perception and use have evolved throughout music history and continue to be manipulated by composers to create diverse musical experiences and emotional effects.

Understanding Musical Rhythm and Meter

Rhythm and meter are fundamental aspects of music. Rhythm is defined as the actual pattern of note events and their duration, and it tends to be much less regular than meter or tactus. It’s the “felt” quality of the regular subdivision of time on paper. Rhythm can be catchy and move us physically.

Meter, on the other hand, is the regular division of time into instants separated by equal intervals, providing what is colloquially called the ‘beat’. It’s the underlying pulse. The numbers at the start of a stave, the time signature, indicate how many notes of a particular duration should appear in each bar, essentially telling us whether to count the rhythm in groups of two, three, four, or more beats. To create a beat from a regular pulse, some pulses need to be emphasized over others, often by making them louder. Our minds tend to impose such groupings even on identical pulses. The grouping of pulses defines the music’s meter. Western music mostly uses simple meters with recurring groups of two, three, or four pulses, or sometimes six.

The tactus is related to but different from meter; it’s the beat we would clap out while listening to music and may be culture-specific. We tend to tap out a slower pulse to familiar music.

The source emphasizes that not all music possesses rhythm in a discernible way, citing compositions by Ligeti and Xenakis as continuous skeins of sound without a clear pulse, and Stockhausen’s Kontakte as being made of disconnected aural events. Gregorian chant is an example of music that can have regularly spaced notes but lack a true meter. Music for the Chinese fretless zither (qin) has rhythm in terms of note lengths, but these are not arranged against a steady underlying pulse.

However, a quasi-regular pulse pervades most of the world’s music. A rhythm is typically created by elaborating the periodic beats. Subdivisions and stresses superimposed on a steady pulse give us a sense of true rhythm, helping us locate ourselves in time much like the tonal hierarchy helps us in pitch space. This orderly and hierarchical structuring of time is found in the rhythmic systems of many musical traditions.

The source notes that the metre is often portrayed as a regular temporal grid on which the rhythm is arrayed, but the real relationship is more complex. Musicians tend subconsciously to distort the metrical grid to bring out accents and groupings implied by the rhythm. This stretching and shrinking of metrical time helps us perceive both meter and rhythm.

Western European music has traditionally chopped up time by binary branching, with melodies broken into phrases grouped in twos or fours, divided into bars, and beats subdivided into halves and quarters. This binary division is reflected in note durations like semibreve, minim, and crochet. However, some Balkan music uses prime numbers of beats in a bar, suggesting that binary division is not universal. Eastern European song may have constantly changing meter due to the rhythmic structure of its poetry.

Creating a true sense of rhythm and avoiding monotony involves not just stressing some beats but an asymmetry of events, similar to the skipping rather than plodding nature of spoken language. The source discusses rhythmic figures like the iamb, trochee, dactyl, and anapest, which are “atoms” from which we build a sense of rhythm and interpret musical events. Repetition of these units is crucial for that coherence to be felt. Our assignment of rhythmic patterns draws on various information beyond note duration, including melody, phrasing, dynamics, harmony, and timbre.

Composers generally want us to perceive the intended rhythm and use various factors to reinforce it. However, they may also seek to confuse our expectations regarding rhythm to introduce tension and emotion, as it is easy to hear when a beat is disrupted. Examples of this include:

• Syncopation, which involves shifting emphasis off the beat.
• Beethoven’s Fifth Symphony starting with a rest on the downbeat.
• Rhythmic ambiguity created by conflicting rhythmic groupings and meter, as in Beethoven’s Piano Sonata No. 13 and Bernstein’s “America”.
• Rhythmic elisions and deceptive rhythmic figures in popular music.
• Unambiguous disruption of meter, creating a jolt, as in Stravinsky’s The Rite of Spring.
• The use of anticipation tones in classical cadences to modulate the expectation of the impending cadence.

The source also points out that our sense of metrical regularity isn’t always strong, especially without musical training, and folk music traditions can exhibit irregular meters. In early polyphonic music, complex crossed rhythms were common, even without explicit metrical notation. Some musical traditions, like African, Indian, and Indonesian music, use cross-rhythms and polyrhythms. The minimalist compositions of Steve Reich utilize phasing, where repetitive riffs played at slightly different tempos create shifting rhythmic patterns.

Ultimately, rhythm provides a way to interpret and make sense of the stream of musical events by apportioning them into coherent temporal units. Composers manipulate rhythm and meter in various ways to create structure, expectation, and emotional impact in their music.

By Amjad Izhar
Contact: amjad.izhar@gmail.com
https://amjadizhar.blog

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Philip Ball’s The Music Instinct explores the multifaceted nature of music, examining its scientific underpinnings and its profound impact on human experience. The book investigates how our brains process sound, perceive melody and harmony, and respond emotionally to music across diverse cultures and historical periods. Ball considers the universality of music, the evolution of musical scales and structures, and the ongoing debate about music’s meaning and purpose. Through explorations of acoustics, psychology, neuroscience, and cultural studies, the book seeks to understand why music is so integral to humanity.

The Science and Art of Music

Music is not simply a kind of mathematics but rather a remarkable blend of art and science, logic and emotion, physics and psychology. The study of how music works involves both scientific investigation and an appreciation for its artistic qualities.

Here are some aspects of the relationship between music and science discussed in the sources:

• The Physics of Sound and Music: Musical notes can be understood in terms of their acoustic frequencies. The relationship between pitch and frequency seems simple, with higher frequencies generally corresponding to higher pitches. However, the selection of discrete notes used in music across cultures is not solely determined by nature. The interaction of nature and culture shapes the diverse palettes of notes found in different musical traditions. Helmholtz combined his knowledge of the ear’s workings with the mathematics of vibration to understand how we hear tones, producing a significant early scientific exposition on music cognition in his 1863 book “On the Sensations of Tone as a Physiological Basis for the Theory of Music”. He also explored the ancient question of consonance, noting the historical preference for intervals with simple frequency ratios.
• The Neuroscience of Music: When we listen to music, our brains perform complex feats of filtering, ordering, and prediction automatically and unconsciously. Neuroscience seeks to identify which brain regions are used for different musical tasks, providing insights into how the brain classifies and interprets music. For example, pitch perception appears to be mostly localized in the right hemisphere. Pitch intervals and melody are processed in areas like Heschl’s gyrus and the planum temporale. The brain also engages in sophisticated streaming and binding of sound to distinguish different musical elements and create a coherent perception. Musical training can alter the brain, leading to more analytical processing in musicians and changes in the corpus callosum and auditory cortex. However, the precise link between the rich experience of music and brain activity remains a significant challenge for neuroscience. The “Mozart Effect,” which suggested a positive effect of listening to Mozart on general intellect, has been qualified by findings showing that children might respond best to their favorite kind of music, leading to the idea of a “Blur Effect” as well.
• Music Cognition and Psychology: The science of music cognition is increasingly exploring the universal aspects of music by breaking it down into basic structural elements like pitch, tone, and rhythm. However, emotional, social, and cultural factors also significantly influence music perception. For instance, the perception of melodic pitch steps shows probability distributions that are fairly universal across Western and many other musical traditions. Music psychologists study how we process melodies, which involves learning expectations about pitch steps. They also investigate how we decode sound, including the streaming and binding of different musical voices. The field of music and emotion has become central to music cognition, moving away from purely atomistic dissections of music to examine responses to actual music. Theories like Meyer’s and Narmour’s attempt to explain emotional responses in terms of expectation, tension, and release.
• Music as Organized Sound: Avant-garde composer Edgar Varèse defined his music as “organized sound,” distinguishing his experimental sonic explorations from conventional music. This definition highlights the role of organization in what we perceive as music, although the listener also actively participates in this organization.
• Music and Language: Some researchers propose an evolutionary link between music and language, suggesting a common ancestral “musilanguage”. This theory posits that musilanguage might have contained features like lexical tone, combinatorial phrases, and expressive phrasing. Even today, non-vocal music seems to share speech-like patterns, such as pitch contours (prosody). Studies suggest that the rhythmic and melodic patterns of language may have shaped the music of composers from the same linguistic background. While there are neurological dissociations between language and music processing (amusia and aphasia), some theories suggest that syntactic processing in both domains might share neural resources.
• The Meaning of Music: The question of whether music has inherent meaning is debated. Some believe music is purely formal and does not “say” anything. Others argue that music can convey and elicit emotions , although the precise relationship is complex. Musical affect might arise from underlying principles that can be analyzed rationally. Composers and musicians intuitively manipulate human characteristics to create musical effects.

In conclusion, the study of music is deeply intertwined with various scientific disciplines. Acoustics provides the foundation for understanding musical sound, neuroscience explores the brain’s engagement with music, and music cognition investigates how we perceive and process musical information. While music is undoubtedly an art form, scientific inquiry continues to shed light on the intricate mechanisms underlying our musical experiences.

The Fundamentals of Musical Scales

Musical scales are fundamental to most musical traditions, serving as the set of pitches from which melodies and harmonies are constructed. They represent a selection of discrete pitches from the continuous spectrum of audible frequencies.

Here are key aspects of musical scales discussed in the sources:

• Definition and Basic Concepts: A musical scale is a set of discrete pitches within the octave that a tradition uses to build its music. Unlike the smoothly varying pitch of a siren, a scale is like a staircase of frequencies. Most musical systems are based on the division of pitch space into octaves, a seemingly fundamental aspect of human pitch perception. Within this octave, different cultures choose a subset of potential notes to form their scales. This selection is not solely determined by nature but arises from an interaction of nature and culture.
• Western Scales and Their Development:
• Pythagorean Scales: One of the earliest theoretical frameworks for Western scales is attributed to Pythagoras, though the knowledge was likely older. Pythagorean scales are derived mathematically from the harmonious interval of a perfect fifth, based on the simple frequency ratio of 3:2. By repeatedly stepping up by a perfect fifth from a tonic and folding the resulting notes back into an octave, the major scale can be generated. This scale has an uneven pattern of whole tones and semitones. The Pythagorean system aimed to place music on a solid mathematical footing, suggesting music was a branch of mathematics embedded in nature. However, the cycle of fifths in Pythagorean tuning does not perfectly close, leading to an infinite number of potential notes, which can be problematic if music modulates between many keys.
• Diatonic Scales: Western music inherited diatonic scales from Greek tradition, characterized by seven tones between each octave. The major and minor scales became the basis of most Western music from the late Renaissance to the early twentieth century. Each note of a diatonic scale has a specific order, with the tonic being the starting and central note.
• Chromatic Scale: In addition to the seven diatonic notes, there are five other notes within an octave (like the black notes on a piano within a C major scale). The scale that includes all twelve semitones is called the chromatic scale, and music that uses notes outside the diatonic scale is considered chromatic.
• Modes: Before diatonic scales became dominant, Western music utilized modes, which can be thought of as scales using the same notes but starting in different places, each with a different sequence of step heights. Medieval modes had anchoring notes called the final and often a reciting tone called the tenor. The Ionian and Aeolian modes introduced later are essentially the major and a modern minor scale, respectively.
• Accidentals, Transposition, and Modulation: Sharps and flats (accidentals) were added to the modal system to preserve pitch steps when transposing melodies to different starting notes (keys). This also enabled modulation, the process of moving smoothly from one key to another, which became central to Western classical music. Transposition and modulation necessitate the introduction of new scales and notes.
• Non-Western Scales: Musical scales vary significantly across cultures.
• Javanese Gamelan: Gamelan music uses non-diatonic scales like pélog and sléndro, which have different interval structures compared to Western scales. The sléndro scale is a rare exception with equal pitch steps.
• Indian Music: The Indian subcontinent has a rich musical tradition with non-diatonic scales that include perfect fifths. North Indian music employs thirty-two different scales (thats) of seven notes per octave, drawn from a palette of twenty-two possible pitches. These scales (ragas) have tunings that can differ significantly from Western scales.
• Arab-Persian Music: This tradition also uses pitch divisions smaller than a semitone, with estimates ranging from fifteen to twenty-four potential notes within an octave. However, some of these might function as embellishments rather than basic scale tones.
• The existence of diverse scale systems demonstrates that the selection of notes is not solely dictated by acoustics or mathematics.
• Number and Distribution of Notes: Most musical systems use melodies constructed from four to twelve distinct notes within an octave. This limitation likely stems from cognitive constraints: too few notes limit melodic complexity, while too many make it difficult for the brain to track and organize the distinctions. The unequal pitch steps found in most scales (with sléndro being an exception) are thought to provide reference points for listeners to perceive the tonal center or key of a piece. Scales with five (pentatonic) or seven (diatonic) notes are particularly widespread, possibly because they allow for simpler interconversion between scales with different tonic notes during modulation.
• Cognitive Processing of Scales: Our brains possess a mental facility for categorizing pitches, allowing us to perceive melodies as coherent even on slightly mistuned instruments. We learn to assign pitches to a small set of categories based on interval sizes, forming mental “boxes”. To comprehend music, we need to discern a hierarchy of status between the notes of a scale, which depends on our ability to intuit the probabilities of different notes occurring.
• Alternative Scales: Some twentieth-century composers explored non-standard scales to create unique sounds, such as Debussy’s whole-tone scale, Messiaen’s octatonic scales, and Scriabin’s “mystic” scales.

In essence, musical scales are carefully chosen sets of pitches that provide the foundational elements for musical expression. Their structure and the specific notes they contain vary greatly across historical periods and cultural traditions, reflecting both acoustic principles and human cognitive and cultural preferences.

The Perception of Melody in Music

Melody perception is a complex cognitive process through which we hear a sequence of musical notes as a unified and meaningful whole, often referred to as a “tune”. However, the sources clarify that “melody” is a more versatile term than “tune,” as not all music has a readily identifiable tune like “Singin’ in the Rain”. For instance, Bach’s fugues feature short, overlapping melodic fragments rather than a continuous, extended tune.

Pitch and Pitch Relationships:

The foundation of melody perception lies in our ability to process pitch, which is processed by pitch-selective neurons in the primary auditory cortex. These neurons have a unique one-to-one mapping for pitch, unlike our perception of other senses. While pitch increases with acoustic frequency, our auditory system creates a cyclical perception where pitches an octave apart sound similar, a phenomenon called octave equivalence. This is a unique perceptual experience in music. However, the sources emphasize that simply having the correct pitch classes in different octaves does not guarantee melody recognition. When listeners were presented with familiar tunes where the octave of each note was randomized, they couldn’t even recognize the melody. This suggests that register or ‘height’ (which octave a note is in) is a crucial dimension of melody perception, alongside chroma (the pitch class).

Our brains possess a remarkable mental facility for categorizing pitches, allowing us to perceive melodies as coherent even if played on slightly mistuned instruments. We learn to assign pitches to mental “boxes” representing intervals like “major second” or “major third,” classifying any pitch close enough to that ideal interval size.

Melodic Contour:

The contour of a melody, or how it rises and falls in pitch, is a vital cue for memory and recognition. Even infants as young as five months respond to changes in melodic contour. Interestingly, both children and untrained adults often think melodies with the same contour but slightly altered intervals are identical, highlighting the primacy of contour in initial recognition. Familiar tunes remain recognizable even when the melodic contour is “compressed”. Composers can create repeating contour patterns to help bind a melody together, even if they are not exact repeats, adapting the contour to fit the specific pitch staircase of a scale. Diana Deutsch refers to these building blocks as “pitch alphabets,” which can be compiled from scales and arpeggios.

Tonal Hierarchy and Expectation:

Our perception of melody is deeply influenced by the tonal hierarchy, which is our subjective evaluation of how well different notes “fit” within a musical context or key. Even listeners without extensive musical training have a mental image of this hierarchy and constantly refer to it to form anticipations and judgments about a tune. This is supported by experiments where listeners consistently rated the “rightness” of notes within a set tonal context. The tonal hierarchy helps us organize and understand music, making it sound like music rather than a random sequence of notes. Music that ignores these hierarchies can be harder to process and may sound bewildering.

Gestalt Principles and Binding:

Underlying melody perception is the brain’s constant search for coherence in the auditory stimuli it receives. We mentally and unconsciously “bind” a string of notes into a unified acoustic entity, a tune. This process aligns with principles of gestalt psychology, where the brain seeks to perceive patterns. For example, large intervals can create a discontinuity, challenging the brain’s ability to perceive the melody as a single “gestalt”. Conversely, repetition of notes or contours can strengthen the perception of a unified melody. The auditory picket-fence effect demonstrates our ability to perceive a continuous tone even when interrupted by noise, highlighting the brain’s tendency to “fill in” gaps to maintain a coherent auditory stream. In sequences with large pitch jumps, listeners may even separate the notes into two distinct melodic streams.

Phrasing and Rhythm:

Phrasing, the way a melody is divided into meaningful segments, is crucial for perception. Click migration experiments show that listeners tend to perceive breaks between notes that delineate musical phrases. Phrasing is closely linked to rhythmic patterns, which provide a natural breathing rhythm to music and help us segment it into manageable chunks. The duration and accentuation of notes contribute to our perception of rhythmic groupings.

Memory and Context:

When we listen to a melody, we hear each note in the context of what we have already heard, including previous notes, the melodic contour, repeated phrases, the established key, and even our memories of other music. This constant referencing and updating of information shapes our perception of the unfolding melody.

Brain Processing:

The brain processes melody through various regions, including the lateral part of Heschl’s gyrus and the planum temporale in the temporal lobe, which are involved in pitch perception and sophisticated auditory attributes. The anterior superior temporal gyrus also handles streams of sound like melodies. Research suggests that the right hemisphere discerns the global pattern of pitch contour, while the left hemisphere processes the detailed aspects of pitch steps.

Atonal Music:

Music that rejects tonal hierarchies can be harder to process because it goes against our learned expectations about note probabilities. While some theories attempt to analyze atonal music through concepts like pitch-class sets, these approaches often don’t explain how such music is actually perceived.

