412edo: Difference between revisions
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== Theory == | == Theory == | ||
412edo has a very accurate [[3/2|perfect fifth]], but it is not quite accurate beyond that. The equal temperament [[tempering out|tempers out]] {{monzo| 32 -7 -9 }} ([[escapade comma]]) and {{monzo| -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 === | === Prime harmonics === | ||
| Line 9: | Line 9: | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
412 factors into 2<sup>2</sup> × 103, with subset edos {{EDOs|2, 4, 103, and 206}}. [[1236edo]], which triples it, gives a good correction to | 412 factors into 2<sup>2</sup> × 103, 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" | {| class="wikitable center-4 center-5 center-6" | ||
! rowspan="2" |[[Subgroup]] | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" |[[Comma list|Comma List]] | ! rowspan="2" | [[Comma list|Comma List]] | ||
! rowspan="2" |[[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" |Optimal<br>8ve Stretch (¢) | ! rowspan="2" | Optimal<br>8ve Stretch (¢) | ||
! colspan="2" |Tuning Error | ! colspan="2" | Tuning Error | ||
|- | |- | ||
![[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
![[TE simple badness|Relative]] (%) | ! [[TE simple badness|Relative]] (%) | ||
|- | |- | ||
|2.3 | | 2.3 | ||
|{{monzo|-653 412}} | | {{monzo|-653 412}} | ||
|{{mapping|412 653}} | | {{mapping| 412 653 }} | ||
| +0.0042 | | +0.0042 | ||
| 0.0042 | | 0.0042 | ||
| 0.14 | | 0.14 | ||
|- | |- | ||
|2.3.5 | | 2.3.5 | ||
|{{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 | | -0.1501 | ||
| 0.2182 | | 0.2182 | ||
| 7.49 | | 7.49 | ||
|- | |- | ||
|2.3.5.7 | | 2.3.5.7 | ||
|6144/6125, 2460375/2458624, | | 6144/6125, 2460375/2458624, 49009212/48828125 | ||
|{{mapping|412 653 957 1157}} | | {{mapping| 412 653 957 1157 }} | ||
| -0.2085 | | -0.2085 | ||
| 0.2143 | | 0.2143 | ||
| Line 48: | Line 48: | ||
|+Table of rank-2 temperaments by generator | |+Table of rank-2 temperaments by generator | ||
! Periods<br>per 8ve | ! Periods<br>per 8ve | ||
! Generator | ! Generator* | ||
! Cents | ! Cents* | ||
! Associated<br>Ratio* | ! Associated<br>Ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
|1 | | 1 | ||
|9\412 | | 9\412 | ||
|26.21 | | 26.21 | ||
|49/48 | | 49/48 | ||
|[[Sfourth]] | | [[Sfourth]] (5-limit) | ||
|- | |- | ||
|1 | | 1 | ||
|19\412 | | 19\412 | ||
|55.34 | | 55.34 | ||
|16875/16384 | | 16875/16384 | ||
|[[Escapade]] | | [[Escapade]] (5-limit) | ||
|- | |- | ||
|1 | | 1 | ||
|171\412 | | 171\412 | ||
|498.06 | | 498.06 | ||
|4/3 | | 4/3 | ||
|[[Counterschismic]] | | [[Counterschismic]]<br>[[Nanic]] | ||
|- | |- | ||
|2 | | 2 | ||
|19\412 | | 19\412 | ||
|55.34 | | 55.34 | ||
|16875/16384 | | 16875/16384 | ||
|[[Septisuperfourth]] | | [[Septisuperfourth]] (7-limit) | ||
|} | |} | ||
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | <nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | ||
Revision as of 10:34, 1 January 2024
| ← 411edo | 412edo | 413edo → |
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 it is distinct