436edo: Difference between revisions

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'''436edo''' is the [[EDO|equal division of the octave]] into 436 parts of 2.7522935780 [[cent]]s each. The patent val has a distinct flat tendency, in the sense that if the [[octave]] is pure, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, and 37 are all flat. It is consistent to the [[23-odd-limit|23-limit]], tempering 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.
'''436edo''' is the [[EDO|equal division of the octave]] into 436 parts of 2.7522935780 [[cent]]s each.  
 
== 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 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.
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 ===
{{Harmonics in equal|436}}
== 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
| {{monzo| -691 436 }}
| [{{val| 436 691 }}]
| +0.0379
| 0.0379
| 1.38
|-
| 2.3.5
| 32805/32768, {{monzo| 1 -68 46 }}
| [{{val| 436 691 1012 }}]
| +0.1678
| 0.1863
| 6.77
|-
| 2.3.5.7
| 32805/32768, 390625/388962, 420175/419904
| [{{val| 436 691 1012 1224 }}]
| +0.1275
| 0.1758
| 6.39
|-
| 2.3.5.7.11
| 1375/1372, 6250/6237, 32805/32768, 41503/41472
| [{{val| 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
| [{{val| 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
| [{{val| 422 669 980 1185 1460 1562 1725 }}]
| +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
| [{{val| 422 669 980 1185 1460 1562 1725 1793 1852 }}]
| +0.1503
| 0.1443
| 5.24
|}
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods<br>per octave
! Generator<br>(reduced)
! Cents<br>(reduced)
! Associated<br>ratio
! Temperaments
|-
| 1
| 51\436
| 140.37
| 243/224
| [[Tsaharuk]]
|-
| 1
| 181\436
| 498.17
| 4/3
| [[Helmholtz]]
|-
| 4
| 181\436<br>(37\436)
| 498.17<br>(101.83)
| 4/3<br>(35/33)
| [[Quadrant]]
|}


[[Category:Equal divisions of the octave]]
[[Category:Equal divisions of the octave]]

Revision as of 13:40, 1 March 2022

436edo is the equal division of the octave into 436 parts of 2.7522935780 cents each.

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 [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 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)

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 [⟨422 669 980 1185 1460 1562 1725]] +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 [⟨422 669 980 1185 1460 1562 1725 1793 1852]] +0.1503 0.1443 5.24

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per octave 		Generator
(reduced) 		Cents
(reduced) 		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
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