351edo: Difference between revisions

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== Theory ==
== Theory ==
351et tempers out 184528125/184473632, 26873856/26796875, [[65625/65536]], 235298/234375 and 40353607/40310784 in the 7-limit; 806736/805255, 1019215872/1019046875, 46656/46585, 1366875/1362944, 759375/758912, 117649/117612, 172032/171875, [[441/440]], 5788125/5767168, 24057/24010, 537109375/536870912, 422576/421875, 184877/184320, 275653125/275365888, 78408/78125, [[41503/41472]], 766656/765625, 35937/35840, 43923/43904, 246071287/246037500 and [[1771561/1769472]] in the 11-limit. It [[support]]s [[snape]].
351et is [[consistent]] to the [[7-odd-limit]] with a reasonable approximation to the 11-limit. The equal temperament [[tempering out|tempers out]] [[19683/19600]], [[65625/65536]], and 235298/234375 in the 7-limit; [[441/440]], 24057/24010, 35937/35840, [[41503/41472]], 43923/43904, and 46656/46585 in the 11-limit. It [[support]]s [[snape]].


=== Odd harmonics ===
=== Odd harmonics ===
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=== Subsets and supersets ===
=== Subsets and supersets ===
351 factors into 3<sup>3</sup> × 13 with subset edos {{EDOs|3, 9, 13, 27, 39, and 117}}.
351 factors into 3<sup>3</sup> × 13 with subset edos {{EDOs| 3, 9, 13, 27, 39, and 117 }}.


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" |[[Subgroup]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" |[[Comma list|Comma List]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! 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]] (¢)
! [[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
! [[TE simple badness|Relative]] (%)
|-
|-
|2.3
| 2.3
|{{monzo|-556 351}}
| {{monzo| -556 351 }}
|{{mapping|351 556}}
| {{mapping| 351 556 }}
| 0.3471
| 0.3471
| 0.3472
| 0.3472
| 10.16
| 10.16
|-
|-
|2.3.5
| 2.3.5
|{{monzo|-36 11 8}}, {{monzo|-11 26 -13}}
| {{monzo| -36 11 8 }}, {{monzo| -11 26 -13 }}
|{{mapping|351 556 815}}
| {{mapping| 351 556 815 }}
| 0.2298
| 0.2298
| 0.3284
| 0.3284
| 9.61
| 9.61
|-
|-
|2.3.5.7
| 2.3.5.7
|19683/19600, 65625/65536, 235298/234375
| 19683/19600, 65625/65536, 235298/234375
|{{mapping|351 556 815 985}}
| {{mapping| 351 556 815 985 }}
| 0.2885
| 0.2885
| 0.3021
| 0.3021
| 8.84
| 8.84
|-
|-
|2.3.5.7.11
| 2.3.5.7.11
|441/440, 19683/19600, 41503/41472, 759375/758912
| 441/440, 19683/19600, 35937/35840, 65625/65536
|{{mapping|351 556 815 985 1214}}
| {{mapping| 351 556 815 985 1214 }}
| 0.2823
| 0.2823
| 0.2705
| 0.2705
Line 60: Line 60:
! Temperaments
! Temperaments
|-
|-
|1
| 1
|116\351
| 116\351
|396.58
| 396.58
|98304/78125
| 98304/78125
|[[Squarschmidt]]
| [[Squarschmidt]]
|}
|}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

Revision as of 18:57, 11 January 2024

← 350edo 351edo 352edo →
Prime factorization 33 × 13
Step size 3.4188 ¢ 
Fifth 205\351 (700.855 ¢)
Semitones (A1:m2) 31:28 (106 ¢ : 95.73 ¢)
Consistency limit 7
Distinct consistency limit 7

Template:EDO intro

Theory

351et is consistent to the 7-odd-limit with a reasonable approximation to the 11-limit. The equal temperament tempers out 19683/19600, 65625/65536, and 235298/234375 in the 7-limit; 441/440, 24057/24010, 35937/35840, 41503/41472, 43923/43904, and 46656/46585 in the 11-limit. It supports snape.

Odd harmonics

Approximation of odd harmonics in 351edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -1.10 +0.01 -1.30 +1.22 -0.89 +0.50 -1.09 +1.03 -0.08 +1.01 +0.79
Relative (%) -32.2 +0.3 -38.2 +35.6 -26.0 +14.6 -31.9 +30.1 -2.3 +29.7 +23.0
Steps
(reduced) 		556
(205) 		815
(113) 		985
(283) 		1113
(60) 		1214
(161) 		1299
(246) 		1371
(318) 		1435
(31) 		1491
(87) 		1542
(138) 		1588
(184)

Subsets and supersets

351 factors into 33 × 13 with subset edos 3, 9, 13, 27, 39, and 117.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3 [-556 351⟩ [⟨351 556]] 0.3471 0.3472 10.16
2.3.5 [-36 11 8⟩, [-11 26 -13⟩ [⟨351 556 815]] 0.2298 0.3284 9.61
2.3.5.7 19683/19600, 65625/65536, 235298/234375 [⟨351 556 815 985]] 0.2885 0.3021 8.84
2.3.5.7.11 441/440, 19683/19600, 35937/35840, 65625/65536 [⟨351 556 815 985 1214]] 0.2823 0.2705 7.91

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
Ratio* 		Temperaments
1 116\351 396.58 98304/78125 Squarschmidt

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct

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