In summary, melody perception is a dynamic process involving the processing of pitch and its relationships, the recognition of melodic contour, the influence of tonal hierarchies and learned expectations, the brain’s ability to bind sequences of notes into coherent units, the segmentation of melodies into phrases guided by rhythmic patterns, and the crucial role of memory and context. These elements work together to allow us to experience a series of discrete musical notes as a meaningful and unified melodic line.

Understanding Harmony and Dissonance in Music

Harmony is about fitting notes together. Conventionally, combinations that fit well are called consonant, and those that fit less well are dissonant. In a reductive formulation, consonance is considered good and pleasing, while dissonance is bad and unsettling. However, these concepts are often misunderstood and misrepresented.

Historical Perspectives on Consonance and Dissonance:

• In tenth-century Europe, a perfect fifth was generally not deemed consonant; only the octave was.
• When harmonizing in fifths became common, fourths were considered equally consonant, which is different from how they are perceived today.
• The major third (C-E), part of the “harmonious” major triad, was rarely used even by the early fourteenth century and was not fully accepted as consonant until the High Renaissance.
• The tritone interval, supposedly dissonant, becomes pleasing and harmonious when part of a dominant seventh chord (e.g., adding a D bass to C-FG).
• The whole polarizing terminology of consonance and dissonance is a rather unfortunate legacy of music theory.

Sensory (or Tonal) Dissonance:

• There is a genuinely physiological aspect of dissonance, distinguished from musical convention, called sensory or tonal dissonance.
• This refers to the rough, rattle-like auditory sensation produced by two tones closely spaced in pitch.
• It is caused by the beating of acoustic waves when two pure tones with slightly different frequencies are played simultaneously. If the beat rate exceeds about 20 Hz, it is heard as roughness.
• The width of the dissonant region depends on the absolute frequencies of the two notes. An interval consonant in a high register may be dissonant in a lower register. Therefore, there is no such thing as a tonally dissonant interval independent of register.
• In the mid-range of the piano, minor thirds generally lie beyond the band of roughness, while even a semitone does not create roughness for high notes. However, in the bass, even a perfect fifth can become dissonant in sensory terms, explaining the “gruffness” of low chords.

Consonance, Dissonance, and Overtones:

• Tones played by musical instruments are complex, containing several harmonics.
• Two simultaneously sounded notes offer many possibilities for overtones to clash and produce sensory dissonance if close enough in frequency.
• Hermann von Helmholtz calculated the total roughness for all overtone combinations, generating a curve of sensory dissonance with dips at various intervals of the chromatic scale. The octave and fifth have particularly deep “consonant” valleys.
• However, the depths of several “consonant” valleys don’t differ much. The modern dissonance curve shows that most intervals between the major second and major seventh lie within a narrow band of dissonance levels, except for the perfect fifth. Even the tritone appears less dissonant than major or minor thirds according to some measurements.
• The greatest sensory dissonance is found close to the unison, particularly the minor second, predicted to sound fairly nasty. However, such intervals can be used for interesting timbral effects.
• The brain is insistent on “binding” overtones into a single perceived pitch. If a harmonic is detuned, the brain tries to find a new fundamental frequency that fits, and only when the detuning is too large does it register the “bad” harmonic as a distinct tone. Percussive instruments often produce inharmonic overtones, resulting in an ambiguous pitch.

Cultural Influences and Learning:

• Whether we experience note combinations as smooth or grating is not solely a matter of convention, but there is a physiological aspect. However, likes and dislikes for certain combinations probably involve very little that is innate and are mostly products of learning.
• What is disliked is probably not the dissonances themselves but how they are combined into music.
• Acculturation can overcome sensory dissonance, as seen in the ganga songs of Bosnia and Herzegovina, where chords of major and minor seconds are considered harmonious.
• People tend to like best what is most familiar. Western listeners, being accustomed to tonal music, will be acclimatized to octaves, fifths, thirds, etc., and hear less common intervals as more odd.
• Studies suggest that cultural tradition exerts a stronger influence than inherent qualities in determining the emotional connotations of music, implying that perceptions of consonance and dissonance can also be culturally influenced.

Harmony in Musical Composition:

• In polyphonic music, harmony fills out the musical landscape. If melody is the path, harmony is the terrain.
• Harmonization is the process of fitting melodic lines to chords. This is often where music comes alive.
• Harmonization is generally more sophisticated in classical music, tending to use voice-leading, where accompanying voices have their own impetus and logic, rather than being monolithic chords.
• Harmonic progressions are sequences of chords. In Western classical music until the mid-nineteenth century, these tended to be formulaic and conservative, involving transitions to closely related chords. Pop and rock music have inherited much of this tradition.
• Modulation is the alteration of the key itself within a harmonic progression.
• Music theorists and psychologists have attempted to create a cartography of chords and keys, trying to map out relationships in harmonic space. Carol Krumhansl’s research suggests that the perceived relatedness of keys aligns with the cycle of fifths.

Harmony, Dissonance, and Musical Style/Emotion:

• Many classical-music traditionalists deny enjoying dissonance, associating it with jarring modern music. However, even composers like Chopin use dissonance extensively.
• The use of dissonance by modernist composers was seen by some as an affront to music itself. However, champions of atonalism argued that aversion to dissonance is culturally learned.
• “Dissonant” intervals like major sixths, sevenths, and ninths can create luxuriant sounds in the hands of composers like Debussy and Ravel.
• Composers may confuse our expectations regarding harmony to introduce tension and emotion.
• Expectations about harmony are crucial for our emotional response to music. Composers manipulate these expectations through devices like cadences, anticipation notes, and suspensions.
• Ambiguity in harmony and tonality can also create a powerful effect, with pleasure arising from the resolution of confusion.
• Different musical genres establish their own harmonic schemas, which they can then use to manipulate tension.

Dissonance in Polyphony:

• In early medieval polyphony, it was considered better to compromise the melody than to incur dissonance. However, composers increasingly prioritized maintaining good melodies in each voice, even if it led to occasional dissonances.
• This led to rules governing permissible dissonances in counterpoint. In Palestrina’s counterpoint, dissonances often occur on “passing tones” leading towards a consonance, and strong consonances are achieved at the beginnings and ends of phrases. The main objective is to maintain horizontal coherence of each voice while enforcing vertical integration through judicious use of consonance and controlled dissonance.
• Streaming of sound can offer a barrier to the perception of dissonance in polyphony. If voices are sufficiently distinct, potentially dissonant intervals may not be registered as jarring. Bach’s fugues, for example, contain striking dissonances that can go unnoticed due to the independence of the voices.
• Harmony can support the mental juggling act of listening to multiple melodies simultaneously, especially when the melodies are in the same key. Harmonic concordance seems to assist cognition.
• The composer doesn’t always want polyphonic voices to be clearly defined. In hymn singing, the focus is on creating a sense of unity through harmonies, resulting in a more homophonic texture where voices combine to carry a single melody, as opposed to the elaborate interweaving of voices in Bach’s contrapuntal music.

In conclusion, harmony and dissonance are fundamental aspects of music that involve both acoustic/physiological phenomena and cultural learning and conventions. Their perception and use have evolved throughout music history and continue to be manipulated by composers to create diverse musical experiences and emotional effects.

Understanding Musical Rhythm and Meter

Rhythm and meter are fundamental aspects of music. Rhythm is defined as the actual pattern of note events and their duration, and it tends to be much less regular than meter or tactus. It’s the “felt” quality of the regular subdivision of time on paper. Rhythm can be catchy and move us physically.

Meter, on the other hand, is the regular division of time into instants separated by equal intervals, providing what is colloquially called the ‘beat’. It’s the underlying pulse. The numbers at the start of a stave, the time signature, indicate how many notes of a particular duration should appear in each bar, essentially telling us whether to count the rhythm in groups of two, three, four, or more beats. To create a beat from a regular pulse, some pulses need to be emphasized over others, often by making them louder. Our minds tend to impose such groupings even on identical pulses. The grouping of pulses defines the music’s meter. Western music mostly uses simple meters with recurring groups of two, three, or four pulses, or sometimes six.

The tactus is related to but different from meter; it’s the beat we would clap out while listening to music and may be culture-specific. We tend to tap out a slower pulse to familiar music.

The source emphasizes that not all music possesses rhythm in a discernible way, citing compositions by Ligeti and Xenakis as continuous skeins of sound without a clear pulse, and Stockhausen’s Kontakte as being made of disconnected aural events. Gregorian chant is an example of music that can have regularly spaced notes but lack a true meter. Music for the Chinese fretless zither (qin) has rhythm in terms of note lengths, but these are not arranged against a steady underlying pulse.

However, a quasi-regular pulse pervades most of the world’s music. A rhythm is typically created by elaborating the periodic beats. Subdivisions and stresses superimposed on a steady pulse give us a sense of true rhythm, helping us locate ourselves in time much like the tonal hierarchy helps us in pitch space. This orderly and hierarchical structuring of time is found in the rhythmic systems of many musical traditions.

The source notes that the metre is often portrayed as a regular temporal grid on which the rhythm is arrayed, but the real relationship is more complex. Musicians tend subconsciously to distort the metrical grid to bring out accents and groupings implied by the rhythm. This stretching and shrinking of metrical time helps us perceive both meter and rhythm.

Western European music has traditionally chopped up time by binary branching, with melodies broken into phrases grouped in twos or fours, divided into bars, and beats subdivided into halves and quarters. This binary division is reflected in note durations like semibreve, minim, and crochet. However, some Balkan music uses prime numbers of beats in a bar, suggesting that binary division is not universal. Eastern European song may have constantly changing meter due to the rhythmic structure of its poetry.

Creating a true sense of rhythm and avoiding monotony involves not just stressing some beats but an asymmetry of events, similar to the skipping rather than plodding nature of spoken language. The source discusses rhythmic figures like the iamb, trochee, dactyl, and anapest, which are “atoms” from which we build a sense of rhythm and interpret musical events. Repetition of these units is crucial for that coherence to be felt. Our assignment of rhythmic patterns draws on various information beyond note duration, including melody, phrasing, dynamics, harmony, and timbre.

Composers generally want us to perceive the intended rhythm and use various factors to reinforce it. However, they may also seek to confuse our expectations regarding rhythm to introduce tension and emotion, as it is easy to hear when a beat is disrupted. Examples of this include:

• Syncopation, which involves shifting emphasis off the beat.
• Beethoven’s Fifth Symphony starting with a rest on the downbeat.
• Rhythmic ambiguity created by conflicting rhythmic groupings and meter, as in Beethoven’s Piano Sonata No. 13 and Bernstein’s “America”.
• Rhythmic elisions and deceptive rhythmic figures in popular music.
• Unambiguous disruption of meter, creating a jolt, as in Stravinsky’s The Rite of Spring.
• The use of anticipation tones in classical cadences to modulate the expectation of the impending cadence.

The source also points out that our sense of metrical regularity isn’t always strong, especially without musical training, and folk music traditions can exhibit irregular meters. In early polyphonic music, complex crossed rhythms were common, even without explicit metrical notation. Some musical traditions, like African, Indian, and Indonesian music, use cross-rhythms and polyrhythms. The minimalist compositions of Steve Reich utilize phasing, where repetitive riffs played at slightly different tempos create shifting rhythmic patterns.

Ultimately, rhythm provides a way to interpret and make sense of the stream of musical events by apportioning them into coherent temporal units. Composers manipulate rhythm and meter in various ways to create structure, expectation, and emotional impact in their music.

By Amjad Izhar
Contact: amjad.izhar@gmail.com
https://amjadizhar.blog

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This academic text, likely from a book or collection of essays on music theory, explores the geometry of musical spaces, particularly focusing on chords and scales. It argues that understanding the mathematical relationships and “voice leading” between different musical objects provides a powerful framework for analyzing music across various historical periods and styles, including classical, jazz, and rock. The text outlines five fundamental components of tonality and four key claims about their interactions, proposing that concepts like macroharmony and centricity can be understood independently. Ultimately, the author suggests that seemingly disparate compositional techniques and harmonic progressions can be explained by underlying geometrical principles and a preference for efficient voice leading.

Harmony and Counterpoint in Music

Based on the sources provided, harmony and counterpoint are fundamental concepts in music, particularly in Western musical tradition. They represent the two principal dimensions of musical coherence: vertical (harmonic) and horizontal (melodic or contrapuntal).

Here’s a breakdown of how the sources discuss these concepts:

• Harmony refers to the use of sonorities or the vertical aspect of music. A key aspect discussed is harmonic consistency, which means that the harmonies used in a passage tend to be structurally similar to one another, using sonorities that resemble each other. This contributes to a sense of smoothness and helps chords feel like they belong together. The sources contrast passages with harmonic consistency (using similar consonant or dissonant chords) with those lacking it, which can sound jarring. In traditional Western styles like functional harmony, chords are further constrained to move according to specific conventions.
• Counterpoint, also referred to as voice leading or the melodic dimension, involves the combination of simultaneous melodies or voices. A key feature related to counterpoint is conjunct melodic motion, where melodies tend to move by short distances from note to note. Another crucial aspect is efficient voice leading, which describes how smoothly notes move from one chord to the next, typically by small distances. Composers need to be able to compare the overall efficiency or “size” of different voice leadings.

The sources emphasize that harmony and counterpoint are not independent but constrain one another.

• Different types of chords suggest different musical uses. For example, chords whose notes are far apart (like {C, E, G}) differ fundamentally from clustered chords ({B, C, Df}).
• Chords that can be linked by efficient voice leading are well-suited for contrapuntal music where harmonies change quickly.
• Chord structure constrains contrapuntal function. Conversely, composers cannot simply write a harmonic progression without considering the melodic movement (voice leading) between the chords. Figure 1.3.3 illustrates how the proximity of notes between chords (like C major and F major) allows for simultaneous melodies moving by small distances.
• The sources propose that the ability to combine harmonic consistency and efficient voice leading depends on using nearly symmetrical chords. Basic symmetries like transposition, inversion, and permutation are fundamental to understanding efficient voice leading between structurally similar chords.
• The basic sonorities of Western tonal music (such as perfect fifths, triads, and seventh chords) are described as optimal because they are both acoustically consonant (stable-sounding) and well-suited for voice leading, allowing efficient connections between themselves and their transformations. Highly consonant chords divide the octave relatively evenly, which relates to their suitability for efficient voice leading.

The combination of harmonic consistency and conjunct melodic motion is identified as a key feature that links diverse styles throughout Western music history, from early counterpoint to twentieth-century tonal music and jazz. The historical development of Western tonal practice can be seen as composers grappling with the problem of effectively combining these two dimensions. Even seemingly different styles utilize fundamentally similar procedures because there are only a few ways to combine harmonic consistency and stepwise melodies.

The book employs geometrical models to understand the interactions between harmony and counterpoint. For instance, voice leadings can be represented as line segments in musical spaces. In a two-dimensional space, horizontal lines can represent parallel motion, and vertical lines can represent contrary motion. Different geometrical spaces are used to model harmony and counterpoint in various styles, such as the two-dimensional Möbius strip for medieval two-note counterpoint or chromatic spaces for nineteenth-century harmony.

The sources also touch upon theoretical debates regarding the relationship between harmony and counterpoint, contrasting traditional harmonic theory with Schenkerian views like Monism (explaining harmonic rules contrapuntally) and Holism (seeing harmony and counterpoint as inseparable). The author argues that it is possible to provide an informative theory of harmonic progressions that is largely independent of specific counterpoint, even though composers and analysts typically consider both together. Ultimately, the sources suggest that harmonic consistency, efficient voice leading, acoustic consonance, macroharmony, and centricity all contribute to the sense of tonality.

Geometry in Music: Structure and Space

Based on the sources, the application of geometry to music is presented as a powerful and central tool for modeling musical structure and understanding fundamental musical principles. The core idea is to represent musical elements and relationships using points, lines, and shapes in various geometrical spaces. This approach allows for a visual and intuitive grasp of abstract musical concepts.

Here’s how the sources discuss music and geometry:

• Modeling Musical Elements:Individual pitches can be represented as points on a continuous line (linear pitch space).
• Pitch classes (notes regardless of octave) can be represented as points on a circle (circular pitch-class space). In this space, transposition corresponds to rotation, and inversion corresponds to reflection.
• Musical objects, initially defined as ordered sequences of pitches, can be modeled using geometrical concepts and symmetry operations. Different classifications of musical objects (like chords, chord types, set classes) correspond to different combinations of symmetry operations (Octave shifts, Permutations, Transpositions, Inversions, Cardinality changes).
• Modeling Voice Leading and Chord Progressions:Voice leadings (motion between notes) can be represented as line segments in geometrical spaces. The length of the line segment corresponds to the “size” or efficiency of the voice leading.
• In a two-dimensional space representing ordered pairs of notes, a voice leading where voices trade notes can be shown as a line segment. Horizontal and vertical segments in this space represent motion in a single voice.
• Chord progressions can be seen as motion through these musical spaces, either as sequences of unordered points or, more abstractly, as paths or line segments linking points representing chords.
• Chord Spaces:Higher-dimensional spaces are introduced where entire chords are represented as single points. For example, a two-note chord space can be a Möbius strip, a three-note chord space can be represented by a triangular prism, and spaces exist for chords of any number of notes.
• The structure of these spaces (containing twists, mirrors, etc.) illustrates musical principles.
• The distance between chords in these spaces corresponds to the size of the minimal voice leading between them. Thus, “near” chords are those that can be linked by efficient voice leading.
• Scales as Rulers and Geometrical Structures:Scales can be modeled as musical “rulers” that measure distances between notes within a specific system.
• Scales themselves can be represented geometrically. For example, the diatonic scale can be mapped onto a crumpled two-note chord space, where its irregularity reflects the varying chromatic sizes of its steps. Redrawing this using “scalar distance” makes the grid regular.
• Voice leading relations among common scales can be modeled using three-dimensional cubic lattices.
• Revealing Musical Relationships:Geometry helps demonstrate the fundamental connection between harmony and counterpoint, acoustic consonance, efficient voice leading, and harmonic consistency. Nearly symmetrical chords are shown to be well-suited for efficient voice leading between structurally similar chords.
• It can make clear relationships that are not immediately obvious from standard musical notation, such as mirror images between musical passages.
• Different styles throughout Western music history, from early counterpoint to jazz, can be understood as variations exploiting the same basic geometrical techniques.
• The geometrical structure can help explain why intuitive musical exploration leads to certain common practices, such as the use of major-third-related triads or minor-third/tritone-related seventh chords in chromatic music.
• Sometimes understanding discrete musical structures (like equal-tempered music) requires considering the continuous geometrical spaces in which they are embedded.
• Geometry in Analysis:Musical passages can be plotted in these geometrical spaces to reveal their structure.
• Voice-leading lattices provide a way to visualize relationships among chords and map compositional possibilities.
• The Goal: The book aims to provide a user-friendly introduction to these musico-geometrical spaces, explaining their function and how they help visualize a wealth of musical possibilities. Ultimately, fluency with both simpler circular models and higher-dimensional spaces is suggested for a deep understanding of music.

