In music pitch space is the modeling of pitch relationships, represented through mathematical models, most often multidimensional, describing how near or far pitches are from each other. These models are often graphs, groups, lattices, or geometrical figures such as helixes. Pitch space strictly speaking includes octaves, and when relationships between pitch classes are represented instead, we have modulatory space, which is the pitch class space wherein modulation is possible. For twelve tone equal temperament, this includes only the twelve pitch classes. Chordal space is a model of some of the chords of the pitch classes of modulatory space. Wikibooks Wikiversity has more about this subject: School of Music Look up Music in Wiktionary, the free dictionary Wikisource, as part of the 1911 Encyclopedia Wikiproject, has original text related to this article: Music Meta has a page about this at: Music markup MusicNovatory: the science of music encyclopedia The... In common usage, the dimensions (from Latin measured out) of an object are the parameters or measurements required to define its shape and size, that is, usually, its height, width, and length. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... See lattice for other meanings of this term, both within and without mathematics. ... In music and music theory a pitch class contains all notes that have the same name; for example, all Es, no matter which octave they are in, are in the same pitch class. ... In music, modulation is most commonly the act or process of changing from one key (tonic, or tonal center) to another, also known as a key change. ... Equal temperament is a scheme of musical tuning in which the octave is divided into a series of equal steps (equal frequency ratios). ... In music and music theory a pitch class contains all notes that have the same name; for example, all Es, no matter which octave they are in, are in the same pitch class. ... In music, chordal space is a mathematical model of relationships between chords in some musical system. ...

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.

Cognitive psychologists including Longuet-Higgins (1978) and Shepard (1982), and composers and theorists including Weber (1824), Riemann, and Schoenberg (1954) created models of pitch space, modulatory space, or chordal space. For pitch space there are generally at least two dimensions, one for pitch class and one for register (i.e., the specific pitch), but there may be any number. (Lerdahl, 1992) Cognitive psychology is the psychological science which studies cognition, the mental processes that are hypothesised to underlie behavior. ... Roger N. Shepard cognitive scientist and aurthor of Toward a Univeral Law of Greneralization for Psychological Science. ... Dr. Hugo Riemann (full name: Karl Wilhelm Julius Hugo Riemann) (July 18, 1849 - July 10, 1919) was a German musicologist. ... Arnold Schoenberg, Los Angeles, 1948 For the American music critic and journalist, see Harold Charles Schonberg. ... In music and music theory a pitch class contains all notes that have the same name; for example, all Es, no matter which octave they are in, are in the same pitch class. ... In music, a register is the relative height or range of a note, set of pitches or pitch classes, melody, part, instrument or group of instruments. ...

M.W. Drobisch (1855) was the first to suggest a helix (i.e. the spiral of fifths) to represent octave equivalency and reoccurrence (Lerdahl, 2001), and hence to give a model of pitch space. Shepard (1982) uses 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 equivalency than in western tonal music (Tenzer, 2000). See also chromatic circle. A helix (pl: helices), from the Greek word έλικας/έλιξ, is a twisted shape like a spring, screw or a spiral staircase. ... For the numerical computation software, see GNU Octave. ... Michael Tenzer (born 1957) is a composer, performer, educator and scholar. ... A gamelan is a musical ensemble of Indonesian origin typically featuring metallophones, xylophone(s), drums, and gongs. ... A pseudo-octave is an interval whose frequency ratio is not 2:1, the definition of an octave, but is treated in some way or ways equivalent to this ratio. ... In music, the total chromatic is the saturation of the diatonic scale and is also commonly used in place of aggregate. ...

The use of a lattice was first proposed by Euler (1739) to model just intonation using an axis of perfect fifths and another of major thirds (Lerdahl, 2001). James Tenney argues for multidimensional lattices, especially for just intonation systems, which contain a dimension for every pitch axis used (Tenney, 1983). Thus if a justly tuned system is based on the octave and fifths it would contain only two dimensions. W. A. Mathieu uses this perfect fifths and major thirds also (Mathieu, 1997) (see sargam). See lattice for other meanings of this term, both within and without mathematics. ... The word axis has several meanings: In mathematics, axis can mean: A straight line around which a geometric figure can be rotated. ... James Tenney (August 10, 1934 in Silver City, NM) is an American composer and influential music theorist. ... See lattice for other meanings of this term, both within and without mathematics. ... Just intonation is any musical tuning in which the frequencies of notes are related by whole number ratios. ... William Allaudin Mathieu is an author and musician. ... Sargam is the Hindustani or North Indian equivalent to the western solfege. ...

