12276edo: Difference between revisions
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12276edo's step size is sometimes called a '''prima''', a term proposed by [[Lillian Hearne]], when used as an interval size unit. | |||
== Theory == | |||
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. It is the smallest [[atomic]] EDO inside its [[5-odd-limit]] [[diamond tradeoff]] tuning range. | 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. It is the smallest [[atomic]] EDO inside its [[5-odd-limit]] [[diamond tradeoff]] tuning range. | ||
=== As an interval size measure === | |||
The prima is useful for measurement of 11-limit intervals and commas. Given that 12276edo factors as 2<sup>2</sup> × 3<sup>2</sup> × 11 × 31, a prima is a whole number division of one degree of [[12edo]], [[22edo]], [[31edo]], [[99edo]] and [[198edo]]. The [[Pythagorean comma]] is represented by 240 prima, and the [[syntonic comma]] by 220 (and the [[schisma]] is therefore represented as 20 prima). A prima is almost exactly three [[tuning unit]]s. As one degree of [[12edo]] is 1023 prima, one [[cent]] is exactly 10.23 prima. | |||
=== Prime harmonics === | === Prime harmonics === | ||
Latest revision as of 21:46, 27 July 2026
| ← 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.
12276edo's step size is sometimes called a prima, a term proposed by Lillian Hearne, when used as an interval size unit.
Theory
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. It is the smallest atomic EDO inside its 5-odd-limit diamond tradeoff tuning range.
As an interval size measure
The prima is useful for measurement of 11-limit intervals and commas. Given that 12276edo factors as 22 × 32 × 11 × 31, a prima is a whole number division of one degree of 12edo, 22edo, 31edo, 99edo and 198edo. The Pythagorean comma is represented by 240 prima, and the syntonic comma by 220 (and the schisma is therefore represented as 20 prima). A prima is almost exactly three tuning units. As one degree of 12edo is 1023 prima, one cent is exactly 10.23 prima.
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.