23edo: Difference between revisions

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23edo 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 23edo so far, as approximations to just intervals goes. 23edo is the 9th [[prime edo]], following [[19edo]] and coming before [[29edo]].

23edo has good approximations for [[5/3]], [[11/7]], 13 and 17, among many others, allowing it to represent the 2.5/3.11/7.13.17 [[just intonation subgroup]]. If to this subgroup is added the commas of no-19's [[23-limit]] [[46edo]], the larger no-19's 23-limit [[k*N subgroups|2*23 subgroup]] 2.9.15.21.35.33.13.17.23 is obtained. This is the largest subgroup on which 23 has the same tuning and commas as does no-19's 23-limit 46edo, and may be regarded as a basis for analyzing the harmony of 23edo so far, as approximations to just intervals goes. If one dares to take advantage of this harmony by using 23edo as a period, you get [[icositritonic]], a [[23rd-octave temperaments|23rd-octave temperament]], so that the harmony of 23edo is adequately explained by what harmonies you can achieve using only periods and zero generators.

23edo is the 9th [[prime edo]], following [[19edo]] and coming before [[29edo]].

23edo was proposed by ethnomusicologist [[Wikipedia: 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.

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 {{Harmonics in equal|23}}
-edo 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 23edo so far, as approximations to just intervals goes. 23edo is the 9th [[prime edo]], following [[19edo]] and coming before [[29edo]].
+edo has good approximations for [[5/3]], [[11/7]], 13 and 17, among many others, allowing it to represent the 2.5/3.11/7.13.17 [[just intonation subgroup]]. If to this subgroup is added the commas of no-19's [[23-limit]] [[46edo]], the larger no-19's 23-limit [[k*N subgroups|2*23 subgroup]] 2.9.15.21.35.33.13.17.23 is obtained. This is the largest subgroup on which 23 has the same tuning and commas as does no-19's 23-limit 46edo, and may be regarded as a basis for analyzing the harmony of 23edo so far, as approximations to just intervals goes. If one dares to take advantage of this harmony by using 23edo as a period, you get [[icositritonic]], a [[23rd-octave temperaments|23rd-octave temperament]], so that the harmony of 23edo is adequately explained by what harmonies you can achieve using only periods and zero generators.
+edo is the 9th [[prime edo]], following [[19edo]] and coming before [[29edo]].
 edo was proposed by ethnomusicologist [[Wikipedia: 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.