Deutsch and Feroe (1981), and Lerdahl and Jackendoff (1983) use a "reductional format" representing pitch relations by "alphabets" or hierarchy of levels such as the chromatic, diatonic, and triadic. Lerdahl's levels include the octave, perfect fifth, major triad, diatonic scale, and the chromatic scale: Diana Deutsch is a perceptual and cognitive psychologist, born in London, England. ... Fred Lerdahl, Fritz Reiner Professor of Musical Composition at Columbia University, is a composer and music theorist, best known for his work on pitch space and cognitive constraints on compositional systems or musical grammars. ... Ray Jackendoff (born 1945) is an influential contemporary linguist who has always straddled the boundary between generative linguistics and cognitive linguistics, committed as he is both to the existence of an innate Universal Grammar (an all-important thesis of generative linguistics) and to giving an account of language that meshes... Reduction or reducing has several meanings: In mathematics, reduction is the process of manipulating a series of equations or matrices into a desired simpler format. ... In music, an octave (sometimes abbreviated 8ve or 8va) is the interval between one musical note and another with half or double the frequency. ... The perfect fifth or diapente is one of three musical intervals that span five diatonic scale degrees; the others being the diminished fifth, which is one semitone smaller, and the augmented fifth, which is one semitone larger. ... Triad (Simplified Chinese: 三合会; Traditional Chinese: 三合會; pinyin: ; literally Triad Society) is a collective term that describes many branches of an underground society and organizations based in Hong Kong and also operating in Mainland China, Macao, and Chinatowns in Europe, North America, Australia and New Zealand. ... In Music theory, the diatonic major scale (also known as the Guido scale), from the Greek diatonikos or to stretch out, is a fundamental building block of the European-influenced musical tradition. ... In music, chromatic indicates the inclusion of notes not in the prevailing scale and is also used for those notes themselves (Shir-Cliff et al 1965, p. ...

(Lerdahl, 1992)

According to David Kopp (2002), "Harmonic space, or tonal space as defined by Fred Lerdahl, is the abstract nexus of possible normative harmonic connections in a system, as opposed to the actual series of temporal connections in a realized work, linear or otherwise." (p.1)

The matrices used in the twelve tone technique are not representations of pitch space as nearness nor farness is not indicated, or even possible since one may not move freely about. In music matrices are used in the visualization of all permutations or forms of a tone row or set in music written using the twelve tone technique or serialism. ... Twelve-tone technique is a system of musical composition devised by Arnold Schoenberg. ...

See also

• Modulatory space
• Chordal space
• Diatonic set theory
• Emancipation of the dissonance
• Vowel space
• Color space

In music, modulation is most commonly the act or process of changing from one key (tonic, or tonal center) to another, also known as a key change. ... In music, chordal space is a mathematical model of relationships between chords in some musical system. ... Diatonic set theory is a subdivision or application of musical set theory which applies the techniques and insights of set theory to properties of the diatonic collection such as maximal evenness, Myhills property, well formedness, the deep scale property, cardinality equals variety, and structure implies multiplicity. ... The emancipation of the dissonance was a concept or goal put forth by Arnold Schoenberg and others, including his pupil Anton Webern, composer of atonal music and the inventor of the twelve tone technique. ... This article is about the alphabet officially used in linguistics. ... A color model is an abstract mathematical model describing the way colors can be represented as tuples of numbers, typically as three or four values or color components (e. ...

External link

• The James Tenney Project: excerpts from John Cage and the Theory of Harmony
• TonalSoft.com: Lattice diagrams
• TonalSoft.com: Tonnetz

Text and diagrams © 1998 by Joseph L. Monzo

• Seven limit lattices
• Tenney space
• Kees space

Sources

• Franklin, John Curtis, (2002). Diatonic Music in Ancient Greece: A Reassessment of its Antiquity, Memenosyne, 56.1 (2002), 669-702.
• Kopp David, (2002). Chromatic Transformations in Nineteenth-Century Music. Cambridge University Press. ISBN 0521804639.
• Lerdahl, Fred (1992). Cognitive Constraints on Compositional Systems, Contemporary Music Review 6 (2), pp. 97-121.
• Lerdahl, Fred (2001). Tonal Pitch Space, pp. 42-43. Oxford: Oxford University Press. ISBN 0195058348.
• Mathieu, W. A. (1997). Harmonic Experience: Tonal Harmony from Its Natural Origins to Its Modern Expression. Inner Traditions Intl Ltd. ISBN 0892815604.
• 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 0226792811.