227edo: Difference between revisions

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-'''227EDO''' is the [[EDO|equal division of the octave]] into 227 parts of 5.2863 [[cent]]s each. It tempers out 15625/15552 (kleisma) and |61 -37 -1&gt; in the 5-limit; 5120/5103, 65625/65536, and 117649/116640 in the 7-limit, so that it supports [[Kleismic_family#Countercata|countercata temperament]]. In the 11-limit, it tempers out 385/384, 2200/2187, 3388/3375, and 12005/11979, so that it provides the [[Optimal_patent_val|optimal patent val]] for 11-limit countercata. In the 13-limit, it tempers out 325/324, 352/351, 625/624, and 847/845. 227EDO is accurate for the 13th harmonic, as the denominator of a convergent to log<sub>2</sub>13, after [[10edo|10]] and before [[5231edo|5231]].
+{{Infobox ET}}
+{{ED intro}}
-EDO is the 49th prime EDO.
+== Theory ==
+et [[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.
-[[Category:countercata]]
+edo 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]].
-[[Category:Equal divisions of the octave]]
-[[Category:Prime EDO]]
+=== Prime harmonics ===
+{{Harmonics in equal|227}}
+=== Subsets and supersets ===
+edo is the 49th [[prime edo]].
+== Intervals ==
+{{Main|Table of 227edo intervals}}
+== Regular temperament properties ==
+{| 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
+| {{monzo| 360 -227 }}
+| {{mapping| 227 360 }}
+| −0.3561
+| 0.3560
+| 6.73
+|-
+| 2.3.5
+| 15625/15552, {{monzo| 61 -37 -1 }}
+| {{mapping| 227 360 527 }}
+| −0.1785
+| 0.3842
+| 7.27
+|-
+| 2.3.5.7
+| 5120/5103, 15625/15552, 117649/116640
+| {{mapping| 227 360 527 637 }}
+| −0.0071
+| 0.4461
+| 8.44
+|-
+| 2.3.5.7.11
+| 385/384, 2200/2187, 3388/3375, 12005/11979
+| {{mapping| 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
+| {{mapping| 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
+| {{mapping| 227 360 527 637 785 840 928 }}
+| +0.0324
+| 0.3821
+| 7.23
+|}
+=== Rank-2 temperaments ===
+{| 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
+| 25\227
+| 132.16
+| 121/112
+| [[Kastro]]
+|-
+| 1
+| 60\227
+| 317.18
+| 6/5
+| [[Countercata]]
+|-
+| 1
+| 94\227
+| 496.92
+| 4/3
+| [[Undecental]]
+|}
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
+== Music ==
+; [[Francium]]
+* "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) – [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) – [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) – [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:Listen]]