255edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
| Line 17: | Line 17: | ||
! rowspan="2" | [[Comma list]] | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br | ! rowspan="2" | Optimal<br>8ve stretch (¢) | ||
! colspan="2" | Tuning error | ! colspan="2" | Tuning error | ||
|- | |- | ||
| Line 24: | Line 24: | ||
|- | |- | ||
| 2.3 | | 2.3 | ||
| {{ | | {{Monzo| -404 255 }} | ||
| {{ | | {{Mapping| 255 404 }} | ||
| +0.246 | | +0.246 | ||
| 0.246 | | 0.246 | ||
| Line 31: | Line 31: | ||
|- | |- | ||
| 2.3.5 | | 2.3.5 | ||
| {{ | | {{Monzo| 8 14 -13 }}, {{monzo| -36 11 8 }} | ||
| {{ | | {{Mapping| 255 404 592 }} | ||
| +0.226 | | +0.226 | ||
| 0.203 | | 0.203 | ||
| Line 39: | Line 39: | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| 16875/16807, 19683/19600, 65625/65536 | | 16875/16807, 19683/19600, 65625/65536 | ||
| {{ | | {{Mapping| 255 404 592 716 }} | ||
| +0.117 | | +0.117 | ||
| 0.257 | | 0.257 | ||
| Line 46: | Line 46: | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 540/539, 1375/1372, 8019/8000, 65625/65536 | | 540/539, 1375/1372, 8019/8000, 65625/65536 | ||
| {{ | | {{Mapping| 255 404 592 716 882 }} | ||
| +0.136 | | +0.136 | ||
| 0.233 | | 0.233 | ||
| Line 56: | Line 56: | ||
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | ||
|- | |- | ||
! Periods<br | ! Periods<br>per 8ve | ||
! Generator* | ! Generator* | ||
! Cents* | ! Cents* | ||
! Associated<br | ! Associated<br>ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| Line 87: | Line 87: | ||
|- | |- | ||
| 3 | | 3 | ||
| 82\255<br | | 82\255<br>(3\255) | ||
| 385.88<br | | 385.88<br>(14.12) | ||
| 5/4<br | | 5/4<br>(126/125) | ||
| [[Mutt]] (7-limit) | | [[Mutt]] (7-limit) | ||
|- | |- | ||
| 5 | | 5 | ||
| 53\255<br | | 53\255<br>(2\255) | ||
| 249.41<br | | 249.41<br>(9.41) | ||
| 81/70<br | | 81/70<br>(176/175) | ||
| [[ | | [[Hemiquintile]] / hemiquint (255) / hemiquintilis (255f) | ||
|- | |- | ||
| 5 | | 5 | ||
| 106\255<br | | 106\255<br>(4\255) | ||
| 498.82<br | | 498.82<br>(18.82) | ||
| 4/3<br | | 4/3<br>(81/80) | ||
| [[ | | [[Quintile]] | ||
|- | |- | ||
| 17 | | 17 | ||
| 53\255<br | | 53\255<br>(7\255) | ||
| 249.41<br | | 249.41<br>(32.94) | ||
| {{ | | {{Monzo| -25 -9 17 }}<br>(1990656/1953125) | ||
| [[Chlorine]] (5-limit) | | [[Chlorine]] (5-limit) | ||
|} | |} | ||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | <nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | ||
[[Category:Mirkat]] | [[Category:Mirkat]] | ||
Latest revision as of 17:53, 19 February 2025
| ← 254edo | 255edo | 256edo → |
255 equal divisions of the octave (abbreviated 255edo or 255ed2), also called 255-tone equal temperament (255tet) or 255 equal temperament (255et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 255 equal parts of about 4.71 ¢ each. Each step represents a frequency ratio of 21/255, or the 255th root of 2.
Theory
The equal temperament tempers out the parakleisma, [8 14 -13⟩, and the septendecima, [-52 -17 34⟩, in the 5-limit. In the 7-limit it tempers out cataharry, 19683/19600, mirkwai, 16875/16807 and horwell, 65625/65536, so that it supports the mirkat temperament, and in fact provides the optimal patent val. It also gives the optimal patent val for mirkat in the 11-limit, tempering out 540/539, 1375/1372, 3025/3024 and 8019/8000. In the 13-limit it tempers out 847/845, 625/624, 1575/1573 and 1716/1715.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.78 | -0.43 | +0.59 | -0.73 | +1.83 | -1.43 | -1.04 | +2.31 | +1.01 | -1.51 |
| Relative (%) | +0.0 | -16.5 | -9.2 | +12.4 | -15.5 | +38.8 | -30.3 | -22.2 | +49.2 | +21.5 | -32.0 | |
| Steps (reduced) |
255 (0) |
404 (149) |
592 (82) |
716 (206) |
882 (117) |
944 (179) |
1042 (22) |
1083 (63) |
1154 (134) |
1239 (219) |
1263 (243) | |
Subsets and supersets
Since 255 factors into 3 × 5 × 17, 255edo has subset edos 3, 5, 15, 17, 51, and 85.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-404 255⟩ | [⟨255 404]] | +0.246 | 0.246 | 5.22 |
| 2.3.5 | [8 14 -13⟩, [-36 11 8⟩ | [⟨255 404 592]] | +0.226 | 0.203 | 4.30 |
| 2.3.5.7 | 16875/16807, 19683/19600, 65625/65536 | [⟨255 404 592 716]] | +0.117 | 0.257 | 5.46 |
| 2.3.5.7.11 | 540/539, 1375/1372, 8019/8000, 65625/65536 | [⟨255 404 592 716 882]] | +0.136 | 0.233 | 4.95 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 39\255 | 183.53 | 10/9 | Mirkat (255f) |
| 1 | 52\255 | 244.71 | 15/13 | Subsemifourth (255) |
| 1 | 67\255 | 315.29 | 6/5 | Parakleismic (5-limit) |
| 1 | 74\255 | 348.24 | 11/9 | Eris (255) |
| 3 | 82\255 (3\255) |
385.88 (14.12) |
5/4 (126/125) |
Mutt (7-limit) |
| 5 | 53\255 (2\255) |
249.41 (9.41) |
81/70 (176/175) |
Hemiquintile / hemiquint (255) / hemiquintilis (255f) |
| 5 | 106\255 (4\255) |
498.82 (18.82) |
4/3 (81/80) |
Quintile |
| 17 | 53\255 (7\255) |
249.41 (32.94) |
[-25 -9 17⟩ (1990656/1953125) |
Chlorine (5-limit) |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct