23edo: Difference between revisions

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{{Infobox ET

| Prime factorization = 23 (prime)

| Step size = 52.~~174¢~~

| Step size = 52.17391¢

| Fifth = 13\23 ~~= 678.26¢~~

| Fifth = 13\23 (678¢)

| Major 2nd = 3\23 = 157¢

| Major 2nd = 3\23 (157¢)

| Minor 2nd = 4\23 = 209¢

| Minor 2nd = 4\23 (209¢)

| Augmented 1sn = -1\23 = -52¢

| Augmented 1sn = -1\23 (-52¢)

}}

== Theory ==

23-TET, or 23-EDO, is a tempered musical system which divides the [[octave]] into 23 equal parts of approximately 52.~~173913~~ cents, which is also called with the neologism Icositriphony ''(Icositrifonía)''. It has good approximations for [[5/3]], [[11/7]], 13 and 17, allowing it to represent the 2.5/3.11/7.13.17 [[just intonation subgroup]]. If to this subgroup is added the commas of [[17-limit]] [[46edo]], the larger 17-limit [[k*N_subgroups|2*23 subgroup]] 2.9.15.21.33.13.17 is obtained. This is the largest subgroup on which 23 has the same tuning and commas as does 17-limit·46edo, and may be regarded as a basis for analyzing the harmony of 23-EDO so far, as approximations to just intervals goes. 23edo is the 9th [[prime numbers|prime]] edo, following [[19edo]] and coming before [[29edo]].

23-TET, or 23-EDO, is a tempered musical system which divides the [[octave]] into 23 equal parts of approximately 52.2 cents, which is also called with the neologism Icositriphony ''(Icositrifonía)''. It has good approximations for [[5/3]], [[11/7]], 13 and 17, allowing it to represent the 2.5/3.11/7.13.17 [[just intonation subgroup]]. If to this subgroup is added the commas of [[17-limit]] [[46edo]], the larger 17-limit [[k*N_subgroups|2*23 subgroup]] 2.9.15.21.33.13.17 is obtained. This is the largest subgroup on which 23 has the same tuning and commas as does 17-limit·46edo, and may be regarded as a basis for analyzing the harmony of 23-EDO so far, as approximations to just intervals goes. 23edo is the 9th [[prime numbers|prime]] edo, following [[19edo]] and coming before [[29edo]].

23-EDO was proposed by ethnomusicologist [http://en.wikipedia.org/wiki/Erich_von_Hornbostel Erich von Hornbostel] as the result of continuing a circle of "blown" fifths of ~678-cent fifths that (he argued) resulted from "overblowing" a bamboo pipe.

@@ Line 7: / Line 7: @@
 {{Infobox ET
 | Prime factorization = 23 (prime)
-| Step size = 52.174¢
+| Step size = 52.17391¢
-| Fifth = 13\23 = 678.26¢
+| Fifth = 13\23 (678¢)
-| Major 2nd = 3\23 = 157¢
+| Major 2nd = 3\23 (157¢)
-| Minor 2nd = 4\23 = 209¢
+| Minor 2nd = 4\23 (209¢)
-| Augmented 1sn = -1\23 = -52¢
+| Augmented 1sn = -1\23 (-52¢)
 }}
 == Theory ==
 {{Primes in edo|23|columns=9}}
-<b>23-TET</b>, or <b>23-EDO</b>, is a tempered musical system which divides the [[octave]] into 23 equal parts of approximately 52.173913 cents, which is also called with the neologism Icositriphony ''(Icositrifonía)''. It has good approximations for [[5/3]], [[11/7]], 13 and 17, allowing it to represent the 2.5/3.11/7.13.17 [[just intonation subgroup]]. If to this subgroup is added the commas of [[17-limit]] [[46edo]], the larger 17-limit [[k*N_subgroups|2*23 subgroup]] 2.9.15.21.33.13.17 is obtained. This is the largest subgroup on which 23 has the same tuning and commas as does 17-limit·46edo, and may be regarded as a basis for analyzing the harmony of 23-EDO so far, as approximations to just intervals goes. 23edo is the 9th [[prime numbers|prime]] edo, following [[19edo]] and coming before [[29edo]].
+<b>23-TET</b>, or <b>23-EDO</b>, is a tempered musical system which divides the [[octave]] into 23 equal parts of approximately 52.2 cents, which is also called with the neologism Icositriphony ''(Icositrifonía)''. It has good approximations for [[5/3]], [[11/7]], 13 and 17, allowing it to represent the 2.5/3.11/7.13.17 [[just intonation subgroup]]. If to this subgroup is added the commas of [[17-limit]] [[46edo]], the larger 17-limit [[k*N_subgroups|2*23 subgroup]] 2.9.15.21.33.13.17 is obtained. This is the largest subgroup on which 23 has the same tuning and commas as does 17-limit·46edo, and may be regarded as a basis for analyzing the harmony of 23-EDO so far, as approximations to just intervals goes. 23edo is the 9th [[prime numbers|prime]] edo, following [[19edo]] and coming before [[29edo]].
 -EDO was proposed by ethnomusicologist [http://en.wikipedia.org/wiki/Erich_von_Hornbostel Erich von Hornbostel] as the result of continuing a circle of "blown" fifths of ~678-cent fifths that (he argued) resulted from "overblowing" a bamboo pipe.