69edo: Difference between revisions
→Music: Add Bryan Deister's ''69edo waltz'' (2025) |
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== Theory == | == 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{{ | 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. | 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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<nowiki />* Some simpler ratios listed | <nowiki />* Some simpler ratios listed and mapped by direct approximation (even if inconsistent). | ||
== Notation == | == 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: | ||
{{Sharpness-sharp4-szg}} | |||
=== 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. | |||
{{Sharpness-sharp4a}} | {{Sharpness-sharp4a}} | ||
=== Sagittal notation === | === Sagittal notation === | ||
This notation uses the same sagittal sequence as | This notation uses the same sagittal sequence as edos [[62edo #Sagittal notation|62]] and [[76edo #Sagittal notation|76]]. | ||
==== Evo flavor ==== | ==== Evo flavor ==== | ||
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</imagemap> | </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 | 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 == | == Regular temperament properties == | ||
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|[[Devichromic chords|Devichromic Octacot]]<ref group="note" name="tempname">Placeholder name, with link to [[Devichromic chords]] article — no general article currently exists for Devichromic temperament, and this particular incarnation of Devichromic temperament is likely to receive a different permanent name.</ref> | |[[Devichromic chords|Devichromic Octacot]]<ref group="note" name="tempname">Placeholder name, with link to [[Devichromic chords]] article — no general article currently exists for Devichromic temperament, and this particular incarnation of Devichromic temperament is likely to receive a different permanent name. The temperament is [https://x31eq.com/pyscript/uv.html?uvs=81%3A80+400%3A399+%5B17%2C-1%2C8%2C0%2C-8%3E&page=0&limit=2_3_5_7_19 14c & 55d], specified by the 2.3.5.7.19 subgroup unison vector list [-4, 4, -1, 0, 0⟩ ([[81/80]]), [4, -1, 2, -1, -1⟩ ([[400/399]]), [17, -1, 8, 0, -8⟩ (51200000000/50950689123), the last of which currently has no article, but equates a stack of eight [[20/19]]s to [[3/2]].</ref> | ||
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