207edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{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. | |||
===Prime harmonics=== | |||
{{Harmonics in equal|207}} | |||
===Subsets and supersets=== | |||
207 factors into 3<sup>2</sup> × 23, with 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|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}} | |||
|{{val|207 328}} | |||
| +0.1595 | |||
| 0.1596 | |||
| 2.75 | |||
|- | |||
|2.3.5 | |||
|32805/32768, {{monzo|2 31 -22}} | |||
|{{val|207 328 481}} | |||
| -0.1942 | |||
| 0.5166 | |||
| 8.91 | |||
|- | |||
|2.3.5.7 | |||
|6144/6125, 19683/19600, 32805/32768 | |||
|{{val|207 328 481 581}} | |||
| -0.0825 | |||
| 0.4874 | |||
|8.41 | |||
|- | |||
|2.3.5.7.11 | |||
|441/440, 3388/3375, 3773/3750, 6144/6125 | |||
|{{val|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 | |||
|{{val|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 | |||
|{{val|207 328 481 581 716 766 846}} | |||
| -0.0034 | |||
| 0.3834 | |||
| 6.61 | |||
|} | |||
=== Rank-2 temperaments === | |||
{| class="wikitable center-all left-5" | |||
|+Table of rank-2 temperaments by generator | |||
! Periods<br>per 8ve | |||
! Generator<br>(reduced) | |||
! Cents<br>(reduced) | |||
! 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]] | |||
|} | |||
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | [[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | ||
Revision as of 19:08, 18 October 2023
| ← 206edo | 207edo | 208edo → |
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 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
| 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
207 factors into 32 × 23, with 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
| Periods per 8ve |
Generator (reduced) |
Cents (reduced) |
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 |