113edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
The '''113 equal divisions of the octave''' ('''113edo'''), or the '''113(-tone) equal temperament''' ('''113tet''', '''113et''') when viewed from a [[regular temperament]] perspective, is the [[EDO|equal division of the octave]] into 113 parts of about 10.6 [[cent]]s each.
{{EDO intro|113}}


== Theory ==
== Theory ==
113edo is distinctly [[consistent]] in the [[13-odd-limit]] with a flat tendency. As a temperament, it [[tempers out]] the [[amity comma]] and the [[ampersand]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and 1071875/1062882 in the [[7-limit]]; [[243/242]], [[385/384]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[325/324]], [[364/363]], [[729/728]], and 1625/1617 in the [[13-limit]]. It notably [[support]]s the 5-limit [[amity]] temperament, 7-limit [[amicable]] temperament, 7- and 11-limit [[miracle]] temperament, and 13-limit [[manna]] temperament.
113edo is distinctly [[consistent]] in the [[13-odd-limit]] with a flat tendency. As a temperament, it [[tempers out]] the [[amity comma]] and the [[ampersand]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and 1071875/1062882 in the [[7-limit]]; [[243/242]], [[385/384]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[325/324]], [[364/363]], [[729/728]], and 1625/1617 in the [[13-limit]]. It notably [[support]]s the 5-limit [[amity]] temperament, 7-limit [[amicable]] temperament, 7- and 11-limit [[miracle]] temperament, and 13-limit [[manna]] temperament.


113edo is the 30th [[prime EDO]].
=== Prime harmonics ===
{{Harmonics in equal|113}}


=== Prime harmonics ===
=== Subsets and supersets ===
{{Primes in edo|113}}
113edo is the 30th [[prime edo]].


== 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]]
! 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]] (¢)
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{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+Table of rank-2 temperaments by generator
! Periods<br>per octave
! Periods<br>per 8ve
! Generator<br>(reduced)
! Generator<br>(Reduced)
! Cents<br>(reduced)
! Cents<br>(Reduced)
! Associated<br>ratio
! Associated<br>Ratio
! Temperaments
! Temperaments
|-
|-
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| 339.82
| 339.82
| 243/200
| 243/200
| [[Amity]] / [[houborizic]]
| [[Houborizic]]
|-
|-
| 1
| 1
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| [[Gaster temperament|Gaster]]
| [[Gaster temperament|Gaster]]
|}
|}
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Prime EDO]]

Revision as of 14:59, 28 August 2023

← 112edo 113edo 114edo →
Prime factorization 113 (prime)
Step size 10.6195 ¢ 
Fifth 66\113 (700.885 ¢)
Semitones (A1:m2) 10:9 (106.2 ¢ : 95.58 ¢)
Consistency limit 13
Distinct consistency limit 13

Template:EDO intro

Theory

113edo is distinctly consistent in the 13-odd-limit with a flat tendency. As a temperament, it tempers out the amity comma and the ampersand in the 5-limit; 225/224, 1029/1024 and 1071875/1062882 in the 7-limit; 243/242, 385/384, 441/440 and 540/539 in the 11-limit; 325/324, 364/363, 729/728, and 1625/1617 in the 13-limit. It notably supports the 5-limit amity temperament, 7-limit amicable temperament, 7- and 11-limit miracle temperament, and 13-limit manna temperament.

Prime harmonics

Approximation of prime harmonics in 113edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -1.07 -4.01 -2.45 +0.89 -1.59 +1.24 -0.17 -1.73 +0.51 +1.87
Relative (%) +0.0 -10.1 -37.8 -23.1 +8.4 -15.0 +11.7 -1.6 -16.3 +4.8 +17.6
Steps
(reduced) 		113
(0) 		179
(66) 		262
(36) 		317
(91) 		391
(52) 		418
(79) 		462
(10) 		480
(28) 		511
(59) 		549
(97) 		560
(108)

Subsets and supersets

113edo is the 30th prime edo.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3 [-179 113⟩ [⟨113 179]] +0.338 0.338 3.18
2.3.5 1600000/1594323, 34171875/33554432 [⟨113 179 262]] +0.801 0.712 6.70
2.3.5.7 225/224, 1029/1024, 1071875/1062882 [⟨113 179 262 317]] +0.820 0.617 5.81
2.3.5.7.11 225/224, 243/242, 385/384, 980000/970299 [⟨113 179 262 317 391]] +0.604 0.700 6.59
2.3.5.7.11.13 225/224, 243/242, 325/324, 385/384, 1875/1859 [⟨113 179 262 317 391 418]] +0.575 0.643 6.05

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator
(Reduced) 		Cents
(Reduced) 		Associated
Ratio 		Temperaments
1 4\113 42.48 40/39 Humorous
1 6\113 63.72 28/27 Sycamore / betic
1 8\113 84.96 21/20 Amicable / pseudoamical / pseudoamorous
1 11\113 116.81 15/14~16/15 Miracle / manna
1 13\113 138.05 27/25 Quartemka
1 22\113 233.63 8/7 Slendric
1 27\113 286.73 13/11 Gamity
1 29\113 307.96 3200/2673 Familia
1 32\113 339.82 243/200 Houborizic
1 34\113 360.06 16/13 Phicordial
1 37\113 392.92 2744/2187 Emmthird
1 47\113 499.12 4/3 Gracecordial
1 56\113 594.69 55/39 Gaster
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