# Pitch space

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In music theory, pitch spaces model relationships between pitches. These models typically use distance to model the degree of relatedness, with closely related pitches placed near one another, and less closely related pitches placed farther apart. Depending on the complexity of the relationships under consideration, the models may be multidimensional. Models of pitch space are often graphs, groups, lattices, or geometrical figures such as helixes. Pitch spaces distinguish octave-related pitches. When octave-related pitches are not distinguished, we have instead pitch class spaces, which represent relationships between pitch classes. (Some of these models are discussed in the entry on modulatory space, though readers should be advised that the term "modulatory space" is not a standard music-theoretical term.) Chordal spaces model relationships between chords.

Music theory is the study of the practices and possibilities of music. The Oxford Companion to Music describes three interrelated uses of the term "music theory":

The first is what is otherwise called 'rudiments', currently taught as the elements of notation, of key signatures, of time signatures, of rhythmic notation, and so on. [...] The second is the study of writings about music from ancient times onwards. [...] The third is an area of current musicological study that seeks to define processes and general principles in music — a sphere of research that can be distinguished from analysis in that it takes as its starting-point not the individual work or performance but the fundamental materials from which it is built.

In physics and mathematics, the dimension of a mathematical space is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it – for example, the point at 5 on a number line. A surface such as a plane or the surface of a cylinder or sphere has a dimension of two because two coordinates are needed to specify a point on it – for example, both a latitude and longitude are required to locate a point on the surface of a sphere. The inside of a cube, a cylinder or a sphere is three-dimensional because three coordinates are needed to locate a point within these spaces.

In mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called vertices and each of the related pairs of vertices is called an edge. Typically, a graph is depicted in diagrammatic form as a set of dots or circles for the vertices, joined by lines or curves for the edges. Graphs are one of the objects of study in discrete mathematics.

## Linear and helical pitch space

The simplest pitch space model is the real line. A fundamental frequency f is mapped to a real number p according to the equation

${\displaystyle p=69+12\cdot \log _{2}{(f/440)}\,}$

This creates a linear space in which octaves have size 12, semitones (the distance between adjacent keys on the piano keyboard) have size 1, and middle C is assigned the number 60, as it is in MIDI. 440 Hz is the standard frequency of 'concert A', which is the note 9 semitones above 'middle C'. Distance in this space corresponds to physical distance on keyboard instruments, orthographical distance in Western musical notation, and psychological distance as measured in psychological experiments and conceived by musicians. The system is flexible enough to include "microtones" not found on standard piano keyboards. For example, the pitch halfway between C (60) and C# (61) can be labeled 60.5.

MIDI is a technical standard that describes a communications protocol, digital interface, and electrical connectors that connect a wide variety of electronic musical instruments, computers, and related audio devices for playing, editing and recording music. A single MIDI link through a MIDI cable can carry up to sixteen channels of information, each of which can be routed to a separate device or instrument. This could be sixteen different digital instruments, for example.

One problem with linear pitch space is that it does not model the special relationship between octave-related pitches, or pitches sharing the same pitch class. This has led theorists such as M. W. Drobish (1855) and Roger Shepard (1982) to model pitch relations using a helix. In these models, linear pitch space is wrapped around a cylinder so that all octave-related pitches lie along a single line. Care must be taken when interpreting these models however, as it is not clear how to interpret "distance" in the three-dimensional space containing the helix; nor is it clear how to interpret points in the three-dimensional space not contained on the helix itself.

In music, a pitch class (p.c. or pc) is a set of all pitches that are a whole number of octaves apart, e.g., the pitch class C consists of the Cs in all octaves. "The pitch class C stands for all possible Cs, in whatever octave position." Important to musical set theory, a pitch class is, "all pitches related to each other by octave, enharmonic equivalence, or both." Thus, using scientific pitch notation, the pitch class "C" is the set

## Higher-dimensional pitch spaces

Other theorists, such as Leonhard Euler (1739), Hermann von Helmholtz (1863/1885), Arthur von Oettingen (1866), Hugo Riemann (who should not be confused with mathematician Bernhard Riemann), and Christopher Longuet-Higgins (1978) have modeled pitch relationships using two-dimensional (or higher-dimensional) lattices, under the name of Tonnetz. In these models, one dimension typically corresponds to acoustically-pure perfect fifths while the other corresponds to major thirds. (Variations are possible in which one axis corresponds to acoustically pure minor thirds.) Additional dimensions can be used to represent additional intervals including—most typically—the octave.

Leonhard Euler was a Swiss mathematician, physicist, astronomer, logician and engineer who made important and influential discoveries in many branches of mathematics, such as infinitesimal calculus and graph theory, while also making pioneering contributions to several branches such as topology and analytic number theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function. He is also known for his work in mechanics, fluid dynamics, optics, astronomy and music theory.

Hermann Ludwig Ferdinand von Helmholtz was a German physician and physicist who made significant contributions in several scientific fields. The largest German association of research institutions, the Helmholtz Association, is named after him.

Arthur Joachim von Oettingen was a Baltic German physicist and music theorist who was born at the Luua Manor, Tartu County, Livonia. He was the brother of theologian Alexander von Oettingen (1827–1905) and ophthalmologist Georg von Oettingen (1824–1916).

 A♯3 — E♯4 — B♯4 — F 5 — C 6 — G 6 | | | | | | F♯3 — C♯4 — G♯4 — D♯5 — A♯5 — E♯6 | | | | | | D3 — A3 — E4 — B4 — F♯5 — C♯6 | | | | | | B♭2 — F3 — C4 — G4 — D5 — A5 | | | | | | G♭2 — D♭3 — A♭3 — E♭4 — B♭4 — F5 | | | | | | E 2 — B 2 — F♭3 — C♭4 — G♭4 — D♭5

All of these models attempt to capture the fact that intervals separated by acoustically pure intervals such as octaves, perfect fifths, and major thirds are thought to be perceptually closely related. However, proximity in these spaces need not represent physical proximity on musical instruments: by moving one's hands a very short distance on a violin string, one can move arbitrarily far in these multiple-dimensional models. For this reason, it is hard to assess[ according to whom? ] the psychological relevance of distance as measured by these lattices.

## History of pitch space

The idea of pitch space goes back at least as far as the ancient Greek music theorists known as the Harmonists[ citation needed ]. To quote one of their number, Bacchius, "And what is a diagram? A representation of a musical system. And we use a diagram so that, for students of the subject, matters which are hard to grasp with the hearing may appear before their eyes." (Bacchius, in Franklin, Diatonic Music in Ancient Greece.) The Harmonists drew geometrical pictures so that the intervals of various scales could be compared visually; they thereby located the intervals in a pitch space.

Higher-dimensional pitch spaces have also long been investigated. The use of a lattice was proposed by Euler (1739) to model just intonation using an axis of perfect fifths and another of major thirds. Similar models were the subject of intense investigation in the nineteenth century, chiefly by theorists such as Oettingen and Riemann (Cohn 1997). Contemporary theorists such as James Tenney (1983) and W.A. Mathieu (1997) carry on this tradition.

M.W. Drobisch (1855) was the first to suggest a helix (i.e. the spiral of fifths) to represent octave equivalence and recurrence (Lerdahl, 2001), and hence to give a model of pitch space. Shepard (1982) regularizes Drobish's helix, and extends it to a double helix of two wholetone scales over a circle of fifths which he calls the "melodic map" (Lerdahl, 2001). Michael Tenzer suggests its use for Balinese gamelan music since the octaves are not 2:1 and thus there is even less octave equivalence than in western tonal music (Tenzer, 2000). See also chromatic circle.

## Instrument design

Since the 19th century there have been many attempts to design isomorphic keyboards based on pitch spaces. The only ones to have caught on so far are several accordion layouts.

