69edo
edited
... of 695.652 cents, cents. It is closer to 2/7-comma meantone than 1/4-comma, and could bet…
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of 695.652 cents,cents. It is closer to 2/7-comma meantone than 1/4-comma, and could betteris nearly identical to "Synch-Meantone", or Wilson's equal beating meantone, wherein the perfect fifth and the major third beat at equal rates. Therefore 69edo can be describedtreated as approximately 2/7-comma meantone. Ina closed system of Synch-Meantone for most purposes.
In the 7-limit
Monthly Tunings
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... Claudi Meneghin · 21edo Chacony, for harpsichord
December 2016: Fibonacci tuning (look on pag…
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Claudi Meneghin · 21edo Chacony, for harpsichord
December 2016: Fibonacci tuning (look on page 22 the doc / 24 of the PDF)
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458.359213... cents.
January 2017: Bohlen-Pierce
A Mean Little Voice by Stephen Weigel
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Modes for Strings #7 by Prent Rodgers
October and November 2017: 46edo
Curiosity Finds a Frown by Stephen Weigel (this is in 23-edo - 23 is half of 46, so it counts!)
December 2017: 15edo
We Wish You a Merry Christmas and a Happy New Year! (online Christmas card) by Stephen Weigel
Monthly Tunings
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... Iridescent Wenge Fugue by Stephen Weigel, accepted into SEAMUS 2018 and EABD 2018
Claudi Mene…
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Iridescent Wenge Fugue by Stephen Weigel, accepted into SEAMUS 2018 and EABD 2018
Claudi Meneghin · 21edo Chacony, for harpsichord
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Fibonacci tuning [add link](look on page 22 the doc / 24 of the PDF)
*The Fibonacci sequence is a series of integers that successively add to create the next one in the series, beginning with 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, etc... and thus a Fibonacci horogram shows this sequence in its rings / moments-of-symmetry. The generator for the Fibonacci horogram is 458.359213... cents.
January 2017: Bohlen-Pierce
A Mean Little Voice by Stephen Weigel
Monthly Tunings
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... Cam Taylor's Monarda Impressions
Tannic Acid (comparison), Chris Vaisvil
Nowhere in Super P…
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Cam Taylor's Monarda Impressions
Tannic Acid (comparison), Chris Vaisvil
Nowhere in Super Particular by Stephen Weigel
ilu by Noah Jordan
April 2018: Polaris
*Polaris refers to a 17-note constant structure in 13-limit JI. It can either function as an untempered JI version, or a tempered version, where 352/351 and 364/363 are most often tempered out (144/143 can be tempered out as well).
polaris falls by Chris Vaisvil
May 2018: 31edo
Contrapunctus 11 tuned to 31edo by Claudi Meneghin (original composer J. S. Bach)
pergen
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... If using single-pair notation, marvel is notated like 5-limit JI, but the 4:5:6:7 chord is spe…
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If using single-pair notation, marvel is notated like 5-limit JI, but the 4:5:6:7 chord is spelled C - Ev - G - A#vv. If double-pair notation is used, /1 = 64/63, and the chord is spelled C - Ev - G - Bb\.
There's a lot of options in rank-3 double-pair notation for what ratio each accidental pair represents. For example, deep reddish is half-8ve, with a d2 comma, like srutal. Using the same notation as srutal, but with ^1 = 81/80, we have P = \A4 = /d5 and E = //d2. The ratio for /1 is (-10,6,-1,1), a descending interval. 7/4 = 5/4 + 7/5 = vM3 + /d5 = v/m7, and the 4:5:6:7 chord is spelled awkwardly as C Ev G Bbv/. However, double-pair notation for 7-limit rank-3 temperaments can be standardized so that ^1 is always 81/80 and /1 is always 64/63. This ensures the 4:5:6:7 chord is always spelled C Ev G Bb\.
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(64/63)2 · (19,-12)-1.(-19,12). This can
This 3-limit comma defines the tipping point. At the tipping point, the 3-limit comma vanishes too. In a rank-2 temperament, the mapping comma must also vanish, because some number of them plus the 3-limit comma must add up to the original comma, which vanishes. However, a rank-3 temperament has two mapping commas, and neither is forced to vanish if the 3-limit comma vanishes. A rank-3 double-pair notation's tipping point is where both mapping commas are tempered out. For deep reddish, this happens when the tuning is exactly 12edo. This tuning is much farther from just than need be, well outside the sweet spot. Therefore deep reddish doesn't tip. Single-pair rank-3 notation has no enharmonic, and thus no tipping point. Double-pair rank-3 notation has 1 enharmonic, but two mapping commas. Rank-3 notations rarely tip.
