369edo: Difference between revisions

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+RTT table and rank-2 temperaments
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=== Prime harmonics ===
=== Prime harmonics ===
{{Primes in edo|369}}
{{Primes in edo|369}}
== 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
| {{monzo| 32 -7 -9 }}, {{monzo| 1 -27 18 }}
| [{{val| 369 585 857 }}]
| -0.1991
| 0.1409
| 4.33
|-
| 2.3.5.7
| 2401/2400, 4375/4374, {{monzo| 32 -7 -9 }}
| [{{val| 369 585 857 1036 }}]
| -0.1743
| 0.1294
| 3.98
|-
| 2.3.5.7.11
| 2401/2400, 4000/3993, 4375/4374, 5632/5625
| [{{val| 369 585 857 1036 1277 }}]
| -0.2277
| 0.1576
| 4.85
|-
| 2.3.5.7.11.13
| 1575/1573, 2080/2079, 2200/2197, 2401/2400, 3584/3575
| [{{val| 369 585 857 1036 1277 1366 }}] (369f)
| -0.2685
| 0.1703
| 5.24
|}
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods<br>per octave
! Generator<br>(reduced)
! Cents<br>(reduced)
! Associated<br>ratio
! Temperaments
|-
| 1
| 17\369
| 339.56
| 33/32
| [[Escapade]]
|-
| 1
| 172\369
| 559.35
| 864/625
| [[Tritriple]] (5-limit)
|-
| 9
| 77\369<br>(5\369)
| 250.41<br>(16.26)
| 140/121<br>(100/99)
| [[Semiennealimmal]]
|-
| 9
| 97\369<br>(15\369)
| 315.45<br>(48.78)
| 6/5<br>(36/35)
| [[Ennealimmal]]
|-
| 9
| 68\369<br>(14\369)
| 221.14<br>(45.53)
| 25/22<br>(77/75)
| [[Quadraennealimmal]]
|-
| 41
| 55\369<br>(1\369)
| 178.86<br>(3.25)
| 567/512<br>(352/351)
| [[Hemicounterpyth]]
|}


[[Category:Equal divisions of the octave]]
[[Category:Equal divisions of the octave]]
[[Category:Semiporwellismic]]
[[Category:Semiporwellismic]]

Revision as of 21:05, 2 January 2022

The 369 equal divisions of the octave divides the octave into 369 equal parts of 3.252 cents each.

Theory

369edo tempers out 2401/2400 and 4375/4374 in the 7-limit, so that it supports the ennealimmal temperament; in the 11-limit, 4000/3993, 5632/5625 and 16384/16335. It provides the optimal patent val for the 11-limit 21&109 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 factors as 32 × 41, with subset edos 3, 9, 41, and 123.

Prime harmonics

Script error: No such module "primes_in_edo".

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3.5 [32 -7 -9⟩, [1 -27 18⟩ [⟨369 585 857]] -0.1991 0.1409 4.33
2.3.5.7 2401/2400, 4375/4374, [32 -7 -9⟩ [⟨369 585 857 1036]] -0.1743 0.1294 3.98
2.3.5.7.11 2401/2400, 4000/3993, 4375/4374, 5632/5625 [⟨369 585 857 1036 1277]] -0.2277 0.1576 4.85
2.3.5.7.11.13 1575/1573, 2080/2079, 2200/2197, 2401/2400, 3584/3575 [⟨369 585 857 1036 1277 1366]] (369f) -0.2685 0.1703 5.24

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per octave 		Generator
(reduced) 		Cents
(reduced) 		Associated
ratio 		Temperaments
1 17\369 339.56 33/32 Escapade
1 172\369 559.35 864/625 Tritriple (5-limit)
9 77\369
(5\369) 		250.41
(16.26) 		140/121
(100/99) 		Semiennealimmal
9 97\369
(15\369) 		315.45
(48.78) 		6/5
(36/35) 		Ennealimmal
9 68\369
(14\369) 		221.14
(45.53) 		25/22
(77/75) 		Quadraennealimmal
41 55\369
(1\369) 		178.86
(3.25) 		567/512
(352/351) 		Hemicounterpyth
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