1323edo: Difference between revisions
Cleanup; clarify the title row of the rank-2 temp table; -redundant categories |
Tristanbay (talk | contribs) →Subsets and supersets: Added mention of 7edo and 27edo as zeta edos Tags: Mobile edit Mobile web edit |
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
1323edo is the smallest edo [[consistency|distinctly consistent]] in the [[29-odd-limit]]. | 1323edo is the smallest edo [[consistency|distinctly consistent]] in the [[29-odd-limit]]. It is [[enfactoring|enfactored]] in the 7-limit, sharing the same excellent 7-limit approximation with [[441edo]], but it makes for a great higher-limit system by splitting each step of 441edo into three. | ||
It provides the [[optimal patent val]] for the 11-limit [[trinealimmal]] temperament, which has a period of 1\27 octave | It provides the [[optimal patent val]] for the 11-limit [[trinealimmal]] temperament, which has a period of 1\27 octave. | ||
=== Prime harmonics === | === Prime harmonics === | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
1323 | Since 1323 factors into {{factorization|1323}}, 1323edo has subset edos {{EDOs| 3, 7, 9, 21, 27, 49, 63, 147, 189, 441 }}, of which 7, 27, and 441edo are members of the [[zeta edo]]s. | ||
== Regular temperament properties == | == Regular temperament properties == | ||
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{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
! Periods<br>per 8ve | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | ||
|- | |||
! Periods<br />per 8ve | |||
! Generator* | ! Generator* | ||
! Cents* | ! Cents* | ||
! Associated<br> | ! Associated<br />ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| 27 | | 27 | ||
| 299\1323<br>(5\1323) | | 299\1323<br />(5\1323) | ||
| 271.201<br>(4.535) | | 271.201<br />(4.535) | ||
| 1375/1176<br>(?) | | 1375/1176<br />(?) | ||
| [[Trinealimmal]] | | [[Trinealimmal]] | ||
|} | |} | ||
<nowiki>* | <nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | ||
Latest revision as of 01:48, 26 August 2025
| ← 1322edo | 1323edo | 1324edo → |
1323 equal divisions of the octave (abbreviated 1323edo or 1323ed2), also called 1323-tone equal temperament (1323tet) or 1323 equal temperament (1323et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1323 equal parts of about 0.907 ¢ each. Each step represents a frequency ratio of 21/1323, or the 1323rd root of 2.
Theory
1323edo is the smallest edo distinctly consistent in the 29-odd-limit. It is enfactored in the 7-limit, sharing the same excellent 7-limit approximation with 441edo, but it makes for a great higher-limit system by splitting each step of 441edo into three.
It provides the optimal patent val for the 11-limit trinealimmal temperament, which has a period of 1\27 octave.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.086 | +0.081 | -0.118 | +0.156 | +0.289 | +0.260 | -0.007 | +0.297 | -0.099 | -0.364 |
| Relative (%) | +0.0 | +9.5 | +8.9 | -13.1 | +17.2 | +31.8 | +28.7 | -0.8 | +32.8 | -10.9 | -40.2 | |
| Steps (reduced) |
1323 (0) |
2097 (774) |
3072 (426) |
3714 (1068) |
4577 (608) |
4896 (927) |
5408 (116) |
5620 (328) |
5985 (693) |
6427 (1135) |
6554 (1262) | |
Subsets and supersets
Since 1323 factors into 33 × 72, 1323edo has subset edos 3, 7, 9, 21, 27, 49, 63, 147, 189, 441, of which 7, 27, and 441edo are members of the zeta edos.
Regular temperament properties
Rank-2 temperaments
Note: 7-limit temperaments supported by 441et are not included.
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 27 | 299\1323 (5\1323) |
271.201 (4.535) |
1375/1176 (?) |
Trinealimmal |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct