227edo: Difference between revisions

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Note on 13-limit countercata tuning
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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|227}}
{{EDO intro|227}}
 
== Theory ==
The equal temperament tempers out 15625/15552 ([[15625/15552|kleisma]]) and {{monzo| 61 -37 -1 }} in the 5-limit; [[5120/5103]], [[65625/65536]], and 117649/116640 in the 7-limit, so that it [[support]]s [[countercata]]. In the 11-limit, it tempers out [[385/384]], 2200/2187, 3388/3375, and 12005/11979, so that it provides the [[optimal patent val]] for 11-limit countercata. In the 13-limit, it tempers out [[325/324]], [[352/351]], [[625/624]], [[676/675]], and [[847/845]], and again supplies a good tuning for 13-limit countercata, although [[140edo]] tunes it better in this case.  
The equal temperament tempers out 15625/15552 ([[15625/15552|kleisma]]) and {{monzo| 61 -37 -1 }} in the 5-limit; [[5120/5103]], [[65625/65536]], and 117649/116640 in the 7-limit, so that it [[support]]s [[countercata]]. In the 11-limit, it tempers out [[385/384]], 2200/2187, 3388/3375, and 12005/11979, so that it provides the [[optimal patent val]] for 11-limit countercata. In the 13-limit, it tempers out [[325/324]], [[352/351]], [[625/624]], [[676/675]], and [[847/845]], and again supplies a good tuning for 13-limit countercata, although [[140edo]] tunes it better in this case.  


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=== Subsets and supersets ===
=== Subsets and supersets ===
227edo is the 49th [[prime edo]].
227edo is the 49th [[prime edo]].
==Regular temperament properties==
{| class="wikitable center-4 center-5 center-6"
! 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
|{{monzo|360 -227}}
|{{val|227 360}}
| -0.3561
| 0.3560
| 6.73
|-
|2.3.5
|15625/15552, {{monzo|61 -37 -1}}
|{{val|227 360 527}}
| -0.1785
| 0.3842
| 7.27
|-
|2.3.5.7
|5120/5103, 15625/15552, 65625/65536
|{{val|227 360 527 637}}
| -0.0071
| 0.4461
| 8.44
|-
|2.3.5.7.11
|385/384, 2200/2187, 3388/3375, 5120/5103
|{{val|227 360 527 637 785}}
| +0.0832
| 0.4380
| 8.29
|-
|2.3.5.7.11.13
|325/324, 352/351, 385/384, 625/624, 847/845
|{{val|227 360 527 637 785 840}}
| +0.0693
| 0.4010
| 7.59
|-
|2.3.5.7.11.13.17
|352/351, 595/594, 715/714, 847/845, 1001/1000, 3185/3179
|{{val|227 360 527 637 785 840 928}}
| +0.0324
| 0.3821
| 7.23
|}
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Generator<br>(reduced)
! Cents<br>(reduced)
! Associated<br>ratio
! Temperaments
|-
|1
|25\227
|132.16
|{{monzo|-38 5 13}}
|[[Astro]] / [[kastro]]
|-
|1
|60\227
|317.18
|6/5
|[[Hanson]] / [[countercata]]
|-
|1
|94\227
|496.92
|4/3
|[[Undecental]]
|}


[[Category:Countercata]]
[[Category:Countercata]]

Revision as of 10:37, 29 October 2023

← 226edo 227edo 228edo →
Prime factorization 227 (prime)
Step size 5.28634 ¢ 
Fifth 133\227 (703.084 ¢)
Semitones (A1:m2) 23:16 (121.6 ¢ : 84.58 ¢)
Consistency limit 7
Distinct consistency limit 7

Template:EDO intro

Theory

The equal temperament tempers out 15625/15552 (kleisma) and [61 -37 -1⟩ in the 5-limit; 5120/5103, 65625/65536, and 117649/116640 in the 7-limit, so that it supports countercata. In the 11-limit, it tempers out 385/384, 2200/2187, 3388/3375, and 12005/11979, so that it provides the optimal patent val for 11-limit countercata. In the 13-limit, it tempers out 325/324, 352/351, 625/624, 676/675, and 847/845, and again supplies a good tuning for 13-limit countercata, although 140edo tunes it better in this case.

227edo is accurate for the 13th harmonic, as the denominator of a convergent to log213, after 10 and before 5231.

Prime harmonics

Approximation of prime harmonics in 227edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +1.13 -0.41 -1.43 -1.54 +0.00 +0.77 -1.48 +0.80 +1.26 +2.10
Relative (%) +0.0 +21.4 -7.8 -27.0 -29.1 +0.0 +14.6 -28.0 +15.1 +23.8 +39.7
Steps
(reduced) 		227
(0) 		360
(133) 		527
(73) 		637
(183) 		785
(104) 		840
(159) 		928
(20) 		964
(56) 		1027
(119) 		1103
(195) 		1125
(217)

Subsets and supersets

227edo is the 49th prime edo.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3 [360 -227⟩ ⟨227 360] -0.3561 0.3560 6.73
2.3.5 15625/15552, [61 -37 -1⟩ ⟨227 360 527] -0.1785 0.3842 7.27
2.3.5.7 5120/5103, 15625/15552, 65625/65536 ⟨227 360 527 637] -0.0071 0.4461 8.44
2.3.5.7.11 385/384, 2200/2187, 3388/3375, 5120/5103 ⟨227 360 527 637 785] +0.0832 0.4380 8.29
2.3.5.7.11.13 325/324, 352/351, 385/384, 625/624, 847/845 ⟨227 360 527 637 785 840] +0.0693 0.4010 7.59
2.3.5.7.11.13.17 352/351, 595/594, 715/714, 847/845, 1001/1000, 3185/3179 ⟨227 360 527 637 785 840 928] +0.0324 0.3821 7.23

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator
(reduced) 		Cents
(reduced) 		Associated
ratio 		Temperaments
1 25\227 132.16 [-38 5 13⟩ Astro / kastro
1 60\227 317.18 6/5 Hanson / countercata
1 94\227 496.92 4/3 Undecental
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