Navigating the Spectrum of Tonality

Based on the sources, the concepts of tonality and atonality are central to understanding Western musical history and theory, though the term “tonal” is described as contested territory.

Traditionally, “tonal” is sometimes used restrictively to describe Western art music primarily from the eighteenth and nineteenth centuries, labeling later music as “post-tonal”. This creates a category that lumps together diverse styles like Arvo Pärt’s consonances with the dissonant music of Varèse and Xenakis, suggesting they are more similar to each other than to earlier composers.

However, the term “tonal” can also be used expansively to include a much broader range of music, such as rock, folk, jazz, impressionism, minimalism, medieval and Renaissance music, and even some non-Western music. In this broader sense, “tonality” is almost synonymous with “non-atonality,” understood in contrast to music deliberately composed to avoid traditional tonal characteristics. This raises questions about what specific features make music sound tonal and whether tonality is a single property or has multiple components.

The sources propose moving beyond the crude opposition of “tonal/atonal” with a more nuanced set of distinctions. The book’s purpose is to provide theoretical tools to discuss music that falls between classical tonality and complete atonality.

According to the sources, five key features jointly contribute to a sense of tonality across various genres and historical periods:

1. Conjunct melodic motion: Melodies tend to move by small intervals.
2. Acoustic consonance: Consonant harmonies are preferred, especially at points of stability.
3. Harmonic consistency: Harmonies in a passage tend to be structurally similar.
4. Limited macroharmony: Music tends to use relatively small collections of notes (often five to eight) over moderate periods.
5. Centricity: One note is felt as more prominent or stable than others over a period.

The book primarily focuses on the theoretical and historical aspects of how composers use and combine these features. It suggests that the basic sonorities of Western tonal music, such as perfect fifths, triads, and seventh chords, are “optimal” because they are both acoustically consonant and well-suited for efficient voice leading, allowing smooth connections between structurally similar chords. Highly consonant chords tend to divide the octave relatively evenly, which relates to their suitability for efficient voice leading.

Atonality, from this perspective, represents music that often rejects or abandons these five features. For example, music might lack acoustic consonance, conjunct melodic motion, harmonic consistency, or centricity, and might not limit itself to a small number of pitch classes over short stretches. The historical development from highly chromatic late nineteenth-century music to atonality can be seen as composers grappling with the saturation of the chromatic space. Chromatic tonality and free atonality can sometimes share similar statistical profiles, including comparable rates of pitch-class circulation and a lack of centricity. Some twentieth-century musical languages, including serialism (like the twelve-tone method), abandoned traditional tonality, sometimes focusing on the order of pitches rather than their unordered content.

The five components of tonality are presented as vectors or dimensions that span a metaphorical “tonality space”. This conceptual space allows for a more granular way to describe and situate different musical styles and pieces by asking questions about the presence and nature of these five features. This framework replaces the simpler tonal/atonal binary with a richer set of categories, acknowledging the spectrum of musical possibilities that exists between traditional tonality and complete atonality.

Principles of Musical Voice Leading

Based on the sources, voice leading is a fundamental concept in music, described as the atomic constituent of musical scores and the basic building block of polyphonic music. It represents the mapping from one collection of pitches to another. Essentially, it describes how the notes in one chord move to those in the next. Voice leading involves melodic motion in all parts of a contrapuntal texture.

Here’s a more detailed discussion of voice leading based on the sources:

• Representation of Voice Leading:
• Voice leadings are visually represented by an arrow (®) connecting two chords.
• They can be described in terms of specific pitches (pitch-space voice leading). Geometrically, a pitch-space voice leading corresponds to a collection of paths in linear pitch space.
• They can also be described in terms of pitch classes, ignoring the specific octave (pitch-class voice leading or octave-free voice leadings). Numbers above the arrow can indicate the paths in pitch-class space (semitones moved), where the specific path matters (e.g., 1 semitone up is distinct from 11 semitones down). Pitch-class voice leadings function as abstract schemas for composers, representing general “routes” from chord to chord. Geometrically, a pitch-class voice leading can be represented as a collection of paths in circular pitch-class space.
• Voice-Leading Size and Efficiency:
• A key aspect of voice leading is its efficiency, meaning voices move by short distances. This is also referred to as small voice leading.
• Although pedagogues have long encouraged small voice leadings, precisely measuring voice-leading size has been a subject of debate, with various methods proposed.
• The sources suggest that “reasonable” measures of voice-leading size should depend only on the collection of distances moved by each voice.
• Two key constraints for reasonable metrics are proposed: measures should not have the counterintuitive consequence that “voice crossings” make a voice leading smaller, and increasing the distance moved by a voice should not make the voice leading smaller (while other voice movements are fixed).
• Removing voice crossings from a voice leading never makes it larger.
• Maximally efficient voice leadings between any two chords are always scalar or interscalar transpositions.
• Voice Leading in Relation to Other Concepts:
• Harmony and Counterpoint: Voice leading is presented as being fundamentally connected to harmony and counterpoint. Acoustically consonant chords are often well-suited for efficient voice leading, allowing smooth connections between structurally similar chords. Efficient voice leading is listed as one of the five components contributing to a sense of tonality.
• Chord Progressions: Chord progressions are sequences of successive voice leadings between chords. They can be modeled as motion through geometric spaces.
• Geometry: Geometry provides a powerful tool for modeling musical structures.
• In a two-dimensional space representing pairs of notes, voice leadings are represented by line segments. Horizontal/vertical segments mean one voice is fixed, while diagonal segments mean both voices move.
• Any voice leading can be decomposed into pure parallel and pure contrary components. Geometrically, pure parallel motion is represented horizontally, while pure contrary motion is represented vertically.
• In higher-dimensional spaces where chords are points, voice leadings are represented as “generalized line segments”. The distance between chords in these spaces corresponds to the size of the minimal voice leading between them.
• Similarity: Voice leading can model intuitions of musical similarity. The distance between chords in musical spaces corresponds to the minimal voice leading size between them.
• Transposition and Inversion: Voice leadings can be related by uniform or individual transposition and inversion. Individually transposed voice leadings often appear in sequential passages. Individually inverted voice leadings have the same distances but reversed directions. Geometrically, individual transposition alters the horizontal component of a voice leading.
• Symmetry: Efficient voice leading between structurally similar chords can be understood using the basic symmetries: transposition, inversion, and permutation. Chords that are acoustically consonant tend to divide the octave relatively evenly, which relates to their suitability for efficient voice leading. Chords that are near transpositionally or inversionally symmetrical can be linked by efficient voice leading to their transpositions or inversions.
• Scales: Certain types of voice leading, specifically scalar or interscalar transpositions, are strongly crossing-free. Voice leading relationships among scales can be modeled using lattices. Modulation, or motion between macroharmonies (collections of notes used over a period), can be represented as a voice leading.
• Compositional Practice: Composers throughout Western music history have exploited voice-leading techniques. Efficient voice leading is a key principle in Renaissance counterpoint, classical functional tonality, nineteenth-century chromaticism, and jazz. Composers tend to privilege crossing-free voice leadings, sometimes using voice crossings as surface-level embellishments. Composers can analyze voice-leading possibilities when composing.

In summary, voice leading, conceptualized as the movement of notes between chords, is presented as a core organizing principle in Western music. Its efficiency, geometric representation, and relationship to harmony, consonance, and symmetry provide a framework for understanding a wide range of musical styles and practices.

Exploring Musical Scales: Concepts and Applications

Based on the sources and our conversation, musical scales are a fundamental concept for understanding music, particularly its structure and organization.

Here’s a discussion of musical scales:

• Scales as Musical Rulers: A scale is fundamentally described as a means of measuring musical distance – a kind of musical ruler whose unit is the “scale step”. Any collection of pitches can, in principle, be a scale, and they don’t need to adhere to strict criteria like having closely spaced notes or repeating after an octave. The primary function is to define how to move up and down by one unit. Scales provide different ways of measuring musical distance, which contributes to the richness of tonal music.
• Scale Degrees and Transformations: Scales define scale-specific notions of basic musical concepts. Scale degrees are numbers assigned to the notes in a scale, providing a way to label positions within that scale. Scales also define scalar transposition and scalar inversion, which are analogous to chromatic transposition and inversion but are measured using scale steps. These scale-specific transformations can even act on notes that are not in the scale itself. Scalar transposition moves a musical pattern along a single scale, while interscalar transposition moves a pattern from one scale to another.
• Properties of Scales:Evenness: Scales can be assessed for their evenness, which relates to how regularly their notes are distributed across the octave. In a nearly even scale, scalar transposition can resemble chromatic transposition, allowing harmonies within the scale to be transposed along it with minimal distortion. Perfect evenness can sometimes make independent melodic motion difficult to perceive.
• Consonance: Composers tend to favor scales that contain many consonant intervals, such as perfect fifths. Octave-repeating scales are particularly saturated with octaves, the most consonant interval. Scales containing many minor thirds and perfect fifths are also common.
• Common Scales: The sources discuss the construction and importance of various common scales, including the pentatonic, diatonic, and chromatic scales, which are described as nearly even scales containing many perfect fifths. Other scales discussed include the harmonic minor, acoustic (melodic minor ascending), harmonic major, octatonic, and whole-tone scales. These scales often reappear in diverse theoretical contexts and musical styles because they tend to divide the octave fairly evenly while also containing a large number of consonances. The harmonic and acoustic scales are presented as mediating between the diatonic and transpositionally symmetrical scales like whole-tone and octatonic.
• Scales and Macroharmony: A macroharmony is defined as the total collection of notes used over small stretches of musical time. Typically, macroharmonies are also scales. Scales provide a “reservoir” of melodic notes to accompany chords that do not contain chromatic clusters. The properties of a scale, such as its evenness, are related to the properties of the corresponding macroharmony, such as its “gaplessness” (how far an out-of-macroharmony note is from a note within the macroharmony).
• Scales and the “General Theory of Keys”: The concepts of scale, macroharmony, and centricity are identified as the three principal components of what the sources refer to as the “general theory of keys”. This theory provides tools for describing music that is broadly tonal, even if it doesn’t strictly follow eighteenth-century conventions.
• Scales, Voice Leading, and Modulation: Scales are intimately linked to voice leading and modulation. Modulation, understood as motion between macroharmonies, can be represented as a voice leading between scales. Desiring to analyze sequential musical structures often requires postulating changes in the underlying scale, which necessitates specific voice leadings between those scales. Efficient voice leading between structurally similar chords or between scales can be understood through basic symmetries. The relationships and voice leading possibilities between common scales can be modeled using geometrical structures like lattices. Interscalar transposition is deeply connected to the problem of identifying efficient voice leading between arbitrary chords.
• Scales in Different Musical Styles: The use and combination of scales have evolved throughout Western musical history. Early Western music explored tonal centers within a largely diatonic macroharmony. Classical music focused primarily on major and minor scales, and modulations involved scale-to-scale voice leadings. Twentieth-century composers have exploited a wider range of macroharmonies and tonal centers. Musical styles like impressionism and jazz make greater use of nondiatonic scales, and contemporary tonal language includes using efficient voice leadings between a full range of diatonic modes. Scales provide a framework for compositional techniques such as “scale-first composition,” where modulation generalizes traditional modulation to a broader range of scales and modes.

Musical Structures and Concepts Study Guide

Musical Structures and Concepts Study Guide

Quiz

1. What is the primary difference between a melodic approach to harmony and a voice-leading approach, as discussed in the text?
2. How can geometric concepts like plotting music on a Möbius strip reveal hidden musical structure?
3. Explain the concept of “efficient voice leading.”
4. What is a voice-leading lattice, and how can it be used to visualize harmonic movement?
5. According to the text, how does a musical scale function similarly to a mathematical metric?
6. What are “scalar transposition” and “scalar inversion”?
7. What is the “subset technique” in composition, as described in the source material?
8. How does polytonality manifest in music according to the text, particularly in examples by Stravinsky or Grieg?
9. What are pitch-class profiles, and how can they be compositionally useful?
10. How does “sidestepping” in jazz, as exemplified by Bill Evans, relate to the concepts of local stability and global instability?

Quiz Answer Key

1. A melodic approach focuses on the relationships between individual notes, often emphasizing stepwise motion and conventional melodic contours. A voice-leading approach, in contrast, prioritizes the smooth movement of multiple musical lines (voices) between chords, often seeking to minimize the overall distance traveled by all voices.
2. Plotting music on a Möbius strip can visually represent musical structure by revealing cyclical patterns and connections between seemingly distant harmonies or melodic fragments. It can show how musical passages might be related through transposition, inversion, or other transformations that are not immediately obvious in linear notation.
3. Efficient voice leading refers to the movement between chords or sonorities where the total distance traveled by all individual musical lines (voices) is minimized. This results in smooth, economical transitions and is often associated with a sense of connectedness between harmonies.
4. A voice-leading lattice is a geometric representation of musical space where nodes represent chords or sonorities, and edges represent specific types of voice leading between them, often single-step movements. It allows for the visualization of possible harmonic pathways and the relationships between different musical structures.
5. A musical scale acts as a metric by providing a method of measuring distance between notes within that scale. It defines allowed steps and intervals, creating a framework for understanding melodic and harmonic relationships that is distinct from chromatic or log-frequency distance.
6. Scalar transposition involves moving an entire musical segment or scale to a different starting degree within the same scale. Scalar inversion involves flipping the order of intervals within a scale or melodic segment while staying within the framework of that scale.
7. The subset technique is a compositional method where a composer uses scales that all contain a common collection of prominent notes. These shared notes remain stable, while the other notes (mobile pitches) are altered to create different scalar collections.
8. Polytonality in the text is described as the juxtaposition of different tonalities or scales simultaneously, creating a clash between independent harmonic streams. Examples include the use of different diatonic collections in separate registers or instruments, as seen in Stravinsky, or the clash between fixed and mobile pitches creating dissonance, as in Grieg.
9. Pitch-class profiles are graphical representations that show the distribution or emphasis of different pitch classes within a musical passage. Composers can use them to plan or analyze the tonal characteristics of their music, creating specific shadings of underlying tonalities by emphasizing certain notes within a scale.
10. Sidestepping, as practiced by musicians like Bill Evans, involves moving abruptly to a harmony or scale a small distance away (like a half step) from the underlying harmony, creating a sense of local stability within the new, temporary key, while maintaining a global instability relative to the original key. The subsequent resolution back to the original key resolves this tension.

Essay Questions

1. Discuss how the geometric models presented in the text (such as chord space, voice-leading lattices, or plotting on a Möbius strip) offer new perspectives on understanding harmonic relationships and musical structure compared to traditional harmonic analysis methods.
2. Analyze the relationship between voice leading and scales as presented in the source material. How do these two concepts interact in the creation and analysis of musical passages, particularly in chromatic and functional tonal music?
3. Explore the various compositional techniques discussed in the text (e.g., chord-first composition, scale-first composition, subset technique, sidestepping) and analyze how they represent different approaches to organizing pitch and harmony, potentially departing from traditional functional tonality.
4. Compare and contrast the concepts of functional tonality and chromaticism as presented in the source material. How do composers utilize chromaticism to expand or challenge the principles of functional harmony, and what are the historical implications of these developments?
5. The text discusses different approaches to analyzing tonal music, including traditional harmonic analysis, Schenkerian analysis, and pluralism. Discuss the strengths and limitations of each approach, and how they offer different insights into the structure and meaning of tonal music.

Glossary of Key Terms

• Voice Leading: The movement of individual musical lines (or voices) between successive chords or sonorities. Often concerned with smoothness and efficiency of motion.
• Efficient Voice Leading: Voice leading where the total distance traveled by all voices is minimized, typically involving stepwise motion or small leaps.
• Voice-Leading Lattice: A geometric graph representing musical space, where nodes are chords or sonorities and edges represent voice leading between them, often single-step movements.
• Chord Space: A multidimensional geometric space where chords or sonorities are represented as points. The distance between points in this space can represent voice-leading distance or other musical relationships.
• Chromaticism: The use of notes outside the prevailing diatonic scale, often resulting in increased harmonic complexity and tension.
• Scale: A collection of pitches organized in a specific order, serving as a framework for melody and harmony. Can also function as a “ruler” or metric for measuring musical distance.
• Scale Degrees: The individual pitches within a scale, typically numbered or named according to their position relative to the tonic.
• Scalar Transposition: Moving a musical segment or scale to a different starting pitch while maintaining the characteristic interval relationships of the original scale, thus staying within the same scale type.
• Scalar Inversion: Flipping the order of intervals within a scale or melodic segment, while still adhering to the intervals defined by the original scale.
• Macroharmony: A collection of pitches that persists over a longer duration than a single chord, providing a broader harmonic context for melodic and harmonic events.
• Pitch-Class Profile: A graphical representation showing the distribution and emphasis of different pitch classes within a musical passage.
• Polytonality: The simultaneous use of two or more different tonalities or scales, creating a clash between independent harmonic streams.
• Subset Technique: A compositional technique where scales are used that all contain a common, stable collection of pitches, while other pitches are mobile and change to form different scalar collections.
• Functional Harmony: A system of harmony based on the relationships and progressions of chords within a key, where chords have specific “functions” or roles in moving towards a tonic.
• Schenkerian Analysis: A method of musical analysis that seeks to understand tonal music by reducing its surface complexity to underlying fundamental structures, often emphasizing contrapuntal relationships.
• Sidestepping: In jazz improvisation, the practice of moving briefly to a harmony or scale a small distance away (often a half step) from the prevailing harmony, creating temporary tension and release.
• Metric (Musical/Mathematical): In a musical context, a system for measuring distance or relationships between musical objects (notes, chords, scales). In a mathematical context, a function that defines a distance between points in a space.

