232edo: Difference between revisions

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The ''232 equal division'' divides the octave into 232 equal parts of 5.172 cents each. It provides the [[Optimal_patent_val|optimal patent val]] for [[13-limit|13-limit]] [[Hemifamity_temperaments#Mystery|mystery temperament]], the rank three [[Hemifamity_family#Pele|pele temperament]] and the rank three [[Trimyna_family#Trimyna-13-limit|trimyna temperament]] and other temperaments tempering out 196/195, for which it gives the optimal patent val for the corresponding rank five temperament.

~~Aside from~~ its patent val, the ~~232d val is worth considering. Both temper~~ out ~~the wuerschmidt comma, 393216~~/~~390625~~, in the 5~~-limit~~.

== Theory ==

232 = 8 × 29, and 232edo shares its [[3/2|fifth]] with [[29edo]]. The equal temperament [[support]]s and provides the [[optimal patent val]] for the [[13-limit]] [[mystery]] temperament, the rank-3 [[pele]] temperament and the rank-3 [[trimyna]] temperament and other temperaments tempering out [[196/195]], for which it gives the optimal patent val for the corresponding rank-5 temperament.

In the 7-limit, the patent val tempers out hemifamity, 5120/5103 and the trimyna comma, 50421/50000; and 232d ~~3125~~/~~3097~~ and ~~245/243,~~ [[~~support|supporting~~]] [[~~Sensamagic_clan#Bohpier|bohpier temperament~~]]. In the 11-limit, the patent val tempers out 441/440 and 896/891, and 232d 540/539, 1375/1372 and 4000/3993~~, supporting octoid~~. In the 13-limit, the patent val tempers out 196/195, 352/351, 364/363, ~~441~~/~~440~~ and ~~676~~/~~675~~, which leads to 13-limit mystery, for which it provides the optimal patent val. 232d also tempers out 352/351 and 676/675, which supports ~~variants~~ of octoid.

Aside from its [[patent val]], the 232d val {{val| 232 368 539 '''652''' 803 859 }} is worth considering. Both temper out the [[würschmidt comma]], 393216/390625, in the 5-limit. In the 7-limit, the patent val tempers out hemifamity, [[5120/5103]] and the trimyna comma, [[50421/50000]]; and 232d [[4375/4374]] and [[16875/16807]], supporting [[octoid]]. In the 11-limit, the patent val tempers out [[441/440]] and [[896/891]], and 232d [[540/539]], 1375/1372 and [[4000/3993]]. In the 13-limit, the patent val tempers out 196/195, [[352/351]], [[364/363]], [[676/675]], and [[847/845]], which leads to 13-limit mystery, for which it provides the optimal patent val. 232d also tempers out 352/351 and 676/675, which supports a variant of octoid.

Considering the 232edo patent val, 13-limit mystery and 13-limit pele, we note that because it tempers out 441/440 it allows [[~~werckismic_chords|~~werckismic chords]], because it tempers out 196/195 it allows [[~~mynucumic_chords|~~mynucumic chords]], because it tempers out 352/351 it allows [[~~minthmic_chords|~~minthmic chords]], and because it tempers out 364/363 it allows [[~~gentle_chords|gentle~~ chords]], and because it tempers out 847/845 it allows the [[~~cuthbert_triad|~~cuthbert ~~triad~~]], making it a very flexible harmonic system.

Considering the 232edo patent val, 13-limit mystery and 13-limit pele, we note that because it tempers out 441/440 it allows [[werckismic chords]], because it tempers out 196/195 it allows [[mynucumic chords]], because it tempers out 352/351 it allows [[major minthmic chords]], and because it tempers out 364/363 it allows [[minor minthmic chords]], and because it tempers out 847/845 it allows the [[cuthbert chords]], making it a very flexible harmonic system. 232edo is also the first edo that approximates [[6/5]] more accurately than [[19edo]].

=== Odd harmonics ===

[[~~Category:Equal divisions of the octave~~|~~###~~]]

[[~~Category:Bohpier~~]]

=== Subsets and supersets ===

Since 232 factors into 23 × 29, 232edo has subset edos {{EDOs| 2, 4, 8, 29, 58, and 116 }}.

== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

|-

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

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

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

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

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

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

|-

| 2.3.5

| 393216/390625, {{monzo| 46 -29 0 }}

| {{mapping| 232 368 539 }}

| −0.5461

| 0.3989

| 7.71

|}

=== Rank-2 temperaments ===

{| class="wikitable center-all left-5"

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

|-

! Periods per 8ve

! Generator*

! Cents*

! Associated ratio*

! Temperaments

|-

| 1

| 61\232

| 315.52

| 6/5

| [[Acrokleismic]] (7-limit, 232d)

|-

| 1

| 75\232

| 387.93

| 5/4

| [[Würschmidt]] (5-limit)

|-

| 8

| 113\232 (3\232)

| 584.48 (15.52)

| 7/5 (100/99)

| [[Octoid]] (232d)

|-

| 29

| 96\232 (3\232)

| 496.55 (15.52)

| 4/3 (105/104)

| [[Mystery]] (232)

|}

<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 ==

* [[Mystery58]]

