4320edo
Theory
4320edo is distinctly consistent in the 23-odd-limit. While this fact is not remarkable in its own right (282edo is the first such EDO), what's remarkable is the relationship that 4320edo offers to fractions of the octave, given that it is also a largely composite EDO. It is the first largely composite EDO with a greater consistency limit since 72edo.
Divisors
4320's divisors are 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 27, 30, 32, 36, 40, 45, 48, 54, 60, 72, 80, 90, 96, 108, 120, 135, 144, 160, 180, 216, 240, 270, 288, 360, 432, 480, 540, 720, 864, 1080, 1440, 2160. In addition to being largely composite, it is highly abundant (although not superabundant). It's abundancy index is 2.5 = exactly 5/2.
Out of the harmonics in the 23-limit approximated by 4320edo, only 3 and 5 have step sizes coprime with the number 4320. The 7th harmonic comes from 135edo, 11th harmonic comes from 864edo, 13th harmonic derives from 2160edo, 17th harmonic derives from 80edo, 19th harmonic derives from 480edo, and the 23rd harmonic comes from 720edo.
Other notable divisors 4320edo has are 12edo, the dominant tuning system in the world today, 15edo, known for use by Easley Blackwood Jr., 72edo, which has found usage in Byzantine chanting and various other applications, 96edo notable for its use by Julian Carrillo, 270edo, notable for its excellent closed representation of the 13-limit relative to its size, 360edo, notable for being a number of degrees in a circle and carrying the interval size measure Dröbisch angle.
Proposal for an interval size measure
Eliora proposes that 1 step of 4320edo be called a click. This is because 4320 kilometers per hour equals 1200 meters per second, and "clicks" or "clicks" is a slang name for kilometers per hour. Therefore if one were to think of cents as meters per second, this would make steps of 4320edo correspond kilometers per hour.
For example, a perfect fifth is 701.955 cents. Since 701.955 m/s = 2527.038 km/h, this means that perfect fifth in 4320edo is 2527 steps. And checking the harmonics table, it is.
Regular temperament theory
4320edo tempers out the Kirnberger's atom, and aside from tuning the atomic temperament, it supports period-60 temperament minutes. It also supports the period-80 temperament mercury.
In the 7-limit, 4320edo tempers out the landscape comma, and in the 11-limit, the kalisma. It is a tuning for the hemiennealimmal temperament and the rank-3 temperament odin.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | -0.011 | +0.075 | +0.063 | +0.071 | +0.028 | +0.045 | -0.013 | +0.059 | -0.133 | -0.036 |
| Relative (%) | +0.0 | -3.8 | +27.1 | +22.7 | +25.5 | +10.0 | +16.1 | -4.7 | +21.2 | -47.8 | -12.8 | |
| Steps (reduced) |
4320 (0) |
6847 (2527) |
10031 (1391) |
12128 (3488) |
14945 (1985) |
15986 (3026) |
17658 (378) |
18351 (1071) |
19542 (2262) |
20986 (3706) |
21402 (4122) | |
Other scales and techniques
Due to being consistent in the 23-limit, 4320edo is capable of consistently supporting the "factor 9 grid". It's quite coincidental that the number 4320 is divisible by 432, the number of Hertz in absolute pitch to which the alleged mystical properties of the scale are ascribed, except this time it is the cardinality of an EDO supporting the scale.
4320edo has a possible usage in Georgian folk music. 4320edo maps the 3/2 interval to 2527 steps, which factors as 7 x 19^2, and thus 4/3 to 1793 steps, factoring as 11 x 163. Since Georgian traditional music is based on dividing 3/2 and 4/3 into an arbitrary number of steps, it is able to support a variety of Kartvelian scales on the patent val, for example a combination of 7edf and 11ed4/3.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal
8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | -6847 4320 | [⟨4320 6847]] | 0.003 | 0.003 | 1.20 |
| 2.3.5 | 60 31 -47, 161 -84 -12 | [⟨4320 6847 10031]] | -0.009 | 0.017 | 6.12 |
| 2.3.5.7 | 250047/250000, [-55 30 2 1⟩, [33 19 -3 -20⟩ | [⟨4320 6847 10031 12128]] | -0.012 | 0.016 | 5.74 |
| 2.3.5.7.11 | 9801/9800, 250047/250000, [24 -10 -5 0 1⟩, [17 19 4 -9 -9⟩ | ||||
| 2.3.5.7.11.13 | 9801/9800, 67392/67375, 151263/151250, 479773125/479756288, 371293/371250 | ||||
Miscellany
4320edo is the 69th highly abundant EDO. Nice.