Geometry and Structure in Music Theory

Briefing Document: Review of Selected Music Theory Concepts

Subject: Key concepts in music theory, focusing on geometry, scales, functional harmony, chromaticism, and jazz improvisation, as presented in the provided excerpts.

I. Overarching Themes:

The excerpts explore various aspects of music theory through a lens that often employs geometric metaphors and systematic analysis. Key themes include:

• Geometric Representation of Musical Space: Chords, scales, and voice leading are frequently visualized and analyzed using geometric concepts like spaces, lattices, and specific shapes (e.g., the Möbius strip, triangles). This approach allows for the quantitative comparison of musical elements and the identification of underlying structures and relationships.
• Voice Leading as a Primary Analytical Tool: The efficiency and characteristics of voice leading (how individual notes move between chords) are presented as fundamental to understanding musical structure and progression, sometimes even overriding traditional harmonic or scalar considerations.
• The Interplay of Different Musical Dimensions: The text examines music from multiple perspectives – harmonic, scalar, melodic, and contrapuntal – and how these dimensions interact and can be represented in various “spaces” (pitch space, pitch-class space, chord space).
• Historical and Theoretical Perspectives: The excerpts delve into both theoretical frameworks for understanding musical phenomena and historical examples from various periods (Bach, Mozart, Brahms, Debussy, Schubert, Shostakovich, jazz musicians) to illustrate these concepts in practice.
• Systematic Exploration of Musical Possibilities: The text investigates how composers explore the “space of possibilities” defined by various tonal and scalar ingredients, highlighting both conventional practices and innovative departures.

II. Key Concepts and Ideas:

A. Geometry in Analysis (3.5 – 3.9):

• Musical Structures as Geometric Objects: The excerpts demonstrate how musical structures can be plotted and visualized geometrically to reveal underlying patterns. Figure 3.5.1 and 3.5.2 illustrate plotting phrases on a Möbius strip to show structural relationships.
• Chord Space: Different types of chords can be mapped to distinct “spaces” with varying dimensions:
• Three-dimensional chord space: Used to analyze triadic relationships (Figure 3.8.3b). Brahms’ systematic movement along a lattice in three-note chord space is highlighted.
• Higher-dimensional chord spaces: Discussed in relation to seventh chords (four dimensions) and other chord types. Schubert’s use of the “major-third system” in triadic music (three-dimensional geometry) is presented as a warm-up to understanding higher-dimensional relationships among seventh chords (8.4).
• Voice-Leading Lattices: These lattices visualize the relationships between chords based on efficient voice leading, showing paths composers can take through musical space (3.11). Movement along these lattices can be systematic, as seen in Brahms (Figure 3.8.3b) or Janáček (3.11, 9.3.2).
• Metrics of Musical Distance: The concept of measuring distance in music is explored, moving beyond traditional harmonic distance (common tones, shared interval content) to emphasize distances based on voice leading, which are considered “extremely versatile” (theory52). Different geometric metrics (taxicab, Euclidean) can yield different measures of distance between collections of pitches (Appendix B).

B. Scales (Chapter 4, §9.1-9.5, §10.4):

• Scales as Rulers: Scales are conceptualized as methods of measuring musical distance, similar to a mathematical metric (4.1, theory176). They provide a framework for understanding relationships between notes.
• Evenness and Scale Construction: The concept of “near evenness” is important in constructing scales like the pentatonic, diatonic, and chromatic scales by compromising between acoustical properties (e.g., perfect fifths) (4.3).
• Scalar Transposition and Inversion: These operations describe how scales can be moved and mirrored while maintaining their internal structure (4.2).
• Scale-First Composition: This approach involves basing musical pieces on specific scales or collections of notes, as seen in works by Debussy, Messiaen, and Shostakovich (9.3).
• Subset Technique: Composers can use scales that share a common set of “fixed” notes, while other “mobile” notes are altered to create different collections and mild polytonality, as illustrated in Grieg’s “Klokkeklang” (9.4.1) and Stravinsky’s “Petit airs” (9.4.3). The Miles Davis Group’s “Freedom Jazz Dance” is given as a seemingly trivial but illustrative example where the shared collection is a perfect fifth or even a single note (9.4.5).
• Different Scales, Different Characters: The choice of scale can significantly impact the character of a piece, as shown by Debussy’s “Voiles” switching between whole tone and pentatonic scales (Figure 5.1.2).
• Polytonality: The juxtaposition or superimposition of multiple scales or tonal centers is discussed as a characteristic of some 20th-century music (332n19, 342, 344, 347n39, 348, 351, 374–378). It is seen as plausible when musical streams do not completely fuse audibly (332n19).

C. Functional Harmony (Chapter 7):

• Functional Tonality as a Probabilistic System: Functional tonal music exhibits regularities in chord progressions that can be described probabilistically, with certain chords being overwhelmingly likely to move to others (7.1). “for the most part, functionally tonal music cycles through the graph in a few stereotypical ways: classical pieces consist largely of progressions such as I–V–I, I–ii–V–I, I–vii°–I, and I–IV–I.”
• Modulation and Key Distance: Functional tonality also involves conventionalized motions between keys, with predictable modulations to related keys (7.4). “just as a V chord is overwhelmingly likely to progress to I, so too is a classical-style major-key piece overwhelmingly likely to modulate to its dominant.”
• The “Down a Third, Up a Step” Sequence: This sequence is discussed as a significant, though sometimes rare, progression in functional tonality (7.3).
• Relationship between Harmony and Counterpoint: The excerpts touch upon the debate regarding whether tonal regularities are primarily harmonic or contrapuntal, presenting different viewpoints: Dualism (harmony and counterpoint are distinct), Monism (harmonic regularities are explained contrapuntally), and Pluralism (both traditional harmonic theory and Schenkerian counterpoint are valid) (7.6). The author leans towards a pluralist perspective, seeing the construction of a harmonic grammar as independent of musical analysis (7.6.2).

D. Chromaticism and Altered Chords (Chapter 6.6, Chapter 8, §10.5-10.6):

• Chromaticism and Gap Filling: Chromaticism can be understood as filling in the “gaps” that exist when moving between chords within a diatonic framework (Figure 6.6.1).
• “Borrowing” and Alternative Explanations: The traditional concept of “borrowing” chords from other keys is viewed with suspicion, as musical keys are not lending libraries, and it can lead to a compartmentalized understanding of chromatic harmony (history and analysis217, history and analysis218). An alternative is to analyze chromaticism through efficient chromatic voice leading (history and analysis217).
• Chromatic Embellishments and Schemas: Common chromatic techniques involve embellishing diatonic progressions with altered chords. Specific schemas (patterns of voice leading) are identified, such as those involving augmented sixths (8.1).
• Thirds-Based Grammar and Schubert: Schubert’s use of the “major-third system” involves efficient chromatic voice leadings between major-third related triads, often liberated from traditional dominant-tonic functionality (8.4).
• Tritone Substitution: This jazz technique is presented as a transformation where a dominant seventh chord is replaced by another dominant seventh chord a tritone away. This affects the notes in the voicing in specific ways (Figure 10.5.5).

E. Jazz Concepts (Chapter 10):

• Macroharmony: In jazz, the term “macroharmony” is used to describe collections of notes that are stable over extended periods, providing a framework for improvisation. These are often related to scales or other collections (history and analysis156).
• “Avoid” Notes: Certain notes within a macroharmony might be considered “avoid” notes, creating dissonance unless treated carefully (354-357).
• Sidestepping: This jazz technique involves shifting abruptly from a passage in one key or harmony to another, often a half step away, before returning to the original key (Figure 10.6.3 illustrates this in Chopin, but the concept is applied to jazz). Wayne Marsh’s solo is given as an example of audacious sidestepping (10.6.2).
• Polyrhythm and Chromaticism in Improvisation: The excerpt on Bill Evans’ solo (10.7.7, 10.7.8, 10.7.12) illustrates the use of polyrhythms and chromatic motion in jazz improvisation, sometimes independent of the underlying chord changes. Evans’ creative variations on stock patterns are highlighted.

III. Important Facts and Quotations:

• “This book is primarily concerned with the theoretical and historical questions.” (theory117) – States the focus of the book.
• “However, we will see that conceptions based on voice leading are extremely versatile and can be useful in a wide range of contexts.” (theory52) – Emphasizes the importance of voice leading as an analytical tool.
• “a musical scale is very similar to what mathematicians call a metric, or a method of measuring musical distance.” (theory176) – Provides a key metaphor for understanding scales.
• “for the most part, functionally tonal music cycles through the graph in a few stereotypical ways: classical pieces consist largely of progressions such as I–V–I, I–ii–V–I, I–vii°–I, and I–IV–I.” (history and analysis227) – Summarizes typical functional progressions.
• “Musical keys are not lending libraries, and there are no borrower’s cards that can be used to verify whether a composer is authorized to use a particular sonority.” (history and analysis217) – Critiques the concept of “borrowing.”
• “It has been applied both to diatonic music lacking harmonic consistency and to diatonic music lacking centricity.” (history and analysis188n31) – Defines “pandiatonic.”
• “The term “polytonal”… seems unobjectionable to me. Some music can be segregated into relatively independent musical streams, each with its own sonic character…” (history and analysis188n32) – Defends the concept of polytonality.
• “Although several eminent composers and theorists have critiqued the notion of polytonality… the term seems unobjectionable to me.” (history and analysis188n32) – Acknowledges and responds to critiques of polytonality.
• “Compositionally, I find pitch-class profi les to be extremely useful devices.” (history and analysis176) – Suggests a practical application of pitch-class analysis.
• “Here the auditory streams do not completely fuse, allowing us to distinguish independent scales, macroharmonies, and even tonal centers in each stream.” (history and analysis188n32) – Explains the perceptual basis for polytonality.
• “The pianist Al Tinney, one of the pioneers of bebop, suggested that dominant seventh chords resolv-ing to predominant sevenths was a hallmark of the bebop harmonic style…” (history and analysis281n23) – Connects a historical observation about bebop harmony to a theoretical concept.
• “I am somewhat suspicious of this metaphor of “borrowing.” Musical keys are not lending libraries, and there are no borrower’s cards that can be used to verify whether a composer is authorized to use a particular sonority.” (history and analysis217) – Reiteration of the critique of “borrowing.”
• “To my mind, the point cannot be emphasized strongly enough: the project of constructing a harmonic grammar is totally independent history and analysis264 of the enterprise of musical analysis—as independent as linguistics is from literary criticism.39” (history and analysis264) – Argues for the independence of theoretical grammar construction from musical analysis.
• “The music thus suggests a kind of polytonal-ity, or clash between independent harmonic streams, in which an upper-register (Afri-can American) “blues scale” contrasts with a lower-register European harmony.” (Jazz 374) – Describes a specific instance of polytonality in jazz.

IV. Areas for Further Exploration (Based on excerpts):

• The detailed mathematical underpinnings of the geometric spaces and metrics discussed (Appendix B).
• The application of these theoretical concepts to a wider range of musical styles and historical periods.
• The perceptual implications of the theoretical frameworks presented (e.g., how listeners actually decode patterns). The linguistic model is mentioned but seen as potentially understating the distinctiveness of human language (theory24).
• More in-depth analysis of specific musical examples used to illustrate the concepts.

V. Conclusion:

These excerpts provide a fascinating glimpse into a theoretical approach to music that emphasizes the use of geometric models and systematic analysis, particularly focusing on voice leading and scalar structures. The author challenges traditional concepts like “borrowing” and offers alternative ways to understand chromaticism and harmonic progression. The inclusion of examples from various musical periods, including jazz, highlights the broad applicability of these ideas. The discussion of different perspectives on the relationship between harmony and counterpoint underscores the complexity of analyzing musical structure.

Geometry, Voice Leading, and Musical Structure

What is “geometry in analysis” and how is it applied to music?

Geometry in analysis, in this musical context, refers to the application of geometric concepts and spaces to understand and visualize musical structures, particularly chords and scales. The source discusses various geometric models, such as two-dimensional chord spaces (like the Möbius strip), three-dimensional chord space (often visualized as a lattice), and even higher dimensional spaces for more complex chords. These geometric representations allow for the analysis of relationships between musical objects (chords, scales) based on concepts of distance and motion, often linked to voice leading. For example, plotting musical phrases on a Möbius strip can reveal underlying musical structure (Figure 3.5.1). The idea is to provide a spatial understanding of musical relationships that can reveal patterns not immediately obvious through traditional notation.

How does the concept of “voice leading” function as a measure of musical distance?

Voice leading, specifically efficient or stepwise voice leading, is presented as a primary way to measure distance between chords. The idea is that the “size” or distance between two chords is determined by the minimal collective movement of individual notes (voices) required to transform one chord into the other. A voice leading that moves just one note by a small interval (like a semitone) is considered “smaller” or closer than one that moves multiple notes by larger intervals. This concept is considered versatile and useful in a wide range of musical contexts, providing a way to compare the efficiency of different harmonic progressions or chord connections. While other concepts of distance exist (harmonic, diatonic), voice leading is highlighted for its broad applicability.

What are “voice-leading lattices” and how are they used in musical analysis?

Voice-leading lattices are geometric structures that represent the relationships between chords based on efficient voice leading. These lattices visualize chord spaces (e.g., three-dimensional for triads, higher dimensional for seventh chords) as interconnected points or nodes, where the lines or edges between the nodes represent single-step or efficient voice leading transformations. Analyzing music through these lattices allows for the visualization of how composers navigate through chord space. For instance, Brahms is described as moving systematically along a lattice in three-note chord space (Figure 3.8.3b). These lattices provide a framework for understanding harmonic movement and can reveal underlying organizational principles in a composer’s work.

How are musical scales conceived of in this context, particularly in relation to geometry and distance?

Scales are presented as analogous to “rulers” or “metrics” in mathematics, providing a method of measuring musical distance. Listeners are described as being aware of both scalar distance (the steps within a specific scale) and log-frequency distance (the absolute distance in semitones). The dual nature of scalar music lies in this simultaneous perception – recognizing notes as being a certain number of scale degrees apart while also having a specific intervallic distance in semitones. The text also discusses how scales introduce “deformations” as chords move through musical space, suggesting a geometrical impact. Different scales are analyzed for their “evenness” and interval content, and geometric representations like lattices can be used to visualize relationships between different scales.

What is “functional tonality” and how is it described through voice leading and probability?

Functional tonality, particularly in classical music, is characterized by conventionalized motions on both the chord and key levels. Chord progressions in functional tonality often follow predictable patterns, such as V moving to I. The source suggests that the regularities in functional harmony, which appear to follow “harmonic rules,” can also be explained through contrapuntal (voice-leading) processes. Probability is used to describe the likelihood of certain chord transitions within functional tonality, with some progressions being overwhelmingly more common than others (Figure 7.1.2, Figure 7.5.1). This view highlights how efficient voice leading underlies and potentially explains the observed regularities and “strong” motions within functional harmony.

How is chromaticism approached in this framework, particularly in relation to altered chords and modulation?

Chromaticism is discussed as involving notes or chords outside the diatonic scale. The text challenges traditional notions of “borrowing” chords from other keys, suggesting instead that chromatic chords can be understood through efficient chromatic voice leading that fills in gaps within the diatonic or chromatic scale. Chromatic techniques are seen as applying to various progressions, embellishing standard patterns. Modulation, the process of changing keys, is also described as involving conventionalized motions, with composers frequently modulating to related keys (Figure 7.4.1). Chromatic voice leading is presented as a mechanism connecting seemingly distant chords and facilitating these modulations, sometimes taking “scenic detours” in tonal space (Figure 8.3.2).

What is the “subset technique” and how is it used by composers?

The subset technique is a compositional approach where a composer uses scales that share a common collection of prominent notes. These shared notes remain fixed and stable across changes in scale, while other notes are altered to create different scales or collections. This creates a sense of consistency amidst scalar variation. Examples include Grieg and Stravinsky using a fixed collection of notes within different diatonic scales, or Miles Davis improvising over a static harmony while exploring different scales that contain the underlying notes. The subset technique allows composers to create complex and varied textures while maintaining a degree of tonal grounding through the shared fixed pitches.

How are geometric and voice-leading concepts applied to analyze music from different periods, including jazz?

The source demonstrates the application of geometric and voice-leading concepts to music across various historical periods and styles. Classical music is analyzed through voice-leading lattices, functional harmony, and modulation patterns. Twentieth-century music is discussed in terms of expanded scalar vocabularies, chord-first composition, and the subset technique. Jazz music is also analyzed, with examples like Bill Evans’ improvisations demonstrating complex chromatic movements, sidestepping, and the use of voice-leading patterns over underlying harmonies. This illustrates the versatility of these analytical tools in uncovering structural principles and compositional choices in a wide range of musical styles, including those that move beyond traditional functional tonality.

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Richard Sorce’s Music Theory for the Music Professional aims to equip musicians with a comprehensive understanding of music’s fundamental elements and compositional tools. The text systematically explores concepts from basic pitch representation and rhythm to advanced topics like harmony, scales, melody, and structure, drawing comparisons between common-practice and popular genres. It emphasizes analyzing existing music to understand theoretical principles, rather than prescribing rules for composition. The book includes numerous musical examples, self-tests, and exercises to reinforce learning, ultimately intending to enhance performance, creativity, and appreciation of music across various styles.

Fundamentals of Music Theory

Based on the provided excerpts, the fundamentals of music theory encompass a wide range of interconnected concepts. These fundamentals provide the bedrock for understanding the structure, function, and creation of music in Western practice. The text assumes a basic prior knowledge of music theory, but Chapter 1 serves as a review of essential concepts.

