369edo: Difference between revisions

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Cleanup; clarify the title row of the rank-2 temp table; -redundant categories
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== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{{comma basis begin}}
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|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
| 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
| &minus;0.1991
| 0.1409
| 0.1409
| 4.33
| 4.33
Line 34: Line 26:
| 2401/2400, 4375/4374, {{monzo| 32 -7 -9 }}
| 2401/2400, 4375/4374, {{monzo| 32 -7 -9 }}
| {{mapping| 369 585 857 1036 }}
| {{mapping| 369 585 857 1036 }}
| -0.1743
| &minus;0.1743
| 0.1294
| 0.1294
| 3.98
| 3.98
Line 41: Line 33:
| 2401/2400, 4000/3993, 4375/4374, 5632/5625
| 2401/2400, 4000/3993, 4375/4374, 5632/5625
| {{mapping| 369 585 857 1036 1277 }}
| {{mapping| 369 585 857 1036 1277 }}
| -0.2277
| &minus;0.2277
| 0.1576
| 0.1576
| 4.85
| 4.85
Line 48: Line 40:
| 1575/1573, 2080/2079, 2200/2197, 2401/2400, 3584/3575
| 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
| &minus;0.2685
| 0.1703
| 0.1703
| 5.24
| 5.24
|}
{{comma basis end}}


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{{rank-2 begin}}
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>Ratio*
! Temperaments
|-
|-
| 1
| 1
Line 81: Line 67:
|-
|-
| 9
| 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]]
| [[Semiennealimmal]]
|-
|-
| 9
| 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]]
|-
|-
| 9
| 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]]
| [[Quadraennealimmal]]
|-
|-
| 41
| 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]]
| [[Hemicountercomp]]
|}
{{rank-2 end}}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
{{orf}}


[[Category:Semiporwellismic]]
[[Category:Semiporwellismic]]

Revision as of 04:19, 16 November 2024

← 368edo 369edo 370edo →
Prime factorization 32 × 41
Step size 3.25203 ¢ 
Fifth 216\369 (702.439 ¢) (→ 24\41)
Semitones (A1:m2) 36:27 (117.1 ¢ : 87.8 ¢)
Consistency limit 11
Distinct consistency limit 11

Template:EDO intro

Theory

369 = 9 × 41, and 369edo shares its fifth with 41edo. It has a sharp tendency, with harmonics 3 through 11 all tuned sharp. The equal temperament 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 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.

Prime harmonics

Approximation of prime harmonics in 369edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.48 +0.68 +0.28 +1.53 -1.50 -0.89 -1.58 -0.63 +1.32 -0.32
Relative (%) +0.0 +14.9 +20.9 +8.6 +47.0 -46.2 -27.4 -48.5 -19.4 +40.5 -9.8
Steps
(reduced) 		369
(0) 		585
(216) 		857
(119) 		1036
(298) 		1277
(170) 		1365
(258) 		1508
(32) 		1567
(91) 		1669
(193) 		1793
(317) 		1828
(352)

Divisors

Since 369 factors into 32 × 41, 369edo has subset edos 3, 9, 41, and 123.

Regular temperament properties

Template:Comma basis begin |- | 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 Template:Comma basis end

Rank-2 temperaments

Template:Rank-2 begin |- | 1 | 17\369 | 55.28 | 33/32 | Escapade |- | 1 | 172\369 | 559.35 | 864/625 | Tritriple (5-limit) |- | 1 | 181\369 | 588.62 | 128/91 | Countritonic |- | 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) | Hemicountercomp Template:Rank-2 end Template:Orf

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