202edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
| Line 12: | Line 12: | ||
== Regular temperament properties == | == 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 | | 2.3 | ||
| {{monzo| -160 101 }} | | {{monzo| -160 101 }} | ||
| {{mapping| 202 320 }} | | {{mapping| 202 320 }} | ||
| 0.3044 | | +0.3044 | ||
| 0.3045 | | 0.3045 | ||
| 5.13 | | 5.13 | ||
| Line 24: | Line 33: | ||
| {{monzo| -13 17 -6 }}, {{monzo| 23 6 -14 }} | | {{monzo| -13 17 -6 }}, {{monzo| 23 6 -14 }} | ||
| {{mapping| 202 320 469 }} | | {{mapping| 202 320 469 }} | ||
| 0.2280 | | +0.2280 | ||
| 0.2710 | | 0.2710 | ||
| 4.56 | | 4.56 | ||
| Line 31: | Line 40: | ||
| 2401/2400, 19683/19600, 65625/65536 | | 2401/2400, 19683/19600, 65625/65536 | ||
| {{mapping| 202 320 469 567 }} | | {{mapping| 202 320 469 567 }} | ||
| 0.2164 | | +0.2164 | ||
| 0.2356 | | 0.2356 | ||
| 3.97 | | 3.97 | ||
| Line 38: | Line 47: | ||
| 243/242, 441/440, 4000/3993, 65625/65536 | | 243/242, 441/440, 4000/3993, 65625/65536 | ||
| {{mapping| 202 320 469 567 699 }} | | {{mapping| 202 320 469 567 699 }} | ||
| 0.1061 | | +0.1061 | ||
| 0.3049 | | 0.3049 | ||
| 5.13 | | 5.13 | ||
|} | |||
=== Rank-2 temperaments === | === 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 | | 1 | ||
| Line 81: | Line 97: | ||
| 27/20<br />(21/20) | | 27/20<br />(21/20) | ||
| [[Harry]] | | [[Harry]] | ||
|} | |||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | |||
== Scales == | == Scales == | ||
| Line 91: | Line 107: | ||
== Music == | == Music == | ||
; [[Mundoworld]] | ; [[Mundoworld]] | ||
* [https://www.youtube.com/watch?v=_bNbb2o5K80 ''Home Planet Nostalgia''] | * [https://www.youtube.com/watch?v=_bNbb2o5K80 ''Home Planet Nostalgia''] – in Oktone scale | ||
[[Category:Harry]] | [[Category:Harry]] | ||
Latest revision as of 14:13, 20 February 2025
| ← 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
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.
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 its subsets.
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, 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