369edo: Difference between revisions

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== Theory ==

369edo shares its [[3/2|fifth]] with [[41edo]]. It has a sharp tendency, with [[harmonic]]s 3 through 11 all tuned sharp. The equal temperament [[tempering out|tempers out]] [[2401/2400]] and [[4375/4374]] in the 7-limit, so that it [[support]]s the [[ennealimmal]] temperament; in the 11-limit, [[4000/3993]], [[5632/5625]] and [[16384/16335]]. It provides the [[optimal patent val]] for the 11-limit {{nowrap|130 & 239}} temperament, {{nowrap|65 & 152}} temperament, and the rank-4 temperament tempering out 16384/16335, the semiporwellisma, as well as semiporwellic, the no-7 subgroup version thereof.

369edo shares its [[3/2|perfect fifth]] with [[41edo]]. It has a sharp tendency, with [[harmonic]]s 3 through 11 all tuned sharp.

Extension to the 13-limit is viable by the 369f val, tempering out [[1575/1573]], [[2080/2079]], [[2200/2197]], and 3584/3575. The optimal tuning of this temperament is [[consistent]] in the 15-integer-limit.

As an equal temperament, it [[tempering out|tempers out]] the [[escapade comma]] and the [[ennealimma]] in the 5-limit; [[2401/2400]] and [[4375/4374]] in the 7-limit, so that it [[support]]s the [[ennealimmal]] temperament; in the 11-limit, [[4000/3993]], [[5632/5625]] and [[16384/16335]], so that it supports [[escapade]] in the [[2.3.5.11 subgroup]] and in fact provides the [[optimal patent val]]. It also provides the optimal patent val for the 11-limit {{nowrap| 152 & 217 }} temperament (an escapade extension), the {{nowrap| 130 & 239 }} temperament (a weak escapade extension), and the rank-4 temperament tempering out 16384/16335, the semiporwellisma, as well as semiporwellic, the no-7 subgroup version thereof.

Extension to the 13-limit is viable by the 369f val, tempering out [[1575/1573]], [[2080/2079]], [[2200/2197]], and 3584/3575. The [[Tenney–Euclidean tuning|TE-optimal tuning]] of this temperament is [[consistent]] in the 15-integer-limit.

=== Prime harmonics ===

=== ~~Divisors~~ ===

=== Subsets and supersets ===

Since 369 factors into {{~~factorization~~|~~369~~}}, 369edo has subset edos {{EDOs| 3, 9, 41, and 123 }}.

Since 369 factors into primes as {{nowrap| 32 × 41 }}, 369edo has subset edos {{EDOs| 3, 9, 41, and 123 }}.

== 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

|-

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|-

| 2.3.5

| {{~~monzo~~| 32 -7 -9 }}, {{monzo| 1 -27 18 }}

| {{Monzo| 32 -7 -9 }}, {{monzo| 1 -27 18 }}

| {{~~mapping~~| 369 585 857 }}

| {{Mapping| 369 585 857 }}

| −0.1991

| 0.1409

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| 2.3.5.7

| 2401/2400, 4375/4374, {{monzo| 32 -7 -9 }}

| {{~~mapping~~| 369 585 857 1036 }}

| {{Mapping| 369 585 857 1036 }}

| −0.1743

| 0.1294

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| 2.3.5.7.11

| 2401/2400, 4000/3993, 4375/4374, 5632/5625

| {{~~mapping~~| 369 585 857 1036 1277 }}

| {{Mapping| 369 585 857 1036 1277 }}

| −0.2277

| 0.1576

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| 2.3.5.7.11.13

| 1575/1573, 2080/2079, 2200/2197, 2401/2400, 3584/3575

| {{~~mapping~~| 369 585 857 1036 1277 1366 }} (369f)

| {{Mapping| 369 585 857 1036 1277 1366 }} (369f)

| −0.2685

| 0.1703

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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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| 588.62

| 128/91

| [[~~Countritonic~~]]

| [[Ragitritonic]]

|-

| 9

| 77\369 (5\369)

| 77\369 (5\369)

| 250.41 (16.26)

| 250.41 (16.26)

| 140/121 (100/99)

| 140/121 (100/99)

| [[~~Semiennealimmal~~]]

| [[Ennealimmapine]]

|-

| 9

| 97\369 (15\369)

| 97\369 (15\369)

| 315.45 (48.78)

| 315.45 (48.78)

| 6/5 (36/35)

| 6/5 (36/35)

| [[Ennealimmal]]

| [[Ennealimmal]] / enneabiotic

|-

| 9

| 68\369 (14\369)

| 68\369 (14\369)

| 221.14 (45.53)

| 221.14 (45.53)

| 25/22 (77/75)

| 25/22 (77/75)

| [[Quadraennealimmal]]

|-

| 41

| 55\369 (1\369)

| 55\369 (1\369)

| 178.86 (3.25)

| 178.86 (3.25)

| 567/512 (352/351)

| 567/512 (352/351)

| [[Hemicountercomp]]

|}

<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

[[Category:Semiporwellismic]]