Here’s a discussion of some key areas within music theory fundamentals as presented in the sources:

• Rhythm: Rhythm, derived from the Greek word for “flow,” is the element of music that deals with duration and is considered a basic and important aspect of musical structure. It operates on micro (small), meso (middle), and macro (large) time segments. Understanding rhythm involves concepts such as tempo (speed of music), pulse (the underlying beat), meter (the grouping of beats), and time signatures (indicating the meter). The sources also mention subdivisions of the beat into duplets and triplets, as well as more complex groupings. The effect of dots and double dots on note and rest values is also fundamental to rhythmic understanding. Concepts like anacrusis (pickup notes) and syncopation (accenting off-beats) are also important rhythmic considerations.
• Pitch Representation: Pitches, which define the highness or lowness of a sound based on vibrations per second, are represented by the first seven letters of the alphabet (A-G) in Western practice. These pitches are displayed on a staff consisting of five lines and four spaces. Clef signs (treble, bass, alto, tenor, and the movable C clef) are placed at the beginning of the staff to determine the location of specific pitches. Ledger lines extend the staff to accommodate pitches outside the normal range. Sharps, flats, and naturals alter the pitch of a note. The distance between pitches is measured in half steps and whole steps, and notes that sound the same but are spelled differently are enharmonic.
• Scales, Keys, and Modes: A scale is a series of pitches in ascending or descending order. The text discusses various scale structures, including the major scale and minor scales (natural, harmonic, and melodic), outlining that the major and minor scale systems have been predominantly used in the last few hundred years. The whole-step/half-step patterns define these scales. Relative minor scales are derived from the sixth degree of the major scale. Keys establish a tonal center based on these scales, and key signatures indicate the sharps or flats present in a key. The circle of fifths visually represents the relationships between keys. Modes are different variations of the diatonic scale, with distinct whole-step/half-step patterns, dating back to antiquity and still used in contemporary styles. The ability to understand the relationships among keys is a fundamental step in conceptualizing music.
• Intervals: An interval is the distance between two pitches, occurring melodically (horizontally) or harmonically (vertically, also called dyads). Interval distance is calculated numerically, counting both the lower and upper pitches. The quality of an interval (major, minor, perfect, augmented, diminished) is often determined by considering the major key of the lower note. Within a major scale, 2nds, 3rds, 6ths, and 7ths are major intervals, while unisons, 4ths, 5ths, and octaves are perfect intervals. Intervals can be altered by half or whole steps, resulting in minor, diminished, or augmented qualities. Certain intervals are considered consonant (stable, not requiring resolution), while others are dissonant (unstable, often implying a need for resolution).
• Chords: A triad is a harmonic structure of three different notes, each belonging to a different letter class, typically consisting of a root, a third, and a fifth. Triads have four qualities: major, minor, diminished, and augmented, determined by the quality of the intervals between the notes (major or minor thirds). Primary triads are built on the tonic (I), subdominant (IV), and dominant (V) degrees of the scale, while secondary triads are built on the supertonic (ii), mediant (iii), submediant (vi), and leading tone (vii°). Seventh chords are formed by adding another third above the triad, resulting in a four-note chord. Understanding the qualities and functions of triads and seventh chords within different scales and keys is crucial for harmonic analysis and composition.
• Voice Leading: Voice leading refers to the linear (horizontal) movement of individual melodic lines (voices) within a vertical context of intervals and chords. It involves considering the pitch tendency of individual scale degrees, their relative stability, and how they progress to other pitches. Principles of good voice leading aim for smooth connections between chords, often utilizing contrary motion (voices moving in opposite directions) and oblique motion (one voice stationary while others move). Limiting leaps in individual voices and avoiding certain parallel intervals (like perfect fifths and octaves) are also important considerations.
• Harmonic Progression: Harmonic progression describes the way chords move from one to the next, creating tension and resolution within a musical work at micro, meso, and macro levels. Chords, like individual notes, have tendencies and affinities within tonal contexts. Concepts like chord substitution (replacing one chord with another of similar function) and understanding root movement by specific intervals (e.g., descending fifths in circle progressions) are important aspects of harmonic progression. Identifying structural harmonies (essential for the framework) and embellishing harmonies (adding color and interest) is also a key skill.

These fundamental concepts, as outlined in the initial chapters of the source, provide the necessary groundwork for understanding more advanced topics in music theory, such as chromatic harmony, modulation, and musical form, which are explored in later chapters. A thorough understanding of these fundamentals is crucial for any aspiring professional musician to improve performance, creative endeavors, and overall comprehension of music.

Fundamentals of Harmony and Chords

Based on the provided excerpts, harmony refers to the simultaneous sounding of two or more notes. The study of harmony encompasses the vertical aspects of music, including chords and the relationships between them. It also involves the forward progressive thrust of music resulting from two or more melodic lines moving simultaneously. Harmonic progression, or harmonic movement, describes the way chords move from one to the next, creating tension and resolution at micro, meso, and macro levels within a musical work.

Chords are fundamental to harmony. The term “chord” applies to the simultaneous sounding of three or more pitches. However, a dyad, or two-note harmonic interval, is sometimes also referred to as a chord. Chords constructed of stacked thirds are known as tertian harmony.

Here are different types and aspects of chords discussed in the sources:

• Triads: A triad is a harmonic structure of three different notes, typically consisting of a root, a third, and a fifth. Triads have four qualities: major, minor, diminished (indicated by a degree sign °), and augmented (indicated by a plus sign +). The quality is determined by the intervals between the notes (major or minor thirds). In tonal music analysis, Roman numerals are used to identify triads built on different scale degrees. Upper-case Roman numerals denote major triads (e.g., I, IV, V), lower-case denote minor triads (e.g., ii, iii, vi), lower-case with a degree sign denote diminished triads (e.g., vii°), and upper-case with a plus sign denote augmented triads (e.g., III+). Primary triads are built on the tonic (I), subdominant (IV), and dominant (V) degrees. Secondary triads are built on the supertonic (ii), mediant (iii), submediant (vi), and leading tone (vii°).
• Seventh Chords: Seventh chords are formed by adding another third above a triad, resulting in a four-note chord. The quality of the seventh chord depends on the quality of the triad and the added seventh interval (major, minor, diminished). The sources mention various seventh chord types such as dominant seventh chords, minor seventh chords, major seventh chords, minor-minor seventh chords, half-diminished seventh chords (viiø7), and fully diminished seventh chords (viiο7). Dominant seventh chords often have a dominant function.
• Upper-Partial Chords: Continuing the process of tertian stacking beyond seventh chords leads to ninth, eleventh, and thirteenth chords. These are also referred to as upper-partial harmonies. These chords contain root, third, fifth, seventh, ninth, eleventh, and thirteenth scale degrees, although not all factors are always necessary. These chords can be diatonic or hybrid (resulting from pitch alteration).
• Secundal, Quartal, and Quintal Harmony: Besides tertian harmony, other systems of harmonic construction exist. Secundal harmony involves chords built in seconds. There are four types of three-note secundal chords: major-major, major-minor, minor-major, and minor-minor. Multinote secundal chords contain more than three different pitches, and those in close voicing are called clusters. Quartal harmony involves chords built in fourths. There are three types: perfect-perfect, perfect-augmented, and augmented-perfect. Quintal harmony involves chords built in fifths. A quintal chord becomes dissonant upon the addition of a third factor.

Harmonic Progression and Movement:

• Chord progression is a general term for chord-to-chord movement. The movement of one chord to another is a matter of personal choice by the composer, but it is often guided by chordal tendencies and the desired level of tension and resolution.
• Circle progression is a chord progression in which the root relationship is a descending fifth.
• Chord movement can be described as progressive (indicating forward motion and often resolution) or retrogressive (indicating backward motion and often increased tension) based on the root movement by specific intervals.
• Chord substitution is the alternate choice of one chord for another, often based on pitch similarities. Substitution can occur by extension of a lower or higher third.

Voice Leading:

• Voice leading is the linear (horizontal) movement of individual melodic lines (voices) within a vertical context of intervals and chords. It considers the pitch tendency of individual scale degrees and aims for smooth connections between chords. Principles of good voice leading include contrary motion (voices moving in opposite directions), oblique motion (one voice stationary), limiting leaps in individual voices, and avoiding parallel perfect fifths and octaves.

Harmonic Function:

• Structural harmonies appear at strategic points in the melody and provide a sense of tonality. Embellishing harmonies are not essential to the tonal establishment and do not provide the primary structural components.
• Chords, like individual notes, possess tendencies and affinities within tonal contexts.

Chromatic Harmony:

• Chromaticism involves the use of notes outside the diatonic scale of the prevailing key.
• Borrowed chords (BCs) are taken from the parallel key of the prevailing music.
• Change-of-quality (CQ) chords are nondiatonic, nonembellishing chords that are not derived from the parallel key.
• Secondary dominants (e.g., V/V) and secondary leading-tone chords (e.g., vii°/V) are chromatic chords that tonicize a diatonic chord other than the tonic.
• The Neapolitan chord (N6) is a chromatic chord, typically a major triad built on the lowered second degree of the scale.
• Augmented sixth chords (German, French, Italian) are chromatic chords containing an augmented sixth interval that resolves outward by half step.

Harmonic Relationships:

• Chord movement can occur through stepwise, mediant, and tritone relationships, which are less conventional than movement by fourths and fifths.
• Planing is the movement of chords in whole or half steps while maintaining a particular voicing. Parallelism maintains chord voicing but is not restricted to whole and half step movement.

Harmonizing a Melody:

• Harmonizing a melody involves selecting appropriate harmonic content to accompany it. This process requires understanding the melody’s style, period, tempo, and intended instrument or voice. Identifying stressed melody notes can help determine suitable structural harmonies, often the tonic (I or i) and dominant (V) chords. Contrary motion between the bass and soprano is often desirable.

Tension and Resolution:

• Harmonic progression is essentially the control of tension and resolution. Consonant intervals and chords tend to be stable, while dissonant intervals and chords often imply a need for resolution.

This overview provides a comprehensive discussion of harmony and chords based on the fundamentals presented in the provided excerpts. The sources delve deeper into each of these concepts in subsequent chapters.

Melody and Musical Structure: An Overview

Melody and Structure

The sources provide a detailed overview of both melody and musical structure, highlighting their components, characteristics, and interrelationships.

Melody

• A melody is defined as a succession of pitches in a linearly and temporally controlled parameter. It normally consists of pitched sounds arranged into units.
• The perception of melody is influenced by cultural conventions, but its quantitative aspects can still be discussed.
• A melody is built upon a hierarchy of units:
• Cells are the tersest groups of notes. An example shows cells with different intervallic modifications.
• Motives are the shortest complete fragments, self-contained melodic ideas that usually occur at least twice and can be modified. It’s not always easy to identify a motive without tracing its occurrences. Examples of motives and their relation to phrases are provided.
• Phrases are units of melodic material comparable to clauses in a sentence. A phrase can sound complete or incomplete and can consist of cells, motives, or both. Cadences, points of arrival achieved by harmonic, melodic, or rhythmic variation, usually delineate phrases. Less significant points of delineation are known as caesuras. Phrases often appear in pairs as antecedent and consequent.
• Melodies appear in various shapes and lengths, often resulting from the culture and epoch of their composition. They can be:
• Derived from conventional scales, modes, and composer-created pitch sequences.
• Terse, moderate, or protracted in length.
• Have wide or compact range (distance between lowest and highest notes) and wide or compact tessitura (location of the majority of notes).
• Involve primarily stepwise (scalar) motion, skipping motion, or primarily leaping motion.
• Diatonic or chromatic.
• Primarily ascending, descending, or level in direction.
• Undulating (wavelike), sawtooth in design, or pointillistic (sparse texture, brief phrases, with angular and widely scattered melodic lines).
• Rhythmically active or calm.
• Combinations of these types.
• Melodic development involves manipulating and developing an initial idea (cell, motive, or phrase) to extend the melodic line. Techniques include:
• Transposition (restatement at a different pitch level).
• Inversion (reversal of melodic intervals).
• Retrograde (statement in reverse order).
• Retrograde inversion (reversal of melody and melodic intervals).
• Augmentation (proportional increase in durational values).
• Diminution (proportional decrease in durational values).
• Interpolation (addition of new melodic material within a motive).
• The accurate identification of melodic components depends on the context. A “wonderful” melody might be difficult to convey autonomously because the performer often relies on other supporting elements. The perception of a group of sounds as a melody depends on fulfilling certain human needs.

Musical Structure

• Musical structure begins with the single note, which possesses characteristics like pitch, duration, intensity, and timbre that can be incorporated into the work’s development.
• Every element of musical notation relates to the structural character of a work, including pitch, time values, chords, rhythm, tempo, texture, dynamics, cells, motives, phrases, periods, and sections. The study of music intrinsically mandates the study of structure.
• Understanding musical structure involves recognizing the relationships among various elements, their similarities or dissimilarities, the repetition or nonrepetition of elements at the micro, meso, and macro levels, and the factors contributing to continuity, cohesiveness, and logic.
• Structural elements exist at different levels:
• Micro level: Includes syntactical components like notes, rests, time values, intervals, chords, and rhythm, as well as signs like ties, slurs, fermatas, repeat signs, and tempo and intensity directives.
• Meso level: Examined in the context of harmonic movement in phrases.
• Macro level: Involves the achievement of the harmonic goal of a section, movement, or entire work.
• Cadences function like punctuation marks in sentences, helping to delineate structural divisions. They contribute to the sense of closure of melodic or thematic segments. Various types of cadences exist.
• Phrases can be combined to form larger structures:
• Periods consist of at least two phrases, often an antecedent and a consequent, evoking a sense of completeness. Types of periods include symmetrical, parallel, double, asymmetrical, and contrasting. The final cadence of a period typically provides resolution.
• Phrase groups consist of at least two similar phrases where no phrase ends with a conclusive cadence.
• Phrase chains consist of three or more dissimilar phrases, none of which ends with a conclusive cadence.
• Two fundamental structures prevalent in Western music are binary and ternary.
• Binary structure is a fundamental foundation for more complex forms. Rounded binary structure is also mentioned.
• Ternary structure satisfies a human condition related to familiarity, defined by the Statement, Departure, Return (SDR) principle.
• Popular song structure typically includes verse, chorus, and bridge.
• Even in improvisation, awareness of structural development is crucial for success.
• The human ear and psyche are determining factors in the conceptual, perceptual, and developmental aspects of music, mandating continuity and coherence in musical structure.

In essence, melody provides the linear progression of musical ideas, built from smaller units into larger phrases, while structure provides the framework and organization for these ideas, grouping phrases into periods and larger sections to create a sense of unity, contrast, and overall form. The interplay between melodic content and structural design is fundamental to the creation and understanding of music.

Understanding Musical Form: An Analysis

Musical form analysis, as suggested by the sources, involves understanding the organization of a musical work at various levels, from the smallest units like notes to larger sections and entire movements. The study of music intrinsically mandates the study of structure.

Key aspects of musical form analysis based on the sources include:

• Identifying structural elements: This involves recognizing and analyzing components such as notes, rests, time values, intervals, chords, rhythm, tempo, texture, dynamics, cells, motives, phrases, periods, and sections. Even a single note, with its characteristics of pitch, duration, intensity, and timbre, can be a structural element.
• Understanding relationships: Form analysis requires recognizing the relationships among these various elements, including their similarities or dissimilarities, and the repetition or nonrepetition of elements at the micro, meso, and macro levels.
• Analyzing melodic structure: Melody is built upon a hierarchy of units: cells (the tersest groups of notes), motives (the shortest complete fragments), and phrases (units comparable to clauses). Identifying how these melodic units are constructed and developed is crucial for understanding form. Melodic development techniques like transposition, inversion, retrograde, retrograde inversion, augmentation, and diminution contribute to the shaping of musical form.
• Recognizing cadences: Cadences, points of arrival delineated by harmonic, melodic, or rhythmic variation, function like punctuation marks, helping to delineate structural divisions and contribute to the sense of closure of melodic or thematic segments.
• Analyzing phrase structure: Phrases can be combined to form larger structures like periods (at least two phrases, often antecedent and consequent, with a sense of completeness) and phrase groups or phrase chains. Understanding the types of periods (symmetrical, parallel, double, asymmetrical, contrasting) and how phrases relate to each other is essential for form analysis.
• Identifying basic formal types: The sources highlight binary and ternary structures as fundamental forms in Western music. Binary structure provides a basic foundation, while ternary structure follows the Statement, Departure, Return (SDR) principle. In popular music, common structures involve verse, chorus, and bridge.
• Considering harmonic progression: The way chords progress (at the micro level), the harmonic movement in phrases (at the meso level), and the achievement of the harmonic goal (at the macro level) are compelling forces in sustaining tension and resolution, thus impacting musical form. Identifying structural harmonies is important.
• Understanding modulation: The process of changing the key of a musical work (modulation) is a significant aspect of form in many compositions. Analyzing the types of modulation (common-chord, chromatic, enharmonic) and their placement helps in understanding the structural design.
• Considering the role of the ear and psyche: The human ear’s need for continuity and coherence shapes musical events and structural organization. The perception of musical structure is influenced by these factors.
• Analyzing structure in improvisation: Even in improvisation, an awareness of structural development is crucial.

Therefore, musical form analysis, according to the sources, is a multifaceted process involving the identification of structural units at different levels, understanding their relationships, and recognizing how these elements contribute to the overall shape and coherence of a musical composition. It considers melodic organization, harmonic movement, cadential punctuation, phrase and period structures, fundamental formal types, and the impact of key changes, all within the context of human perception and the need for musical logic.

Common Practice Music: Foundations and Influence

Drawing on the information provided in the sources, let’s discuss common practice music.

Definition and Historical Context:

Common practice music encompasses musical works written approximately between 1650 and 1875. This period, which includes the Baroque, Classical, and Romantic eras [84, note 1], provided the fundamental principles and practices that have significantly influenced the course of Western music and continue to do so. The majority of discussions and examples in the source material are drawn from this era.

