207edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|207}}
{{ED intro}}


== Theory ==
== Theory ==
Line 9: Line 9:


=== Subsets and supersets ===
=== Subsets and supersets ===
Since 207 factors into 3<sup>2</sup> × 23, 207edo has subset edos {{EDOs| 3, 9, 23, and 69 }}.
Since 207 factors into {{factorisation|207}}, 207edo has subset edos {{EDOs| 3, 9, 23, and 69 }}.


== Regular temperament properties ==
== Regular temperament properties ==
{{comma basis begin}}
{| 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
| 2.3
| {{monzo| -328 207 }}
| {{Monzo| -328 207 }}
| {{mapping| 207 328 }}
| {{Mapping| 207 328 }}
| +0.1595
| +0.1595
| 0.1596
| 0.1596
Line 23: Line 32:
| 2.3.5
| 2.3.5
| 32805/32768, {{monzo| 2 31 -22 }}
| 32805/32768, {{monzo| 2 31 -22 }}
| {{mapping| 207 328 481 }}
| {{Mapping| 207 328 481 }}
| &minus;0.1942
| −0.1942
| 0.5166
| 0.5166
| 8.91
| 8.91
Line 30: Line 39:
| 2.3.5.7
| 2.3.5.7
| 6144/6125, 19683/19600, 50421/50000
| 6144/6125, 19683/19600, 50421/50000
| {{mapping| 207 328 481 581 }}
| {{Mapping| 207 328 481 581 }}
| &minus;0.0825
| −0.0825
| 0.4874
| 0.4874
| 8.41
| 8.41
Line 37: Line 46:
| 2.3.5.7.11
| 2.3.5.7.11
| 441/440, 3388/3375, 6144/6125, 19683/19600
| 441/440, 3388/3375, 6144/6125, 19683/19600
| {{mapping| 207 328 481 581 716 }}
| {{Mapping| 207 328 481 581 716 }}
| &minus;0.0317
| −0.0317
| 0.4477
| 0.4477
| 7.72
| 7.72
Line 44: Line 53:
| 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
| {{mapping| 207 328 481 581 716 766 }}
| {{Mapping| 207 328 481 581 716 766 }}
| &minus;0.0287
| −0.0287
| 0.4087
| 0.4087
| 7.05
| 7.05
Line 51: Line 60:
| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 351/350, 441/440, 561/560, 676/675, 847/845, 1089/1088
| 351/350, 441/440, 561/560, 676/675, 847/845, 1089/1088
| {{mapping| 207 328 481 581 716 766 846 }}
| {{Mapping| 207 328 481 581 716 766 846 }}
| &minus;0.0034
| −0.0034
| 0.3834
| 0.3834
| 6.61
| 6.61
{{comma basis end}}
|}


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{{rank-2 begin}}
{| 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
| 20\207
| 115.94
| 77/72
| [[Countermiracle]] / counterbenediction
|-
|-
| 1
| 1
Line 76: Line 98:
| 498.55
| 498.55
| 4/3
| 4/3
| [[Helmholtz]]
| [[Helmholtz (temperament)|Helmholtz]]
{{rank-2 end}}
|}
{{orf}}
<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]]

Latest revision as of 10:09, 28 June 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 20\207 115.94 77/72 Countermiracle / counterbenediction
1 25\207 144.93 49/45 Swetneus
1 43\207 249.28 15/13 Hemischis
1 86\207 498.55 4/3 Helmholtz

* In minimal-generator form

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