Diamond function
The diamond function creates a set of intervals or pitches D from a given set of pitches S. It can be applied, for example, to generate scales, tonality diamonds, or targeted intervals of an RTT tuning scheme.
Definition
Given a collection of pitches S, the diamond of S, D = diamond(S), is found as the set of intervals between those pitches, taking the intervals in direct and inverted form, reduced to an octave, and from there D can be used either as those intervals (as is the case with targeted intervals of RTT tuning scheme) or interpreted as pitches themselves (as is the case with tonality diamonds). For instance, given the pitches {1, 3, 5}, diamond({1, 3, 5}) is {1/1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3}). The diamond of a set is usually considered in connection with just intonation, in which case S is a set of rational numbers, but it applies to any collection; for instance diamond({0, 400, 700}) where the notes are expressed in cents, is {0, 300, 400, 500, 700, 800, 900}. The important special case where S is the set of odd integers less than or equal to an odd n is called the tonality diamond, and is often taken as the set of theoretical consonances in the n odd limit. This can be justified on the grounds that these are just the intervals appearing in the "chord of nature", or harmonic series, hence objecting to 17/16, for example, on the grounds it isn't actually very consonant doesn't take account of the fact that the integers up to 17, a "chord of nature", contain this interval.
The above definition is based on sets, but it is also possible to define diamonds in terms of multisets, which can lead to different results. If S is a multiset, then diamult(S) is the multiset of intervals between those notes, taking the intervals in direct and inverted form, reduced to an octave. The underlying set of notes may contain more notes than the diamond of the underlying set of the multiset S.
Creating scales
The scale steps of the tonality diamond are superparticular ratios, but they are not very evenly distributed. Filling in the gaps, as Harry Partch did with the 11-limit diamond to create a constant structure for his famous Genesis scale, is one way to go about constructing a just intonation scale. A constant structure is where each occurrence of a ratio will always have the same number of scale steps. While this is not completely possible with the 11-limit diamond, Partch was able to do so except in two places. This makes his 43 tone scale related to a 41 tone constant structure with two alternates.
The diamond construction can be iterated, giving diamond(diamond(S)), diamond(diamond(diamond(S))), and so forth, and the same is true of diamult. As scales, the results are too large for many applications, but iterating the tonality diamonds, or taking its scale powers, provides a convenient means of obtaining p-limit intervals, or intervals in a desired JI subgroup, in abundance.