Influence on Later Music:

One of the principal themes observed is that Western music has changed minimally throughout the past several hundred years. The book aims to demonstrate that there is no vast difference between the syntaxes of traditional (common practice) practice and popular music. The elements and syntax of music have remained essentially the same for many centuries. Every element of syntax available in 1700 is still used today; what has changed is the style in which these syntactical elements are employed. The methods and procedures established during the common practice period, when vertical harmony was first introduced, have been the basis for nearly all Western composition in various styles, including Baroque, Classical, Romantic, Impressionist, contemporary, popular, jazz, and rock.

Similarities with Popular Music:

A comparison reveals little, if any, difference in fundamental musical syntax between common practice and popular styles. The structural harmonic content that contributes most to establishing and reinforcing the key center has remained largely unchanged, with the tonic-dominant relationship being prevalent. While chromatic embellishment appears more frequently in popular works, much of it can be traced to common-practice syntax and technique, as seen in the example of tritone substitution, which shows similarities to the Neapolitan chord used in common practice. Similarly, the functional augmented 6th chord in popular music often appears as a half-step embellishment above the dominant, mirroring common practice. Even the uses of voice leading in both styles are quite similar, although modern application might appear less rigid. Suspensions in popular music can also appear in the same manner as in the common-practice genre, though the preparation step may sometimes be absent.

Characteristics and Practices:

• Harmony: Harmonic relationship is a primary structural element in the development of common practice music. The tonic-dominant relationship is a unifying force.
• Voice Leading: The study of voice leading, often presented in chorale style (SATB) that was brought to its most developed stage by J.S. Bach, is essential for understanding music composition, regardless of style. The principles extend beyond chorale style to linear and vertical contexts.
• Melody: Melodies in common practice music, like in other styles, consist of pitched sounds arranged into units like cells, motives, and phrases.
• Structure: Basic structural terms like cell, motive, phrase, period, cadence, section, and movement are prevalent in nearly every style of Western music, with much of this terminology emanating from the common-practice period.
• Cadences: Cadences serve to delineate structural divisions.
• Diatonicism and Chromaticism: While many successful common practice works are primarily diatonic, music has generally evolved chromatically since this period.

Nuances and Differences:

While the fundamental syntax is similar, common practice music is sometimes considered more rigid than jazz, though performers still had the ability to convey personal ideas within the established syntax. Traditional common practice permitted less freedom in the movement of individual voices compared to more modern usage. Modern harmonic structures can exhibit a greater level of dissonance and more complex tertian harmony than typically found in common practice.

In conclusion, common practice music represents a foundational period in Western music history, establishing principles of harmony, voice leading, melody, and structure that continue to resonate in various genres today, including popular music. While musical styles have evolved, the underlying syntactical elements largely have their roots in the common practice era. Understanding common practice is crucial for any aspiring musician as it provides the theoretical background that serves as a foundation for all musical genres.

Harmony and Rhythm Fundamentals: A Study Guide

Quiz

1. Define pitch and explain how it is represented in musical notation.
2. Describe the function of a clef in musical notation and name the four clefs in current use.
3. Explain the difference between simple and compound meter, and provide an example of each time signature.
4. Define nonharmonic tones and list three common types of nonharmonic tones discussed in the text.
5. Describe the construction of a major scale, including the whole and half step pattern.
6. What is a triad, and what are the four qualities of triads discussed in the text?
7. Explain the function of Roman numerals in harmonic analysis within a major key.
8. Define a seventh chord and list the three qualities of seventh intervals mentioned in the text.
9. Explain the concept of tonicization and how secondary dominant chords function within a key.
10. Define modulation in music and describe the difference between modulation and tonicization.

Answer Key

1. Pitch is the highness or lowness of a sound, based on the number of vibrations per second. In musical notation, pitches are represented by note symbols placed on, above, or below a staff.
2. A clef sign determines the location of a particular pitch on the musical staff. The four clefs in current use are the treble (G) clef, bass (F) clef, alto (C) clef, and tenor (C) clef.
3. Simple meter has beats that are divided into two equal parts, while compound meter has beats that are divided into three equal parts. An example of simple meter is 4/4, and an example of compound meter is 6/8.
4. Nonharmonic tones are pitches that are not part of the underlying chord at a given moment. Three common types are passing tones (stepwise motion between chord tones), neighboring tones (stepwise motion away from and back to a chord tone), and appoggiaturas (approached by leap, resolved by step).
5. A major scale consists of a specific pattern of whole and half steps: whole, whole, half, whole, whole, whole, half. For example, the C major scale is C-D-E-F-G-A-B-C.
6. A triad is a harmonic structure consisting of three pitches, each belonging to a different letter class, typically formed by stacking two intervals of a third. The four qualities are major, minor, diminished, and augmented.
7. In harmonic analysis of a major key, Roman numerals represent chords built on each scale degree. Upper-case numerals indicate major triads, lower-case indicate minor triads, lower-case with a degree sign indicate diminished triads, and upper-case with a plus sign indicate augmented triads.
8. A seventh chord is a chord consisting of a triad with an added seventh interval above the root. The three qualities of seventh intervals mentioned are major seventh, minor seventh, and diminished seventh.
9. Tonicization is the momentary emphasis of a key center without fully establishing a new key. Secondary dominant chords are dominant chords (V or V7) built on scale degrees other than the tonic, used to create a temporary pull towards the chord they precede.
10. Modulation is a change of key center within a musical work that is firmly established. Tonicization is a temporary emphasis on a chord other than the tonic, while modulation involves a more permanent shift to a new tonal center.

Essay Format Questions

1. Discuss the interconnectedness of rhythm and melody as fundamental elements in musical structure, providing specific examples of how rhythmic patterns can influence the perception and character of a melodic line.
2. Trace the evolution of harmonic language from the basic triad to more complex seventh and extended chords, explaining how these developments expanded the expressive possibilities in Western music.
3. Analyze the functions and resolutions of diminished chords (vii°, vii°7, viiø7) within tonal harmony, detailing their role in creating tension and directing harmonic motion towards points of stability.
4. Compare and contrast the concepts of modulation and key regionalization, discussing the techniques composers employ to create a sense of a new tonal area and the factors that determine whether a true modulation has occurred.
5. Explore the various types and functions of nonharmonic tones in melodic and harmonic contexts, explaining how they contribute to musical interest, embellishment, and the creation of melodic contour.

Glossary of Key Terms

• Accidental: A symbol (sharp, flat, natural) that alters the pitch of a note.
• Alto Clef (C Clef): A clef that places middle C on the third line of the staff.
• Anacrusis: One or more unstressed notes at the beginning of a musical phrase that lead into the downbeat.
• Appoggiatura: A nonharmonic tone approached by a leap and resolved by a step, often occurring on a strong beat.
• Augmented Triad: A triad consisting of a root, a major third, and an augmented fifth.
• Bass Clef (F Clef): A clef that places F below middle C on the fourth line of the staff.
• Beat: The basic rhythmic unit in music, often felt as a pulse.
• Borrowed Chord: A chord from a parallel key (major or minor with the same tonic) used in the current key.
• Cadence: A harmonic or melodic point of rest or closure at the end of a musical phrase, section, or piece.
• Chromaticism: The use of notes outside the diatonic scale of the prevailing key.
• Clef: A musical symbol placed at the beginning of a staff to indicate the pitches of the lines and spaces.
• Common Time: Another term for 4/4 time signature.
• Compound Meter: A meter in which the beat is divisible by three.
• Consonance: Intervals or chords that sound stable and agreeable.
• Diatonic: Notes, intervals, or chords that belong to the prevailing key.
• Diminished Triad: A triad consisting of a root, a minor third, and a diminished fifth.
• Dissonance: Intervals or chords that sound unstable and create tension, often requiring resolution.
• Dominant: The fifth degree of a scale. Also refers to the chord built on the fifth degree.
• Enharmonic: Two different spellings of the same pitch (e.g., C# and Db).
• Grand Staff: A combination of the treble and bass staves, commonly used for piano music.
• Half Step: The smallest interval in Western music, the distance between two adjacent keys on a piano keyboard (including black keys).
• Harmonic Minor Scale: A minor scale with a raised seventh degree.
• Key: The tonal center of a piece of music, identified by a tonic pitch and a characteristic scale.
• Ledger Lines: Short lines added above or below the staff to extend its range.
• Major Scale: A diatonic scale with the pattern whole-whole-half-whole-whole-whole-half steps.
• Major Triad: A triad consisting of a root, a major third, and a perfect fifth.
• Measure: A segment of music contained between two bar lines.
• Melody: A sequence of single notes that is musically satisfying.
• Meter: The organization of musical time into regular patterns of stressed and unstressed beats.
• Minor Scale: A diatonic scale with several variations; the natural minor has the pattern whole-half-whole-whole-half-whole-whole steps.
• Minor Triad: A triad consisting of a root, a minor third, and a perfect fifth.
• Modulation: The process of changing from one key to another in a musical composition.
• Motive: A short, recurring musical idea.
• Natural Minor Scale: A minor scale with the pattern whole-half-whole-whole-half-whole-whole steps.
• Nonharmonic Tone: A note that is not a member of the chord occurring at a particular moment.
• Passing Tone: A nonharmonic tone that moves stepwise between two chord tones.
• Pedal Tone: A sustained note, typically in the bass, over which harmonies change.
• Phrase: A musical unit that forms a complete musical thought, often ending with a cadence.
• Pitch: The highness or lowness of a sound.
• Relative Minor: The minor key that shares the same key signature as a major key; its tonic is the sixth degree of the major scale.
• Rest: A symbol indicating a period of silence in music.
• Retardation: A nonharmonic tone that is held over from a consonant chord and resolves upward by step to another consonant chord.
• Rhythm: The organization of musical sounds and silences in time.
• Scale: A series of pitches arranged in ascending or descending order.
• Secondary Dominant: A dominant chord (V or V7) that functions to tonicize a chord other than the tonic.
• Seventh Chord: A chord consisting of a triad plus a seventh interval above the root.
• Sharp: An accidental (#) that raises a note by a half step.
• Simple Meter: A meter in which the beat is divisible by two.
• Staff: A set of five horizontal lines and four spaces on which musical notes are written.
• Subdominant: The fourth degree of a scale. Also refers to the chord built on the fourth degree.
• Submediant: The sixth degree of a scale. Also refers to the chord built on the sixth degree.
• Supertonic: The second degree of a scale. Also refers to the chord built on the second degree.
• Suspension: A nonharmonic tone that is held over from a consonant chord and resolves downward by step to another consonant chord.
• Tempo: The speed at which music is performed.
• Tenor Clef (C Clef): A clef that places middle C on the fourth line of the staff.
• Tetrachord: A series of four notes with a specific pattern of whole and half steps.
• Tonic: The first and most stable degree of a scale; the tonal center of a key. Also refers to the chord built on the first degree.
• Tonicization: The momentary emphasis of a key center by the use of secondary dominants or leading-tone chords.
• Treble Clef (G Clef): A clef that places G above middle C on the second line of the staff.
• Triad: A three-note chord consisting of a root, a third, and a fifth.
• Whole Step: An interval consisting of two half steps, the distance between two white keys on a piano keyboard with one black key in between.

Briefing Document: Music Theory Concepts

This briefing document summarizes key concepts from the provided text, focusing on fundamental elements of music theory including pitch, rhythm, scales, harmony (triads, seventh chords), melodic construction, musical structure, chromaticism (secondary dominants, diminished chords, Neapolitan and augmented sixth chords), extended harmony (ninth, eleventh, and thirteenth chords), harmonic movement, and modulation.

I. Fundamentals of Pitch Notation and Rhythm:

• Musical Alphabet and Pitch: The Western musical alphabet consists of A-B-C-D-E-F-G, each representing a pitch, which is the “highness or lowness of a sound, based on the number of vibrations per second.”
• Staff and Clefs: Pitches are placed on a five-line staff with four spaces. Ledger lines extend the staff. A clef sign determines the location of specific pitches. The four main clefs are treble (G), bass (F), alto (C on the third line), and tenor (C on the fourth line). The grand staff combines the treble and bass clefs.
• Note Values and Rhythm: Rhythm encompasses all elements of duration in music and is considered the “most basic and important element in the structure of music.” Note heads, stems, and flags represent different note values. Eighth notes and shorter durations can be beamed or flagged.
• Meter: Meter refers to the grouping of beats in a measure, indicated by a time signature. Simple meters have beats divisible by two, while compound meters have beats divisible by three (often felt as groupings of three within a larger beat unit). For example, “in the literal interpretation of 6/8, for example, six beats occur in each measure and the eighth note receives one beat. However, at fast tempi it is common to group sets of three beats into beat units. In 6/8, then, beats 1, 2, and 3 are combined to form the first beat unit, and beats 4, 5, and 6 form the second beat unit. Thus, 6/8 is interpreted in ‘two’ (compound duple meter).”
• Tempo and Pulse: These terms relate to the speed and underlying beat of the music, respectively.
• The Half-Beat Concept: A beat can be divided into two equal halves, a “down” and an “up” part, representing a binary event in the temporal organization of music. “For any of these methods a beat consists of only two parts—the ‘down’ part of the beat and the ‘up’ part; thus, a beat is a binary event.”
• Syncopation and Anacrusis: Syncopation involves the displacement of the normal accent, while an anacrusis is a pickup note or notes before the first strong beat of a phrase.

II. Scales, Keys, and Modes:

• Scales: A scale is a series of pitches in ascending or descending order, usually within an alphabetical distance of three letters between successive pitches. Major and minor scales are predominant.
• Major and Minor Scales: The text outlines the whole and half step patterns for major, natural minor (1 – ½ – 1 – 1 – ½ – 1 – 1), harmonic minor (1 – ½ – 1 – 1 – ½ – 1½ – ½), and melodic minor (1 – ½ – 1 – 1 – 1 – 1 – ½).
• Relative Minor: Every major scale has a relative minor that begins on the sixth degree of the major scale and shares the same key signature.
• Other Scales: The text briefly mentions the blues scale (major scale with lowered third and seventh, often with a raised fourth), diminished scale (alternating whole and half steps), whole-tone scale, and pentatonic scale.
• Tetrachords: Major scales can be divided into two tetrachords (four-note groups) which are crucial in understanding key relationships and key signatures. The second tetrachord of a major scale is the first tetrachord of the major scale a perfect fifth above.
• Key Signatures: Sharps or flats at the beginning of the staff indicate the key of the music, defining which notes are consistently raised or lowered.
• Enharmonic Equivalents: Different note names that represent the same pitch (e.g., E# and F natural).

III. Intervals:

• Determining Intervals: Intervals are the distance between two pitches. They are calculated by considering the alphabetical distance and then the specific number of half steps. “When determining an interval, it is helpful to calculate the distance between the notes by considering the major key of the lower note.”
• Interval Qualities: Intervals can be major, minor, perfect, augmented, or diminished, depending on their size in half steps relative to the major scale of the lower note.

IV. Triads:

• Definition: A triad is a “harmonic structure consisting of three pitches each belonging to a different letter class.” It is typically formed by stacking two intervallic thirds. The notes are the root, third, and fifth.
• Triad Qualities: There are four types of triads:
• Major: Root, major third, perfect fifth.
• Minor: Root, minor third, perfect fifth. “If the first 3rd (which is a major 3rd) of a major triad is decreased by a half step, thus resulting in a minor 3rd, the triad becomes minor.”
• Diminished: Root, minor third, diminished fifth. “If the second 3rd is also decreased, thereby creating another minor 3rd, the triad is smaller still, hence resulting in a diminished triad.”
• Augmented: Root, major third, augmented fifth. “Reverting back to the major triad, if the second 3rd is increased by a half step, thereby creating two major 3rds, the resultant triad is larger in intervallic distance than the major triad and hence is augmented.”
• Numerical Designation (Roman Numerals): Roman numerals are used in analysis to represent triads built on each scale degree. Upper-case indicates major, lower-case indicates minor, lower-case with a degree sign (°) indicates diminished, and upper-case with a plus sign (+) indicates augmented.
• Primary and Secondary Triads: Primary triads are built on the tonic (I), subdominant (IV), and dominant (V). Secondary triads are built on the supertonic (ii), mediant (iii), submediant (vi), and leading tone (vii°).
• Inversions and Spacing: The order of the notes in a triad (root position, first inversion, second inversion) and their spacing affect the sound.
• Chord Symbols in Popular Music: Letters and symbols (e.g., Cmaj, Dmin, G7) are used to indicate chords in popular music.

V. Seventh Chords:

• Definition: A seventh chord consists of a triad with an added seventh above the root.
• Types of Seventh Chords: The quality of the triad and the quality of the seventh determine the type of seventh chord (e.g., major seventh, minor seventh, dominant seventh, half-diminished seventh, fully diminished seventh). “Essentially, three qualities of 7th intervals above a given root are possible: a major seventh interval… a minor seventh interval… and a diminished seventh interval…”
• Numerical Designation (Arabic Numerals): Arabic numerals are added to the Roman numerals to indicate inversions of seventh chords (e.g., V⁷ in root position, V⁶⁵ in first inversion, V⁴³ in second inversion, V² in third inversion).
• Dominant and Nondominant Seventh Chords: Dominant seventh chords (typically V⁷) have a specific function leading to the tonic. Nondominant seventh chords occur on other scale degrees.
• Tritone Interval: The interval between the third and seventh of a dominant seventh chord (and other seventh chords) creates dissonance and a strong tendency to resolve. “The interval that creates the dominant character is the tritone.”

VI. Pitch Tendency and Voice Leading:

• Pitch Tendency: Notes within a scale have a natural tendency to move towards other notes, particularly towards the tonic. “Upon hearing a major scale played from tonic… the first and last note, and especially the last, reinforce the gravitational pull and the final repose of the other notes; the tonic provides the resolution of the entire scalar sequence.” The leading tone has a strong upward tendency, while the seventh of a dominant chord has a strong downward tendency.
• Resolution of Dominant Seventh Chords: Specific voice-leading guidelines exist for resolving dominant seventh chords to the tonic chord to create smooth and effective voice leading.
• Resolution of Leading-Tone Diminished Chords: Leading-tone diminished chords also have specific resolution tendencies, usually moving to the tonic chord.

