207edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|207}}
{{EDO intro|207}}
==Theory==
 
It tempers out 32805/32768 in the 5-limit, 6144/6125 and 19683/19600 in the 7-limit, 441/440 and 43923/43904 in the 11-limit, and 351/350, 847/845, 676/675, 729/728, 1716/1715 in the 13-limit. It serves as the patent val in the 11- and 13-limits for [[Cataharry_temperaments#Swetneus|swetneus temperament]]. It is significantly more accurate on the 2.3.7.11.13 subgroup, a favorite of many people, and one which contains both 729/728 and 10648/10647, which it tempers out.
== Theory ==
===Prime harmonics===
207et [[tempering out|tempers out]] 32805/32768 ([[schisma]]) in the 5-limit, [[6144/6125]] and [[19683/19600]] in the 7-limit, [[441/440]] and 43923/43904 in the 11-limit, and [[351/350]], [[676/675]], [[729/728]], [[847/845]], [[1716/1715]] in the 13-limit. It serves as a tuning in the 11- and 13-limit for the [[swetneus]] temperament. It is significantly more accurate on the 2.3.7.11.13 [[subgroup]], a favorite of many people, and one which contains both 729/728 and [[10648/10647]], which it tempers out.
 
=== Prime harmonics ===
{{Harmonics in equal|207}}
{{Harmonics in equal|207}}
===Subsets and supersets===
 
207 factors into 3<sup>2</sup> × 23, with subset edos {{EDOs|3, 9, 23, and 69}}.
=== Subsets and supersets ===
==Regular temperament properties==
Since 207 factors into 3<sup>2</sup> × 23, 207edo has subset edos {{EDOs| 3, 9, 23, and 69 }}.
 
== 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|Comma List]]
! rowspan="2" | [[Comma list|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
|-
|-
![[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
! [[TE simple badness|Relative]] (%)
|-
|-
|2.3
| 2.3
|{{monzo|-328 207}}
| {{monzo|-328 207}}
|{{val|207 328}}
| {{val|207 328}}
| +0.1595
| +0.1595
| 0.1596
| 0.1596
| 2.75
| 2.75
|-
|-
|2.3.5
| 2.3.5
|32805/32768, {{monzo|2 31 -22}}
| 32805/32768, {{monzo|2 31 -22}}
|{{val|207 328 481}}
| {{val|207 328 481}}
| -0.1942
| -0.1942
| 0.5166
| 0.5166
| 8.91
| 8.91
|-
|-
|2.3.5.7
| 2.3.5.7
|6144/6125, 19683/19600, 32805/32768
| 6144/6125, 19683/19600, 32805/32768
|{{val|207 328 481 581}}
| {{val|207 328 481 581}}
| -0.0825
| -0.0825
| 0.4874
| 0.4874
|8.41
| 8.41
|-
|-
|2.3.5.7.11
| 2.3.5.7.11
|441/440, 3388/3375, 3773/3750, 6144/6125
| 441/440, 3388/3375, 3773/3750, 6144/6125
|{{val|207 328 481 581 716}}
| {{val|207 328 481 581 716}}
| -0.0317
| -0.0317
| 0.4477
| 0.4477
| 7.72
| 7.72
|-
|-
|2.3.5.7.11.13
| 2.3.5.7.11.13
|351/350, 441/440, 676/675, 847/845, 3584/3575
| 351/350, 441/440, 676/675, 847/845, 3584/3575
|{{val|207 328 481 581 716 766}}
| {{val|207 328 481 581 716 766}}
| -0.0287
| -0.0287
| 0.4087
| 0.4087
| 7.05
| 7.05
|-
|-
|2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
|441/440, 561/560, 676/675, 936/935, 1632/1625, 8624/8619
| 441/440, 561/560, 676/675, 936/935, 1632/1625, 8624/8619
|{{val|207 328 481 581 716 766 846}}
| {{val|207 328 481 581 716 766 846}}
| -0.0034
| -0.0034
| 0.3834
| 0.3834
| 6.61
| 6.61
|}
|}
=== 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
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator<br>(reduced)
! Generator*
! Cents<br>(reduced)
! Cents*
! Associated<br>ratio
! Associated<br>Ratio*
! Temperaments
! Temperaments
|-
|-
|1
| 1
|25\207
| 25\207
|144.93
| 144.93
|49/45
| 49/45
|[[Swetneus]]
| [[Swetneus]]
|-
|-
|1
| 1
|43\207
| 43\207
|249.28
| 249.28
|15/13
| 15/13
|[[Hemischis]]
| [[Hemischis]]
|-
|-
|1
| 1
|86\207
| 86\207
|498.55
| 498.55
|4/3
| 4/3
|[[Helmholtz]]
| [[Helmholtz]]
|}
|}
 
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->

Revision as of 10:34, 13 April 2024

← 206edo 207edo 208edo →
Prime factorization 32 × 23
Step size 5.7971 ¢ 
Fifth 121\207 (701.449 ¢)
Semitones (A1:m2) 19:16 (110.1 ¢ : 92.75 ¢)
Consistency limit 7
Distinct consistency limit 7

Template:EDO intro

Theory

207et tempers out 32805/32768 (schisma) in the 5-limit, 6144/6125 and 19683/19600 in the 7-limit, 441/440 and 43923/43904 in the 11-limit, and 351/350, 676/675, 729/728, 847/845, 1716/1715 in the 13-limit. It serves as a tuning in the 11- and 13-limit for the swetneus temperament. It is significantly more accurate on the 2.3.7.11.13 subgroup, a favorite of many people, and one which contains both 729/728 and 10648/10647, which it tempers out.

Prime harmonics

Approximation of prime harmonics in 207edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -0.51 +2.09 -0.71 -0.59 +0.05 -0.61 -1.86 -2.19 +2.31 +2.79
Relative (%) +0.0 -8.7 +36.1 -12.2 -10.2 +0.9 -10.5 -32.1 -37.7 +39.8 +48.1
Steps
(reduced) 		207
(0) 		328
(121) 		481
(67) 		581
(167) 		716
(95) 		766
(145) 		846
(18) 		879
(51) 		936
(108) 		1006
(178) 		1026
(198)

Subsets and supersets

Since 207 factors into 32 × 23, 207edo has subset edos 3, 9, 23, and 69.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3 [-328 207⟩ ⟨207 328] +0.1595 0.1596 2.75
2.3.5 32805/32768, [2 31 -22⟩ ⟨207 328 481] -0.1942 0.5166 8.91
2.3.5.7 6144/6125, 19683/19600, 32805/32768 ⟨207 328 481 581] -0.0825 0.4874 8.41
2.3.5.7.11 441/440, 3388/3375, 3773/3750, 6144/6125 ⟨207 328 481 581 716] -0.0317 0.4477 7.72
2.3.5.7.11.13 351/350, 441/440, 676/675, 847/845, 3584/3575 ⟨207 328 481 581 716 766] -0.0287 0.4087 7.05
2.3.5.7.11.13.17 441/440, 561/560, 676/675, 936/935, 1632/1625, 8624/8619 ⟨207 328 481 581 716 766 846] -0.0034 0.3834 6.61

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
Ratio* 		Temperaments
1 25\207 144.93 49/45 Swetneus
1 43\207 249.28 15/13 Hemischis
1 86\207 498.55 4/3 Helmholtz

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

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