369edo: Difference between revisions

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== 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~~.

369 = 9 × 41, and 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 130 & 239 temperament, the 65 & 152 temperament, 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 optimal tuning of this temperament is [[consistent]] in the 15-integer-limit.

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=== Divisors ===

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

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

== Regular temperament properties ==

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| 2.3.5

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

| [{{~~val~~| 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 }}

| [{{~~val~~| 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

| [{{~~val~~| 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

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

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

| -0.2685

| 0.1703

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|+Table of rank-2 temperaments by generator

! Periods per 8ve

! Generator~~ (Reduced)~~

! Generator*

! Cents~~ (Reduced)~~

! Cents*

! Associated Ratio

! Associated Ratio*

! Temperaments

|-

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| [[Hemicountercomp]]

|}

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

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

[[Category:Semiporwellismic]]

@@ Line 3: / Line 3: @@
 == 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.
+= 9 × 41, and 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 130 &amp; 239 temperament, the 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.
 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.
@@ Line 11: / Line 11: @@
 === Divisors ===
-Since 369 factors into 3<sup>2</sup> × 41, 369edo has subset edos {{EDOs| 3, 9, 41, and 123 }}.
+Since 369 factors into {{factorization|369}}, 369edo has subset edos {{EDOs| 3, 9, 41, and 123 }}.
 == Regular temperament properties ==
@@ Line 26: / Line 26: @@
 | 2.3.5
 | {{monzo| 32 -7 -9 }}, {{monzo| 1 -27 18 }}
-| [{{val| 369 585 857 }}]
+| {{mapping| 369 585 857 }}
 | -0.1991
 | 0.1409
@@ Line 33: / Line 33: @@
 | 2.3.5.7
 | 2401/2400, 4375/4374, {{monzo| 32 -7 -9 }}
-| [{{val| 369 585 857 1036 }}]
+| {{mapping| 369 585 857 1036 }}
 | -0.1743
 | 0.1294
@@ Line 40: / Line 40: @@
 | 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
 | 0.1576
@@ Line 47: / Line 47: @@
 | 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
 | 0.1703
@@ Line 57: / Line 57: @@
 |+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 104: / Line 104: @@
 | [[Hemicountercomp]]
 |}
+<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
 [[Category:Semiporwellismic]]