436edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
The '''436 equal divisions of the octave''' ('''436edo'''), or the '''436(-tone) equal temperament''' ('''436tet''', '''436et''') when viewed from a [[regular temperament]] perspective, is the [[EDO|equal division of the octave]] into 436 parts of about 2.75 [[cent]]s each.
{{ED intro}}


== Theory ==
== Theory ==
The [[patent val]] of 436edo has a distinct flat tendency, in the sense that if the [[octave]] is pure, harmonics from 3 to 37 are all flat. It is consistent to the [[23-odd-limit]], tempering out 32805/32768 and {{monzo| 1 -68 4 }} in the 5-limit; [[390625/388962]], 420175/419904, and 2100875/2097152 in the 7-limit; 1375/1372, 6250/6237, 41503/41472, and 322102/321489 in the 11-limit; [[625/624]], [[1716/1715]], [[2080/2079]], [[10648/10647]], and 15379/15360 in the 13-limit; [[715/714]], [[1089/1088]], [[1225/1224]], 1275/1274, [[2025/2023]], and 11271/11264 in the 17-limit; 1331/1330, [[1445/1444]], [[1521/1520]], 1540/1539, [[1729/1728]], 4394/4389, and 4875/4864 in the 19-limit; 875/874, 897/896, 1105/1104, 1863/1862, 2024/2023, 2185/2184, 2300/2299, and 2530/2527 in the 23-limit.
436edo is [[consistent]] to the [[23-odd-limit]]. The [[patent val]] of 436edo has a distinct flat tendency, in the sense that if the [[octave]] is pure, [[harmonic]]s from 3 to 37 are all flat.  


436edo is accurate for some intervals including [[3/2]], [[7/4]], [[11/10]], [[13/10]], [[18/17]], and [[19/18]], so it is especially suitable for the 2.3.7.11/5.13/5.17.19 subgroup.
It [[tempering out|tempers out]] [[32805/32768]] and {{monzo| 1 -68 46 }} in the 5-limit; [[390625/388962]], 420175/419904, and [[2100875/2097152]] in the 7-limit; 1375/1372, [[6250/6237]], [[41503/41472]], and 322102/321489 in the 11-limit; [[625/624]], [[1716/1715]], [[2080/2079]], [[10648/10647]], and 15379/15360 in the 13-limit; [[715/714]], [[1089/1088]], [[1225/1224]], [[1275/1274]], [[2025/2023]], and 11271/11264 in the 17-limit; 1331/1330, [[1445/1444]], [[1521/1520]], 1540/1539, [[1729/1728]], 4394/4389, and 4875/4864 in the 19-limit; 875/874, 897/896, 1105/1104, 1863/1862, 2024/2023, 2185/2184, 2300/2299, and 2530/2527 in the 23-limit. It [[support]]s and gives a good tuning to [[quadrant]]. It also supports [[tsaharuk]], but [[171edo]] is better suited for that purpose.
 
436edo is accurate for some intervals including [[3/2]], [[7/4]], [[11/10]], [[13/10]], [[18/17]], and [[19/18]], so it is especially suitable for the 2.3.7.11/5.13/5.17.19 [[subgroup]].  


=== Prime harmonics ===
=== Prime harmonics ===
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=== Subsets and supersets ===
=== Subsets and supersets ===
406edo has subset edos {{EDOs|2, 4, 109, 218}}.
Since 436 factors into {{factorization|436}}, 436edo has subset edos {{EDOs| 2, 4, 109, and 218 }}.


[[1308edo]], which divides edostep into three, is a [[zeta gap edo]] and is consistent in the 21-odd-limit.
[[1308edo]], which divides its edostep into three, is a [[zeta gap edo]] and is consistent in the 21-odd-limit.


== 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]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning error
|-
|-
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| 2.3
| 2.3
| {{monzo| -691 436 }}
| {{monzo| -691 436 }}
| [{{val| 436 691 }}]
| {{mapping| 436 691 }}
| +0.0379
| +0.0379
| 0.0379
| 0.0379
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| 2.3.5
| 2.3.5
| 32805/32768, {{monzo| 1 -68 46 }}
| 32805/32768, {{monzo| 1 -68 46 }}
| [{{val| 436 691 1012 }}]
| {{mapping| 436 691 1012 }}
| +0.1678
| +0.1678
| 0.1863
| 0.1863
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| 2.3.5.7
| 2.3.5.7
| 32805/32768, 390625/388962, 420175/419904
| 32805/32768, 390625/388962, 420175/419904
| [{{val| 436 691 1012 1224 }}]
| {{mapping| 436 691 1012 1224 }}
| +0.1275
| +0.1275
| 0.1758
| 0.1758
Line 49: Line 52:
| 2.3.5.7.11
| 2.3.5.7.11
| 1375/1372, 6250/6237, 32805/32768, 41503/41472
| 1375/1372, 6250/6237, 32805/32768, 41503/41472
| [{{val| 436 691 1012 1224 1508 }}]
| {{mapping| 436 691 1012 1224 1508 }}
| +0.1517
| +0.1517
| 0.1645
| 0.1645
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| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 625/624, 1375/1372, 2080/2079, 10648/10647, 15379/15360
| 625/624, 1375/1372, 2080/2079, 10648/10647, 15379/15360
| [{{val| 436 691 1012 1224 1508 1613 }}]
| {{mapping| 436 691 1012 1224 1508 1613 }}
| +0.1749
| +0.1749
| 0.1589
| 0.1589
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| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 625/624, 715/714, 1089/1088, 1225/1224, 2431/2430, 10648/10647
| 625/624, 715/714, 1089/1088, 1225/1224, 2431/2430, 10648/10647
| [{{val| 436 691 1012 1224 1508 1613 1782 }}]
| {{mapping| 436 691 1012 1224 1508 1613 1782 }}
| +0.1628
| +0.1628
| 0.1501
| 0.1501
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| 2.3.5.7.11.13.17.19
| 2.3.5.7.11.13.17.19
| 625/624, 715/714, 1089/1088, 1225/1224, 1331/1330, 1445/1444, 1729/1728
| 625/624, 715/714, 1089/1088, 1225/1224, 1331/1330, 1445/1444, 1729/1728
| [{{val| 436 691 1012 1224 1508 1613 1782 1852 }}]
| {{mapping| 436 691 1012 1224 1508 1613 1782 1852 }}
| +0.1503
| +0.1503
| 0.1443
| 0.1443
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=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods<br>per octave
|-
! Generator<br>(reduced)
! Periods<br />per 8ve
! Cents<br>(reduced)
! Generator*
! Associated<br>ratio
! Cents*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
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| 498.17
| 498.17
| 4/3
| 4/3
| [[Helmholtz]]
| [[Helmholtz (temperament)|Helmholtz]]
|-
|-
| 4
| 4
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| [[Quadrant]]
| [[Quadrant]]
|}
|}
 
