69edo: Difference between revisions
→Music: Add Bryan Deister's ''microtonal improvisation in 69edo'' (2025) |
→Notation: SZG notation |
||
| (8 intermediate revisions by 2 users not shown) | |||
| Line 2: | Line 2: | ||
{{ED intro}} | {{ED intro}} | ||
== 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. | |||
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. In the 11-limit it tempers out [[99/98]], and supports the {{nowrap|31 & 69}} variant of mohajira, identical to the standard 11-limit mohajira in [[31edo]] but not in 69. | 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. | ||
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. In the 11-limit it tempers out [[99/98]], and supports the {{nowrap|31 & 69}} variant of mohajira, identical to the standard 11-limit mohajira in [[31edo]] but not in 69. | |||
The [[concoctic scale]] for 69edo is 22\69, and the corresponding rank two temperament is {{nowrap|22 & 69}}, defined by tempering out the [-41, 1, 17⟩ comma in the 5-limit. | The [[concoctic scale]] for 69edo is 22\69, and the corresponding rank two temperament is {{nowrap|22 & 69}}, defined by tempering out the [-41, 1, 17⟩ comma in the 5-limit. | ||
| Line 445: | Line 447: | ||
| 0.000 | | 0.000 | ||
|} | |} | ||
<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 ==== | ||
| Line 489: | Line 494: | ||
</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 == | ||
| Line 542: | Line 547: | ||
| 2\69 | | 2\69 | ||
| [[Gammy]] (69de) | | [[Gammy]] (69de) | ||
|- | |||
|1 | |||
|5\69 | |||
|[[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> | |||
|- | |- | ||
| 1 | | 1 | ||
| Line 571: | Line 580: | ||
| [[Nessafof]] (69e) | | [[Nessafof]] (69e) | ||
|} | |} | ||
<references group="note" /> | |||
== Scales == | == Scales == | ||
| Line 578: | Line 588: | ||
* Lithium[9], [[3L 6s]] – 11 6 6 11 6 6 11 6 6 | * Lithium[9], [[3L 6s]] – 11 6 6 11 6 6 11 6 6 | ||
* Lithium[12], [[9L 3s]] – 5 6 6 6 5 6 6 6 5 6 6 6 | * Lithium[12], [[9L 3s]] – 5 6 6 6 5 6 6 6 5 6 6 6 | ||
== Instruments == | |||
A [[Lumatone mapping for 69edo]] is available. | |||
== Music == | == Music == | ||
; [[Bryan Deister]] | ; [[Bryan Deister]] | ||
* [https://www.youtube.com/watch?v=ZAqPonAHuUM ''microtonal improvisation in 69edo''] (2025) | * [https://www.youtube.com/watch?v=ZAqPonAHuUM ''microtonal improvisation in 69edo''] (2025) | ||
* [https://www.youtube.com/shorts/4XBELeySMPk ''Compass - Mili (microtonal cover in 69edo)''] (2025) | |||
* [https://www.youtube.com/shorts/GOjgAm6uwOY ''69edo waltz''] (2025) | |||
; [[Eliora]] | ; [[Eliora]] | ||