12276edo: Difference between revisions
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12276 is a strong 11-limit system, with a lower 11-limit relative error than any division aside from [[6691edo|6691]]. | 12276 is a strong 11-limit system, with a lower 11-limit relative error than any lower division aside from [[6691edo|6691]]. 12276 tempers out the [[Kirnberger's atom|atom]] and the [[septimal ruthenia]], so that the Pythagorean and syntonic commas an be approximated by 12 and 11 schismas, 240 and 220 steps respectively, and septimal comma is represented by 1/44 of the octave, 279 steps. | ||
[[ | === Prime harmonics === | ||
[[Category: | {{Harmonics in equal|12276|columns=11}} | ||
=== Subsets and supersets === | |||
12276edo factors into primes as {{nowrap| 2<sup>2</sup> × 3<sup>2</sup> × 11 × 31 }}, and among its divisors are [[12edo|12]], [[22edo|22]], [[31edo|31]], [[99edo|99]] and [[198edo|198]]. This creates a unit known as the ''[[prima]]'', useful for measurement of 11-limit intervals and commas. A prima is almost exactly three [[tuning unit]]s. | |||
[[Category:3-limit record edos|#####]] <!-- 5-digit number --> | |||
Latest revision as of 16:34, 28 July 2025
| ← 12275edo | 12276edo | 12277edo → |
(semiconvergent)
12276 equal divisions of the octave (abbreviated 12276edo or 12276ed2), also called 12276-tone equal temperament (12276tet) or 12276 equal temperament (12276et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 12276 equal parts of about 0.0978 ¢ each. Each step represents a frequency ratio of 21/12276, or the 12276th root of 2.
12276 is a strong 11-limit system, with a lower 11-limit relative error than any lower division aside from 6691. 12276 tempers out the atom and the septimal ruthenia, so that the Pythagorean and syntonic commas an be approximated by 12 and 11 schismas, 240 and 220 steps respectively, and septimal comma is represented by 1/44 of the octave, 279 steps.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0000 | +0.0000 | +0.0010 | -0.0087 | +0.0017 | +0.0393 | +0.0299 | +0.0432 | -0.0241 | +0.0416 | +0.0280 |
| Relative (%) | +0.0 | +0.0 | +1.1 | -8.9 | +1.7 | +40.2 | +30.6 | +44.2 | -24.7 | +42.5 | +28.6 | |
| Steps (reduced) |
12276 (0) |
19457 (7181) |
28504 (3952) |
34463 (9911) |
42468 (5640) |
45427 (8599) |
50178 (1074) |
52148 (3044) |
55531 (6427) |
59637 (10533) |
60818 (11714) | |
Subsets and supersets
12276edo factors into primes as 22 × 32 × 11 × 31, and among its divisors are 12, 22, 31, 99 and 198. This creates a unit known as the prima, useful for measurement of 11-limit intervals and commas. A prima is almost exactly three tuning units.