720edo: Difference between revisions
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== Theory == | == Theory == | ||
720edo is only [[consistent]] to the [[5-odd-limit]], but it has a reasonable approximation of the full 17-limit using the [[patent val]]. It [[tempering out|tempers out]] the [[schisma]] in the 5-limit. It [[support]]s [[octant]] up to the 11-limit and [[tetraicosic]] up to the 19-limit. | |||
720edo is | |||
The patent val can also be thought of as a 2.3.17.23.31.43 [[subgroup]]-suited val, because these harmonics have error of less than 1 standard deviaiton away from step. In it, it supports the 195 & 720 temperament, period 15 with comma basis 1377/1376, 19683/19652, 67797/67712, 177147/176824. | |||
=== Prime harmonics === | |||
{{Harmonics in equal|720}} | |||
=== | === Subsets and supersets === | ||
720edo is the 14th [[superabundant edo]], and also the 6th factorial edo ({{nowrap|720 {{=}} 1 × 2 × 3 × 4 × 5 × 6 {{=}} 6!}}), which means it contains a massive amount of subsets, limited modes of transposition, and fraction-octave [[mos]]ses. With 720edo, it is better to use various vals mimicking smaller edos instead of the patent val, because it sounds as if the patent val is ''creating'' commas, not tempering them out (for example, the 421\720 patent val fifth vs. the 420\720 fifth stemming from [[12edo]]). | |||
However, the patent val's 5/4 of 720edo comes from [[90edo]], and not 72edo. | Since {{nowrap|720 {{=}} 72 × 10}}, its possible to conceptualize it as a superset of [[72edo]] and [[10edo]], which are interesting in their own right. However, the patent val's 5/4 of 720edo comes from [[90edo]], and not 72edo. | ||
=== | == 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 | |||
| 421\720 | |||
| 701.667 | |||
| 4/3 | |||
| [[Helmholtz (temperament)|Helmholtz]] | |||
|- | |||
| 8 | |||
| 421\720<br />(61\720) | |||
| 701.667<br />(101.667) | |||
| 4/3<br />(36/35) | |||
| [[Octant]] | |||
|- | |||
| 80 | |||
| 421\720<br />(7\720) | |||
| 701.667<br />(11.667) | |||
| 4/3<br />(?) | |||
| [[Octogintic]] | |||
|- | |||
| 80 | |||
| 283\720<br />(4\720) | |||
| 471.667<br />(6.667) | |||
| 130/99<br />(?) | |||
| [[Tetraicosic]] | |||
|} | |||
<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: | [[Category:Schismic]] | ||
[[Category:Octant]] | |||
Latest revision as of 13:33, 13 March 2026
| ← 719edo | 720edo | 721edo → |
720 equal divisions of the octave (abbreviated 720edo or 720ed2), also called 720-tone equal temperament (720tet) or 720 equal temperament (720et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 720 equal parts of about 1.67 ¢ each. Each step represents a frequency ratio of 21/720, or the 720th root of 2.
Theory
720edo is only consistent to the 5-odd-limit, but it has a reasonable approximation of the full 17-limit using the patent val. It tempers out the schisma in the 5-limit. It supports octant up to the 11-limit and tetraicosic up to the 19-limit.
The patent val can also be thought of as a 2.3.17.23.31.43 subgroup-suited val, because these harmonics have error of less than 1 standard deviaiton away from step. In it, it supports the 195 & 720 temperament, period 15 with comma basis 1377/1376, 19683/19652, 67797/67712, 177147/176824.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | -0.288 | +0.353 | -0.493 | +0.349 | -0.528 | +0.045 | +0.820 | +0.059 | +0.423 | -0.036 |
| Relative (%) | +0.0 | -17.3 | +21.2 | -29.6 | +20.9 | -31.7 | +2.7 | +49.2 | +3.5 | +25.4 | -2.1 | |
| Steps (reduced) |
720 (0) |
1141 (421) |
1672 (232) |
2021 (581) |
2491 (331) |
2664 (504) |
2943 (63) |
3059 (179) |
3257 (377) |
3498 (618) |
3567 (687) | |
Subsets and supersets
720edo is the 14th superabundant edo, and also the 6th factorial edo (720 = 1 × 2 × 3 × 4 × 5 × 6 = 6!), which means it contains a massive amount of subsets, limited modes of transposition, and fraction-octave mosses. With 720edo, it is better to use various vals mimicking smaller edos instead of the patent val, because it sounds as if the patent val is creating commas, not tempering them out (for example, the 421\720 patent val fifth vs. the 420\720 fifth stemming from 12edo).
Since 720 = 72 × 10, its possible to conceptualize it as a superset of 72edo and 10edo, which are interesting in their own right. However, the patent val's 5/4 of 720edo comes from 90edo, and not 72edo.
Regular temperament properties
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 421\720 | 701.667 | 4/3 | Helmholtz |
| 8 | 421\720 (61\720) |
701.667 (101.667) |
4/3 (36/35) |
Octant |
| 80 | 421\720 (7\720) |
701.667 (11.667) |
4/3 (?) |
Octogintic |
| 80 | 283\720 (4\720) |
471.667 (6.667) |
130/99 (?) |
Tetraicosic |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct