202edo: Difference between revisions
Wikispaces>genewardsmith **Imported revision 213412414 - Original comment: ** |
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{{Infobox ET}} | |||
{{ED intro}} | |||
== Theory == | |||
202edo is [[consistent]] to the [[9-odd-limit]] with a flat tendency in harmonics [[3/1|3]], [[5/1|5]], and [[7/1|7]]. It also has a decent harmonic [[11/1|11]], though it is sharp unlike the previous harmonics, with [[11/9]] barely exceeding 50% [[relative interval error|relative error]]. Despite this, it is most notable in the [[11-limit]], providing the [[optimal patent val]] for many temperaments tempering out [[243/242]]. | |||
Using the patent val, 202et [[tempering out|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]], equating a stack of five [[33/32]] quartertones with [[7/6]]. 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, which also tempers out [[540/539]], and the rank-4 [[rastmic]] temperament, which tempers out 243/242. | |||
It extends less well to the [[13-limit]], with harmonic [[13/1|13]] being about halfway between its steps. Nonetheless, the patent val tempers out [[351/350]], [[364/363]], [[676/675]], [[729/728]], and [[2080/2079]], supporting [[breed family #Jovial|jovial]] and [[breed family #Jovis|jovis]], as well as 13-limit harry. Primes [[17/1|17]] and [[23/1|23]] are quite sharp, but prime [[19/1|19]] is accurate. 202edo can thus be considered a 2.3.5.7.11.13.19-subgroup temperament with a mostly flat tendency, with the exception of prime 11. The intervals [[11/9]], [[13/11]], and their octave complements are the only inconsistencies in the no-17 [[21-odd-limit]], and the no-11 no-17 21-odd limit is completely consistent, though one may also want to exclude prime 13 given its inaccuracy, giving us the 2.3.5.7.19 subgroup. | |||
=== Prime harmonics === | |||
{{Harmonics in equal|202}} | |||
=== Subsets and supersets === | |||
Since 202 factors into {{nowrap| 2 × 101 }}, 202edo contains [[2edo]] and [[101edo]] as subset edos. | |||
== 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]] (%) | |||
|- | |||
| 2.3.5 | |||
| {{Monzo| -13 17 -6 }}, {{monzo| 23 6 -14 }} | |||
| {{Mapping| 202 320 469 }} | |||
| +0.2280 | |||
| 0.2710 | |||
| 4.56 | |||
|- | |||
| 2.3.5.7 | |||
| 2401/2400, 19683/19600, 65625/65536 | |||
| {{Mapping| 202 320 469 567 }} | |||
| +0.2164 | |||
| 0.2356 | |||
| 3.97 | |||
|- | |||
| 2.3.5.7.11 | |||
| 243/242, 441/440, 4000/3993, 65625/65536 | |||
| {{Mapping| 202 320 469 567 699 }} | |||
| +0.1061 | |||
| 0.3049 | |||
| 5.13 | |||
|} | |||
=== 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 | |||
| 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 | |||
| [[Larry]] | |||
|- | |||
| 2 | |||
| 12\202 | |||
| 71.29 | |||
| 25/24 | |||
| [[Narayana]] | |||
|- | |||
| 2 | |||
| 87\202<br>(14\202) | |||
| 516.83<br>(83.17) | |||
| 27/20<br>(21/20) | |||
| [[Harry]] | |||
|} | |||
<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 | |||
== Scales == | |||
* [[Jove1]], [[jove2]], [[jove3]], [[jove4]], [[jove5]], [[jove6]] | |||
* [[Elfjove7]], [[elfjove8d]], [[elfjove10]], [[elfjove11c]], [[elfjove12]] | |||
* [[Oktone]] | |||
== Music == | |||
; [[Mundoworld]] | |||
* [https://www.youtube.com/watch?v=_bNbb2o5K80 ''Home Planet Nostalgia''] – in Oktone scale | |||
[[Category:Harry]] | |||
[[Category:Tertiaseptal]] | |||
[[Category:Jove]] | |||
[[Category:Rastmic]] | |||
[[Category:Listen]] | |||
Latest revision as of 13:30, 13 March 2026
| ← 201edo | 202edo | 203edo → |
202 equal divisions of the octave (abbreviated 202edo or 202ed2), also called 202-tone equal temperament (202tet) or 202 equal temperament (202et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 202 equal parts of about 5.94 ¢ each. Each step represents a frequency ratio of 21/202, or the 202nd root of 2.
Theory
202edo is consistent to the 9-odd-limit with a flat tendency in harmonics 3, 5, and 7. It also has a decent harmonic 11, though it is sharp unlike the previous harmonics, with 11/9 barely exceeding 50% relative error. Despite this, it is most notable in the 11-limit, providing the optimal patent val for many temperaments tempering out 243/242.
Using the patent val, 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, equating a stack of five 33/32 quartertones with 7/6. 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, which also tempers out 540/539, and the rank-4 rastmic temperament, which tempers out 243/242.
It extends less well to the 13-limit, with harmonic 13 being about halfway between its steps. Nonetheless, the patent val tempers out 351/350, 364/363, 676/675, 729/728, and 2080/2079, supporting jovial and jovis, as well as 13-limit harry. Primes 17 and 23 are quite sharp, but prime 19 is accurate. 202edo can thus be considered a 2.3.5.7.11.13.19-subgroup temperament with a mostly flat tendency, with the exception of prime 11. The intervals 11/9, 13/11, and their octave complements are the only inconsistencies in the no-17 21-odd-limit, and the no-11 no-17 21-odd limit is completely consistent, though one may also want to exclude prime 13 given its inaccuracy, giving us the 2.3.5.7.19 subgroup.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.96 | -0.18 | -0.51 | +1.16 | -2.90 | +1.98 | -0.48 | +1.43 | -1.85 | +1.50 |
| Relative (%) | +0.0 | -16.2 | -2.9 | -8.6 | +19.5 | -48.9 | +33.3 | -8.1 | +24.0 | -31.2 | +25.2 | |
| Steps (reduced) |
202 (0) |
320 (118) |
469 (65) |
567 (163) |
699 (93) |
747 (141) |
826 (18) |
858 (50) |
914 (106) |
981 (173) |
1001 (193) | |
Subsets and supersets
Since 202 factors into 2 × 101, 202edo contains 2edo and 101edo as subset edos.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 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, 4000/3993, 65625/65536 | [⟨202 320 469 567 699]] | +0.1061 | 0.3049 | 5.13 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | 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 | Larry |
| 2 | 12\202 | 71.29 | 25/24 | Narayana |
| 2 | 87\202 (14\202) |
516.83 (83.17) |
27/20 (21/20) |
Harry |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct
Scales
- Jove1, jove2, jove3, jove4, jove5, jove6
- Elfjove7, elfjove8d, elfjove10, elfjove11c, elfjove12
- Oktone
Music
- Home Planet Nostalgia – in Oktone scale