VII. Chord Relationships and Substitution:

• Chord Relationships: Chords can be related if they share common tones or if one chord can imply another. For example, the B diminished triad contains notes found in the D minor and G dominant seventh chords.
• Chord Substitution: Composers can substitute one chord for another if they share a similar function or contain common tones. This adds variety and interest to harmonic progressions.

VIII. Musical Function and Progression:

• Three Levels of Musical Function: These are not explicitly defined in the excerpts but relate to the role of chords in creating tension, stability, and direction within a musical piece.
• Chord Selection: Composers choose chords based on factors such as melodic contour, desired harmonic color, and the overall structure of the piece. “How does a composer decide on chord selection? The gravitational tendencies of chords is a result of their ____________.” (The answer would be “constituent pitches” or similar based on the context of pitch tendency).
• Progressive and Retrogressive Movement: These terms describe the direction of root movement in chord progressions based on intervals like descending fifths/ascending fourths (progressive) and ascending fifths/descending fourths (retrogressive).

IX. Nonharmonic Tones:

• Definition: Nonharmonic tones are pitches that do not belong to the underlying chord. They are used to create melodic interest and embellishment. “Describe nonharmonic tones in general. Mention several determining factors in the assessment of whether or not a tone is nonharmonic. A note or tone cannot be labeled as harmonic or nonharmonic unless it is supported by or is in a ____________.” (The answer would be “chord”).
• Types of Nonharmonic Tones: The text lists and briefly describes nine types:
• Passing Tone (PT): Moves stepwise between two chord tones.
• Neighboring Tone (NT): Steps away from and back to a chord tone.
• Appoggiatura (APP): Approached by leap and resolves stepwise. “‘Leaning,’ and this nonharmonic tone is characterized by its tendency to lean toward the note of resolution. The appoggiatura is normally approached by leap… and resolves by step.”
• Escape Tone (ET): Approached stepwise and resolves by leap in the opposite direction.
• Suspension (SUS): Prepared by being held over from a previous chord tone, creating dissonance, and then resolving downward stepwise.
• Retardation (RET): Similar to a suspension but resolves upward stepwise.
• Anticipation (ANT): Appears before the chord tone it anticipates.
• Free Tone (FT): Not approached or left by step.
• Pedal Tone: A sustained or repeated note, usually in the bass, while harmonies above change.
• Organ Point: Similar to a pedal tone but can occur in any voice.
• Cambiata (Changing Tones): A pattern of two nonharmonic tones moving stepwise and then by a skip.

X. Melody:

• Definition: Melody is a linear succession of pitches that create a musical idea.
• Melodic Components: The hierarchy of melodic construction includes cells (the smallest identifiable musical idea), motives (the “shortest complete fragment” that recurs and can be modified), and phrases (larger musical units often delineated by cadences). “A motive is defined as the shortest complete fragment.”
• Phrases and Cadences: Phrases often come in antecedent-consequent pairs. Cadences are “points of arrival in the flow of the music, achieved by harmonic, melodic, or rhythmic variation.”
• Sequence: A melodic pattern repeated at different pitch levels.
• Thematic Development: Melodies can be developed through techniques like repetition, variation, and augmentation.

XI. Musical Structure:

• Micro, Meso, and Macro Levels: These terms refer to the small-scale (motives, phrases), medium-scale (sections), and large-scale (movements, entire pieces) structural levels of music.
• Cadences: Different types of cadences (perfect authentic, imperfect authentic, semicadence, plagal, deceptive, Phrygian, Landini) create varying degrees of closure. “Which cadence provides the highest degree of closure? Perfect authentic cadence.”
• Musical Forms: The text mentions binary (two-part), rounded binary, and ternary (three-part) structures.
• Modulation: A change of key center within a musical work.

XII. Chromaticism and Altered Chords:

• Secondary Dominants: Chromatic chords that function as dominant chords leading to diatonic chords other than the tonic. “To facilitate an understanding of the derivation of a secondary dominant, one should consider the resolution of the secondary dominant as tonicizing the resultant chord of resolution. The term tonicize means to suggest a new key center without actually establishing one.” They are labeled as V/X (dominant of X).
• Diminished Chords: Possess a strong tendency for motion due to the tritone interval(s) they contain. They can function as leading-tone chords to various diatonic chords. “Diminished-chord structures are perhaps the most active and unstable elements in the harmonic vocabulary.”
• Neapolitan Chord (N⁶): A major triad built on the lowered second degree of a major or minor scale, typically appearing in first inversion.
• Augmented Sixth Chords (Italian 6th, French 6th, German 6th): Chromatic chords containing an augmented sixth interval, with a strong tendency to resolve to a dominant chord. The different types have distinct intervallic structures. “The augmented 6th interval in the augmented 6th chord resolves ____________.” (The answer is “outward by a half step”).

XIII. Extended Harmony (Ninth, Eleventh, and Thirteenth Chords):

• Construction: These chords are formed by adding further tertian intervals above the seventh.
• Qualities and Usage: Different qualities of ninths, elevenths, and thirteenths exist, with varying degrees of common usage. Altered upper partials (sharpened or flattened) create additional chord possibilities. “Upper-partial chords that utilize altered scale degrees are referred to as ____________ types.” (The answer is “altered”).
• Chord Symbols: Specific symbols are used to denote these extended chords.
• Embedded Harmonic Structures: Extended chords can contain implied triads or seventh chords within their upper partials.

XIV. Harmonic Movement:

• Stepwise, Mediant, and Symmetrical Movement: These describe different types of root movement between chords.
• Planing and Parallelism: Techniques involving the parallel movement of voices or harmonic structures.

XV. Modulation:

• Definition: A “change of key center.” This is distinct from tonicization (suggesting a key) and key regionalization (prolonging a non-tonic key).
• Types of Modulation: The text discusses common-chord (diatonic), chromatic, and enharmonic modulation.
• Pivot Chords: Chords common to both the original and the new key used as a link in modulation.
• Augmented Sixths as Pivots: Augmented sixth chords can be respelled enharmonically to function as dominant seventh chords, facilitating modulation to different keys.

XVI. Tension and Resolution:

• Consonance and Dissonance: These concepts relate to the stability and instability of musical sounds, affecting the perception of tension and resolution.
• Secundal, Quartal, and Quintal Harmony: Alternatives to traditional tertian harmony, built on intervals of seconds, fourths, and fifths, respectively, often creating different types of tension.

XVII. Remote Key Relationships:

• Diminished Seventh Chords and Modulation: Fully diminished seventh chords have a unique ability to modulate to remote keys due to their symmetrical structure and multiple enharmonic spellings. “The Dο7 spelling can be altered to accommodate any inclusive pitch as a leading tone, and, also, any pitch altered enharmonically.”
• Implied Roots: Through enharmonic respelling and considering intervallic relationships, remotely related implied roots can be discovered, leading to unexpected harmonic connections.

This briefing document provides a foundational overview of the music theory concepts presented in the source material. Further detailed study of the examples and exercises would be necessary for a complete understanding of each topic.

Fundamentals of Pitch and Notation

1. What is the fundamental definition of pitch in Western music, and how are pitches visually represented on a musical staff? Pitch refers to the highness or lowness of a sound, determined by the frequency of vibrations per second. In Western music, pitches are represented by note symbols placed on, above, or below a staff, which consists of five horizontal lines and four spaces. The musical alphabet (A-G) is used to name these pitches. Ledger lines are used to extend the range of the staff.
2. Explain the purpose and function of clef signs on a musical staff, and describe the four main clefs currently in use. A clef sign is placed at the beginning of a musical staff to determine the specific pitches represented by the lines and spaces. The four clefs in current use are: the treble (or G) clef, which locates the pitch G on the second line from the bottom; the bass (or F) clef, which locates F on the fourth line from the bottom; the alto (or C) clef, which places middle C on the third line; and the tenor (or C) clef, which places middle C on the fourth line. The C clef is movable. The grand staff combines the treble and bass staves, typically used for piano music.
3. How do sharps, flats, double sharps, double flats, and natural signs alter the pitch of a note? These symbols, known as accidentals, are placed before a notehead to alter its pitch. A sharp (#) raises the pitch of a note by a half step, while a flat (♭) lowers it by a half step. A double sharp (## or x) raises the pitch by a whole step, and a double flat (♭♭) lowers it by a whole step. A natural sign (♮) cancels the effect of a previously applied sharp or flat, returning the note to its original pitch.

Rhythm and Meter

1. Define rhythm in music, and explain its fundamental importance. What are some basic elements of musical duration discussed in the chapter? Rhythm, derived from the Greek word for “flow,” is the broad term encompassing all elements of musical duration. It is arguably the most basic and important element of music, as music is conceived, perceived, and experienced in time. The chapter examines the “micro” portion of rhythm, including concepts like beat, tempo (the speed of the pulse), meter (the grouping of beats), and pulse (the underlying steady beat).
2. Distinguish between simple and compound meter. How are time signatures used to indicate these different types of meter? Simple meter features beats that are normally divided into two equal parts (e.g., quarter note divides into two eighth notes). Common time signatures in simple meter include 2/4, 3/4, and 4/4. Compound meter features beats that are normally divided into three equal parts (e.g., a dotted quarter note divides into three eighth notes). Time signatures in compound meter often appear with an upper number of 6, 9, or 12 (e.g., 6/8, 9/8, 12/8), where the upper number indicates the number of divisions per measure, and the lower number indicates the note value of one division.
3. Explain the concept of the “half-beat” and its significance in understanding rhythm. What is the binary nature of a beat? The “half-beat” concept, as proposed by Tobias Matthay, highlights the binary nature of a single beat. A complete beat can be divided into two equal halves: the “down” part and the “up” part. This binary understanding is fundamental to how we count and feel rhythmic subdivisions within a beat, whether done verbally, by tapping, with a metronome, or through a conductor’s gestures.

Harmony: Triads and Seventh Chords

1. Define a triad and describe its basic structure. What are the four qualities of triads, and how are they determined? A triad is a harmonic structure consisting of three different pitches, each belonging to a different letter class, typically stacked in intervals of a third. The lowest note is the root, the second is the third, and the highest is the fifth. The four triad qualities are major, minor, diminished, and augmented. These qualities are determined by the specific intervals (major or minor thirds) between the root, third, and fifth of the triad.
2. What is a seventh chord, and how does it differ from a triad? Describe the basic types of seventh chords that can be formed on the degrees of a major scale. A seventh chord is a harmonic structure consisting of four pitches stacked in intervals of a third. It includes a root, a third, a fifth, and a seventh above the root. Unlike a triad, it has an additional seventh factor. On the degrees of a major scale, various types of seventh chords can be formed, including major seventh (I7, IV7), minor seventh (ii7, iii7, vi7), dominant seventh (V7), and half-diminished seventh (viiø7). The specific quality of each seventh chord is determined by the qualities of the constituent thirds and the overall interval of the seventh from the root.

By Amjad Izhar
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Philip Ball’s The Music Instinct explores the multifaceted nature of music, examining its scientific underpinnings and its profound impact on human experience. The book investigates how our brains process sound, perceive melody and harmony, and respond emotionally to music across diverse cultures and historical periods. Ball considers the universality of music, the evolution of musical scales and structures, and the ongoing debate about music’s meaning and purpose. Through explorations of acoustics, psychology, neuroscience, and cultural studies, the book seeks to understand why music is so integral to humanity.

The Science and Art of Music

Music is not simply a kind of mathematics but rather a remarkable blend of art and science, logic and emotion, physics and psychology. The study of how music works involves both scientific investigation and an appreciation for its artistic qualities.

Here are some aspects of the relationship between music and science discussed in the sources:

• The Physics of Sound and Music: Musical notes can be understood in terms of their acoustic frequencies. The relationship between pitch and frequency seems simple, with higher frequencies generally corresponding to higher pitches. However, the selection of discrete notes used in music across cultures is not solely determined by nature. The interaction of nature and culture shapes the diverse palettes of notes found in different musical traditions. Helmholtz combined his knowledge of the ear’s workings with the mathematics of vibration to understand how we hear tones, producing a significant early scientific exposition on music cognition in his 1863 book “On the Sensations of Tone as a Physiological Basis for the Theory of Music”. He also explored the ancient question of consonance, noting the historical preference for intervals with simple frequency ratios.
• The Neuroscience of Music: When we listen to music, our brains perform complex feats of filtering, ordering, and prediction automatically and unconsciously. Neuroscience seeks to identify which brain regions are used for different musical tasks, providing insights into how the brain classifies and interprets music. For example, pitch perception appears to be mostly localized in the right hemisphere. Pitch intervals and melody are processed in areas like Heschl’s gyrus and the planum temporale. The brain also engages in sophisticated streaming and binding of sound to distinguish different musical elements and create a coherent perception. Musical training can alter the brain, leading to more analytical processing in musicians and changes in the corpus callosum and auditory cortex. However, the precise link between the rich experience of music and brain activity remains a significant challenge for neuroscience. The “Mozart Effect,” which suggested a positive effect of listening to Mozart on general intellect, has been qualified by findings showing that children might respond best to their favorite kind of music, leading to the idea of a “Blur Effect” as well.
• Music Cognition and Psychology: The science of music cognition is increasingly exploring the universal aspects of music by breaking it down into basic structural elements like pitch, tone, and rhythm. However, emotional, social, and cultural factors also significantly influence music perception. For instance, the perception of melodic pitch steps shows probability distributions that are fairly universal across Western and many other musical traditions. Music psychologists study how we process melodies, which involves learning expectations about pitch steps. They also investigate how we decode sound, including the streaming and binding of different musical voices. The field of music and emotion has become central to music cognition, moving away from purely atomistic dissections of music to examine responses to actual music. Theories like Meyer’s and Narmour’s attempt to explain emotional responses in terms of expectation, tension, and release.
• Music as Organized Sound: Avant-garde composer Edgar Varèse defined his music as “organized sound,” distinguishing his experimental sonic explorations from conventional music. This definition highlights the role of organization in what we perceive as music, although the listener also actively participates in this organization.
• Music and Language: Some researchers propose an evolutionary link between music and language, suggesting a common ancestral “musilanguage”. This theory posits that musilanguage might have contained features like lexical tone, combinatorial phrases, and expressive phrasing. Even today, non-vocal music seems to share speech-like patterns, such as pitch contours (prosody). Studies suggest that the rhythmic and melodic patterns of language may have shaped the music of composers from the same linguistic background. While there are neurological dissociations between language and music processing (amusia and aphasia), some theories suggest that syntactic processing in both domains might share neural resources.
• The Meaning of Music: The question of whether music has inherent meaning is debated. Some believe music is purely formal and does not “say” anything. Others argue that music can convey and elicit emotions , although the precise relationship is complex. Musical affect might arise from underlying principles that can be analyzed rationally. Composers and musicians intuitively manipulate human characteristics to create musical effects.

In conclusion, the study of music is deeply intertwined with various scientific disciplines. Acoustics provides the foundation for understanding musical sound, neuroscience explores the brain’s engagement with music, and music cognition investigates how we perceive and process musical information. While music is undoubtedly an art form, scientific inquiry continues to shed light on the intricate mechanisms underlying our musical experiences.

The Fundamentals of Musical Scales

Musical scales are fundamental to most musical traditions, serving as the set of pitches from which melodies and harmonies are constructed. They represent a selection of discrete pitches from the continuous spectrum of audible frequencies.

Here are key aspects of musical scales discussed in the sources:

• Definition and Basic Concepts: A musical scale is a set of discrete pitches within the octave that a tradition uses to build its music. Unlike the smoothly varying pitch of a siren, a scale is like a staircase of frequencies. Most musical systems are based on the division of pitch space into octaves, a seemingly fundamental aspect of human pitch perception. Within this octave, different cultures choose a subset of potential notes to form their scales. This selection is not solely determined by nature but arises from an interaction of nature and culture.
• Western Scales and Their Development:
• Pythagorean Scales: One of the earliest theoretical frameworks for Western scales is attributed to Pythagoras, though the knowledge was likely older. Pythagorean scales are derived mathematically from the harmonious interval of a perfect fifth, based on the simple frequency ratio of 3:2. By repeatedly stepping up by a perfect fifth from a tonic and folding the resulting notes back into an octave, the major scale can be generated. This scale has an uneven pattern of whole tones and semitones. The Pythagorean system aimed to place music on a solid mathematical footing, suggesting music was a branch of mathematics embedded in nature. However, the cycle of fifths in Pythagorean tuning does not perfectly close, leading to an infinite number of potential notes, which can be problematic if music modulates between many keys.
• Diatonic Scales: Western music inherited diatonic scales from Greek tradition, characterized by seven tones between each octave. The major and minor scales became the basis of most Western music from the late Renaissance to the early twentieth century. Each note of a diatonic scale has a specific order, with the tonic being the starting and central note.
• Chromatic Scale: In addition to the seven diatonic notes, there are five other notes within an octave (like the black notes on a piano within a C major scale). The scale that includes all twelve semitones is called the chromatic scale, and music that uses notes outside the diatonic scale is considered chromatic.
• Modes: Before diatonic scales became dominant, Western music utilized modes, which can be thought of as scales using the same notes but starting in different places, each with a different sequence of step heights. Medieval modes had anchoring notes called the final and often a reciting tone called the tenor. The Ionian and Aeolian modes introduced later are essentially the major and a modern minor scale, respectively.
• Accidentals, Transposition, and Modulation: Sharps and flats (accidentals) were added to the modal system to preserve pitch steps when transposing melodies to different starting notes (keys). This also enabled modulation, the process of moving smoothly from one key to another, which became central to Western classical music. Transposition and modulation necessitate the introduction of new scales and notes.
• Non-Western Scales: Musical scales vary significantly across cultures.
• Javanese Gamelan: Gamelan music uses non-diatonic scales like pélog and sléndro, which have different interval structures compared to Western scales. The sléndro scale is a rare exception with equal pitch steps.
• Indian Music: The Indian subcontinent has a rich musical tradition with non-diatonic scales that include perfect fifths. North Indian music employs thirty-two different scales (thats) of seven notes per octave, drawn from a palette of twenty-two possible pitches. These scales (ragas) have tunings that can differ significantly from Western scales.
• Arab-Persian Music: This tradition also uses pitch divisions smaller than a semitone, with estimates ranging from fifteen to twenty-four potential notes within an octave. However, some of these might function as embellishments rather than basic scale tones.
• The existence of diverse scale systems demonstrates that the selection of notes is not solely dictated by acoustics or mathematics.
• Number and Distribution of Notes: Most musical systems use melodies constructed from four to twelve distinct notes within an octave. This limitation likely stems from cognitive constraints: too few notes limit melodic complexity, while too many make it difficult for the brain to track and organize the distinctions. The unequal pitch steps found in most scales (with sléndro being an exception) are thought to provide reference points for listeners to perceive the tonal center or key of a piece. Scales with five (pentatonic) or seven (diatonic) notes are particularly widespread, possibly because they allow for simpler interconversion between scales with different tonic notes during modulation.
• Cognitive Processing of Scales: Our brains possess a mental facility for categorizing pitches, allowing us to perceive melodies as coherent even on slightly mistuned instruments. We learn to assign pitches to a small set of categories based on interval sizes, forming mental “boxes”. To comprehend music, we need to discern a hierarchy of status between the notes of a scale, which depends on our ability to intuit the probabilities of different notes occurring.
• Alternative Scales: Some twentieth-century composers explored non-standard scales to create unique sounds, such as Debussy’s whole-tone scale, Messiaen’s octatonic scales, and Scriabin’s “mystic” scales.

