Compton

Compton is a 5-limit regular temperament similar to 12edo, except that instead of being mapped to one of 12edo's intervals, the fifth harmonic is given its own generator. Equivalently, it is the rank-2 temperament which tempers out the Pythagorean comma 531441/524288. This equates Pythagorean comma-flat or -sharp intervals with their simpler counterparts (for example, the comma-flat major third 8192/6561 with the standard major third 81/64), and if the comma-flat third is seen as a diminished fourth, it can be seen as tempering together the two kinds of Pythagorean semitones, diatonic 256/243 and chromatic 2187/2048, into a single interval of 1/12 octave, which serves as the period. The generator can then be seen as any ptolemaic interval (the alteration of a Pythagorean interval by a syntonic comma), but is most usefully 5/4 or 81/80. The generator does not have any explicit constraints, unlike in many other temperaments.

As such, in terms of equal temperaments, compton is only supported by equal temperaments that are a multiple of 12, with 60edo, 72edo, 84edo, 96edo, 108edo, and 240edo perhaps being best.

For technical data, see Compton family#Compton.

Higher-limit extensions

If 7-limit intervals are desired, one may observe that simple septimal intervals are usually about twice as far away from 12edo intervals as simple classical intervals are. As such, the septimal comma 64/63 can be mapped to two syntonic commas. This tempers out 413343/409600, and can also be seen as tempering out 225/224. This implies that 72edo is a good tuning, as it tempers both the 5-limit and 7-limit intervals of compton close to their just counterparts. For the 2.3.7 subgroup, 36edo is a good tuning.

Assuming one has chosen to approximate the 7-limit, the comma generator will be around 15-17 cents. This means that the 11th harmonic can be reached either by going up or down three steps. Going down three steps results in the canonical extension of compton. Stepping down one more comma, depending on the tuning, can lead to the 13th harmonic, and results in the canonical tridecimal compton. This works best with tunings of the comma around 15 cents.

Alternatively, any prime may be merged into its 12edo mapping, making the smallest available prime the generator. This is done in catler (which shares 12edo's 5-limit and is generated by 7) and duodecim (which shares 12edo's 7-limit and is generated by 11). This is a natural choice for 17 and 19, as 12edo tunes those primes especially well, so compton can be seen as a 19-limit temperament.

Interval chain

Todo: verify Is this 11- and 13-limit mapping correct?

Generator steps	Cents	Intervals (down from 400c)	Intervals (up from 400c)	Intervals (up from 0c)	Harmonics
1	0	81/64	81/64	1/1	3/2, 17/16, 19/16
2	15.4	5/4		81/80	5/4
3	30.8		9/7	64/63	7/4
4	46.2	11/9	13/10		11/8
5	61.5	39/32			13/8