202edo: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Correct comma bases; formatting; +links to more scales
m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"
 
(15 intermediate revisions by 4 users not shown)
Line 1: Line 1:
{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|202}}
{{ED intro}}


== Theory ==
== 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.
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 ===
=== Prime harmonics ===
{{Harmonics in equal|202}}
{{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 ==
== 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]] (%)
|-
| 2.3
| {{monzo| -160 101 }}
| {{mapping| 202 320 }}
| 0.3044
| 0.3045
| 5.13
|-
|-
| 2.3.5
| 2.3.5
| {{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 35: Line 36:
| 2.3.5.7
| 2.3.5.7
| 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 42: Line 43:
| 2.3.5.7.11
| 2.3.5.7.11
| 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
Line 50: Line 51:
=== 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<br>(Reduced)
! Generator*
! Cents<br>(Reduced)
! Cents*
! Associated<br>Ratio
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
|-
Line 93: Line 95:
| [[Harry]]
| [[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 ==
== Scales ==
Line 107: Line 110:
[[Category:Jove]]
[[Category:Jove]]
[[Category:Rastmic]]
[[Category:Rastmic]]
[[Category:Quartismic]]
[[Category:Listen]]

Latest revision as of 13:30, 13 March 2026

← 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

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

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 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

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
Retrieved from "https://en.xen.wiki/index.php?title=202edo&oldid=225712"