202edo: Difference between revisions
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202et tempers out [[2401/2400]], 19683/19600 and 65625/65536 in the 7-limit, and [[243/242]], [[441/440]], [[4000/3993]] in the 11-limit. It also notably tempers out the [[quartisma]]. It is the [[optimal patent val]] for the 11-limit rank-2 temperaments [[harry]] and [[tertiaseptal]], the rank-3 temperament [[jove]] tempering out 243/242 and 441/440, and the rank-4 rastmic temperament tempering out 243/242. | 202et tempers out [[2401/2400]], 19683/19600 and 65625/65536 in the 7-limit, and [[243/242]], [[441/440]], [[4000/3993]] in the 11-limit. It also notably tempers out the [[quartisma]]. It is the [[optimal patent val]] for the 11-limit rank-2 temperaments [[harry]] and [[tertiaseptal]], the rank-3 temperament [[jove]] tempering out 243/242 and 441/440, and the rank-4 rastmic temperament tempering out 243/242. | ||
{{harmonics in equal|202|start=2|prec=3}} | {{harmonics in equal|202|start=2|prec=3}} | ||
==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|-160 101}} | |||
|{{val|202 320}} | |||
| 0.3044 | |||
| 0.3045 | |||
| 5.13 | |||
|- | |||
|2.3.5 | |||
|{{monzo|-13 17 -6}}, {{monzo|23 6 -14}} | |||
|{{val|202 320 469}} | |||
| 0.2280 | |||
| 0.2710 | |||
| 4.56 | |||
|- | |||
|2.3.5.7 | |||
|2401/2400, 19683/19600, 65625/65536 | |||
|{{val|202 320 469 567}} | |||
| 0.2164 | |||
| 0.2356 | |||
| 3.97 | |||
|- | |||
|2.3.5.7.11 | |||
|243/242, 441/440, 540/539, 2401/2400 | |||
|{{val|202 320 469 567 699}} | |||
| 0.1061 | |||
| 0.3049 | |||
| 5.13 | |||
|} | |||
=== 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 | |||
|13\202 | |||
|77.23 | |||
|256/245 | |||
|[[Tertiaseptal]] | |||
|- | |||
|1 | |||
|51\202 | |||
|302.97 | |||
|25/21 | |||
|[[Quinmite]] | |||
|- | |||
|1 | |||
|85\202 | |||
|504.95 | |||
|104976/78125 | |||
|[[Countermeantone]] | |||
|- | |||
|1 | |||
|87\202 | |||
|516.83 | |||
|27/20 | |||
|[[Gravity]] | |||
|- | |||
|2 | |||
|12\202 | |||
|71.29 | |||
|25/24 | |||
|[[Vishnu]] | |||
|- | |||
|2 | |||
|87\202<br>(14\202) | |||
|516.83<br>(83.17) | |||
|27/20<br>(21/20) | |||
|[[Harry]] | |||
|} | |||
== Scales == | == Scales == | ||
Revision as of 14:38, 12 October 2023
| ← 201edo | 202edo | 203edo → |
The 202 equal temperament divides the octave into 202 equal parts of 5.941 cents each.
202et tempers out 2401/2400, 19683/19600 and 65625/65536 in the 7-limit, and 243/242, 441/440, 4000/3993 in the 11-limit. It also notably tempers out the quartisma. It is the optimal patent val for the 11-limit rank-2 temperaments harry and tertiaseptal, the rank-3 temperament jove tempering out 243/242 and 441/440, and the rank-4 rastmic temperament tempering out 243/242.
| Harmonic | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.965 | -0.175 | -0.509 | +1.157 | -2.904 | +1.975 | -0.483 | +1.429 | -1.854 | +1.499 | -1.839 |
| Relative (%) | -16.2 | -2.9 | -8.6 | +19.5 | -48.9 | +33.3 | -8.1 | +24.0 | -31.2 | +25.2 | -31.0 | |
| Steps (reduced) |
320 (118) |
469 (65) |
567 (163) |
699 (93) |
747 (141) |
826 (18) |
858 (50) |
914 (106) |
981 (173) |
1001 (193) |
1052 (42) | |
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-160 101⟩ | ⟨202 320] | 0.3044 | 0.3045 | 5.13 |
| 2.3.5 | [-13 17 -6⟩, [23 6 -14⟩ | ⟨202 320 469] | 0.2280 | 0.2710 | 4.56 |
| 2.3.5.7 | 2401/2400, 19683/19600, 65625/65536 | ⟨202 320 469 567] | 0.2164 | 0.2356 | 3.97 |
| 2.3.5.7.11 | 243/242, 441/440, 540/539, 2401/2400 | ⟨202 320 469 567 699] | 0.1061 | 0.3049 | 5.13 |
Rank-2 temperaments
| Periods per 8ve |
Generator (reduced) |
Cents (reduced) |
Associated ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 13\202 | 77.23 | 256/245 | Tertiaseptal |
| 1 | 51\202 | 302.97 | 25/21 | Quinmite |
| 1 | 85\202 | 504.95 | 104976/78125 | Countermeantone |
| 1 | 87\202 | 516.83 | 27/20 | Gravity |
| 2 | 12\202 | 71.29 | 25/24 | Vishnu |
| 2 | 87\202 (14\202) |
516.83 (83.17) |
27/20 (21/20) |
Harry |
Scales
Music
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