69edo: Difference between revisions

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== 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{{c}}. 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.

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{{C}}. 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, where the frequencies of the root, third, and fifth of a major triad are equidistant (in a linear scale); in other words, where the [[4:5:6]] triad is [[delta-rational]] (sometimes abbreviated DR). Therefore 69edo can be treated as a closed system of Synch-Meantone for most purposes.

The alternative sharp fifth of 713.043{{C}} in the 69b [[val]] supports a tuning of [[archy]] where the [[6:7:8]] triad, which divides the fourth instead of the fifth, is tuned close to DR.

69edo offers two kinds of meantone 12-tone scales. One is the raw meantone scale, which has a 7:4 step ratio, and other is period-3 [[Meantone family#Lithium|lithium]] scale, which has a 6:5 step ratio and stems from a temperament tempering out [[3125/3087]] along with [[81/80]]. It should be noted that while the lithium scale has a meantone fifth, it produces a [[3L 6s|tcherepnin]] scale instead of traditional diatonic.

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 {{ED intro}}
 == 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{{c}}. 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.
+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{{C}}. 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, where the frequencies of the root, third, and fifth of a major triad are equidistant (in a linear scale); in other words, where the [[4:5:6]] triad is [[delta-rational]] (sometimes abbreviated DR). Therefore 69edo can be treated as a closed system of Synch-Meantone for most purposes.
+The alternative sharp fifth of 713.043{{C}} in the 69b [[val]] supports a tuning of [[archy]] where the [[6:7:8]] triad, which divides the fourth instead of the fifth, is tuned close to DR.
 edo offers two kinds of meantone 12-tone scales. One is the raw meantone scale, which has a 7:4 step ratio, and other is period-3 [[Meantone family#Lithium|lithium]] scale, which has a 6:5 step ratio and stems from a temperament tempering out [[3125/3087]] along with [[81/80]]. It should be noted that while the lithium scale has a meantone fifth, it produces a [[3L 6s|tcherepnin]] scale instead of traditional diatonic.