207edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
The 207 equal division of the octave, 207edo, divides the octave into 207 equal parts of size 5.797 cents each. 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.
{{ED intro}}


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
== Theory ==
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}}
 
=== Subsets and supersets ===
Since 207 factors into {{factorisation|207}}, 207edo has subset edos {{EDOs| 3, 9, 23, and 69 }}.
 
== 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| -328 207 }}
| {{mapping| 207 328 }}
| +0.1595
| 0.1596
| 2.75
|-
| 2.3.5
| 32805/32768, {{monzo| 2 31 -22 }}
| {{mapping| 207 328 481 }}
| −0.1942
| 0.5166
| 8.91
|-
| 2.3.5.7
| 6144/6125, 19683/19600, 50421/50000
| {{mapping| 207 328 481 581 }}
| −0.0825
| 0.4874
| 8.41
|-
| 2.3.5.7.11
| 441/440, 3388/3375, 6144/6125, 19683/19600
| {{mapping| 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
| {{mapping| 207 328 481 581 716 766 }}
| −0.0287
| 0.4087
| 7.05
|-
| 2.3.5.7.11.13.17
| 351/350, 441/440, 561/560, 676/675, 847/845, 1089/1088
| {{mapping| 207 328 481 581 716 766 846 }}
| −0.0034
| 0.3834
| 6.61
|}
 
=== Rank-2 temperaments ===
{| 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
| 25\207
| 144.93
| 49/45
| [[Swetneus]]
|-
| 1
| 43\207
| 249.28
| 15/13
| [[Hemischis]]
|-
| 1
| 86\207
| 498.55
| 4/3
| [[Helmholtz (temperament)|Helmholtz]]
|}
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

Latest revision as of 13:31, 13 March 2026

← 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

207 equal divisions of the octave (abbreviated 207edo or 207ed2), also called 207-tone equal temperament (207tet) or 207 equal temperament (207et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 207 equal parts of about 5.8 ¢ each. Each step represents a frequency ratio of 21/207, or the 207th root of 2.

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, 50421/50000 [⟨207 328 481 581]] −0.0825 0.4874 8.41
2.3.5.7.11 441/440, 3388/3375, 6144/6125, 19683/19600 [⟨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
2.3.5.7.11.13.17 351/350, 441/440, 561/560, 676/675, 847/845, 1089/1088 [⟨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 distinct

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