[[Category:Mystery]]

[[Category:Pele]]

[[Category:Trimyna]]

[[Category:Mynucumic]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-The ''232 equal division'' divides the octave into 232 equal parts of 5.172 cents each. It provides the [[Optimal_patent_val|optimal patent val]] for [[13-limit|13-limit]] [[Hemifamity_temperaments#Mystery|mystery temperament]], the rank three [[Hemifamity_family#Pele|pele temperament]] and the rank three [[Trimyna_family#Trimyna-13-limit|trimyna temperament]] and other temperaments tempering out 196/195, for which it gives the optimal patent val for the corresponding rank five temperament.
+{{ED intro}}
-Aside from its patent val, the 232d val is worth considering. Both temper out the wuerschmidt comma, 393216/390625, in the 5-limit.
+== Theory ==
+= 8 × 29, and 232edo shares its [[3/2|fifth]] with [[29edo]]. The equal temperament [[support]]s and provides the [[optimal patent val]] for the [[13-limit]] [[mystery]] temperament, the rank-3 [[pele]] temperament and the rank-3 [[trimyna]] temperament and other temperaments tempering out [[196/195]], for which it gives the optimal patent val for the corresponding rank-5 temperament.
-In the 7-limit, the patent val tempers out hemifamity, 5120/5103 and the trimyna comma, 50421/50000; and 232d 3125/3097 and 245/243, [[support|supporting]] [[Sensamagic_clan#Bohpier|bohpier temperament]]. In the 11-limit, the patent val tempers out 441/440 and 896/891, and 232d 540/539, 1375/1372 and 4000/3993, supporting octoid. In the 13-limit, the patent val tempers out 196/195, 352/351, 364/363, 441/440 and 676/675, which leads to 13-limit mystery, for which it provides the optimal patent val. 232d also tempers out 352/351 and 676/675, which supports variants of octoid.
+Aside from its [[patent val]], the 232d val {{val| 232 368 539 '''652''' 803 859 }} is worth considering. Both temper out the [[würschmidt comma]], 393216/390625, in the 5-limit. In the 7-limit, the patent val tempers out hemifamity, [[5120/5103]] and the trimyna comma, [[50421/50000]]; and 232d [[4375/4374]] and [[16875/16807]], supporting [[octoid]]. In the 11-limit, the patent val tempers out [[441/440]] and [[896/891]], and 232d [[540/539]], 1375/1372 and [[4000/3993]]. In the 13-limit, the patent val tempers out 196/195, [[352/351]], [[364/363]], [[676/675]], and [[847/845]], which leads to 13-limit mystery, for which it provides the optimal patent val. 232d also tempers out 352/351 and 676/675, which supports a variant of octoid.
-Considering the 232edo patent val, 13-limit mystery and 13-limit pele, we note that because it tempers out 441/440 it allows [[werckismic_chords|werckismic chords]], because it tempers out 196/195 it allows [[mynucumic_chords|mynucumic chords]], because it tempers out 352/351 it allows [[minthmic_chords|minthmic chords]], and because it tempers out 364/363 it allows [[gentle_chords|gentle chords]], and because it tempers out 847/845 it allows the [[cuthbert_triad|cuthbert triad]], making it a very flexible harmonic system.
+Considering the 232edo patent val, 13-limit mystery and 13-limit pele, we note that because it tempers out 441/440 it allows [[werckismic chords]], because it tempers out 196/195 it allows [[mynucumic chords]], because it tempers out 352/351 it allows [[major minthmic chords]], and because it tempers out 364/363 it allows [[minor minthmic chords]], and because it tempers out 847/845 it allows the [[cuthbert chords]], making it a very flexible harmonic system. 232edo is also the first edo that approximates [[6/5]] more accurately than [[19edo]].
+=== Odd harmonics ===
 {{Harmonics in equal|232}}
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
-[[Category:Bohpier]]
+=== Subsets and supersets ===
+Since 232 factors into 2<sup>3</sup> × 29, 232edo has subset edos {{EDOs| 2, 4, 8, 29, 58, and 116 }}.
+== 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
+| 393216/390625, {{monzo| 46 -29 0 }}
+| {{mapping| 232 368 539 }}
+| −0.5461
+| 0.3989
+| 7.71
+|}
+=== 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
+| 61\232
+| 315.52
+| 6/5
+| [[Acrokleismic]] (7-limit, 232d)
+|-
+| 1
+| 75\232
+| 387.93
+| 5/4
+| [[Würschmidt]] (5-limit)
+|-
+| 8
+| 113\232<br>(3\232)
+| 584.48<br>(15.52)
+| 7/5<br>(100/99)
+| [[Octoid]] (232d)
+|-
+| 29
+| 96\232<br>(3\232)
+| 496.55<br>(15.52)
+| 4/3<br>(105/104)
+| [[Mystery]] (232)
+|}
+<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 ==
+* [[Mystery58]]
 [[Category:Mystery]]
+[[Category:Pele]]
+[[Category:Trimyna]]
+[[Category:Mynucumic]]