150edo: Difference between revisions
Wikispaces>FREEZE No edit summary |
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" Tags: Mobile edit Mobile web edit |
||
| (16 intermediate revisions by 8 users not shown) | |||
| Line 1: | Line 1: | ||
{{Infobox ET}} | |||
[[ | {{ED intro}} | ||
== Theory == | |||
150edo is [[contorted]] in the 5-limit, [[tempering out]] the same commas as [[75edo]], including [[20000/19683]] and [[2109375/2097152]]. However, every 11th step of 150edo is equal to the [[88cET]] nonoctave tuning, which is also represented as [[octacot]] through a regular temperament theory perspective. It provides a good tuning for octacot, for which 88{{c}} provides a generator. | |||
The equal temperament tempers out [[245/243]], [[2401/2400]], and [[4000/3969]] in the 7-limit, [[385/384]], [[896/891]], and [[1375/1372]] in the 11-limit, and [[352/351]], [[364/363]], [[676/675]] and [[1575/1573]] in the 13-limit. | |||
=== Odd harmonics === | |||
{{Harmonics in equal|150|columns=10}} | |||
=== Subsets and supersets === | |||
Since 150 factors into 2 × 3 × 5<sup>2</sup>, 150edo has subset edos {{EDOs| 2, 3, 5, 6, 10, 15, 25, 30, 50, and 75 }}. | |||
== Regular temperament properties == | |||
=== Rank-2 temperaments === | |||
{| class="wikitable center-all left-5" | |||
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | |||
|- | |||
! Periods<br />per 8ve | |||
! Generator* | |||
! Cents* | |||
! Associated<br />ratio* | |||
! Temperaments | |||
|- | |||
| 1 | |||
| 11\150 | |||
| 88.00 | |||
| 21/20 | |||
| [[Octacot]] (150e) / [[october]] (150) | |||
|- | |||
| 1 | |||
| 29\150 | |||
| 232.00 | |||
| 8/7 | |||
| [[Quadrawell]] | |||
|- | |||
| 10 | |||
| 31\150<br />(1\150) | |||
| 248.00<br />(8.00) | |||
| 15/13<br />(176/175) | |||
| [[Decoid]] (150e) | |||
|} | |||
<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 | |||
Latest revision as of 13:31, 13 March 2026
| ← 149edo | 150edo | 151edo → |
150 equal divisions of the octave (abbreviated 150edo or 150ed2), also called 150-tone equal temperament (150tet) or 150 equal temperament (150et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 150 equal parts of exactly 8 ¢ each. Each step represents a frequency ratio of 21/150, or the 150th root of 2.
Theory
150edo is contorted in the 5-limit, tempering out the same commas as 75edo, including 20000/19683 and 2109375/2097152. However, every 11th step of 150edo is equal to the 88cET nonoctave tuning, which is also represented as octacot through a regular temperament theory perspective. It provides a good tuning for octacot, for which 88 ¢ provides a generator.
The equal temperament tempers out 245/243, 2401/2400, and 4000/3969 in the 7-limit, 385/384, 896/891, and 1375/1372 in the 11-limit, and 352/351, 364/363, 676/675 and 1575/1573 in the 13-limit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +2.04 | -2.31 | -0.83 | -3.91 | +0.68 | -0.53 | -0.27 | -0.96 | -1.51 | +1.22 |
| Relative (%) | +25.6 | -28.9 | -10.3 | -48.9 | +8.5 | -6.6 | -3.4 | -11.9 | -18.9 | +15.2 | |
| Steps (reduced) |
238 (88) |
348 (48) |
421 (121) |
475 (25) |
519 (69) |
555 (105) |
586 (136) |
613 (13) |
637 (37) |
659 (59) | |
Subsets and supersets
Since 150 factors into 2 × 3 × 52, 150edo has subset edos 2, 3, 5, 6, 10, 15, 25, 30, 50, and 75.
Regular temperament properties
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 11\150 | 88.00 | 21/20 | Octacot (150e) / october (150) |
| 1 | 29\150 | 232.00 | 8/7 | Quadrawell |
| 10 | 31\150 (1\150) |
248.00 (8.00) |
15/13 (176/175) |
Decoid (150e) |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct