69edo: Difference between revisions

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The '''69 equal ~~division~~''' or '''69-~~EDO~~''', ~~which~~ divides the octave into 69 equal parts of 17.~~391~~ [[cent]]s each. Nice.

{{Infobox ET

| Prime factorization = 3 × 23

| Step size = 17.3913¢

| Fifth = 40\69 (695.6¢)

| Semitones = 4:7 (69.6¢ : 121.7¢)

| Consistency = 5

}}

The '''69 equal divisions of the octave''' ('''69edo'''), or '''69-tone equal temperament''' ('''69tet''', '''69et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 69 [[equal]] parts of about 17.4 [[cent]]s each. Nice.

== Theory ==

69edo has been called "the love-child of [[23edo]] and [[quarter-comma meantone]]". As a meantone system, it is on the flat side, with a fifth of 695.652 cents. Such a fifth is closer to 2/7-comma meantone than 1/4-comma, and is nearly identical to that of "Synch-Meantone", or Wilson's equal beating meantone, wherein the perfect fifth and the major third beat at equal rates. Therefore 69edo can be treated as a closed system of Synch-Meantone for most purposes.

In the [[7-limit]] it is a [[mohajira]] system, tempering out 6144/6125, but not a septimal meantone system, as [[126/125]] maps to one step. It also [[support]]s the 12&69 temperament tempering out 3125/3087 along with [[81/80]]. In the 11-limit it tempers out [[99/98]], and supports the 31&69 variant of mohajira, identical to the standard 11-limit mohajira in [[31edo~~|31EDO~~]] but not in 69.

In the [[7-limit]] it is a [[mohajira]] system, tempering out 6144/6125, but not a septimal meantone system, as [[126/125]] maps to one step. It also [[support]]s the 12&69 temperament tempering out 3125/3087 along with [[81/80]]. In the 11-limit it tempers out [[99/98]], and supports the 31&69 variant of mohajira, identical to the standard 11-limit mohajira in [[31edo]] but not in 69.

=== Odd harmonics ===

@@ Line 1: / Line 1: @@
-The '''69 equal division''' or '''69-EDO''', which divides the octave into 69 equal parts of 17.391 [[cent]]s each. Nice.
+{{Infobox ET
+| Prime factorization = 3 × 23
+| Step size = 17.3913¢
+| Fifth = 40\69 (695.6¢)
+| Semitones = 4:7 (69.6¢ : 121.7¢)
+| Consistency = 5
+}}
+The '''69 equal divisions of the octave''' ('''69edo'''), or '''69-tone equal temperament''' ('''69tet''', '''69et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 69 [[equal]] parts of about 17.4 [[cent]]s each. Nice.
 == Theory ==
 edo has been called "the love-child of [[23edo]] and [[quarter-comma meantone]]". As a meantone system, it is on the flat side, with a fifth of 695.652 cents. Such a fifth is closer to 2/7-comma meantone than 1/4-comma, and is nearly identical to that of "Synch-Meantone", or Wilson's equal beating meantone, wherein the perfect fifth and the major third beat at equal rates. Therefore 69edo can be treated as a closed system of Synch-Meantone for most purposes.
-In the [[7-limit]] it is a [[mohajira]] system, tempering out 6144/6125, but not a septimal meantone system, as [[126/125]] maps to one step. It also [[support]]s the 12&amp;69 temperament tempering out 3125/3087 along with [[81/80]]. In the 11-limit it tempers out [[99/98]], and supports the 31&amp;69 variant of mohajira, identical to the standard 11-limit mohajira in [[31edo|31EDO]] but not in 69.
+In the [[7-limit]] it is a [[mohajira]] system, tempering out 6144/6125, but not a septimal meantone system, as [[126/125]] maps to one step. It also [[support]]s the 12&amp;69 temperament tempering out 3125/3087 along with [[81/80]]. In the 11-limit it tempers out [[99/98]], and supports the 31&amp;69 variant of mohajira, identical to the standard 11-limit mohajira in [[31edo]] but not in 69.
 === Odd harmonics ===