472edo: Difference between revisions
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472edo is [[ | == Theory == | ||
472edo is [[enfactoring|enfactored]] in the 5-limit, with the same tuning as [[118edo]], defined by [[tempering out]] the [[schisma]] and the [[parakleisma]], but the approximation to higher harmonics are much improved. It is a [[zeta peak integer edo]], [[consistent]] to the [[11-odd-limit]] or the no-13 [[29-odd-limit]]. | |||
In the 7-limit, the equal temperament tempers out [[2401/2400]], 2460375/2458624, and 30623756184/30517578125; in the 11-limit, [[9801/9800]], 46656/46585, 117649/117612, and 234375/234256, [[support]]ing the [[maviloid]] temperament, the [[Schismatic family #Bisesqui|bisesqui]] temperament, and the [[octant]] temperament. Using the [[patent val]], it tempers out [[729/728]], [[1575/1573]], [[2200/2197]], [[4096/4095]], and 21168/21125 in the 13-limit, so it also supports the 13-limit octant. | |||
=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|472}} | {{Harmonics in equal|472}} | ||
[[ | === Subsets and supersets === | ||
[[ | Since 472 factors into {{factorization|472}}, 472edo has subset edos {{EDOs| 2, 4, 8, 59, 118, and 236 }}. | ||
== Regular temperament properties == | |||
{| class="wikitable center-4 center-5 center-6" | |||
|- | |||
! rowspan="2" | [[Subgroup]] | |||
! rowspan="2" | [[Comma list]] | |||
! rowspan="2" | [[Mapping]] | |||
! rowspan="2" | Optimal<br />8ve stretch (¢) | |||
! colspan="2" | Tuning error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |||
| 2.3.5.7 | |||
| 2401/2400, 32805/32768, {{monzo| 8 14 -13 }} | |||
| [{{val| 472 748 1096 1325 }}] | |||
| +0.0435 | |||
| 0.0814 | |||
| 3.20 | |||
|- | |||
| 2.3.5.7.11 | |||
| 2401/2400, 9801/9800, 32805/32768, 46656/46585 | |||
| [{{val| 472 748 1096 1325 1633 }}] | |||
| +0.0130 | |||
| 0.0950 | |||
| 3.74 | |||
|- | |||
| 2.3.5.7.11.13 | |||
| 729/728, 1575/1573, 2200/2197, 2401/2400, 4096/4095 | |||
| [{{val| 472 748 1096 1325 1633 1747 }}] | |||
| −0.0341 | |||
| 0.1365 | |||
| 5.37 | |||
|} | |||
=== Rank-2 temperaments === | |||
Note: 5-limit temperaments supported by [[118edo|118et]] are not included. | |||
{| 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 | |||
| 69\472 | |||
| 175.42 | |||
| 448/405 | |||
| [[Sesquiquartififths]] | |||
|- | |||
| 1 | |||
| 137\472 | |||
| 348.31 | |||
| 57344/46875 | |||
| [[Subneutral]] | |||
|- | |||
| 1 | |||
| 205\472 | |||
| 521.19 | |||
| 875/648 | |||
| [[Maviloid]] | |||
|- | |||
| 2 | |||
| 69\472 | |||
| 175.42 | |||
| 448/405 | |||
| [[Bisesqui]] | |||
|- | |||
| 8 | |||
| 196\472<br />(19\472) | |||
| 498.31<br />(48.31) | |||
| 4/3<br />(36/35) | |||
| [[Octant]] | |||
|} | |||
<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 22:38, 20 February 2025
| ← 471edo | 472edo | 473edo → |
472 equal divisions of the octave (abbreviated 472edo or 472ed2), also called 472-tone equal temperament (472tet) or 472 equal temperament (472et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 472 equal parts of about 2.54 ¢ each. Each step represents a frequency ratio of 21/472, or the 472nd root of 2.
Theory
472edo is enfactored in the 5-limit, with the same tuning as 118edo, defined by tempering out the schisma and the parakleisma, but the approximation to higher harmonics are much improved. It is a zeta peak integer edo, consistent to the 11-odd-limit or the no-13 29-odd-limit.
In the 7-limit, the equal temperament tempers out 2401/2400, 2460375/2458624, and 30623756184/30517578125; in the 11-limit, 9801/9800, 46656/46585, 117649/117612, and 234375/234256, supporting the maviloid temperament, the bisesqui temperament, and the octant temperament. Using the patent val, it tempers out 729/728, 1575/1573, 2200/2197, 4096/4095, and 21168/21125 in the 13-limit, so it also supports the 13-limit octant.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.26 | +0.13 | -0.18 | +0.38 | +1.00 | -0.72 | -0.06 | -0.31 | +0.08 | -0.97 |
| Relative (%) | +0.0 | -10.2 | +5.0 | -7.2 | +14.8 | +39.2 | -28.2 | -2.2 | -12.1 | +3.3 | -38.1 | |
| Steps (reduced) |
472 (0) |
748 (276) |
1096 (152) |
1325 (381) |
1633 (217) |
1747 (331) |
1929 (41) |
2005 (117) |
2135 (247) |
2293 (405) |
2338 (450) | |
Subsets and supersets
Since 472 factors into 23 × 59, 472edo has subset edos 2, 4, 8, 59, 118, and 236.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5.7 | 2401/2400, 32805/32768, [8 14 -13⟩ | [⟨472 748 1096 1325]] | +0.0435 | 0.0814 | 3.20 |
| 2.3.5.7.11 | 2401/2400, 9801/9800, 32805/32768, 46656/46585 | [⟨472 748 1096 1325 1633]] | +0.0130 | 0.0950 | 3.74 |
| 2.3.5.7.11.13 | 729/728, 1575/1573, 2200/2197, 2401/2400, 4096/4095 | [⟨472 748 1096 1325 1633 1747]] | −0.0341 | 0.1365 | 5.37 |
Rank-2 temperaments
Note: 5-limit temperaments supported by 118et are not included.
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 69\472 | 175.42 | 448/405 | Sesquiquartififths |
| 1 | 137\472 | 348.31 | 57344/46875 | Subneutral |
| 1 | 205\472 | 521.19 | 875/648 | Maviloid |
| 2 | 69\472 | 175.42 | 448/405 | Bisesqui |
| 8 | 196\472 (19\472) |
498.31 (48.31) |
4/3 (36/35) |
Octant |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct