369edo: Difference between revisions

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== Theory ==

~~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.

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.

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

| {{mapping| 369 585 857 }}

| ~~−0~~.1991

| −0.1991

| 0.1409

| 4.33

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

| {{mapping| 369 585 857 1036 }}

| ~~−0~~.1743

| −0.1743

| 0.1294

| 3.98

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

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

| ~~−0~~.2277

| −0.2277

| 0.1576

| 4.85

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

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

| ~~−0~~.2685

| −0.2685

| 0.1703

| 5.24

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| [[Hemicountercomp]]

|}

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

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

[[Category:Semiporwellismic]]

@@ Line 3: / Line 3: @@
 == Theory ==
-= 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.
+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.
 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 28: / Line 28: @@
 | {{monzo| 32 -7 -9 }}, {{monzo| 1 -27 18 }}
 | {{mapping| 369 585 857 }}
-| &minus;0.1991
+| −0.1991
 | 0.1409
 | 4.33
@@ Line 35: / Line 35: @@
 | 2401/2400, 4375/4374, {{monzo| 32 -7 -9 }}
 | {{mapping| 369 585 857 1036 }}
-| &minus;0.1743
+| −0.1743
 | 0.1294
 | 3.98
@@ Line 42: / Line 42: @@
 | 2401/2400, 4000/3993, 4375/4374, 5632/5625
 | {{mapping| 369 585 857 1036 1277 }}
-| &minus;0.2277
+| −0.2277
 | 0.1576
 | 4.85
@@ Line 49: / Line 49: @@
 | 1575/1573, 2080/2079, 2200/2197, 2401/2400, 3584/3575
 | {{mapping| 369 585 857 1036 1277 1366 }} (369f)
-| &minus;0.2685
+| −0.2685
 | 0.1703
 | 5.24
@@ Line 106: / Line 106: @@
 | [[Hemicountercomp]]
 |}
-<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 [[Category:Semiporwellismic]]