32:
178:. 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.
20:
193:(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.
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233:. 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.
775:
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
766:
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
47:
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
767:
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.
174:
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
169:
791:
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
817:(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).
780:). 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.
181:
One problem with linear pitch space is that it does not model the special relationship between octave-related pitches, or pitches sharing the same
813:(i.e. the spiral of fifths) to represent octave equivalence and recurrence (Lerdahl, 2001), and hence to give a model of pitch space.
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are not 2:1 and thus there is even less octave equivalence than in western tonal music (Tenzer, 2000). See also
924:
Cohn, Richard. (1997). Neo
Riemannian Operations, Parsimonious Trichords, and Their "Tonnetz" representations.
873:
222:
80:, though readers should be advised that the term "modulatory space" is not a standard music-theoretical term.)
53:
788:
1062:
1104:
931:
Franklin, John Curtis, (2002). Diatonic Music in
Ancient Greece: A Reassessment of its Antiquity,
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186:
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8:
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225:(1978) have modeled pitch relationships using two-dimensional (or higher-dimensional)
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68:-related pitches. When octave-related pitches are not distinguished, we have instead
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Harmonic
Experience: Tonal Harmony from Its Natural Origins to Its Modern Expression
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Higher-dimensional pitch spaces have also long been investigated. The use of a
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based on pitch spaces. The only ones to have caught on so far are several
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The simplest pitch space model is the real line. A fundamental frequency
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the hearing may appear before their eyes" (Bacchius, in
Franklin,
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Gamelan Gong Kebyar: The Art of
Twentieth-Century Balinese Music
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Since the 19th century there have been many attempts to design
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was proposed by Euler (1739) to model just intonation using an
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810:
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The Music of James Tenney, Volume 1: Contexts and
Paradigms
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The Music of James Tenney, Volume 1: Contexts and
Paradigms
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35:
The circle of fifths is another example of pitch space.
76:. (Some of these models are discussed in the entry on
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942:, pp. 42–43. Oxford: Oxford University Press.
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164:{\displaystyle p=69+12\cdot \log _{2}{(f/440)}\,}
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915:(University of Illinois Press, 2021), 81-84.
795:(Cohn 1997). Contemporary theorists such as
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978:. Chicago: University of Chicago Press.
72:, which represent relationships between
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217:(not to be confused with mathematician
16:Model for relationships between pitches
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1017:(University of Illinois Press, 2021).
27:space is an example of a pitch space.
970:John Cage and the Theory of Harmony.
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84:model relationships between chords.
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998:Introduction to Post Tonal Theory.
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809:(1846) was the first to suggest a
52:. Models of pitch space are often
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185:. This has led theorists such as
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803:(1997) carry on this tradition.
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197:Higher-dimensional pitch spaces
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88:Linear and helical pitch space
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957:. Inner Traditions Intl Ltd.
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223:Christopher Longuet-Higgins
96:is mapped to a real number
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100:according to the equation
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1105:Fokker periodicity blocks
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909:" in Wannamaker, Robert,
201:Other theorists, such as
974:Tenzer, Michael (2000).
996:Straus, Joseph. (2004)
953:Mathieu, W. A. (1997).
935:, 56.1 (2002), 669-702.
926:Journal of Music Theory
807:Moritz Wilhelm Drobisch
187:Moritz Wilhelm Drobisch
968:Tenney, James (1983).
938:Lerdahl, Fred (2001).
771:History of pitch space
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1120:Pitch constellation
869:Diatonic set theory
1125:Spiral array model
864:Spiral array model
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70:pitch class spaces
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1165:Pitch space
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799:(1983) and
189:(1846) and
183:pitch class
25:pitch class
1038:Kees space
933:Memenosyne
895:References
849:layouts.
847:accordion
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126:⋅
1159:Category
853:See also
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361:♯
349:♯
266:♯
254:♯
242:♯
227:lattices
213:(1866),
205:(1739),
62:lattices
1135:Tonnetz
859:Tonnetz
827:octaves
823:gamelan
793:Riemann
785:lattice
231:Tonnetz
221:), and
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66:octave
58:groups
54:graphs
811:helix
1002:ISBN
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176:MIDI
655:F5
560:A5
554:D5
548:G4
542:C4
536:F3
465:B4
459:E4
453:A3
447:D3
154:440
130:log
39:In
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263:B
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239:A
157:)
150:/
146:f
143:(
134:2
120:+
114:=
111:p
98:p
94:f
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