227edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|227}}
{{ED intro}}


== Theory ==
== Theory ==
The equal temperament [[tempering out|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.  
227et [[tempering out|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.  


227edo is accurate for the [[13/1|13th harmonic]], as the denominator of a convergent to log<sub>2</sub>13, after [[10edo|10]] and before [[5231edo|5231]].
227edo is accurate for the [[13/1|13th harmonic]], as the denominator of a convergent to log<sub>2</sub>13, after [[10edo|10]] and before [[5231edo|5231]].
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== Regular temperament properties ==
== Regular temperament properties ==
{{comma basis begin}}
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
|-
| 2.3
| 2.3
| {{monzo| 360 -227 }}
| {{monzo| 360 -227 }}
| {{mapping| 227 360 }}
| {{mapping| 227 360 }}
| &minus;0.3561
| −0.3561
| 0.3560
| 0.3560
| 6.73
| 6.73
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| 15625/15552, {{monzo| 61 -37 -1 }}
| 15625/15552, {{monzo| 61 -37 -1 }}
| {{mapping| 227 360 527 }}
| {{mapping| 227 360 527 }}
| &minus;0.1785
| −0.1785
| 0.3842
| 0.3842
| 7.27
| 7.27
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| 5120/5103, 15625/15552, 117649/116640
| 5120/5103, 15625/15552, 117649/116640
| {{mapping| 227 360 527 637 }}
| {{mapping| 227 360 527 637 }}
| &minus;0.0071
| −0.0071
| 0.4461
| 0.4461
| 8.44
| 8.44
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| 0.3821
| 0.3821
| 7.23
| 7.23
{{comma basis end}}
|}


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{{rank-2 begin}}
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br />per 8ve
! Generator*
! Cents*
! Associated<br />ratio*
! Temperaments
|-
|-
| 1
| 1
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| 4/3
| 4/3
| [[Undecental]]
| [[Undecental]]
{{rank-2 end}}
|}
{{orf}}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct


== Music ==
== Music ==
; [[Francium]]
; [[Francium]]
* "little hope" from ''hope in dark times'' (2024) &ndash; [https://open.spotify.com/track/1J6VKSGTkTRFs75WlEb6GP Spotify] | [https://francium223.bandcamp.com/track/little-hope Bandcamp] | [https://www.youtube.com/watch?v=juMfOpUu25I YouTube]
* "little hope" from ''hope in dark times'' (2024) – [https://open.spotify.com/track/1J6VKSGTkTRFs75WlEb6GP Spotify] | [https://francium223.bandcamp.com/track/little-hope Bandcamp] | [https://www.youtube.com/watch?v=juMfOpUu25I YouTube]
* "Cuckoo Cucumber" from ''Cursed Cuckoo Creations'' (2024) &ndash; [https://open.spotify.com/track/4WSc7cTf1ctWIiOXjTSAmc Spotify] | [https://francium223.bandcamp.com/track/cuckoo-cucumber Bandcamp] | [https://www.youtube.com/watch?v=po7hrgzSeb8 YouTube]
* "Cuckoo Cucumber" from ''Cursed Cuckoo Creations'' (2024) – [https://open.spotify.com/track/4WSc7cTf1ctWIiOXjTSAmc Spotify] | [https://francium223.bandcamp.com/track/cuckoo-cucumber Bandcamp] | [https://www.youtube.com/watch?v=po7hrgzSeb8 YouTube]
* "Did You Put Resistors In My Brain?" from ''Questions'' (2024) &ndash; [https://open.spotify.com/track/3QS6mj3GAMSmfJuQSsOE7Y Spotify] | [https://francium223.bandcamp.com/track/did-you-put-resistors-in-my-brain Bandcamp] | [https://www.youtube.com/watch?v=-FzOGzpxPv4 YouTube]
* "Did You Put Resistors In My Brain?" from ''Questions'' (2024) – [https://open.spotify.com/track/3QS6mj3GAMSmfJuQSsOE7Y Spotify] | [https://francium223.bandcamp.com/track/did-you-put-resistors-in-my-brain Bandcamp] | [https://www.youtube.com/watch?v=-FzOGzpxPv4 YouTube]
* "Too Bad Homeboy" from ''Abbreviations Gone Wrong'' (2024) &ndash; [https://open.spotify.com/track/6VPup7pwSC10c0VzsBU4PG Spotify] | [https://francium223.bandcamp.com/track/too-bad-homeboy Bandcamp] | [https://www.youtube.com/watch?v=Y246sdIRbwQ YouTube]
* "Too Bad Homeboy" from ''Abbreviations Gone Wrong'' (2024) – [https://open.spotify.com/track/6VPup7pwSC10c0VzsBU4PG Spotify] | [https://francium223.bandcamp.com/track/too-bad-homeboy Bandcamp] | [https://www.youtube.com/watch?v=Y246sdIRbwQ YouTube]


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

Latest revision as of 14:20, 20 February 2025

← 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

227 equal divisions of the octave (abbreviated 227edo or 227ed2), also called 227-tone equal temperament (227tet) or 227 equal temperament (227et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 227 equal parts of about 5.29 ¢ each. Each step represents a frequency ratio of 21/227, or the 227th root of 2.

Theory

227et 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.

Intervals

Main article: Table of 227edo intervals

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, 117649/116640 [⟨227 360 527 637]] −0.0071 0.4461 8.44
2.3.5.7.11 385/384, 2200/2187, 3388/3375, 12005/11979 [⟨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, 12005/11979 [⟨227 360 527 637 785 840]] +0.0693 0.4010 7.59
2.3.5.7.11.13.17 325/324, 352/351, 385/384, 595/594, 625/624, 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* Cents* Associated
ratio* 		Temperaments
1 25\227 132.16 121/112 Kastro
1 60\227 317.18 6/5 Countercata
1 94\227 496.92 4/3 Undecental

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

Music

Francium
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