Modulatory space
Encyclopedia
The spaces described in this article are pitch class space
s 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 represents pitches rather than pitch classes, and chordal space
, which models relationships between chords.
The simplest pitch space model is the real line. 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 middle C 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
that mathematicians call Z/12Z.
, attempt to describe the special relationship between pitch classes related by perfect fifth. In equal temperament
, twelve successive fifths equate to seven octaves exactly, and hence in terms of pitch classes closes back to itself, forming a circle. Abstractly, this circle is a cyclic group
of order twelve, and may be identified with the residue classes
modulo twelve.
If we divide the octave into n equal parts, and choose an integer mpolygon
.
"r" generator or an "s" generator (the so-called Cayley graph
of
with generators r and s). The result is a graph of genus one, which is to say, a graph with a donut or torus
shape. Such a graph is called a toroidal graph
.
An example is equal temperament
; 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.
of rank two generated by the octave and another interval, commonly called "the" generator. The most familiar example by far is meantone temperament
, 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.
For example, diaschismic temperament is the temperament which tempers out the diaschisma
, 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.)
just intonation
has a modulatory space based on the fact that its pitch classes can be represented by 3a 5b, where a and b are integers. It is therefore a free abelian group
with the two generators 3 and 5, and can be represented in terms of a square lattice
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
instead; this is the Tonnetz
of Hugo Riemann
, 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
, 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
. 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 A3, which is equivalent to the face centered cubic lattice, or D3. Abstractly, it can be defined as the integer triples (a, b, c), associated to 3a 5b 7c, 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
are arranged around 1 in the shape of a cuboctahedron
.
Pitch class space
In music theory, pitch class space is the circular space representing all the notes in a musical octave.In this space, there is no distinction between tones that are separated by an integral number of octaves...
s which model the relationships between pitch classes in some musical system. These models are often graphs
Graph (mathematics)
In mathematics, a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges...
, groups
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
or lattices
Lattice (group)
In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. Every lattice in Rn can be generated from a basis for the vector space by forming all linear combinations with integer coefficients...
. Closely related to pitch class space is pitch space
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 apart. Depending on the complexity of the relationships...
, which represents pitches rather than pitch classes, and chordal space
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....
, which models relationships between chords.
Circular Pitch Class Space
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. 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
Pitch class space
In music theory, pitch class space is the circular space representing all the notes in a musical octave.In this space, there is no distinction between tones that are separated by an integral number of octaves...
that mathematicians call Z/12Z.
Circles of generators
Other models of pitch class space, such as the circle of fifthsCircle of fifths
In music theory, the circle of fifths shows the relationships among the 12 tones of the chromatic scale, their corresponding key signatures, and the associated major and minor keys...
, attempt to describe the special relationship between pitch classes related by perfect fifth. In equal temperament
Equal temperament
An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...
, twelve successive fifths equate to seven octaves exactly, and hence in terms of pitch classes closes back to itself, forming a circle. Abstractly, this circle is a cyclic group
Cyclic group
In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...
of order twelve, and may be identified with the residue classes
Modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
modulo twelve.
If we divide the octave into n equal parts, and choose an integer m
Polygon
In geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...
.
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 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
Cayley graph
In mathematics, a Cayley graph, also known as a Cayley colour graph, Cayley diagram, group diagram, or colour group is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem and uses a specified, usually finite, set of generators for the group...
of
with generators r and s). The result is a graph of genus one, which is to say, a graph with a donut or torusTorus
In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
shape. Such a graph is called a toroidal graph
Toroidal graph
In mathematics, a graph G is toroidal if it can be embedded on the torus. In other words, the graph's vertices can be placed on a torus such that no edges cross...
.
An example is equal temperament
Equal temperament
An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...
; 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.
Chains of generators
A linear temperament is a regular temperamentRegular 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...
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
Meantone temperament is a musical temperament, which is a system of musical tuning. In general, a meantone is constructed the same way as Pythagorean tuning, as a stack of perfect fifths, but in meantone, each fifth is narrow compared to the ratio 27/12:1 in 12 equal temperament, the opposite of...
, 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 a 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
Diaschisma
The diaschisma is a small musical interval defined as the difference between three octaves and four perfect fifths plus two major thirds . It can be represented by the ratio 2048:2025 and is about 19.5 cents...
, 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 limitLimit (music)
In music theory, limit or harmonic limit is a way of characterizing the harmony found in a piece or genre of music, or the harmonies that can be made using a particular scale. The term was introduced by Harry Partch, who used it to give an upper bound on the complexity of harmony; hence the name...
just intonation
Just intonation
In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...
has a modulatory space based on the fact that its pitch classes can be represented by 3a 5b, where a and b are integers. It is therefore a free abelian group
Free abelian group
In abstract algebra, a free abelian group is an abelian group that has a "basis" in the sense that every element of the group can be written in one and only one way as a finite linear combination of elements of the basis, with integer coefficients. Hence, free abelian groups over a basis B are...
with the two generators 3 and 5, and can be represented in terms of a square lattice
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. It is one of the five types of two-dimensional lattices as classified by their symmetry groups; its symmetry group is known symbolically as p4m.Two...
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
Hexagonal lattice
The hexagonal lattice or equilateral triangular lattice is one of the five 2D lattice types.Three nearby points form an equilateral triangle. In images four orientations of such a triangle are by far the most common...
instead; this is the Tonnetz
Tonnetz
In musical tuning and harmony, the Tonnetz is a conceptual lattice diagram representing tonal space first described by Leonhard Euler in 1739....
of Hugo Riemann
Hugo Riemann
Karl Wilhelm Julius Hugo Riemann was a German music theorist.-Biography:Riemann was born at Grossmehlra, Schwarzburg-Sondershausen. He was educated in theory by Frankenberger, studied the piano with Barthel and Ratzenberger, studied law, and finally philosophy and history at Berlin and Tübingen...
, discovered independently around the same time by Shohé Tanaka
Shohé Tanaka
was a Japanese physicist, music theorist, and inventor. He graduated from Tokyo University in 1882 as a science student. On an imperial scholarship, he was sent to Germany for doctoral studies in 1884, together with Mori Ōgai...
. 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
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...
, 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 3a 5b 7c in terms of a corresponding cubic latticeCubic lattice
Cubic lattice may refer to:*Cubic crystal system*Cubic honeycomb*Integer 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 A3, which is equivalent to the face centered cubic lattice, or D3. Abstractly, it can be defined as the integer triples (a, b, c), associated to 3a 5b 7c, 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
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...
are arranged around 1 in the shape of a cuboctahedron
Cuboctahedron
In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such it is a quasiregular polyhedron,...
.
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