4320edo
| ← 4319edo | 4320edo | 4321edo → |
4320 equal divisions of the octave (abbreviated 4320edo or 4320ed2), also called 4320-tone equal temperament (4320tet) or 4320 equal temperament (4320et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 4320 equal parts of about 0.278 ¢ each. Each step represents a frequency ratio of 21/4320, or the 4320th root of 2.
Theory
4320edo is distinctly consistent in the 23-odd-limit and it is an excellent no-29s 37-limit tuning. While the consistency 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.
Subsets
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), with an abundancy index of 2.5 = exactly 5/2, as well as highly factorable EDO, with a total of 382 ways of being split into subset EDOs.
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. Beyond that, 31st harmonic comes from 240edo, and the 37th comes from 864edo.
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. A cent is equal to 3.6 steps of 4320edo, just as 1 m/s = 3.6 km/h. 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 does match the actual value.
A semitone therefore is 360 clicks, a quartertone is 180 clicks, minutes period is 72 clicks, a morion is 60 clicks, mercury period is 54 clicks, the Dröbisch angle is 12 clicks.
Since 4320edo is consistent in the 23-odd-limit, this means that the values of the 23-odd-limit intervals in clicks can be found by simply applying the patent val.
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 provides the optimal patent val for the period-80 temperament mercury.
In the 7-limit, 4320edo tempers out the landscape comma, and in the 11-limit, the kalisma, and as such it is a tuning for the rank-3 temperament odin tempering out both of them. In the 13-limit, it tempers out 6656/6655, 67392/67375, 151263/151250. In the 17-limit, it tempers out 12376/12375, 14400/14399, 28561/28560, and also commas associated with 80edo, such as 80-17-comma and 80-11/10-comma, that is [-91 0 -80 0 80⟩.
Higher harmonics it represents well past the 23-limit are 31, 37, 47, 59, 61, 71.
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⟩ | [⟨4320 6847 10031 12128 14945]] | -0.014 | 0.015 | 5.28 |
| 2.3.5.7.11.13 | 9801/9800, 67392/67375, 151263/151250, 479773125/479756288, 371293/371250 | [⟨4320 6847 10031 12128 14945 15986]] |
-0.013 | 0.014 | 4.89 |
| 2.3.5.7.11.13.17 | 9801/9800, 12376/12375, 194481/194480, 11275335/11275264, 63922176/63903125, 152649728/152628125 | [⟨4320 6847 10031 12128 14945 15986 17658]] |
-0.012 | 0.013 | 4.53 |
Rank-2 temperaments
| Periods per 8ve |
Generator (reduced) |
Cents (reduced) |
Associated ratio |
Temperaments |
|---|---|---|---|---|
| 12 | 2527\4320 (7\2460) |
498.056 (1.944) |
4/3 (32805/32768) |
Atomic |
| 60 | 2527\4320 (7\2460) |
498.056 (1.944) |
4/3 (32805/32768) |
Minutes |
| 80 | 1337\4320 (41\4320) |
371.389 (11.389) |
2275/1836 (?) |
Mercury |
Miscellany
4320edo is the 69th highly abundant EDO. Nice.
When it comes to interval size measures, a curious observation is also that 4320 km/h is close enough to whole integer to equal to 2684 mph, and 2684edo is a zeta peak EDO.