Geometrical Models of Musical Structure Chords and Scales

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.

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

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