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The spaces described in this article are
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 model the 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 in some musical system. These models 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
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
or
lattices. Closely related to pitch class space is
pitch space
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 apa ...
, which represents pitches rather than pitch classes, and
chordal space
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.
His ...
, which models relationships between chords.
Circular pitch class space
The simplest pitch space model is the real line. In the
MIDI Tuning Standard, for example, fundamental frequencies ''f'' are mapped to numbers ''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
A440 A440 or A-440 may refer to:
* A440 (pitch standard)
* A440 highway (Australia), a road in Victoria, Australia
* Quebec Autoroute 440 (Laval)
* Quebec Autoroute 440 (Quebec City)
See also
* Apollo 440
Apollo 440 (also known as Apollo Four Fo ...
is assigned the number 69 (meaning
middle C
C or Do is the first note and semitone of the C major scale, the third note of the A minor scale (the relative minor of C major), and the fourth note (G, A, B, C) of the Guidonian hand, commonly pitched around 261.63 Hz. The actual frequen ...
is assigned the number 60). To create circular pitch class space we identify or "glue together" pitches ''p'' and ''p'' + 12. The result is a continuous, circular
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, ...
that mathematicians call Z/12Z.
Circles of generators
Other models of pitch class space, such as the
circle of fifths
In music theory, the circle of fifths is a way of organizing the 12 chromatic pitches as a sequence of perfect fifths. (This is strictly true in the standard 12-tone equal temperament system — using a different system requires one interval ...
, attempt to describe the special relationship between pitch classes related by perfect fifth. In
equal temperament
An equal temperament is a musical temperament or tuning system, which approximates just intervals by dividing an octave (or other interval) into equal steps. This means the ratio of the frequencies of any adjacent pair of notes is the same, wh ...
, twelve successive fifths equate to seven octaves exactly, and hence in terms of pitch classes closes back to itself, forming a circle. We say that the pitch class of the fifth generates – or is a
generator
Generator may refer to:
* Signal generator, electronic devices that generate repeating or non-repeating electronic signals
* Electric generator, a device that converts mechanical energy to electrical energy.
* Generator (circuit theory), an eleme ...
of – the space of twelve pitch classes.
By dividing the octave into n equal parts, and choosing an integer m
relatively prime
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
– that is, have no common
divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
– we obtain similar circles, which all have the structure of finite cyclic groups. By drawing a line between two pitch classes when they differ by a generator, we can depict the circle of generators as a
cycle graph
In graph theory, a cycle graph or circular graph is a graph that consists of a single cycle, or in other words, some number of vertices (at least 3, if the graph is simple) connected in a closed chain. The cycle graph with vertices is called ...
, in the shape of a regular
polygon
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...
.
Toroidal modulatory spaces
If we divide the octave into n parts, where n = rs is the product of two relatively prime integers r and s, we may represent every element of the tone space as the product of a certain number of "r" generators times a certain number of "s" generators; in other words, as the
direct sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more ...
of two cyclic groups of orders r and s. We may now define a graph with n vertices on which the group acts, by adding an edge between two pitch classes whenever they differ by either an
"r" generator or an "s" generator (the so-called
Cayley graph
In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem (named after Arthur Cay ...
of
with generators ''r'' and ''s''). The result is a graph of
genus
Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...
one, which is to say, a graph with a donut or
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not tou ...
shape. Such a graph is called a
toroidal graph
In the mathematical field of graph theory, a toroidal graph is a graph that can be embedded on a torus. In other words, the graph's vertices can be placed on a torus such that no edges cross.
Examples
Any graph that can be embedded in a plane ...
.
An example is
equal temperament
An equal temperament is a musical temperament or tuning system, which approximates just intervals by dividing an octave (or other interval) into equal steps. This means the ratio of the frequencies of any adjacent pair of notes is the same, wh ...
; twelve is the product of 3 and 4, and we may represent any pitch class as a combination of thirds of an octave, or major thirds, and fourths of an octave, or minor thirds, and then draw a toroidal graph by drawing an edge whenever two pitch classes differ by a major or minor third.
We may generalize immediately to any number of relatively prime factors, producing
graphs can be drawn in a regular manner on an
n-torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not to ...
.
Chains of generators
A
linear temperament
Regular temperament is any tempered system of musical tuning such that each frequency ratio is obtainable as a product of powers of a finite number of generators, or generating frequency ratios. For instance, in 12-TET, the system of music most c ...
is a
regular temperament
Regular temperament is any tempered system of musical tuning such that each frequency ratio is obtainable as a product of powers of a finite number of generators, or generating frequency ratios. For instance, in 12-TET, the system of music most ...
of rank two generated by the octave and another interval, commonly called "the" generator. The most familiar example by far is
meantone temperament
Meantone temperament is a musical temperament, that is a tuning system, obtained by narrowing the fifths so that their ratio is slightly less than 3:2 (making them ''narrower'' than a perfect fifth), in order to push the thirds closer to pure. M ...
, whose generator is a flattened, meantone fifth. The pitch classes of any linear temperament can be represented as lying along an infinite chain of generators; in meantone for instance this would be -F-C-G-D-A-
etc. This defines a linear modulatory space.
Cylindrical modulatory spaces
A temperament of rank two which is not linear has one generator which is a fraction of an octave, called the period. We may represent the modulatory space of such a temperament as n chains of generators in a circle, forming a cylinder. Here n is the number of periods in an octave.
