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In
music theory 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 the " rudiments", that are needed to understand music notation (k ...
, 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 Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties * Graph (topology), a topological space resembling a graph in the sense of discr ...
, 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 space In music theory, pitch-class space is the circular space representing all the notes (pitch classes) 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, ...
s, which represent relationships between
pitch class In music, a pitch class (p.c. or pc) is a set of all pitches that are a whole number of octaves apart; for example, the pitch class C consists of the Cs in all octaves. "The pitch class C stands for all possible Cs, in whatever octave posit ...
es. (Some of these models are discussed in the entry on
modulatory space 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 ...
, though readers should be advised that the term "modulatory space" is not a standard music-theoretical term.) Chordal spaces model relationships between chords.


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 : p = 69 + 12\cdot\log_2 \, 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 MIDI (; Musical Instrument Digital Interface) 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 rel ...
. 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. 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 In music, a pitch class (p.c. or pc) is a set of all pitches that are a whole number of octaves apart; for example, the pitch class C consists of the Cs in all octaves. "The pitch class C stands for all possible Cs, in whatever octave posit ...
. This has led theorists such as
Moritz Wilhelm Drobisch Moritz Wilhelm Drobisch (16 August 1802 – 30 September 1896) was a German mathematician, logician, psychologist and philosopher. His brother was the composer Karl Ludwig Drobisch (1803–1854). Life Drobisch studied mathematics and philosophy a ...
(1846) and
Roger Shepard Roger Newland Shepard (January 30, 1929 – May 30, 2022) was an American cognitive scientist and author of the " universal law of generalization" (1987). He was considered a father of research on spatial relations. He studied mental rotation, a ...
(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.


Higher-dimensional pitch spaces

Other theorists, such as
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ...
(1739),
Hermann von Helmholtz Hermann Ludwig Ferdinand von Helmholtz (31 August 1821 – 8 September 1894) was a German physicist and physician who made significant contributions in several scientific fields, particularly hydrodynamic stability. The Helmholtz Association, ...
(1863/1885),
Arthur von Oettingen Arthur Joachim von Oettingen ( – 5 September 1920) was a Baltic German physicist and music theorist. He was the brother of theologian Alexander von Oettingen (1827–1905) and ophthalmologist Georg von Oettingen (1824–1916). Biography ...
(1866),
Hugo Riemann Karl Wilhelm Julius Hugo Riemann (18 July 1849 – 10 July 1919) was a German musicologist and composer who was among the founders of modern musicology. The leading European music scholar of his time, he was active and influential as both a mus ...
(who should not be confused with mathematician
Bernhard Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rig ...
), 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. 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 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. 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 An axis (plural ''axes'') is an imaginary line around which an object rotates or is symmetrical. Axis may also refer to: Mathematics * Axis of rotation: see rotation around a fixed axis *Axis (mathematics), a designator for a Cartesian-coordinate ...
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 Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rig ...
(Cohn 1997). Contemporary theorists such as
James Tenney James Tenney (August 10, 1934 – August 24, 2006) was an American composer and music theorist. He made significant early musical contributions to plunderphonics, sound synthesis, algorithmic composition, process music, spectral music, microtonal ...
(1983) and W.A. Mathieu (1997) carry on this tradition.
Moritz Wilhelm Drobisch Moritz Wilhelm Drobisch (16 August 1802 – 30 September 1896) was a German mathematician, logician, psychologist and philosopher. His brother was the composer Karl Ludwig Drobisch (1803–1854). Life Drobisch studied mathematics and philosophy a ...
(1846) was the first to suggest a
helix A helix () is a shape like a corkscrew or spiral staircase. It is a type of smooth space curve with tangent lines at a constant angle to a fixed axis. Helices are important in biology, as the DNA molecule is formed as two intertwined helices ...
(i.e. the spiral of fifths) to represent octave equivalence and recurrence (Lerdahl, 2001), and hence to give a model of pitch space.
Roger Shepard Roger Newland Shepard (January 30, 1929 – May 30, 2022) was an American cognitive scientist and author of the " universal law of generalization" (1987). He was considered a father of research on spatial relations. He studied mental rotation, a ...
(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 Gamelan () ( jv, ꦒꦩꦼꦭꦤ꧀, su, ᮌᮙᮨᮜᮔ᮪, ban, ᬕᬫᭂᬮᬦ᭄) is the traditional ensemble music of the Javanese, Sundanese, and Balinese peoples of Indonesia, made up predominantly of percussive instruments. T ...
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 The chromatic circle is a clock diagram for displaying relationships among the 12 equal-tempered pitch classes making up the familiar chromatic scale on a circle. Explanation If one starts on any equal-tempered pitch and repeatedly ascends by t ...
.


Instrument design

Since the 19th century there have been many attempts to design
isomorphic keyboard An isomorphic keyboard is a musical input device consisting of a two-dimensional grid of note-controlling elements (such as buttons or keys) on which any given sequence and/or combination of musical intervals has the "same shape" on the keyboard w ...
s based on pitch spaces. The only ones to have caught on so far are several
accordion Accordions (from 19th-century German ''Akkordeon'', from ''Akkord''—"musical chord, concord of sounds") are a family of box-shaped musical instruments of the bellows-driven free-reed aerophone type (producing sound as air flows past a reed ...
layouts.


See also

* Tonnetz *
Spiral array model In music theory, the spiral array model is an extended type of pitch space. A mathematical model involving concentric helices (an "array of spirals"), it represents human perceptions of pitches, chords, and keys in the same geometric space. It ...
*
Diatonic set theory Diatonic set theory is a subdivision or application of musical set theory which applies the techniques and analysis of discrete mathematics to properties of the diatonic collection such as maximal evenness, Myhill's property, well formedness, th ...
*
Emancipation of the dissonance The emancipation of the dissonance was a concept or goal put forth by composer Arnold Schoenberg and others, including his pupil Anton Webern. The phrase first appears in Schoenberg's 1926 essay "Opinion or Insight?" . It may be described as a m ...
*
Unified field In music, unified field is the 'unity of musical space' created by the free use of melodic material as harmonic material and vice versa. The technique is most associated with the twelve-tone technique, created by its 'total thematicism' where a ...
*
Vowel space A vowel is a syllabic speech sound pronounced without any stricture in the vocal tract. Vowels are one of the two principal classes of speech sounds, the other being the consonant. Vowels vary in quality, in loudness and also in quantity (leng ...
*
Color space A color space is a specific organization of colors. In combination with color profiling supported by various physical devices, it supports reproducible representations of colorwhether such representation entails an analog or a digital represen ...


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. . *Mathieu, W. A. (1997). ''Harmonic Experience: Tonal Harmony from Its Natural Origins to Its Modern Expression''. Inner Traditions Intl Ltd. . *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. .


Further reading

*Straus, Joseph. (2004) ''Introduction to Post Tonal Theory.'' Prentice Hall. .


External links


Seven limit latticesÜber die mathematische Bestimmung der musikalischen Intervalle, von M.W. Drobisch
{{DEFAULTSORT:Pitch Space