202edo: Difference between revisions

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== Theory ==

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]]. 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 ===

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=== Subsets and supersets ===

Since 202 factors into {{~~factorization~~|~~202~~}}, 202edo contains [[2edo]] and [[101edo]] as ~~its subsets~~.

Since 202 factors into {{nowrap| 2 × 101 }}, 202edo contains [[2edo]] and [[101edo]] as subset edos.

== Regular temperament properties ==

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! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve stretch (¢)

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

! [[TE simple badness|Relative]] (%)

|-

~~| 2.3~~

~~| {{monzo| -160 101 }}~~

~~| {{mapping| 202 320 }}~~

~~| +0.3044~~

~~| 0.3045~~

~~| 5.13~~

|-

| 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

| 0.2710

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| 2.3.5.7

| 2401/2400, 19683/19600, 65625/65536

| {{~~mapping~~| 202 320 469 567 }}

| {{Mapping| 202 320 469 567 }}

| +0.2164

| 0.2356

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| 2.3.5.7.11

| 243/242, 441/440, 4000/3993, 65625/65536

| {{~~mapping~~| 202 320 469 567 699 }}

| {{Mapping| 202 320 469 567 699 }}

| +0.1061

| 0.3049

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|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

|-

! Periods per 8ve

! Periods per 8ve

! Generator*

! Cents*

! Associated ratio*

! Associated ratio*

! Temperaments

|-

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|-

| 2

| 87\202 (14\202)

| 87\202 (14\202)

| 516.83 (83.17)

| 516.83 (83.17)

| 27/20 (21/20)

| 27/20 (21/20)

| [[Harry]]

|}

<nowiki />* [[Normal ~~lists~~|Octave-reduced form]], reduced to the first half-octave, and [[~~Normal lists~~|minimal form]] in parentheses if distinct

<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 ==

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro}}
+{{ED intro}}
 == Theory ==
-et [[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]]. 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.
+edo 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 ===
@@ Line 9: / Line 13: @@
 === Subsets and supersets ===
-Since 202 factors into {{factorization|202}}, 202edo contains [[2edo]] and [[101edo]] as its subsets.
+Since 202 factors into {{nowrap| 2 × 101 }}, 202edo contains [[2edo]] and [[101edo]] as subset edos.
 == Regular temperament properties ==
@@ Line 17: / Line 21: @@
 ! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br />8ve stretch (¢)
+! rowspan="2" | Optimal<br>8ve stretch (¢)
 ! colspan="2" | Tuning error
 |-
 ! [[TE error|Absolute]] (¢)
 ! [[TE simple badness|Relative]] (%)
-|-
-| 2.3
-| {{monzo| -160 101 }}
-| {{mapping| 202 320 }}
-| +0.3044
-| 0.3045
-| 5.13
 |-
 | 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
 | 0.2710
@@ Line 39: / Line 36: @@
 | 2.3.5.7
 | 2401/2400, 19683/19600, 65625/65536
-| {{mapping| 202 320 469 567 }}
+| {{Mapping| 202 320 469 567 }}
 | +0.2164
 | 0.2356
@@ Line 46: / Line 43: @@
 | 2.3.5.7.11
 | 243/242, 441/440, 4000/3993, 65625/65536
-| {{mapping| 202 320 469 567 699 }}
+| {{Mapping| 202 320 469 567 699 }}
 | +0.1061
 | 0.3049
@@ Line 56: / Line 53: @@
 |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
 |-
-! Periods<br />per 8ve
+! Periods<br>per 8ve
 ! Generator*
 ! Cents*
-! Associated<br />ratio*
+! Associated<br>ratio*
 ! Temperaments
 |-
@@ Line 93: / Line 90: @@
 |-
 | 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]]
 |}
-<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
+<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 ==