407edo: Difference between revisions
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Created page with "{{Infobox ET}} {{EDO intro|407}} == Theory == 407et tempers out 32805/32768 in the 5-limit; 4096000/4084101, 134217728/133984375, 26873856/26796875, 78125000/781218..." |
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 == | ||
407edo is a strong 5-limit system and 2.3.5.11.13.19.23 [[subgroup]] system. The equal temperament [[tempering out|tempers out]] [[32805/32768]] in the 5-limit; using the [[patent val]], [[16875/16807]], [[4096000/4084101]], and 26873856/26796875 in the 7-limit. It [[support]]s and provides the [[optimal patent val]] for the [[subsemifourth]] temperament in the 7- and 11-limit. [[Essentially tempered chord]]s available in 407et include [[pinkanberry chords]]. | |||
=== Prime harmonics === | === Prime harmonics === | ||
| Line 9: | Line 9: | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
407 factors into 11 × 37, with [[11edo]] and [[37edo]] as its subset edos. | 407 factors into {{nowrap|11 × 37}}, with [[11edo]] and [[37edo]] as its subset edos. [[814edo]], which doubles it, gives a good correction to harmonics 7 and 17, and is a notable full 23-limit temperament. | ||
== Regular temperament properties == | == Regular temperament properties == | ||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
|- | |- | ||
|2.3 | ! rowspan="2" | [[Subgroup]] | ||
|{{monzo|-645 407}} | ! rowspan="2" | [[Comma list]] | ||
|{{mapping|407 645}} | ! rowspan="2" | [[Mapping]] | ||
| 0.0742 | ! rowspan="2" | Optimal<br />8ve stretch (¢) | ||
! colspan="2" | Tuning error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |||
| 2.3 | |||
| {{monzo| -645 407 }} | |||
| {{mapping| 407 645 }} | |||
| +0.0742 | |||
| 0.0742 | | 0.0742 | ||
| 2.52 | | 2.52 | ||
|- | |- | ||
|2.3.5 | | 2.3.5 | ||
|32805/32768, {{monzo|30 47 -45}} | | 32805/32768, {{monzo| 30 47 -45 }} | ||
|{{mapping|407 645 945}} | | {{mapping| 407 645 945 }} | ||
| 0.0599 | | +0.0599 | ||
| 0.0638 | | 0.0638 | ||
| 2.16 | | 2.16 | ||
| Line 39: | Line 40: | ||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
|+Table of rank-2 temperaments by generator | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | ||
! Periods<br>per 8ve | |- | ||
! Generator | ! Periods<br />per 8ve | ||
! Cents | ! Generator* | ||
! Associated<br> | ! Cents* | ||
! Associated<br />ratio* | |||
! Temperaments | ! Temperaments | ||
|- | |- | ||
|1 | | 1 | ||
|63\407 | | 63\407 | ||
|185.75 | | 185.75 | ||
|{{monzo|24 4 -13}} | | {{monzo| 24 4 -13 }} | ||
|[[Pirate]] | | [[Pirate]] | ||
|- | |||
| 1 | |||
| 83\407 | |||
| 244.72 | |||
| 15/13 | |||
| [[Subsemifourth]] (407f) | |||
|- | |- | ||
|1 | | 1 | ||
|169\407 | | 169\407 | ||
|498.28 | | 498.28 | ||
|4/3 | | 4/3 | ||
|[[Helmholtz]] | | [[Helmholtz (temperament)|Helmholtz]] | ||
|} | |} | ||
<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 | |||
[[Category:Subsemifourth]] | |||
Latest revision as of 13:32, 13 March 2026
| ← 406edo | 407edo | 408edo → |
407 equal divisions of the octave (abbreviated 407edo or 407ed2), also called 407-tone equal temperament (407tet) or 407 equal temperament (407et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 407 equal parts of about 2.95 ¢ each. Each step represents a frequency ratio of 21/407, or the 407th root of 2.
Theory
407edo is a strong 5-limit system and 2.3.5.11.13.19.23 subgroup system. The equal temperament tempers out 32805/32768 in the 5-limit; using the patent val, 16875/16807, 4096000/4084101, and 26873856/26796875 in the 7-limit. It supports and provides the optimal patent val for the subsemifourth temperament in the 7- and 11-limit. Essentially tempered chords available in 407et include pinkanberry chords.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.24 | -0.07 | +1.20 | +0.03 | -0.23 | +1.19 | +0.28 | -0.26 | -0.58 | -1.06 |
| Relative (%) | +0.0 | -8.0 | -2.5 | +40.7 | +1.1 | -7.9 | +40.3 | +9.4 | -9.0 | -19.8 | -35.8 | |
| Steps (reduced) |
407 (0) |
645 (238) |
945 (131) |
1143 (329) |
1408 (187) |
1506 (285) |
1664 (36) |
1729 (101) |
1841 (213) |
1977 (349) |
2016 (388) | |
Subsets and supersets
407 factors into 11 × 37, with 11edo and 37edo as its subset edos. 814edo, which doubles it, gives a good correction to harmonics 7 and 17, and is a notable full 23-limit temperament.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-645 407⟩ | [⟨407 645]] | +0.0742 | 0.0742 | 2.52 |
| 2.3.5 | 32805/32768, [30 47 -45⟩ | [⟨407 645 945]] | +0.0599 | 0.0638 | 2.16 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 63\407 | 185.75 | [24 4 -13⟩ | Pirate |
| 1 | 83\407 | 244.72 | 15/13 | Subsemifourth (407f) |
| 1 | 169\407 | 498.28 | 4/3 | Helmholtz |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct