In
music theory
Music theory is the study of theoretical frameworks for understanding the practices and possibilities of music. ''The Oxford Companion to Music'' describes three interrelated uses of the term "music theory": The first is the "Elements of music, ...
, 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 farther apart. Depending on the complexity of the relationships under consideration, the models may be
multidimensional
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or Mathematical object, object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a ...
. 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 discre ...
,
groups,
lattices, or geometrical figures such as helixes. Pitch spaces distinguish
octave
In music, an octave (: eighth) or perfect octave (sometimes called the diapason) is an interval between two notes, one having twice the frequency of vibration of the other. The octave relationship is a natural phenomenon that has been referr ...
-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 Musical note, notes (pitch classes) in a musical octave. In this space, there is no distinction between tones 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 positio ...
es. (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.
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
:
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
Musical Instrument Digital Interface (; MIDI) is an American-Japanese technical standard that describes a communication protocol, digital interface, and electrical connectors that connect a wide variety of electronic musical instruments, ...
. 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 positio ...
. This has led theorists such as
Moritz Wilhelm Drobisch (1846) and
Roger Shepard
Roger Newland Shepard (January 30, 1929 – May 30, 2022) was an American cognitive science, cognitive scientist and author of the "universal law of generalization" (1987). He was considered a father of research on spatial relations. He studied m ...
(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, 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 polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...
(1739),
Hermann von Helmholtz
Hermann Ludwig Ferdinand von Helmholtz (; ; 31 August 1821 – 8 September 1894; "von" since 1883) was a German physicist and physician who made significant contributions in several scientific fields, particularly hydrodynamic stability. The ...
(1863/1885),
Arthur von Oettingen (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 ...
(not to be confused with mathematician
Bernhard Riemann
Georg Friedrich Bernhard Riemann (; ; 17September 182620July 1866) was a German mathematician who made profound contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the f ...
), 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 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. But 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 them, 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 (: axes) may refer to:
Mathematics
*A specific line (often a directed line) that plays an important role in some contexts. In particular:
** Coordinate axis of a coordinate system
*** ''x''-axis, ''y''-axis, ''z''-axis, common names ...
of perfect fifths and another of major thirds. Similar models were the subject of intense investigation in the 19th century, chiefly by theorists such as Oettingen and
Riemann (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, microt ...
(1983)
["Harmonic Space (CDC-1)" in Wannamaker, Robert, ]
The Music of James Tenney, Volume 1: Contexts and Paradigms
' (University of Illinois Press, 2021), 81-84. and
W.A. Mathieu (1997) carry on this tradition.
Moritz Wilhelm Drobisch (1846) was the first to suggest a
helix
A helix (; ) is a shape like a cylindrical coil spring or the thread of a machine screw. 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 for ...
(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 science, cognitive scientist and author of the "universal law of generalization" (1987). He was considered a father of research on spatial relations. He studied m ...
(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 (; ; , ; ) is the traditional musical ensemble, ensemble music of the Javanese people, Javanese, Sundanese people, Sundanese, and Balinese people, Balinese peoples of Indonesia, made up predominantly of percussion instrument, per ...
music since the
octave
In music, an octave (: eighth) or perfect octave (sometimes called the diapason) is an interval between two notes, one having twice the frequency of vibration of the other. The octave relationship is a natural phenomenon that has been referr ...
s 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 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 ...
s based on pitch spaces. The only ones to have caught on so far are several
accordion
Accordions (from 19th-century German language, German ', from '—"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 (mou ...
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 pitch (music), pitches, chord (music), chords, and key (music ...
*
Diatonic set theory
Diatonic set theory is a subdivision or application of set theory (music), musical set theory which applies the techniques and musical analysis, analysis of discrete mathematics to properties of the diatonic collection such as maximal evenness, Myh ...
*
Emancipation of the dissonance
*
Unified field
*
Vowel space
*
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 represe ...
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. .
*Wannamaker, Robert.
The Music of James Tenney, Volume 1: Contexts and Paradigms' (University of Illinois Press, 2021).
External links
Über die mathematische Bestimmung der musikalischen Intervalle, von M.W. Drobisch
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