User:Lhearne/15edo: Difference between revisions

==Introduction==

15 Equal Divisions of the Octave (15edo) is a tuning which divides the octave into 15 equally spaced pitches. It can be thought of as three sets of 5edo connected to each other by intervals of 3edo.

From [http://en.wikipedia.org/wiki/15_equal_temperament Wikipedia]:

First implemented by R.M.A Kusumadinata (Sundra) somewhere between the 1930s and the 1960s and Augusto Novaro in 1951, 15edo saw it's most notable early use in 1991 by Easley Blackwood Jr. in his Opus 28, ''Twelve Microtonal Etudes for Electronic Music Media'' and Opus 33, ''Suite for Guitar in 15-note Equal Tuning''.  

=Intervals=

Relative to 12edo, 15edo maintains some categorically-similar intervals, particularly the 3rds, 4ths, 5ths, and 6ths, but is quite different in the categories of 2nds and 7ths.  The closest intervals it has to a 12edo [[wikipedia:Major_second|whole-tone]] are both 40 cents sharp or flat of the 200-cent 12edo whole-tone.  This makes it rather difficult to translate traditional diatonic melodic approaches into 15edo, and also means that things like 7th, 9th, and 11th chords will behave very differently, even though major and minor triads are still relatively familiar-sounding.

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[[File:15ed2-001.svg|border|link=https://en.xen.wiki/w/File:15ed2-001.svg]]

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=Rank two temperaments=

[[List of 15et rank two temperaments by badness]]

| |[[Blackwood]]/[[blacksmith]]

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=Notation=

Because 15edo's best fifth is the same as that of [[5edo]], the circle of fifths fully closes with only five notes, meaning the [[256/243|Pythagorean limma]] is tempered out.  This slightly breaks traditional fifth-based notation, because E and F, as well as B and C, will merge into singular pitches.  This also has the effect of making the [[2187/2048|Pythagorean apotome]] equal to the [[9/8|Pythagorean whole-tone]], such that the sharp and flat accidentals will indicate raising or lowering by one step of 5edo, or 240 cents.  There are two main approaches to dealing with this issue: the use of ups and downs in conjunction with the traditional A through G nominals based on the 5edo circle of fifths, and the use of Porcupine temperament instead of the circle of fifths as a notational basis.

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0-4-9-13 = D F^ A C^ = D.^m7 = "D dot up minor-seven", or D F^ A B^ = D.^m6 = "D dot up minor-six"

==Porcupine[7] Notation==

''See the main [[Porcupine Notation|porcupine notation]] page.''

...A3 - A4 - A5 - A6 - A7 - A1 - A2 - M3 - M4 - M5 - M6 - P7 - P1 - P2 - m3 - m4 - m5 - m6 - d7 -- d8 - d2 - d3 -- d4 - d5 -- d6...

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An alternative porcupine notation is based on the porcupine[8] LLLLLLLs scale using eight nominals α β χ δ ε φ γ η. Others have proposed ABCDEFGHA, but conflicts with European notation have caused many to reject this approach. Thus greek letters can be used in place with a close resemblance to the spelling of ABCDEFGHA.

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=Theory=

=Practical Theory / Books=

[http://www.swordguitars.com/ Sword, Ronald. "Pendecaphonic Scales for Guitar" IAAA Press, UK-USA. First Ed: June 2009.] - A large repository of all known scales and temperament families in the 15-edo system. 300+ examples /w chord-scale progressions.