Regular Temperaments

The p-limit rational numbers Np are an abelian group under multiplication, so that Np acts on itself as a transformation group of a musical space, and constitutes a generalized interval system in the sense of David Lewin. It is a free abelian group of rank m=pi(p). An epimorphic map onto an abstract free abelian group of rank r less than m we will call an icon of dimension r-1; where we subtract one from the rank in order to conform to historical usage. A monomorphism from the image of the icon map to the positive reals under multiplication (or by the log map, equivalently to the reals under addition) we will call a tuning map. This is a map to an abelian group acting on itself as an interval space, with 1 in Np going to the real number 1. If we are to translate anything into actual pitches we need to have a further mapping of 1 to a base frequency, but we are better off staying at the higher level of abstraction.

We have therefore a sequence of mappings

Np --> Ic --> T

Here Ic is a note group, the image under the icon map of Np, which we will also call the icon. It is an abstract abelian group of rank r, and hence its group members can be written as r-tuples of integers, with the icon mapping being given by an r-tuple of vals. More concretely, since vals may be thought of as m-dimensional column vectors of integers, the icon map may be written as an mxr matrix of integers. The image of Ic is T, a tone group; this is again a free abelian group, but this time expressed as a subgroup of either the positive reals under multiplication, or the reals under addition (since these two abelian groups are themselves isomorphic.)

If given an icon we can find a homomorphic mapping to a tone group which is reasonable, not allowing too great a distortion nor transmogrifying the music into something quite different, we will say the icon defines regular temperament of dimension r, with the mapping to the tone group being the tuning of that temperament. This is, of course, not a mathematical definition since we have left determining what is reasonable up to the discretion of the one making use of this definition; nonetheless, it does seem to fit what we mean by it in practice. We say it defines a regular temperament, rather than is a regular temperament, since the same temperament can have more than one icon. If we write the icon as an mxr matrix, then we may perform the elementary column operations on this matrix of exchanging any two columns, changing signs (multiplication by -1) for any column, and multiplying any column by an integer and adding it to another column. Equivalently, we may left multiply the mxr matrix with an rxr unimodular matrix, leading to another mxr matrix which is an icon for the same temperament.

To give an example, consider the 5-limit group N5. An icon for this is the 2-tuple of vals [h12, h7], where h12 = <12 19 28|and h7 = <7 11 16|. The image of this is a note group, easily checked to satisfy the condition of being free of rank two; since the rank is two, the dimension is one. This is therefore a linear temperament. If we left multiply the icon matrix

[12 7]

[19 11]

[28 16]

by the unimodular matrix

[-4 7]

[ 7 -12]

we obtain

[1 0]

[1 1]

[0 4]

This is an icon for the same temperament (in this case, 5-limit meantone) which we may also write [<1 1 0|, <0 1 4|].