## Related Research Articles

In western music theory, a diatonic scale is a heptatonic scale that includes five whole steps and two half steps (semitones) in each octave, in which the two half steps are separated from each other by either two or three whole steps, depending on their position in the scale. This pattern ensures that, in a diatonic scale spanning more than one octave, all the half steps are maximally separated from each other.

In music theory, a scale is any set of musical notes ordered by fundamental frequency or pitch. A scale ordered by increasing pitch is an ascending scale, and a scale ordered by decreasing pitch is a descending scale. Some scales contain different pitches when ascending than when descending, for example, the melodic minor scale.

In modern musical notation and tuning, an enharmonic equivalent is a note, interval, or key signature that is equivalent to some other note, interval, or key signature but "spelled", or named differently. Thus, the enharmonic spelling of a written note, interval, or chord is an alternative way to write that note, interval, or chord. For example, in twelve-tone equal temperament, the notes C and D are enharmonic notes. Namely, they are the same key on a keyboard, and thus they are identical in pitch, although they have different names and different roles in harmony and chord progressions. Arbitrary amounts of accidentals can produce further enharmonic equivalents, such as B, although these are much rarer and have less practical use.

Musical set theory provides concepts for categorizing musical objects and describing their relationships. Howard Hanson first elaborated many of the concepts for analyzing tonal music. Other theorists, such as Allen Forte, further developed the theory for analyzing atonal music, drawing on the twelve-tone theory of Milton Babbitt. The concepts of musical set theory are very general and can be applied to tonal and atonal styles in any equal temperament tuning system, and to some extent more generally than that.

The spaces described in this article are pitch class spaces which model the relationships between pitch classes in some musical system. These models are often graphs, groups or lattices. Closely related to pitch class space is pitch space, which represents pitches rather than pitch classes, and chordal space, which models relationships between chords.

The chromatic circle is a geometrical space that shows relationships among the 12 equal-tempered pitch classes making up the familiar chromatic scale on a circle.

In diatonic set theory, maximal evenness is a quality of a collection or scale in which every generic interval has either one or two consecutive (adjacent) specific intervals—in other words a scale that is "spread out as much as possible." This property was first described by music theorist John Clough and mathematician Jack Douthett in "Maximally Even Sets" (1991).

Music theorists have often used graphs, tilings, and geometrical spaces to represent the relationship between chords. We can describe these spaces as chord spaces or chordal spaces, though the terms are relatively recent in origin.

In music theory, pitch-class space is the circular space representing all the notes in a musical octave. In this space, there is no distinction between tones that are separated by an integral number of octaves. For example, C4, C5, and C6, though different pitches, are represented by the same point in pitch class space.

In musical tuning and harmony, the Tonnetz is a conceptual lattice diagram representing tonal space first described by Leonhard Euler in 1739. Various visual representations of the Tonnetz can be used to show traditional harmonic relationships in European classical music.

Music theory has no axiomatic foundation in modern mathematics, yet the basis of musical sound can be described mathematically and exhibits "a remarkable array of number properties". Elements of music such as its form, rhythm and metre, the pitches of its notes and the tempo of its pulse can be related to the measurement of time and frequency, offering ready analogies in geometry.

Transformational theory is a branch of music theory David Lewin developed in the 1980s, and formally introduced in his 1987 work, Generalized Musical Intervals and Transformations. The theory—which models musical transformations as elements of a mathematical group—can be used to analyze both tonal and atonal music.

Diatonic and chromatic are terms in music theory that are most often used to characterize scales, and are also applied to musical instruments, intervals, chords, notes, musical styles, and kinds of harmony. They are very often used as a pair, especially when applied to contrasting features of the common practice music of the period 1600–1900.

In music theory, the spiral array model is an extended type of pitch space. A mathematical model involving concentric helices, it represents human perceptions of pitches, chords and keys in the same geometric space. It was proposed in 2000 by Prof. Elaine Chew in her MIT doctoral thesis Toward a Mathematical Model of Tonality. Further research by Chew and others have produced modifications of the spiral array model, and, applied it to various problems in music theory and practice, such as key finding, pitch spelling, tonal segmentation, similarity assessment, and musical humor. The extensions and applications are described in Mathematical and Computational Modeling of Tonality: Theory and Applications.

Neo-Riemannian theory is a loose collection of ideas present in the writings of music theorists such as David Lewin, Brian Hyer, Richard Cohn, and Henry Klumpenhouwer. What binds these ideas is a central commitment to relating harmonies directly to each other, without necessary reference to a tonic. Initially, those harmonies were major and minor triads; subsequently, neo-Riemannian theory was extended to standard dissonant sonorities as well. Harmonic proximity is characteristically gauged by efficiency of voice leading. Thus, C major and E minor triads are close by virtue of requiring only a single semitonal shift to move from one to the other. Motion between proximate harmonies is described by simple transformations. For example, motion between a C major and E minor triad, in either direction, is executed by an "L" transformation. Extended progressions of harmonies are characteristically displayed on a geometric plane, or map, which portrays the entire system of harmonic relations. Where consensus is lacking is on the question of what is most central to the theory: smooth voice leading, transformations, or the system of relations that is mapped by the geometries. The theory is often invoked when analyzing harmonic practices within the Late Romantic period characterized by a high degree of chromaticism, including work of Schubert, Liszt, Wagner and Bruckner.

In musical tuning, a lattice "is a way of modeling the tuning relationships of a just intonation system. It is an array of points in a periodic multidimensional pattern. Each point on the lattice corresponds to a ratio. The lattice can be two-, three-, or n-dimensional, with each dimension corresponding to a different prime-number partial [pitch class]." When listed in a spreadsheet a lattice may be referred to as a tuning table.

A generative theory of tonal music (GTTM) is a theory of music conceived by American composer and music theorist Fred Lerdahl and American linguist Ray Jackendoff and presented in the 1983 book of the same title. It constitutes a "formal description of the musical intuitions of a listener who is experienced in a musical idiom" with the aim of illuminating the unique human capacity for musical understanding.

Vogel's Tonnetz is a graphical and mathematical representation of the scale range of just intonation, introduced by German music theorist Martin Vogel 1976 in his book Die Lehre von den Tonbeziehungen. The graphical representation is based on Euler's Tonnetz, adding a third dimension for just sevenths to the two dimensions for just fifths and just thirds. It serves to illustrate and analyze chords and their relations. The four-dimensional mathematical representation including octaves allows the Evaluation of the congruency of harmonics of chords depending on the tonal material. It can thus also serve to determine the optimal tonal material for a certain chord.

In music, a common tone is a pitch class that is a member of, or common to two or more scales or sets.

## References

• Cohn, Richard. (1997). Neo Riemannian Operations, Parsimonious Trichords, and Their "Tonnetz" representations. Journal of Music Theory, 41.1: 1-66.
• Franklin, John Curtis, (2002). Diatonic Music in Ancient Greece: A Reassessment of its Antiquity, Memenosyne, 56.1 (2002), 669-702.
• Lerdahl, Fred (2001). Tonal Pitch Space, pp. 42–43. Oxford: Oxford University Press. ISBN   0-19-505834-8.
• Mathieu, W. A. (1997). Harmonic Experience: Tonal Harmony from Its Natural Origins to Its Modern Expression. Inner Traditions Intl Ltd. ISBN   0-89281-560-4.
• Tenney, James (1983). John Cage and the Theory of Harmony.
• Tenzer, Michael (2000). Gamelan Gong Kebyar: The Art of Twentieth-Century Balinese Music. Chicago: University of Chicago Press. ISBN   0-226-79281-1.
• Straus, Joseph. (2004) Introduction to Post Tonal Theory. Prentice Hall. ISBN   0-13-189890-6.