210edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{EDO intro|210}} | |||
==Theory== | |||
210et tempers out 67108864/66430125 (misty comma) and 30958682112/30517578125 (trisedodge comma) in the 5-limit; 3136/3125, 5120/5103, and 118098/117649 in the 7-limit. It is consistent to the 7-limit, but there is a sharp tendency for harmonics 3, 5, and 7. Using the patent val, it tempers out 176/175, 1375/1372, 8019/8000, and 41503/41472 in the 11-limit; 351/350, 352/351, 847/845, 2197/2187, and 16900/16807 in the 13-limit. Using the 210e val, it tempers out 540/539, 4000/3993, 6912/6875, and 15488/15435 in the 11-limit; 351/350, 364/363, 1001/1000, 2197/2187, and 3584/3575 in the 13-limit. | |||
===Odd harmonics=== | |||
{{Harmonics in equal|210}} | |||
===Subsets and supersets=== | |||
210edo factors into 2 × 3 × 5 × 7, with subset edos {{EDOs|2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, and 105}}. | |||
==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|111 -70}} | |||
|{{val|210 333}} | |||
| -0.2846 | |||
| 0.2845 | |||
| 4.98 | |||
|- | |||
|2.3.5 | |||
|{{monzo|26 -12 -3}}, {{monzo|19 10 -15}} | |||
|{{val|210 333 488}} | |||
| -0.5138 | |||
| 0.3987 | |||
| 6.98 | |||
|- | |||
|2.3.5.7 | |||
|3136/3125, 5120/5103, 118098/117649 | |||
|{{val|210 333 488 590}} | |||
| -0.6170 | |||
| 0.3888 | |||
| 6.80 | |||
|} | |||
=== 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 | |||
|- | |||
|3 | |||
|123\210<br>(17\210) | |||
|702.86<br>(97.14) | |||
|3/2<br>(135\128) | |||
|[[Misty]] | |||
|- | |||
|5 | |||
|98\210<br>(13\210) | |||
|560.00<br>(74.29) | |||
|864/625<br>(25/24) | |||
|[[Trisendodge]] | |||
|} | |||
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | [[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | ||
Revision as of 05:11, 19 October 2023
| ← 209edo | 210edo | 211edo → |
Theory
210et tempers out 67108864/66430125 (misty comma) and 30958682112/30517578125 (trisedodge comma) in the 5-limit; 3136/3125, 5120/5103, and 118098/117649 in the 7-limit. It is consistent to the 7-limit, but there is a sharp tendency for harmonics 3, 5, and 7. Using the patent val, it tempers out 176/175, 1375/1372, 8019/8000, and 41503/41472 in the 11-limit; 351/350, 352/351, 847/845, 2197/2187, and 16900/16807 in the 13-limit. Using the 210e val, it tempers out 540/539, 4000/3993, 6912/6875, and 15488/15435 in the 11-limit; 351/350, 364/363, 1001/1000, 2197/2187, and 3584/3575 in the 13-limit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.90 | +2.26 | +2.60 | +1.80 | -2.75 | -0.53 | -2.55 | -2.10 | -0.37 | -2.21 | +0.30 |
| Relative (%) | +15.8 | +39.5 | +45.5 | +31.6 | -48.1 | -9.2 | -44.7 | -36.7 | -6.5 | -38.7 | +5.2 | |
| Steps (reduced) |
333 (123) |
488 (68) |
590 (170) |
666 (36) |
726 (96) |
777 (147) |
820 (190) |
858 (18) |
892 (52) |
922 (82) |
950 (110) | |
Subsets and supersets
210edo factors into 2 × 3 × 5 × 7, with subset edos 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, and 105.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [111 -70⟩ | ⟨210 333] | -0.2846 | 0.2845 | 4.98 |
| 2.3.5 | [26 -12 -3⟩, [19 10 -15⟩ | ⟨210 333 488] | -0.5138 | 0.3987 | 6.98 |
| 2.3.5.7 | 3136/3125, 5120/5103, 118098/117649 | ⟨210 333 488 590] | -0.6170 | 0.3888 | 6.80 |
Rank-2 temperaments
| Periods per 8ve |
Generator (reduced) |
Cents (reduced) |
Associated ratio |
Temperaments |
|---|---|---|---|---|
| 3 | 123\210 (17\210) |
702.86 (97.14) |
3/2 (135\128) |
Misty |
| 5 | 98\210 (13\210) |
560.00 (74.29) |
864/625 (25/24) |
Trisendodge |