For example,
diaschismic temperament is the temperament which tempers out the
diaschisma
The diaschisma (or diacisma) is a small musical interval defined as the difference between three octaves and four perfect fifths plus two major thirds (in just intonation). It can be represented by the ratio 2048:2025 and is about 19.5 cents. T ...
, or 2048/2025. It can be represented as two chains of slightly (3.25 to 3.55 cents) sharp fifths a half-octave apart, which can be depicted as two chains perpendicular to a circle and at opposite side of it. The cylindrical appearance of this sort of modulatory space becomes more apparent when the period is a smaller fraction of an octave; for example,
ennealimmal temperament has a modulatory space consisting of nine chains of minor thirds in a circle (where the thirds may be only 0.02 to 0.03 cents sharp.)
Five-limit modulatory space
Five limit just intonation
In music, just intonation or pure intonation is the tuning of musical intervals
Interval may refer to:
Mathematics and physics
* Interval (mathematics), a range of numbers
** Partially ordered set#Intervals, its generalization from numbers to ...
has a modulatory space based on the fact that its pitch classes can be represented by 3
a 5
b, where a and b are integers. It is therefore a
free abelian group
In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subse ...
with the two generators 3 and 5, and can be represented in terms of a
square lattice
In mathematics, the square lattice is a type of lattice in a two-dimensional Euclidean space. It is the two-dimensional version of the integer lattice, denoted as . It is one of the five types of two-dimensional lattices as classified by their ...
with fifths along the horizontal axis, and major thirds along the vertical axis.
In many ways a more enlightening picture emerges if we represent it in terms of a
hexagonal lattice
The hexagonal lattice or triangular lattice is one of the five two-dimensional Bravais lattice types. The symmetry category of the lattice is wallpaper group p6m. The primitive translation vectors of the hexagonal lattice form an angle of 120° ...
instead; this is the
Tonnetz
In musical tuning and harmony, the (German for 'tone network') 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 traditi ...
of
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 musi ...
, discovered independently around the same time by
Shohé Tanaka. The fifths are along the horizontal axis, and the major thirds point off to the right at an angle of sixty degrees. Another sixty degrees gives us the axis of major sixths, pointing off to the left. The non-unison elements of the 5-limit
tonality diamond
In music theory and tuning, a tonality diamond is a two-dimensional diagram of ratios in which one dimension is the Otonality and one the Utonality.Rasch, Rudolph (2000). "A Word or Two on the Tunings of Harry Partch", ''Harry Partch: An Anthol ...
, 3/2, 5/4, 5/3, 4/3, 8/5, 6/5 are now arranged in a regular hexagon around 1. The triads are the equilateral triangles of this lattice, with the upwards-pointing triangles being major triads, and downward-pointing triangles being minor triads.
This picture of five-limit modulatory space is generally preferable since it treats the consonances in a uniform way, and does not suggest that, for instance, a major third is more of a consonance than a major sixth. When two lattice points are as close as possible, a unit distance apart, then and only then are they separated by a consonant interval. Hence the hexagonal lattice provides a superior picture of the structure of the five-limit modulatory space.
In more abstract mathematical terms, we can describe this lattice as the integer
pairs (a, b), where instead of the usual Euclidean distance we have a Euclidean distance defined in terms of the vector space norm
:
Seven-limit modulatory space
In similar fashion, we can define a modulatory space for
seven-limit just intonation, by representing 3
a 5
b 7
c in terms of a corresponding
cubic lattice. Once again, however, a more enlightening picture emerges if we represent it instead in terms of the three-dimensional analog of the hexagonal lattice, a lattice called A
3, which is equivalent to the
face centered cubic lattice, or D
3. Abstractly, it can be defined as the integer triples (a, b, c), associated to 3
a 5
b 7
c, where the distance measure is not the usual Euclidean distance but rather the Euclidean distance deriving from the vector space norm
:
In this picture, the twelve non-unison elements of the seven-limit
tonality diamond
In music theory and tuning, a tonality diamond is a two-dimensional diagram of ratios in which one dimension is the Otonality and one the Utonality.Rasch, Rudolph (2000). "A Word or Two on the Tunings of Harry Partch", ''Harry Partch: An Anthol ...
are arranged around 1 in the shape of a
cuboctahedron
A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it ...
.
See also
*
Pitch space
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 apa ...
*
Chordal space
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.
His ...
References
*Riemann, Hugo, ''Ideen zu einer Lehre von den Tonvorstellungen'', Jahrbuch der Musikbibliothek Peters, (1914/15), Leipzig 1916, pp. 1–26
*Tanaka, Shohé, ''Studien im Gebiete der reinen Stimmung'', Vierteljahrsschrift für Musikwissenschaft vol. 6 no. 1, Friedrich Chrysander, Philipp Spitta, Guido Adler (eds.), Breitkopf und Härtel, Leipzig, pp. 1–90
Further reading
*Cohn, Richard, ''Introduction to Neo-Riemannian Theory: A Survey and a Historical Perspective'', The Journal of Music Theory, (1998) 42(2), pp. 167–80
*Lerdahl, Fred (2001). ''Tonal Pitch Space'', pp. 42–43. Oxford: Oxford University Press. {{ISBN, 0-19-505834-8.
*Lubin, Steven, 1974, ''Techniques for the Analysis of Development in Middle-Period Beethoven'', Ph. D. diss., New York University, 1974
External links
Seven-limit modulatory spaceTonnetz and generalizations
Pitch space
Post-tonal music theory