12276edo: Difference between revisions
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-section title for now since there's no other sections; +prime error table; +categories |
not sure why reduplicate how much are commas in steps in different sections, write that more concisely |
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{{EDO intro|12276}} | {{EDO intro|12276}} | ||
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]], so that the Pythagorean and syntonic commas an be approximated by 12 and 11 schismas respectively | 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 === | === Prime harmonics === | ||
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=== Interval size measure === | === Interval size measure === | ||
12276edo factors as 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 | 12276edo factors as 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. | ||
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[[Category:12276edo| ]] <!-- main article --> | [[Category:12276edo| ]] <!-- main article --> | ||
Revision as of 12:56, 10 June 2023
| ← 12275edo | 12276edo | 12277edo → |
(semiconvergent)
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) | |
Interval size measure
12276edo factors 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.