369edo: Difference between revisions

(10 intermediate revisions by 2 users not shown)

Line 1:

The '''369 equal divisions of the octave''' ('''369edo'''), or the '''369(-tone) equal temperament''' ('''369tet''', '''369et''') when viewed from a [[regular temperament]] perspective, divides the octave into 369 [[equal]] parts of about 3.25 [[cent]]s each.

== Theory ==

~~369 = 9 × 41, and it~~ shares ~~the~~ fifth with [[41edo]]. It has a sharp tendency, with [[harmonic]]s 3 through 11 all tuned sharp. It 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 130&239 temperament, the 65&152 temperament, and the rank-4 temperament tempering out 16384/16335, the semiporwellisma, as well as the no-7 subgroup version of it.

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 32 × 41, 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 ==

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

|-

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

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

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

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

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

! colspan="2" | Tuning ~~Error~~

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

Line 25:

Line 28:

|-

| 2.3.5

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

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

| [{{~~val~~| 369 585 857 }}]

| {{Mapping| 369 585 857 }}

| -0.1991

| −0.1991

| 0.1409

| 4.33

Line 33:

Line 36:

| 2.3.5.7

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

| [{{~~val~~| 369 585 857 1036 }}]

| {{Mapping| 369 585 857 1036 }}

| -0.1743

| −0.1743

| 0.1294

| 3.98

Line 40:

Line 43:

| 2.3.5.7.11

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

| [{{~~val~~| 369 585 857 1036 1277 }}]

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

| -0.2277

| −0.2277

| 0.1576

| 4.85

Line 47:

Line 50:

| 2.3.5.7.11.13

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

| [{{~~val~~| 369 585 857 1036 1277 1366 }}] (369f)

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

| -0.2685

| −0.2685

| 0.1703

| 5.24

Line 55:

Line 58:

=== Rank-2 temperaments ===

{| 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 per 8ve

! Generator~~ (Reduced)~~

! Generator*

! Cents~~ (Reduced)~~

! Cents*

! Associated ~~Ratio~~

! Associated ratio*

! Temperaments

|-

Line 73:

Line 77:

| 864/625

| [[Tritriple]] (5-limit)

|-

| 1

| 181\369

| 588.62

| 128/91

| [[Ragitritonic]]

|-

| 9

Line 78:

Line 88:

| 250.41 (16.26)

| 140/121 (100/99)

| [[~~Semiennealimmal~~]]

| [[Ennealimmapine]]

|-

| 9

Line 84:

Line 94:

| 315.45 (48.78)

| 6/5 (36/35)

| [[Ennealimmal]]

| [[Ennealimmal]] / enneabiotic

|-

| 9

Line 98:

Line 108:

| [[Hemicountercomp]]

|}

<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:Equal divisions of the octave|###]] ~~

[[Category:Semiporwellismic]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-The '''369 equal divisions of the octave''' ('''369edo'''), or the '''369(-tone) equal temperament''' ('''369tet''', '''369et''') when viewed from a [[regular temperament]] perspective, divides the octave into 369 [[equal]] parts of about 3.25 [[cent]]s each.
+{{ED intro}}
 == Theory ==
-= 9 × 41, and it shares the fifth with [[41edo]]. It has a sharp tendency, with [[harmonic]]s 3 through 11 all tuned sharp. It 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 130&amp;239 temperament, the 65&amp;152 temperament, and the rank-4 temperament tempering out 16384/16335, the semiporwellisma, as well as the no-7 subgroup version of it.
+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 3<sup>2</sup> × 41, 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 ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! rowspan="2" | [[Subgroup]]
 ! rowspan="2" | [[Comma list]]
 ! 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]] (¢)
@@ Line 25: / Line 28: @@
 |-
 | 2.3.5
-| {{monzo| 32 -7 -9 }}, {{monzo| 1 -27 18 }}
+| {{Monzo| 32 -7 -9 }}, {{monzo| 1 -27 18 }}
-| [{{val| 369 585 857 }}]
+| {{Mapping| 369 585 857 }}
-| -0.1991
+| −0.1991
 | 0.1409
 | 4.33
@@ Line 33: / Line 36: @@
 | 2.3.5.7
 | 2401/2400, 4375/4374, {{monzo| 32 -7 -9 }}
-| [{{val| 369 585 857 1036 }}]
+| {{Mapping| 369 585 857 1036 }}
-| -0.1743
+| −0.1743
 | 0.1294
 | 3.98
@@ Line 40: / Line 43: @@
 | 2.3.5.7.11
 | 2401/2400, 4000/3993, 4375/4374, 5632/5625
-| [{{val| 369 585 857 1036 1277 }}]
+| {{Mapping| 369 585 857 1036 1277 }}
-| -0.2277
+| −0.2277
 | 0.1576
 | 4.85
@@ Line 47: / Line 50: @@
 | 2.3.5.7.11.13
 | 1575/1573, 2080/2079, 2200/2197, 2401/2400, 3584/3575
-| [{{val| 369 585 857 1036 1277 1366 }}] (369f)
+| {{Mapping| 369 585 857 1036 1277 1366 }} (369f)
-| -0.2685
+| −0.2685
 | 0.1703
 | 5.24
@@ Line 55: / Line 58: @@
 === Rank-2 temperaments ===
 {| 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
-! Generator<br>(Reduced)
+! Generator*
-! Cents<br>(Reduced)
+! Cents*
-! Associated<br>Ratio
+! Associated<br>ratio*
 ! Temperaments
 |-
@@ Line 73: / Line 77: @@
 | 864/625
 | [[Tritriple]] (5-limit)
+|-
+| 1
+| 181\369
+| 588.62
+| 128/91
+| [[Ragitritonic]]
 |-
 | 9
@@ Line 78: / Line 88: @@
 | 250.41<br>(16.26)
 | 140/121<br>(100/99)
-| [[Semiennealimmal]]
+| [[Ennealimmapine]]
 |-
 | 9
@@ Line 84: / Line 94: @@
 | 315.45<br>(48.78)
 | 6/5<br>(36/35)
-| [[Ennealimmal]]
+| [[Ennealimmal]] / enneabiotic
 |-
 | 9
@@ Line 98: / Line 108: @@
 | [[Hemicountercomp]]
 |}
+<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:Equal divisions of the octave|###]] <!-- 3-digit number -->
 [[Category:Semiporwellismic]]