227edo: Difference between revisions

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== 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.

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 13th harmonic, as the denominator of a convergent to log213, after [[10edo|10]] and before [[5231edo|5231]].

227edo is accurate for the [[13/1|13th harmonic]], as the denominator of a convergent to log213, after [[10edo|10]] and before [[5231edo|5231]].

=== Prime harmonics ===

Line 12:

Line 13:

227edo is the 49th [[prime edo]].

==Regular temperament properties==

== Intervals ==

== 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 8ve Stretch (¢)~~

~~! colspan="2" |Tuning Error~~

|-

~~![[TE error|Absolute]] (¢)~~

~~![[TE simple badness|Relative]] (%)~~

|-

|2.3

! rowspan="2" | [[Subgroup]]

|{{monzo|360 -227}}

! rowspan="2" | [[Comma list]]

|{{~~val~~|227 360}}

! rowspan="2" | [[Mapping]]

| -0.3561

! rowspan="2" | Optimal 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

| 2.3.5

|15625/15552, {{monzo|61 -37 -1}}

| 15625/15552, {{monzo| 61 -37 -1 }}

|{{~~val~~|227 360 527}}

| {{mapping| 227 360 527 }}

| -0.1785

| −0.1785

| 0.3842

| 7.27

|-

|2.3.5.7

| 2.3.5.7

|5120/5103, 15625/15552, ~~65625~~/~~65536~~

| 5120/5103, 15625/15552, 117649/116640

|{{~~val~~|227 360 527 637}}

| {{mapping| 227 360 527 637 }}

| -0.0071

| −0.0071

| 0.4461

| 8.44

|-

|2.3.5.7.11

| 2.3.5.7.11

|385/384, 2200/2187, 3388/3375, ~~5120~~/~~5103~~

| 385/384, 2200/2187, 3388/3375, 12005/11979

|{{~~val~~|227 360 527 637 785}}

| {{mapping| 227 360 527 637 785 }}

| +0.0832

| 0.4380

| 8.29

|-

|2.3.5.7.11.13

| 2.3.5.7.11.13

|325/324, 352/351, 385/384, 625/624, ~~847~~/~~845~~

| 325/324, 352/351, 385/384, 625/624, 12005/11979

|{{~~val~~|227 360 527 637 785 840}}

| {{mapping| 227 360 527 637 785 840 }}

| +0.0693

| 0.4010

| 7.59

|-

|2.3.5.7.11.13.17

| 2.3.5.7.11.13.17

|352/351, 595/594, ~~715/714, 847/845, 1001~~/~~1000~~, 3185/3179

| 325/324, 352/351, 385/384, 595/594, 625/624, 3185/3179

|{{~~val~~|227 360 527 637 785 840 928}}

| {{mapping| 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

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

! Periods per 8ve

|-

! Generator~~ (reduced)~~

! Periods per 8ve

! Cents~~ (reduced)~~

! Generator*

! Associated ratio

! Cents*

! Associated ratio*

! Temperaments

|-

|1

| 1

|25\227

| 25\227

|132.16

| 132.16

|~~{{monzo|-38 5 13}}~~

| 121/112

|[[~~Astro]] / [[kastro~~]]

| [[Kastro]]

|-

|1

| 1

|60\227

| 60\227

|317.18

| 317.18

|6/5

| 6/5

|[[~~Hanson]] / [[countercata~~]]

| [[Countercata]]

|-

|1

| 1

|94\227

| 94\227

|496.92

| 496.92

|4/3

| 4/3

|[[Undecental]]

| [[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]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|227}}
+{{ED intro}}
 == 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.
+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.
-edo 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]].
+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]].
 === Prime harmonics ===
@@ Line 12: / Line 13: @@
 edo is the 49th [[prime edo]].
-==Regular temperament properties==
+== Intervals ==
+{{Main|Table of 227edo intervals}}
+== 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
+! rowspan="2" | [[Subgroup]]
-|{{monzo|360 -227}}
+! rowspan="2" | [[Comma list]]
-|{{val|227 360}}
+! rowspan="2" | [[Mapping]]
-| -0.3561
+! 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
+| 2.3.5
-|15625/15552, {{monzo|61 -37 -1}}
+| 15625/15552, {{monzo| 61 -37 -1 }}
-|{{val|227 360 527}}
+| {{mapping| 227 360 527 }}
-| -0.1785
+| −0.1785
 | 0.3842
 | 7.27
 |-
-|2.3.5.7
+| 2.3.5.7
-|5120/5103, 15625/15552, 65625/65536
+| 5120/5103, 15625/15552, 117649/116640
-|{{val|227 360 527 637}}
+| {{mapping| 227 360 527 637 }}
-| -0.0071
+| −0.0071
 | 0.4461
 | 8.44
 |-
-|2.3.5.7.11
+| 2.3.5.7.11
-|385/384, 2200/2187, 3388/3375, 5120/5103
+| 385/384, 2200/2187, 3388/3375, 12005/11979
-|{{val|227 360 527 637 785}}
+| {{mapping| 227 360 527 637 785 }}
 | +0.0832
 | 0.4380
 | 8.29
 |-
-|2.3.5.7.11.13
+| 2.3.5.7.11.13
-|325/324, 352/351, 385/384, 625/624, 847/845
+| 325/324, 352/351, 385/384, 625/624, 12005/11979
-|{{val|227 360 527 637 785 840}}
+| {{mapping| 227 360 527 637 785 840 }}
 | +0.0693
 | 0.4010
 | 7.59
 |-
-|2.3.5.7.11.13.17
+| 2.3.5.7.11.13.17
-|352/351, 595/594, 715/714, 847/845, 1001/1000, 3185/3179
+| 325/324, 352/351, 385/384, 595/594, 625/624, 3185/3179
-|{{val|227 360 527 637 785 840 928}}
+| {{mapping| 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
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
-! Periods<br>per 8ve
+|-
-! Generator<br>(reduced)
+! Periods<br />per 8ve
-! Cents<br>(reduced)
+! Generator*
-! Associated<br>ratio
+! Cents*
+! Associated<br />ratio*
 ! Temperaments
 |-
-|1
+| 1
-|25\227
+| 25\227
-|132.16
+| 132.16
-|{{monzo|-38 5 13}}
+| 121/112
-|[[Astro]] / [[kastro]]
+| [[Kastro]]
 |-
-|1
+| 1
-|60\227
+| 60\227
-|317.18
+| 317.18
-|6/5
+| 6/5
-|[[Hanson]] / [[countercata]]
+| [[Countercata]]
 |-
-|1
+| 1
-|94\227
+| 94\227
-|496.92
+| 496.92
-|4/3
+| 4/3
-|[[Undecental]]
+| [[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]]