202edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|202}}
{{ED intro}}


== Theory ==
== Theory ==
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== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning Error
! colspan="2" | Tuning error
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
! [[TE simple badness|Relative]] (%)
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| {{monzo| -160 101 }}
| {{monzo| -160 101 }}
| {{mapping| 202 320 }}
| {{mapping| 202 320 }}
| 0.3044
| +0.3044
| 0.3045
| 0.3045
| 5.13
| 5.13
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| {{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
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| 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
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| 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
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=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods<br>per 8ve
|-
! Periods<br />per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br>Ratio*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
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|-
|-
| 2
| 2
| 87\202<br>(14\202)
| 87\202<br />(14\202)
| 516.83<br>(83.17)
| 516.83<br />(83.17)
| 27/20<br>(21/20)
| 27/20<br />(21/20)
| [[Harry]]
| [[Harry]]
|}
|}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct


== Scales ==
== Scales ==

Latest revision as of 14:13, 20 February 2025

← 201edo 202edo 203edo →
Prime factorization 2 × 101
Step size 5.94059 ¢ 
Fifth 118\202 (700.99 ¢) (→ 59\101)
Semitones (A1:m2) 18:16 (106.9 ¢ : 95.05 ¢)
Consistency limit 9
Distinct consistency limit 9

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

Approximation of prime harmonics in 202edo
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

Table of rank-2 temperaments by generator
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

Music

Mundoworld
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