Ploidacot

Ploidacot is a naming scheme for rank-2 tuning/temperament structures based on the number of periods per octave and the number of periods and independent generators per fifth, created by Praveen Venkataramana.

The general form is x-ploid y-sheared z-cot, where x is the number of periods in the octave, and where y periods down and z generators up are needed to reach ~3/2. Equivalently, z generators make 3/2 plus y/x octaves.

The numbers x and z may be expressed as numerical prefixes (e.g. mono, di, tri, tetra) instead of regular numbers, and "y-sheared" may be replaced with the number y in Greek gematria (e.g. 1 = alpha, 2 = beta, 3 = gamma, 10 = iota). When x = 1, haploid is used, not monoploid, and it can optionally be omitted.

A temperament with a ~3/2 of an exact number of periods is acot or 0-cot. In this situation, the shear amount tends to be left out of the ploidacot name, but it can be seen as -1 times however many periods there are in the fifth.

Examples

Blackwood has a fifth-octave period with the perfect fifth at exactly 3\5, making it a pentaploid (-3-sheared) acot temperament.

Meantone and Schismic have a full-octave period and a perfect fifth generator, making them (haploid) monocot temperaments.

Diaschismic can be interpreted as having a half-octave period and a perfect fifth generator, so it is a diploid monocot temperament.

Harry has a half-octave period and splits ~6/1 into 6 equal generator steps, so ~3/2 can be reached by going 4 periods down from there. This makes it a diploid delta-hexacot temperament.

Relationship to pergens

TODO: present omega as official

Pergens are an alternative notation system to ploidacot which notates 3-limit intervals in terms of diatonic scale degrees, created by Kite Giedraitis. While ploidacot describes how to get to ~3/2, pergens describe what 3-limit interval is divided into how many parts to get the generator. To use an above example, Harry, which is a diploid delta-hexacot temperament, would have the pergen (P8/2, P19/6), which simplifies to (P8/2, P4/6). Note the use of the perfect fourth. This does not appear to be in the "official" ploidacot standard, although Osmium has proposed the use of omega to refer to -1-shearing, corresponding to ~4/3 for single-period-octave (monoploid) temperaments.