@@ Line 3: / Line 3: @@
 == Theory ==
-edo shares its [[3/2|fifth]] with [[41edo]]. It has a sharp tendency, with [[harmonic]]s 3 through 11 all tuned sharp. The equal temperament [[tempering out|tempers out]] [[2401/2400]] and [[4375/4374]] in the 7-limit, so that it [[support]]s the [[ennealimmal]] temperament; in the 11-limit, [[4000/3993]], [[5632/5625]] and [[16384/16335]]. It provides the [[optimal patent val]] for the 11-limit {{nowrap|130 &amp; 239}} temperament, {{nowrap|65 &amp; 152}} temperament, and the rank-4 temperament tempering out 16384/16335, the semiporwellisma, as well as semiporwellic, the no-7 subgroup version thereof.
+edo shares its [[3/2|perfect fifth]] with [[41edo]]. It has a sharp tendency, with [[harmonic]]s 3 through 11 all tuned sharp.
-Extension to the 13-limit is viable by the 369f val, tempering out [[1575/1573]], [[2080/2079]], [[2200/2197]], and 3584/3575. The optimal tuning of this temperament is [[consistent]] in the 15-integer-limit.
+As an equal temperament, it [[tempering out|tempers out]] the [[escapade comma]] and the [[ennealimma]] in the 5-limit; [[2401/2400]] and [[4375/4374]] in the 7-limit, so that it [[support]]s the [[ennealimmal]] temperament; in the 11-limit, [[4000/3993]], [[5632/5625]] and [[16384/16335]], so that it supports [[escapade]] in the [[2.3.5.11 subgroup]] and in fact provides the [[optimal patent val]]. It also provides the optimal patent val for the 11-limit {{nowrap| 152 & 217 }} temperament (an escapade extension), the {{nowrap| 130 & 239 }} temperament (a weak escapade extension), and the rank-4 temperament tempering out 16384/16335, the semiporwellisma, as well as semiporwellic, the no-7 subgroup version thereof.
+Extension to the 13-limit is viable by the 369f val, tempering out [[1575/1573]], [[2080/2079]], [[2200/2197]], and 3584/3575. The [[Tenney–Euclidean tuning|TE-optimal tuning]] of this temperament is [[consistent]] in the 15-integer-limit.
 === Prime harmonics ===
 {{Harmonics in equal|369|columns=11}}
-=== Divisors ===
+=== Subsets and supersets ===
-Since 369 factors into {{factorization|369}}, 369edo has subset edos {{EDOs| 3, 9, 41, and 123 }}.
+Since 369 factors into primes as {{nowrap| 3<sup>2</sup> × 41 }}, 369edo has subset edos {{EDOs| 3, 9, 41, and 123 }}.
 == Regular temperament properties ==
@@ Line 19: / 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
 |-
@@ Line 26: / Line 28: @@
 |-
 | 2.3.5
-| {{monzo| 32 -7 -9 }}, {{monzo| 1 -27 18 }}
+| {{Monzo| 32 -7 -9 }}, {{monzo| 1 -27 18 }}
-| {{mapping| 369 585 857 }}
+| {{Mapping| 369 585 857 }}
 | −0.1991
 | 0.1409
@@ Line 34: / Line 36: @@
 | 2.3.5.7
 | 2401/2400, 4375/4374, {{monzo| 32 -7 -9 }}
-| {{mapping| 369 585 857 1036 }}
+| {{Mapping| 369 585 857 1036 }}
 | −0.1743
 | 0.1294
@@ Line 41: / Line 43: @@
 | 2.3.5.7.11
 | 2401/2400, 4000/3993, 4375/4374, 5632/5625
-| {{mapping| 369 585 857 1036 1277 }}
+| {{Mapping| 369 585 857 1036 1277 }}
 | −0.2277
 | 0.1576
@@ Line 48: / Line 50: @@
 | 2.3.5.7.11.13
 | 1575/1573, 2080/2079, 2200/2197, 2401/2400, 3584/3575
-| {{mapping| 369 585 857 1036 1277 1366 }} (369f)
+| {{Mapping| 369 585 857 1036 1277 1366 }} (369f)
 | −0.2685
 | 0.1703
@@ Line 58: / Line 60: @@
 |+ 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 80: / Line 82: @@
 | 588.62
 | 128/91
-| [[Countritonic]]
+| [[Ragitritonic]]
 |-
 | 9
-| 77\369<br />(5\369)
+| 77\369<br>(5\369)
-| 250.41<br />(16.26)
+| 250.41<br>(16.26)
-| 140/121<br />(100/99)
+| 140/121<br>(100/99)
-| [[Semiennealimmal]]
+| [[Ennealimmapine]]
 |-
 | 9
-| 97\369<br />(15\369)
+| 97\369<br>(15\369)
-| 315.45<br />(48.78)
+| 315.45<br>(48.78)
-| 6/5<br />(36/35)
+| 6/5<br>(36/35)
-| [[Ennealimmal]]
+| [[Ennealimmal]] / enneabiotic
 |-
 | 9
-| 68\369<br />(14\369)
+| 68\369<br>(14\369)
-| 221.14<br />(45.53)
+| 221.14<br>(45.53)
-| 25/22<br />(77/75)
+| 25/22<br>(77/75)
 | [[Quadraennealimmal]]
 |-
 | 41
-| 55\369<br />(1\369)
+| 55\369<br>(1\369)
-| 178.86<br />(3.25)
+| 178.86<br>(3.25)
-| 567/512<br />(352/351)
+| 567/512<br>(352/351)
 | [[Hemicountercomp]]
 |}
-<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
 [[Category:Semiporwellismic]]