472edo: Difference between revisions
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472edo is [[consistent]] to the [[11-odd-limit]]. It is [[Enfactoring|enfactored]] in the 5-limit, with the same tuning as 118edo, defined by tempering out the [[schisma]] and the [[parakleisma]]. In the 7-limit, it tempers out [[2401/2400]], 2460375/2458624, and 30623756184/30517578125; in the 11-limit, [[9801/9800]], 46656/46585, 117649/117612, and 234375/234256 , [[Support|supporting]] the [[Breedsmic temperaments #Maviloid|maviloid]] temperament, the [[Schismatic family #Bisesqui|bisesqui temperament]], and the [[Schismatic family #Octant|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. | 472edo is [[consistent]] to the [[11-odd-limit]]. It is [[Enfactoring|enfactored]] in the 5-limit, with the same tuning as 118edo, defined by tempering out the [[schisma]] and the [[parakleisma]]. In the 7-limit, it tempers out [[2401/2400]], 2460375/2458624, and 30623756184/30517578125; in the 11-limit, [[9801/9800]], 46656/46585, 117649/117612, and 234375/234256 , [[Support|supporting]] the [[Breedsmic temperaments #Maviloid|maviloid]] temperament, the [[Schismatic family #Bisesqui|bisesqui temperament]], and the [[Schismatic family #Octant|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. | ||
472edo is a [[zeta peak integer edo]]. | |||
=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|472}} | {{Harmonics in equal|472|columns=11}} | ||
== Regular temperament properties == | == Regular temperament properties == | ||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
! rowspan="2" | Subgroup | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" | [[Comma list]] | ! rowspan="2" | [[Comma list|Comma List]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br>8ve | ! rowspan="2" | Optimal<br>8ve Stretch (¢) | ||
! colspan="2" | Tuning | ! colspan="2" | Tuning Error | ||
|- | |- | ||
! [[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
| Line 54: | Line 54: | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
|+Table of rank-2 temperaments by generator | |+Table of rank-2 temperaments by generator | ||
! Periods<br>per | ! Periods<br>per Octave | ||
! Generator<br>( | ! Generator<br>(Reduced) | ||
! Cents<br>( | ! Cents<br>(Reduced) | ||
! Associated<br> | ! Associated<br>Ratio | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| Line 74: | Line 74: | ||
| 1 | | 1 | ||
| 205\472 | | 205\472 | ||
| | | 521.19 | ||
| 875/648 | | 875/648 | ||
| [[Maviloid]] | | [[Maviloid]] | ||
Revision as of 00:07, 30 August 2022
| ← 471edo | 472edo | 473edo → |
The 472 equal divisions of the octave (472edo), or the 472(-tone) equal temperament (472tet, 472et) when viewed from a regular temperament perspective, is the equal division of the octave into 472 parts of about 2.54 cents each.
Theory
472edo is consistent to the 11-odd-limit. It is enfactored in the 5-limit, with the same tuning as 118edo, defined by tempering out the schisma and the parakleisma. In the 7-limit, it 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.
472edo is a zeta peak integer edo.
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) | |
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 Octave |
Generator (Reduced) |
Cents (Reduced) |
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 |