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->

Latest revision as of 02:29, 17 April 2025

← 435edo 436edo 437edo →
Prime factorization 22 × 109
Step size 2.75229 ¢ 
Fifth 255\436 (701.835 ¢)
Semitones (A1:m2) 41:33 (112.8 ¢ : 90.83 ¢)
Consistency limit 23
Distinct consistency limit 23

436 equal divisions of the octave (abbreviated 436edo or 436ed2), also called 436-tone equal temperament (436tet) or 436 equal temperament (436et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 436 equal parts of about 2.75 ¢ each. Each step represents a frequency ratio of 21/436, or the 436th root of 2.

Theory

436edo is consistent to the 23-odd-limit. The patent val of 436edo has a distinct flat tendency, in the sense that if the octave is pure, harmonics from 3 to 37 are all flat.

It tempers out 32805/32768 and [1 -68 46⟩ in the 5-limit; 390625/388962, 420175/419904, and 2100875/2097152 in the 7-limit; 1375/1372, 6250/6237, 41503/41472, and 322102/321489 in the 11-limit; 625/624, 1716/1715, 2080/2079, 10648/10647, and 15379/15360 in the 13-limit; 715/714, 1089/1088, 1225/1224, 1275/1274, 2025/2023, and 11271/11264 in the 17-limit; 1331/1330, 1445/1444, 1521/1520, 1540/1539, 1729/1728, 4394/4389, and 4875/4864 in the 19-limit; 875/874, 897/896, 1105/1104, 1863/1862, 2024/2023, 2185/2184, 2300/2299, and 2530/2527 in the 23-limit. It supports and gives a good tuning to quadrant. It also supports tsaharuk, but 171edo is better suited for that purpose.

436edo is accurate for some intervals including 3/2, 7/4, 11/10, 13/10, 18/17, and 19/18, so it is especially suitable for the 2.3.7.11/5.13/5.17.19 subgroup.

Prime harmonics

Approximation of prime harmonics in 436edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -0.12 -0.99 -0.02 -0.86 -1.08 -0.37 -0.27 -0.75 -0.22 -0.08
Relative (%) +0.0 -4.4 -36.1 -0.7 -31.2 -39.2 -13.4 -9.6 -27.3 -8.0 -3.0
Steps
(reduced) 		436
(0) 		691
(255) 		1012
(140) 		1224
(352) 		1508
(200) 		1613
(305) 		1782
(38) 		1852
(108) 		1972
(228) 		2118
(374) 		2160
(416)

Subsets and supersets

Since 436 factors into 22 × 109, 436edo has subset edos 2, 4, 109, and 218.

1308edo, which divides its edostep into three, is a zeta gap edo and is consistent in the 21-odd-limit.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-691 436⟩ [⟨436 691]] +0.0379 0.0379 1.38
2.3.5 32805/32768, [1 -68 46⟩ [⟨436 691 1012]] +0.1678 0.1863 6.77
2.3.5.7 32805/32768, 390625/388962, 420175/419904 [⟨436 691 1012 1224]] +0.1275 0.1758 6.39
2.3.5.7.11 1375/1372, 6250/6237, 32805/32768, 41503/41472 [⟨436 691 1012 1224 1508]] +0.1517 0.1645 5.98
2.3.5.7.11.13 625/624, 1375/1372, 2080/2079, 10648/10647, 15379/15360 [⟨436 691 1012 1224 1508 1613]] +0.1749 0.1589 5.77
2.3.5.7.11.13.17 625/624, 715/714, 1089/1088, 1225/1224, 2431/2430, 10648/10647 [⟨436 691 1012 1224 1508 1613 1782]] +0.1628 0.1501 5.45
2.3.5.7.11.13.17.19 625/624, 715/714, 1089/1088, 1225/1224, 1331/1330, 1445/1444, 1729/1728 [⟨436 691 1012 1224 1508 1613 1782 1852]] +0.1503 0.1443 5.24

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 51\436 140.37 243/224 Tsaharuk
1 181\436 498.17 4/3 Helmholtz
4 181\436
(37\436) 		498.17
(101.83) 		4/3
(35/33) 		Quadrant

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

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