412edo: Difference between revisions
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m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" |
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
| Line 9: | Line 9: | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
412 factors into | 412 factors into {{factorisation|412}}, with subset edos {{EDOs|2, 4, 103, and 206}}. [[1236edo]], which triples it, gives a good correction to harmonics 5, 7, and 11. | ||
== Regular temperament properties == | == Regular temperament properties == | ||
{ | {| class="wikitable center-4 center-5 center-6" | ||
|- | |||
! rowspan="2" | [[Subgroup]] | |||
! rowspan="2" | [[Comma list]] | |||
! rowspan="2" | [[Mapping]] | |||
! rowspan="2" | Optimal<br />8ve stretch (¢) | |||
! colspan="2" | Tuning error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |- | ||
| 2.3 | | 2.3 | ||
| Line 24: | Line 33: | ||
| {{monzo| 32 -7 -9 }}, {{monzo| -5 31 -19 }} | | {{monzo| 32 -7 -9 }}, {{monzo| -5 31 -19 }} | ||
| {{mapping| 412 653 957 }} | | {{mapping| 412 653 957 }} | ||
| | | −0.1501 | ||
| 0.2182 | | 0.2182 | ||
| 7.49 | | 7.49 | ||
| Line 31: | Line 40: | ||
| 6144/6125, 2460375/2458624, 49009212/48828125 | | 6144/6125, 2460375/2458624, 49009212/48828125 | ||
| {{mapping| 412 653 957 1157 }} | | {{mapping| 412 653 957 1157 }} | ||
| | | −0.2085 | ||
| 0.2143 | | 0.2143 | ||
| 7.36 | | 7.36 | ||
|} | |||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{ | {| class="wikitable center-all left-5" | ||
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | |||
|- | |||
! Periods<br />per 8ve | |||
! Generator* | |||
! Cents* | |||
! Associated<br />ratio* | |||
! Temperaments | |||
|- | |- | ||
| 1 | | 1 | ||
| Line 62: | Line 78: | ||
| 16875/16384 | | 16875/16384 | ||
| [[Septisuperfourth]] (7-limit) | | [[Septisuperfourth]] (7-limit) | ||
|} | |||
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct | |||
Latest revision as of 13:31, 13 March 2026
| ← 411edo | 412edo | 413edo → |
412 equal divisions of the octave (abbreviated 412edo or 412ed2), also called 412-tone equal temperament (412tet) or 412 equal temperament (412et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 412 equal parts of about 2.91 ¢ each. Each step represents a frequency ratio of 21/412, or the 412th root of 2.
Theory
412edo has a very accurate perfect fifth, but it is not quite accurate beyond that. The equal temperament tempers out [32 -7 -9⟩ (escapade comma) and [-69 45 -1⟩ (counterschisma) in the 5-limit; 6144/6125, 118098/117649, 2460375/2458624, 49009212/48828125, and notably the nanisma in the 7-limit. It supports nanic and counterschismic.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.01 | +1.06 | +1.08 | -0.83 | +1.22 | -0.10 | -0.43 | +0.85 | -1.42 | -0.38 |
| Relative (%) | +0.0 | -0.5 | +36.6 | +37.0 | -28.6 | +41.9 | -3.5 | -14.6 | +29.2 | -48.8 | -12.9 | |
| Steps (reduced) |
412 (0) |
653 (241) |
957 (133) |
1157 (333) |
1425 (189) |
1525 (289) |
1684 (36) |
1750 (102) |
1864 (216) |
2001 (353) |
2041 (393) | |
Subsets and supersets
412 factors into 22 × 103, with subset edos 2, 4, 103, and 206. 1236edo, which triples it, gives a good correction to harmonics 5, 7, and 11.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-653 412⟩ | [⟨412 653]] | +0.0042 | 0.0042 | 0.14 |
| 2.3.5 | [32 -7 -9⟩, [-5 31 -19⟩ | [⟨412 653 957]] | −0.1501 | 0.2182 | 7.49 |
| 2.3.5.7 | 6144/6125, 2460375/2458624, 49009212/48828125 | [⟨412 653 957 1157]] | −0.2085 | 0.2143 | 7.36 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 9\412 | 26.21 | 49/48 | Sfourth (5-limit) |
| 1 | 19\412 | 55.34 | 16875/16384 | Escapade (5-limit) |
| 1 | 171\412 | 498.06 | 4/3 | Counterschismic Nanic |
| 2 | 19\412 | 55.34 | 16875/16384 | Septisuperfourth (7-limit) |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct