251edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|251}}
{{ED intro}}


== Theory ==
== Theory ==
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== 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]]
! 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]] (%)
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| {{monzo| 398 -251 }}
| {{monzo| 398 -251 }}
| {{mapping| 251 398 }}
| {{mapping| 251 398 }}
| -0.2630
| −0.2630
| 0.2630
| 0.2630
| 5.50
| 5.50
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| {{monzo| 9 -13 5 }}, {{monzo| 49 -6 -17 }}
| {{monzo| 9 -13 5 }}, {{monzo| 49 -6 -17 }}
| {{mapping| 251 398 583 }}
| {{mapping| 251 398 583 }}
| -0.3099
| −0.3099
| 0.2247
| 0.2247
| 4.70
| 4.70
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| 4375/4374, 5120/5103, 40500000/40353607
| 4375/4374, 5120/5103, 40500000/40353607
| {{mapping| 251 398 583 705 }}
| {{mapping| 251 398 583 705 }}
| -0.3830
| −0.3830
| 0.2322
| 0.2322
| 4.86
| 4.86
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=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods<br>per 8ve
|-
! Periods<br />per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br>Ratio*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
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| [[Maquila]]
| [[Maquila]]
|}
|}
<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 />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct


== Music ==
== Music ==

Latest revision as of 19:28, 20 February 2025

← 250edo 251edo 252edo →
Prime factorization 251 (prime)
Step size 4.78088 ¢ 
Fifth 147\251 (702.789 ¢)
Semitones (A1:m2) 25:18 (119.5 ¢ : 86.06 ¢)
Consistency limit 9
Distinct consistency limit 9

251 equal divisions of the octave (abbreviated 251edo or 251ed2), also called 251-tone equal temperament (251tet) or 251 equal temperament (251et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 251 equal parts of about 4.78 ¢ each. Each step represents a frequency ratio of 21/251, or the 251st root of 2.

Theory

251et tempers out 1600000/1594323 (amity comma) and [49 -6 -17⟩ (maquila comma) in the 5-limit; 4375/4374, 5120/5103, and 40500000/40353607 in the 7-limit, supporting amity, supermajor, and acrokleismic.

Using the patent val ⟨251 398 583 705 868], it tempers out 1331/1323, 1375/1372, 16896/16807, and 24057/24010 in the 11-limit; 352/351, 676/675, 847/845, and 1573/1568 in the 13-limit.

Using the 251e val ⟨251 398 583 705 869], it tempers out 540/539, 5632/5625, 6250/6237, and 12005/11979 in the 11-limit; 364/363, 676/675, 1716/1715, and 3584/3575 in the 13-limit.

Odd harmonics

Approximation of odd harmonics in 251edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +0.83 +0.94 +1.69 +1.67 -1.52 +0.91 +1.77 +0.22 -1.10 -2.26 -1.98
Relative (%) +17.4 +19.6 +35.4 +34.9 -31.7 +19.0 +37.0 +4.7 -23.0 -47.2 -41.4
Steps
(reduced) 		398
(147) 		583
(81) 		705
(203) 		796
(43) 		868
(115) 		929
(176) 		981
(228) 		1026
(22) 		1066
(62) 		1102
(98) 		1135
(131)

Subsets and supersets

251edo is the 54th prime edo.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [398 -251⟩ [⟨251 398]] −0.2630 0.2630 5.50
2.3.5 [9 -13 5⟩, [49 -6 -17⟩ [⟨251 398 583]] −0.3099 0.2247 4.70
2.3.5.7 4375/4374, 5120/5103, 40500000/40353607 [⟨251 398 583 705]] −0.3830 0.2322 4.86

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 66\251 315.54 6/5 Acrokleismic
1 71\251 339.44 243/200 Amity
1 91\251 435.06 9/7 Supermajor
1 96\251 458.96 125/96 Majvam
1 112\251 535.46 512/375 Maquila

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

Music

Francium
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