I am working with two different tunings, one consisting of the first 32 prime partials (“prime tuning”), the other consisting of 13-limit constellations over the 29-limit prime partials – and throwing 31 in for good measure, resulting in a 33-pitch scale (“constellation tuning”). Prime tuning is, for now, focused around my solo keyboard work. For this project of building a just intonation ensemble, I am focused on constellation tuning. However, I may also be considering working in a tuning that either borrows higher primes for use in constellation tuning or that combines both sets of pitches for a total of 54 pitches in the scale, although some of the unique pitches are so close as to be synonymous (“prime-constellation tuning”).

I wanted to find all approximations of low-limit harmony within these scales. For example, 2233/2048 is within one cent of 12/11. Having a catalog of all possible approximations of low-limit intervals would help me as I design harmony. I started out cataloguing these “by hand” with some calculator help, but it became apparent I wasn’t getting paid enough to spend the next five years on that project. So I wrote a program to calculate all the approximations for me.

This chart shows all approximations within five cents of 31-limit ratios for constellation tuning. Within each larger column, the left column gives the base partial, or the partial up from which the program measured the interval given in the Ratio column. The following two columns show octave-reduced forms of each number in the ratio. The second-from-the-right column indicates which pitch in the scale approximates that ratio above the given base column, and the right column shows the approximation of the interval in cents. The columns are sorted first by base partial, then by denominator, which groups together all pitches (or approximations) of the overtone series over that base partial.

low-level-consonances