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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== Notation ==

=== ~~Ups~~ and downs notation ===

=== Stein–Zimmermann–Gould notation ===

69edo can be notated with [[ups and downs]], spoken as up, dup, downsharp, sharp, upsharp etc. and down, dud, upflat etc. Note that dup is equivalent to dudsharp and dud is equivalent to dupflat.

[[Stein–Zimmermann–Gould notation]] uses sharps and flats along with quartertone accidentals and arrows:

=== Kite's ups and downs notation ===

69edo can also be notated with [[Kite's ups and downs notation|Kite's ups and downs]], spoken as up, dup, downsharp, sharp, upsharp etc. and down, dud, upflat etc. Note that dup is equivalent to dudsharp and dud is equivalent to dupflat.

[[Alternative symbols for ups and downs notation]] uses sharps and flats along with Stein–Zimmerman [[24edo#Notation|quarter-tone]] accidentals, combined with arrows, borrowed from extended [[Helmholtz–Ellis notation]]:

~~{{Sharpness-sharp4}}~~

=== Sagittal notation ===

This notation uses the same sagittal sequence as ~~EDOs~~ [[62edo#Sagittal notation|62]] and [[76edo#Sagittal notation|76]].

This notation uses the same sagittal sequence as edos [[62edo #Sagittal notation|62]] and [[76edo #Sagittal notation|76]].

==== Evo flavor ====

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</imagemap>

In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this ~~EDO~~.

In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation #Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this edo.

== Regular temperament properties ==

@@ Line 2: / Line 2: @@
 {{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.
@@ Line 448: / Line 450: @@
 == Notation ==
-=== Ups and downs notation ===
+=== Stein–Zimmermann–Gould notation ===
-edo can be notated with [[ups and downs]], spoken as up, dup, downsharp, sharp, upsharp etc. and down, dud, upflat etc. Note that dup is equivalent to dudsharp and dud is equivalent to dupflat.
+[[Stein–Zimmermann–Gould notation]] uses sharps and flats along with quartertone accidentals and arrows:
+{{Sharpness-sharp4-szg}}
+=== Kite's ups and downs notation ===
+edo can also be notated with [[Kite's ups and downs notation|Kite's ups and downs]], spoken as up, dup, downsharp, sharp, upsharp etc. and down, dud, upflat etc. Note that dup is equivalent to dudsharp and dud is equivalent to dupflat.
 {{Sharpness-sharp4a}}
-[[Alternative symbols for ups and downs notation]] uses sharps and flats along with Stein–Zimmerman [[24edo#Notation|quarter-tone]] accidentals, combined with arrows, borrowed from extended [[Helmholtz–Ellis notation]]:
-{{Sharpness-sharp4}}
 === Sagittal notation ===
-This notation uses the same sagittal sequence as EDOs [[62edo#Sagittal notation|62]] and [[76edo#Sagittal notation|76]].
+This notation uses the same sagittal sequence as edos [[62edo #Sagittal notation|62]] and [[76edo #Sagittal notation|76]].
 ==== Evo flavor ====
@@ Line 489: / Line 494: @@
 </imagemap>
-In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO.
+In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation #Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this edo.
 == Regular temperament properties ==