Unlike the previous examples, Demeter's gen2 can't be expressed as a mapping comma. It divides 5/4 into three 15/14 generators, and 7/6 into two generators. Its pergen is (P8, P5, vm3/2). It could also be called (P8, P5, vM3/3), but the pergen with a smaller fraction is preferred. Because the 8ve and 5th are unsplit, single-pair notation is possible, with gen2 = ^m2 and no E. But the 4:5:6:7 chord would be spelled C -- Fbbb^^^ -- G -- Bbb^^, very awkward! Standard double-pair notation is better. Gen2 = v/A1, E = ^^\\\dd3, and C^^\\\ = A##. Genchain2 is C -- C#v/ -- Eb\ -- Ev -- Gb\\ -- Gv\ -- G#vv=Bbb\\\ -- Bbv\\... Unlike other genchains we've seen, the additional accidentals get progressively more complex. Whenever an accidental has its own enharmonic, with no other accidentals in it, it always adds up to something simpler eventually. If it doesn't have its own enharmonic, it's infinitely stackable. A case can be made for a convention that colors are used only for infinitely stackable accidentals, and ups/downs/highs/lows only for the other kind of accidentals.
pergen
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... 20, 28
The edos that support the fewest pergens are prime-number edos like 11edo or 13edo. Th…
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20, 28
The edos that support the fewest pergens are prime-number edos like 11edo or 13edo. The most "pergen-friendly" edos tend to be ones in which the circle of 5ths doesn't reach every edostep. For example, 24edo supports all half-split pergens, since both P8 and P5 map to an even number of edosteps. 72edo supports all half-splits and all third-splits. 15, 21 and 36 edo support many but not all third-splits (not those with m = 2 or n = 2).
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in the nextSupplemental Materials section for
EDO-pair names
Just as a pair of edos and a prime subgroup can specify a rank-2 temperament, a pair of edos can specify a rank-2 pergen. For N-edo & N'-edo, m = GCD (N,N'). The period P equals both (N/m)\N and (N'/m)\N'. For example, for 12edo and 16edo, m = 4, and the period is both 3\12 and 4\16.
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(P8/4, P4/2)
A specific pergen can be converted to an edo pair by finding the range of its generator cents in the arbitrary generator table, looking up that cents in the scale tree, and finding a conveniently-sized parent-child pair of edos in that range. For example, half-5th has a generator in the 320-360¢ range, and that part of the scale tree has among others 2\7, 3\10 and 5\17. Any two of those three edos defines (P8, P5/2).
Array Keyboards (unfinished)
Array keyboards have a 2-dimensional layout of keys, and are very appropriate for rank-2 tunings. A good layout can be found from the tuning's pergen. First find an edo N-edo that is compatible with the pergen, then arrange the keys in N columns to the 8ve, with each row usually containing the multigen interval. The unsplit pergen in 7 columns:
D#
D
E
F#
G#
A#
Db
Eb
F
G
A
B
C#
D#
Gb
Ab
Bb
C
D
Db
Higher notes are at the top of each column. The rows would actually be angled so that the two D's are at the same horizontal level. The vertical steps are A1 and the horizontal steps are M2, and the keyboard is defined as 7(A1, M2).
The third-4th keyboard is 7(A1/3 = ^1 = vvA1, P4/3 = vM2 = ^^m2).
D#v
E^
F#
G#v
A^
B
C#v
D^
D^
E
F#v
G^
A
Bv
C^
D
D
Ev
F^
G
Av
B^
C
Dv
Dv
Eb^
F
Gv
Ab^
Bb
Cv
Db^
Hypothesis: Let the 5th's keyspan (i.e. column-span) be F. In order for the keyboard to have the pitches in order, the fifth must fall between the two Stern-Brocot ancestors of F\N (simplified if possible). For example, an 8-column keyboard has F = 5, the ancestors of 5\8 are 3\5 and 2\3, and the 5th must be between 720¢ and 800¢. Thus the most musically useful N values are 5, 7, 10, 12 and 14.
(more to come)
Supplemental materials
Notation guide PDF
Munit
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A munit is a template of melodic steps combined with an expectation for the size & harmonic qua…
A munit is a template of melodic steps combined with an expectation for the size & harmonic quality of the interval they subdivide. For example, in meantone some important munits are "5/4 LL", the "do re mi" munit of 5/4 being divided into two equal parts; "4/3 LLs", which says if you hear two large steps and a small step you expect the outer interval to be 4/3; and "7/5 LLL", which means that if you hear three equal steps in a row you expect the outer interval to be, if not 7/5 specifically, then much less strongly consonant than 4/3.
See http://launch.groups.yahoo.com/group/tuning/message/102052http://groups.yahoo.com/group/tuning/message/102052