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<span class="" style="text-align: right; display: block;" data-ve-style="display: block; text-align: right;" data-ve-attributes="{"style":"text-align: right; display: block;"}">[[:de:15edo|Deutsch]] - [[15平均律|日本語]]</span> | <span class="" style="text-align: right; display: block;" data-ve-style="display: block; text-align: right;" data-ve-attributes="{"style":"text-align: right; display: block;"}">[[:de:15edo|Deutsch]] - [[15平均律|日本語]]</span> | ||
==Introduction== | ==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.[[category:alternative pages]] | 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. | ||
[[category:alternative pages]] | |||
From [http://en.wikipedia.org/wiki/15_equal_temperament Wikipedia]: | From [http://en.wikipedia.org/wiki/15_equal_temperament Wikipedia]: | ||
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Blackwood[10]: 2121212121 (Blackwood Decatonic) | Blackwood[10]: 2121212121 (Blackwood Decatonic) | ||
Augmented[12]: 211121112111 | |||
=== Other Scales === | === Other Scales === | ||
Zarlino/Ptolemy | Zarlino/Ptolemy diatonic, "just" major, Ma grama - 3213231 | ||
inverse of Zarlino/Ptolemy diatonic, natural minor - 3123132 | |||
tetrachordal major, Sa grama - 3213321 | |||
inverse of tetrachordal major, "just"/tetrachordal minor - 3123123 | |||
Porcupine bright major #7 - Porcupine[7] 6|0 #7 - 3222231 | |||
Porcupine bright major #6 #7 - Porcupine[7] 6|0 #7 - 3222321 | |||
Porcupine bright minor #2 - Porcupine[7] 4|2 #2 3132222 (mode of bright major #7) | |||
Porcupine dark minor #2 - Porcupine[7] 3|3 #2 3123222 (inverse of bright major #6 #7) | |||
Porcupine bright harmonic 11th mode - Porcupine[7] 6|0 b7 3222213 | |||
"just" harmonic minor - 3123141 | |||
"just" harmonic major - 3213141 | |||
"just" melodic minor ascending - 3123231 | |||
"just" double harmonic - 1413141 | |||
=Notation= | =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. | 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. | ||
==Ups and | ==Ups and downs notation== | ||
In [[Ups and | In [[Ups and downs notation#Summary of EDO notation-.22Pentatonic.22 EDOs|ups and downs notation]], which is fifth-generated, every 15edo note has at least three names. | ||
{| class="wikitable" | {| class="wikitable" | ||
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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" | 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" | ||
For a more complete list, see [[Ups and | For a more complete list, see [[Ups and downs notation#Chord names in other EDOs|Ups and downs notation - Chord names in other EDOs]]. | ||
==Porcupine[7] Notation== | ==Porcupine[7] Notation== | ||
''See the main [[Porcupine Notation|porcupine notation]] page.'' | ''See the main [[Porcupine Notation|porcupine notation]] page.'' | ||
15edo can also be notated using the [[Ups and | 15edo can also be notated using the [[Ups and downs notation#Natural Generators|natural generator]], which is not the 9\15 5th but the 2\15 2nd. For 15edo, this is also known as porcupine notation. The 15edo porcupine genchain in both relative and absolute notation: | ||
...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... | ...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... | ||