User:Lhearne/15edo: Difference between revisions
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==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. | 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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The following | The following diagram shows how closely some prominent JI intervals are approximated in 15edo, using their best direct mappings, even if they lead to inconsistencies. | ||
[[File:15ed2-001.svg|border|link=https://en.xen.wiki/w/File:15ed2-001.svg]] | [[File:15ed2-001.svg|border|link=https://en.xen.wiki/w/File:15ed2-001.svg]] | ||
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=Rank two temperaments= | =Rank two temperaments= | ||
[[List of 15et rank two temperaments by badness]] | [[List of 15et rank two temperaments by badness]] | ||
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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. | ||
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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... | ||
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==Porcupine[8] Notation | ==Porcupine[8] Notation== | ||
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. | 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= | =Theory= | ||
[http:// | [http://www.webcitation.org/5xZyzKBEW The 15-Tone Scale System] by [[Ivor_Darreg|Ivor Darreg]] (Originally at [http://sonic-arts.org/darreg/dar35.htm], now broken) | ||
[https://web.archive.org/web/20110713044141/http://www.inteas.com/Penta01.htm The Pentadecaphonic System] (Originally at [http://www.inteas.com/Penta01.htm], now broken) | |||
[http://www. | [http://www.webcitation.org/5xeJYBsDg 15-EDO Tutorial] by [[Brent_Carson|Brent Carson]] (Originally at [http://home.comcast.net/%7Ebrentishere/15noteequaltempermenttutorial.html], now broken) | ||
=Practical Theory / Books= | =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. | [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. | ||