In essence, musical scales are carefully chosen sets of pitches that provide the foundational elements for musical expression. Their structure and the specific notes they contain vary greatly across historical periods and cultural traditions, reflecting both acoustic principles and human cognitive and cultural preferences.

The Perception of Melody in Music

Melody perception is a complex cognitive process through which we hear a sequence of musical notes as a unified and meaningful whole, often referred to as a “tune”. However, the sources clarify that “melody” is a more versatile term than “tune,” as not all music has a readily identifiable tune like “Singin’ in the Rain”. For instance, Bach’s fugues feature short, overlapping melodic fragments rather than a continuous, extended tune.

Pitch and Pitch Relationships:

The foundation of melody perception lies in our ability to process pitch, which is processed by pitch-selective neurons in the primary auditory cortex. These neurons have a unique one-to-one mapping for pitch, unlike our perception of other senses. While pitch increases with acoustic frequency, our auditory system creates a cyclical perception where pitches an octave apart sound similar, a phenomenon called octave equivalence. This is a unique perceptual experience in music. However, the sources emphasize that simply having the correct pitch classes in different octaves does not guarantee melody recognition. When listeners were presented with familiar tunes where the octave of each note was randomized, they couldn’t even recognize the melody. This suggests that register or ‘height’ (which octave a note is in) is a crucial dimension of melody perception, alongside chroma (the pitch class).

Our brains possess a remarkable mental facility for categorizing pitches, allowing us to perceive melodies as coherent even if played on slightly mistuned instruments. We learn to assign pitches to mental “boxes” representing intervals like “major second” or “major third,” classifying any pitch close enough to that ideal interval size.

Melodic Contour:

The contour of a melody, or how it rises and falls in pitch, is a vital cue for memory and recognition. Even infants as young as five months respond to changes in melodic contour. Interestingly, both children and untrained adults often think melodies with the same contour but slightly altered intervals are identical, highlighting the primacy of contour in initial recognition. Familiar tunes remain recognizable even when the melodic contour is “compressed”. Composers can create repeating contour patterns to help bind a melody together, even if they are not exact repeats, adapting the contour to fit the specific pitch staircase of a scale. Diana Deutsch refers to these building blocks as “pitch alphabets,” which can be compiled from scales and arpeggios.

Tonal Hierarchy and Expectation:

Our perception of melody is deeply influenced by the tonal hierarchy, which is our subjective evaluation of how well different notes “fit” within a musical context or key. Even listeners without extensive musical training have a mental image of this hierarchy and constantly refer to it to form anticipations and judgments about a tune. This is supported by experiments where listeners consistently rated the “rightness” of notes within a set tonal context. The tonal hierarchy helps us organize and understand music, making it sound like music rather than a random sequence of notes. Music that ignores these hierarchies can be harder to process and may sound bewildering.

Gestalt Principles and Binding:

Underlying melody perception is the brain’s constant search for coherence in the auditory stimuli it receives. We mentally and unconsciously “bind” a string of notes into a unified acoustic entity, a tune. This process aligns with principles of gestalt psychology, where the brain seeks to perceive patterns. For example, large intervals can create a discontinuity, challenging the brain’s ability to perceive the melody as a single “gestalt”. Conversely, repetition of notes or contours can strengthen the perception of a unified melody. The auditory picket-fence effect demonstrates our ability to perceive a continuous tone even when interrupted by noise, highlighting the brain’s tendency to “fill in” gaps to maintain a coherent auditory stream. In sequences with large pitch jumps, listeners may even separate the notes into two distinct melodic streams.

Phrasing and Rhythm:

Phrasing, the way a melody is divided into meaningful segments, is crucial for perception. Click migration experiments show that listeners tend to perceive breaks between notes that delineate musical phrases. Phrasing is closely linked to rhythmic patterns, which provide a natural breathing rhythm to music and help us segment it into manageable chunks. The duration and accentuation of notes contribute to our perception of rhythmic groupings.

Memory and Context:

When we listen to a melody, we hear each note in the context of what we have already heard, including previous notes, the melodic contour, repeated phrases, the established key, and even our memories of other music. This constant referencing and updating of information shapes our perception of the unfolding melody.

Brain Processing:

The brain processes melody through various regions, including the lateral part of Heschl’s gyrus and the planum temporale in the temporal lobe, which are involved in pitch perception and sophisticated auditory attributes. The anterior superior temporal gyrus also handles streams of sound like melodies. Research suggests that the right hemisphere discerns the global pattern of pitch contour, while the left hemisphere processes the detailed aspects of pitch steps.

Atonal Music:

Music that rejects tonal hierarchies can be harder to process because it goes against our learned expectations about note probabilities. While some theories attempt to analyze atonal music through concepts like pitch-class sets, these approaches often don’t explain how such music is actually perceived.

In summary, melody perception is a dynamic process involving the processing of pitch and its relationships, the recognition of melodic contour, the influence of tonal hierarchies and learned expectations, the brain’s ability to bind sequences of notes into coherent units, the segmentation of melodies into phrases guided by rhythmic patterns, and the crucial role of memory and context. These elements work together to allow us to experience a series of discrete musical notes as a meaningful and unified melodic line.

Understanding Harmony and Dissonance in Music

Harmony is about fitting notes together. Conventionally, combinations that fit well are called consonant, and those that fit less well are dissonant. In a reductive formulation, consonance is considered good and pleasing, while dissonance is bad and unsettling. However, these concepts are often misunderstood and misrepresented.

Historical Perspectives on Consonance and Dissonance:

• In tenth-century Europe, a perfect fifth was generally not deemed consonant; only the octave was.
• When harmonizing in fifths became common, fourths were considered equally consonant, which is different from how they are perceived today.
• The major third (C-E), part of the “harmonious” major triad, was rarely used even by the early fourteenth century and was not fully accepted as consonant until the High Renaissance.
• The tritone interval, supposedly dissonant, becomes pleasing and harmonious when part of a dominant seventh chord (e.g., adding a D bass to C-FG).
• The whole polarizing terminology of consonance and dissonance is a rather unfortunate legacy of music theory.

Sensory (or Tonal) Dissonance:

• There is a genuinely physiological aspect of dissonance, distinguished from musical convention, called sensory or tonal dissonance.
• This refers to the rough, rattle-like auditory sensation produced by two tones closely spaced in pitch.
• It is caused by the beating of acoustic waves when two pure tones with slightly different frequencies are played simultaneously. If the beat rate exceeds about 20 Hz, it is heard as roughness.
• The width of the dissonant region depends on the absolute frequencies of the two notes. An interval consonant in a high register may be dissonant in a lower register. Therefore, there is no such thing as a tonally dissonant interval independent of register.
• In the mid-range of the piano, minor thirds generally lie beyond the band of roughness, while even a semitone does not create roughness for high notes. However, in the bass, even a perfect fifth can become dissonant in sensory terms, explaining the “gruffness” of low chords.

Consonance, Dissonance, and Overtones:

• Tones played by musical instruments are complex, containing several harmonics.
• Two simultaneously sounded notes offer many possibilities for overtones to clash and produce sensory dissonance if close enough in frequency.
• Hermann von Helmholtz calculated the total roughness for all overtone combinations, generating a curve of sensory dissonance with dips at various intervals of the chromatic scale. The octave and fifth have particularly deep “consonant” valleys.
• However, the depths of several “consonant” valleys don’t differ much. The modern dissonance curve shows that most intervals between the major second and major seventh lie within a narrow band of dissonance levels, except for the perfect fifth. Even the tritone appears less dissonant than major or minor thirds according to some measurements.
• The greatest sensory dissonance is found close to the unison, particularly the minor second, predicted to sound fairly nasty. However, such intervals can be used for interesting timbral effects.
• The brain is insistent on “binding” overtones into a single perceived pitch. If a harmonic is detuned, the brain tries to find a new fundamental frequency that fits, and only when the detuning is too large does it register the “bad” harmonic as a distinct tone. Percussive instruments often produce inharmonic overtones, resulting in an ambiguous pitch.

Cultural Influences and Learning:

• Whether we experience note combinations as smooth or grating is not solely a matter of convention, but there is a physiological aspect. However, likes and dislikes for certain combinations probably involve very little that is innate and are mostly products of learning.
• What is disliked is probably not the dissonances themselves but how they are combined into music.
• Acculturation can overcome sensory dissonance, as seen in the ganga songs of Bosnia and Herzegovina, where chords of major and minor seconds are considered harmonious.
• People tend to like best what is most familiar. Western listeners, being accustomed to tonal music, will be acclimatized to octaves, fifths, thirds, etc., and hear less common intervals as more odd.
• Studies suggest that cultural tradition exerts a stronger influence than inherent qualities in determining the emotional connotations of music, implying that perceptions of consonance and dissonance can also be culturally influenced.

Harmony in Musical Composition:

• In polyphonic music, harmony fills out the musical landscape. If melody is the path, harmony is the terrain.
• Harmonization is the process of fitting melodic lines to chords. This is often where music comes alive.
• Harmonization is generally more sophisticated in classical music, tending to use voice-leading, where accompanying voices have their own impetus and logic, rather than being monolithic chords.
• Harmonic progressions are sequences of chords. In Western classical music until the mid-nineteenth century, these tended to be formulaic and conservative, involving transitions to closely related chords. Pop and rock music have inherited much of this tradition.
• Modulation is the alteration of the key itself within a harmonic progression.
• Music theorists and psychologists have attempted to create a cartography of chords and keys, trying to map out relationships in harmonic space. Carol Krumhansl’s research suggests that the perceived relatedness of keys aligns with the cycle of fifths.

Harmony, Dissonance, and Musical Style/Emotion:

• Many classical-music traditionalists deny enjoying dissonance, associating it with jarring modern music. However, even composers like Chopin use dissonance extensively.
• The use of dissonance by modernist composers was seen by some as an affront to music itself. However, champions of atonalism argued that aversion to dissonance is culturally learned.
• “Dissonant” intervals like major sixths, sevenths, and ninths can create luxuriant sounds in the hands of composers like Debussy and Ravel.
• Composers may confuse our expectations regarding harmony to introduce tension and emotion.
• Expectations about harmony are crucial for our emotional response to music. Composers manipulate these expectations through devices like cadences, anticipation notes, and suspensions.
• Ambiguity in harmony and tonality can also create a powerful effect, with pleasure arising from the resolution of confusion.
• Different musical genres establish their own harmonic schemas, which they can then use to manipulate tension.

Dissonance in Polyphony:

• In early medieval polyphony, it was considered better to compromise the melody than to incur dissonance. However, composers increasingly prioritized maintaining good melodies in each voice, even if it led to occasional dissonances.
• This led to rules governing permissible dissonances in counterpoint. In Palestrina’s counterpoint, dissonances often occur on “passing tones” leading towards a consonance, and strong consonances are achieved at the beginnings and ends of phrases. The main objective is to maintain horizontal coherence of each voice while enforcing vertical integration through judicious use of consonance and controlled dissonance.
• Streaming of sound can offer a barrier to the perception of dissonance in polyphony. If voices are sufficiently distinct, potentially dissonant intervals may not be registered as jarring. Bach’s fugues, for example, contain striking dissonances that can go unnoticed due to the independence of the voices.
• Harmony can support the mental juggling act of listening to multiple melodies simultaneously, especially when the melodies are in the same key. Harmonic concordance seems to assist cognition.
• The composer doesn’t always want polyphonic voices to be clearly defined. In hymn singing, the focus is on creating a sense of unity through harmonies, resulting in a more homophonic texture where voices combine to carry a single melody, as opposed to the elaborate interweaving of voices in Bach’s contrapuntal music.

In conclusion, harmony and dissonance are fundamental aspects of music that involve both acoustic/physiological phenomena and cultural learning and conventions. Their perception and use have evolved throughout music history and continue to be manipulated by composers to create diverse musical experiences and emotional effects.

Understanding Musical Rhythm and Meter

Rhythm and meter are fundamental aspects of music. Rhythm is defined as the actual pattern of note events and their duration, and it tends to be much less regular than meter or tactus. It’s the “felt” quality of the regular subdivision of time on paper. Rhythm can be catchy and move us physically.

Meter, on the other hand, is the regular division of time into instants separated by equal intervals, providing what is colloquially called the ‘beat’. It’s the underlying pulse. The numbers at the start of a stave, the time signature, indicate how many notes of a particular duration should appear in each bar, essentially telling us whether to count the rhythm in groups of two, three, four, or more beats. To create a beat from a regular pulse, some pulses need to be emphasized over others, often by making them louder. Our minds tend to impose such groupings even on identical pulses. The grouping of pulses defines the music’s meter. Western music mostly uses simple meters with recurring groups of two, three, or four pulses, or sometimes six.

The tactus is related to but different from meter; it’s the beat we would clap out while listening to music and may be culture-specific. We tend to tap out a slower pulse to familiar music.

The source emphasizes that not all music possesses rhythm in a discernible way, citing compositions by Ligeti and Xenakis as continuous skeins of sound without a clear pulse, and Stockhausen’s Kontakte as being made of disconnected aural events. Gregorian chant is an example of music that can have regularly spaced notes but lack a true meter. Music for the Chinese fretless zither (qin) has rhythm in terms of note lengths, but these are not arranged against a steady underlying pulse.

However, a quasi-regular pulse pervades most of the world’s music. A rhythm is typically created by elaborating the periodic beats. Subdivisions and stresses superimposed on a steady pulse give us a sense of true rhythm, helping us locate ourselves in time much like the tonal hierarchy helps us in pitch space. This orderly and hierarchical structuring of time is found in the rhythmic systems of many musical traditions.

The source notes that the metre is often portrayed as a regular temporal grid on which the rhythm is arrayed, but the real relationship is more complex. Musicians tend subconsciously to distort the metrical grid to bring out accents and groupings implied by the rhythm. This stretching and shrinking of metrical time helps us perceive both meter and rhythm.

Western European music has traditionally chopped up time by binary branching, with melodies broken into phrases grouped in twos or fours, divided into bars, and beats subdivided into halves and quarters. This binary division is reflected in note durations like semibreve, minim, and crochet. However, some Balkan music uses prime numbers of beats in a bar, suggesting that binary division is not universal. Eastern European song may have constantly changing meter due to the rhythmic structure of its poetry.

Creating a true sense of rhythm and avoiding monotony involves not just stressing some beats but an asymmetry of events, similar to the skipping rather than plodding nature of spoken language. The source discusses rhythmic figures like the iamb, trochee, dactyl, and anapest, which are “atoms” from which we build a sense of rhythm and interpret musical events. Repetition of these units is crucial for that coherence to be felt. Our assignment of rhythmic patterns draws on various information beyond note duration, including melody, phrasing, dynamics, harmony, and timbre.

Composers generally want us to perceive the intended rhythm and use various factors to reinforce it. However, they may also seek to confuse our expectations regarding rhythm to introduce tension and emotion, as it is easy to hear when a beat is disrupted. Examples of this include:

• Syncopation, which involves shifting emphasis off the beat.
• Beethoven’s Fifth Symphony starting with a rest on the downbeat.
• Rhythmic ambiguity created by conflicting rhythmic groupings and meter, as in Beethoven’s Piano Sonata No. 13 and Bernstein’s “America”.
• Rhythmic elisions and deceptive rhythmic figures in popular music.
• Unambiguous disruption of meter, creating a jolt, as in Stravinsky’s The Rite of Spring.
• The use of anticipation tones in classical cadences to modulate the expectation of the impending cadence.

The source also points out that our sense of metrical regularity isn’t always strong, especially without musical training, and folk music traditions can exhibit irregular meters. In early polyphonic music, complex crossed rhythms were common, even without explicit metrical notation. Some musical traditions, like African, Indian, and Indonesian music, use cross-rhythms and polyrhythms. The minimalist compositions of Steve Reich utilize phasing, where repetitive riffs played at slightly different tempos create shifting rhythmic patterns.

Ultimately, rhythm provides a way to interpret and make sense of the stream of musical events by apportioning them into coherent temporal units. Composers manipulate rhythm and meter in various ways to create structure, expectation, and emotional impact in their music.

By Amjad Izhar
Contact: amjad.izhar@gmail.com
https://amjadizhar.blog

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