128edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} It is notable for being the equal division corresponding to a standard [[MIDI]] piano roll of 128 notes. | ||
== Theory == | == Theory == | ||
The equal temperament [[tempering out|tempers out]] 2109375/2097152 ([[semicomma]]) in the [[5-limit]]; [[245/243]], [[1029/1024]] and [[5120/5103]] in the 7-limit; [[385/384]] and [[441/440]] in the 11-limit. It provides the [[optimal patent val]] for [[7-limit]] [[rodan]], the {{nowrap|41 & 87}} temperament, as well as for 7-limit [[fourfives]], the {{nowrap|60 & 68}} temperament. | |||
See also [https://www.youtube.com/watch?v=lGa66qHzKME 128 notes per octave on Alto Saxophone] (Demo by Philipp Gerschlauer) | See also [https://www.youtube.com/watch?v=lGa66qHzKME 128 notes per octave on Alto Saxophone] (Demo by Philipp Gerschlauer) | ||
=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|128 | {{Harmonics in equal|128}} | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
Since 128 factors into 2<sup>7</sup>, 128edo has subset edos {{EDOs| 2, 4, 8, 16, 32, and 64 }}. | Since 128 factors into 2<sup>7</sup>, 128edo has subset edos {{EDOs| 2, 4, 8, 16, 32, and 64 }}. | ||
== Regular temperament properties == | == Regular temperament properties == | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
|+ | |+ 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 | ||
|- | |- | ||
| Line 68: | Line 66: | ||
|- | |- | ||
| 4 | | 4 | ||
| 53\128<br>(11\128) | | 53\128<br />(11\128) | ||
| 496.875<br>(103.125) | | 496.875<br />(103.125) | ||
| 4/3 | | 4/3 | ||
| [[Undim]] (7-limit) | | [[Undim]] (7-limit) | ||
|} | |} | ||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | |||
== Scales == | == Scales == | ||
Latest revision as of 11:18, 11 April 2025
| ← 127edo | 128edo | 129edo → |
128 equal divisions of the octave (abbreviated 128edo or 128ed2), also called 128-tone equal temperament (128tet) or 128 equal temperament (128et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 128 equal parts of about 9.38 ¢ each. Each step represents a frequency ratio of 21/128, or the 128th root of 2. It is notable for being the equal division corresponding to a standard MIDI piano roll of 128 notes.
Theory
The equal temperament tempers out 2109375/2097152 (semicomma) in the 5-limit; 245/243, 1029/1024 and 5120/5103 in the 7-limit; 385/384 and 441/440 in the 11-limit. It provides the optimal patent val for 7-limit rodan, the 41 & 87 temperament, as well as for 7-limit fourfives, the 60 & 68 temperament.
See also 128 notes per octave on Alto Saxophone (Demo by Philipp Gerschlauer)
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +1.17 | -1.94 | -3.20 | +1.81 | +3.22 | -1.83 | +2.49 | -0.15 | +1.67 | -1.29 |
| Relative (%) | +0.0 | +12.5 | -20.7 | -34.1 | +19.3 | +34.4 | -19.5 | +26.5 | -1.6 | +17.8 | -13.7 | |
| Steps (reduced) |
128 (0) |
203 (75) |
297 (41) |
359 (103) |
443 (59) |
474 (90) |
523 (11) |
544 (32) |
579 (67) |
622 (110) |
634 (122) | |
Subsets and supersets
Since 128 factors into 27, 128edo has subset edos 2, 4, 8, 16, 32, and 64.
Regular temperament properties
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 25\128 | 234.375 | 8/7 | Rodan |
| 1 | 29\128 | 271.875 | 75/64 | Orson |
| 1 | 33\128 | 309.375 | 448/375 | Triwell |
| 1 | 53\128 | 496.875 | 4/3 | Undecental |
| 2 | 13\128 | 121.875 | 15/14 | Lagaca |
| 2 | 15\128 | 140.625 | 27/25 | Fifive |
| 4 | 15\128 | 140.625 | 27/25 | Fourfives |
| 4 | 53\128 (11\128) |
496.875 (103.125) |
4/3 | Undim (7-limit) |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct