12edo: Difference between revisions
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== Theory == | == Theory == | ||
{| class="wikitable center-all" | {| class="wikitable center-all" | ||
! | |+ Approximation of prime intervals in 12-EDO | ||
! | ! Prime number | ||
! | ! 2 | ||
! | ! 3 | ||
! | ! 5 | ||
! | ! 7 | ||
! | ! 11 | ||
! | ! 13 | ||
! | ! 17 | ||
! 19 | |||
|- | |- | ||
! Error (¢) | ! Error (¢) | ||
| 0 | | 0.0 | ||
| -2.0 | | -2.0 | ||
| +13.7 | | +13.7 | ||
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| +2.5 | | +2.5 | ||
|- | |- | ||
! [[Patent val| | ! [[Patent val|Nearest edomapping]] | ||
| 12 | | 12 | ||
| 7 | | 7 | ||
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<nowiki>*</nowiki> based on treating 12-EDO as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible. | <nowiki>*</nowiki> based on treating 12-EDO as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible. | ||
== | == JI approximation == | ||
=== 15-odd-limit interval mappings === | |||
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. | The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. | ||
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|} | |} | ||
=== Selected 19-limit intervals === | |||
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]] | [[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]] | ||
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[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]] | [[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]] | ||
== | == Regular temperament properties == | ||
{| class="wikitable center-all" | {| class="wikitable center-all" | ||
! colspan="2" | | ! colspan="2" | | ||
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| 4.53 | | 4.53 | ||
|} | |} | ||
12et (12f val) is lower in relative error than any previous edos in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroup are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72. | |||
== | === Commas === | ||
12-EDO [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}. | 12-EDO [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}. | ||
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|} | |} | ||
<references/> | <references/> | ||
=== Rank-2 temperaments === | |||
* [[List of 12et rank two temperaments by badness]] | |||
* [[List of 12et rank two temperaments by complexity]] | |||
* [[List of edo-distinct 12f rank two temperaments]] | |||
* [[Schismic-Pythagorean equivalence continuum]] | |||
{| class="wikitable center-1 center-2" | |||
|- | |||
! Periods <br> per octave | |||
! Generator | |||
! Temperaments | |||
|- | |||
| 1 | |||
| 1\12 | |||
| [[Ripple]] | |||
|- | |||
| 1 | |||
| 5\12 | |||
| [[Meantone]]/[[dominant]] | |||
|- | |||
| 2 | |||
| 1\12 | |||
| [[Srutal]]/[[pajara]]/[[injera]] | |||
|- | |||
| 3 | |||
| 1\12 | |||
| [[Augmented]] | |||
|- | |||
| 4 | |||
| 1\12 | |||
| [[Diminished]] | |||
|- | |||
| 6 | |||
| 1\12 | |||
| [[Hexe]] | |||
|} | |||
== Scales == | |||
{{main| MOS in 12edo }} | |||
The two most common 12-EDO MOS scales are meantone[5] and meantone[7]. | |||
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12) | |||
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12) | |||
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12) | |||
=== Pathological Modes === | |||
2 1 1 1 1 2 1 1 1 1 [[2L 8s]] MOS | |||
3 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS | |||
2 1 1 1 1 1 1 1 1 1 1 [[1L 10s]] MOS | |||
== See also == | == See also == | ||