472edo: Difference between revisions

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'''472edo''' is the [[EDO|equal division of the octave]] into 472 parts of 2.54237 cents each. It is consistent to the 11-limit, tempering out 32805/32768 (schisma) and 1224440064/1220703125 (parakleisma) in the 5-limit; 2401/2400, 2460375/2458624, and 30623756184/30517578125 in the 7-limit; 9801/9800, 46656/46585, 117649/117612, and 234375/234256 in the 11-limit, supporting the [[Breedsmic temperaments|maviloid temperament]], the [[Schismatic family|bisesqui temperament]], and the [[Schismatic family|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.
{{Infobox ET}}
{{ED intro}}


[[Category:Edo]]
== 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 ===
{{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 →
Prime factorization 23 × 59
Step size 2.54237 ¢ 
Fifth 276\472 (701.695 ¢) (→ 69\118)
Semitones (A1:m2) 44:36 (111.9 ¢ : 91.53 ¢)
Consistency limit 11
Distinct consistency limit 11

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

Approximation of prime harmonics in 472edo
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.

Table of rank-2 temperaments by generator
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

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