15edo: Difference between revisions

* [[Metallic harmony]] - harmony involving stacking sevenths instead of thirds; 15edo is one of the systems it is intended for.

{{See also|15edo-interval names}}

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 [[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. One step of 15edo almost exactly equals the reduced 67th harmonic, [[67/64]].

|[[File:piano_1_1edo.mp3]]

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For a more complete list, see [[Ups and downs notation#Chords and Chord Progressions]].

There are many ways to notate 15edo, and the choice of notation depends heavily on which temperament or scale one wishes to focus on.

In these notations, the nominals form a chain of perfect 2nds, each of which are two edosteps wide. From the last note of the chain up to the first there is an augmented 2nd of three edosteps. Accidentals raise or lower by one edostep.  

Porcupine notation can be based on the Porcupine[7] Lssssss scale. By representing the 3|3 mode (sssLsss) with a chain of seconds (D E F G A B C D) and using sharps and flats (#/b) to denote an edostep up or down respectively, 15edo can be notated using standard notation. Its intervals are here named with respect to diatonic intervals, i.e., as if fifth-generated. Thus the 4th and 5th are called perfect even though they are not generators, and the 2nd and 7th are not called perfect even though they are generators.

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{{Uniform map|edo=15}}

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

* [[List of edo-distinct 15et rank two temperaments]]

15edo's [[prime]]s 3, 5, 11 and 13 are all tuned sharp, so it can benefit from [[octave shrinking]]. Shrunk-octaves versions of 15edo include [[equal tuning|50ed10]], [[47zpi]] and [[ed12|54ed12]].

Some scales commonly used in 15edo, written in a common mode, in steps of 15edo:

* Augene[6] [[3L 3s]] (period = 5\15, gen = 1\15): 4 1 4 1 4 1

*Augene[9] [[3L 6s]] (period = 5\15, gen = 1\15): 3 1 1 3 1 1 3 1 1

Blackwood decatonic, major mode, in 15edo

*[[Zarlino]]/Ptolemy diatonic, "just" major (Porcupine[7] 6|0 b4 #7): 3 2 1 3 2 3 1[[File:15edo nicetone.mp3|thumb|Zarlino/Ptolemy diatonic/Nicetone major scale 3 2 1 3 2 3 1 in 15edo tuning]]

*inverse of [[Zarlino]]/Ptolemy diatonic, natural minor (Porcupine[7] 3|3 #2 b6): 3 1 2 3 1 3 2

*[[5- to 10-tone scales in 47zpi]] (slightly stretched 15edo)

[[File:Screen Shot 2020-04-24 at 12.04.03 AM.png|none|thumb|986x986px|2\15 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 7L 1s]]

[[File:Screen Shot 2020-04-24 at 12.04.45 AM.png|none|thumb|735x735px|4\15 MOS using 1L 1s, 1L 2s, 3L 1s, 4L 3s, 4L 7s]]

[[File:Screen Shot 2020-04-24 at 12.05.29 AM.png|none|thumb|927x927px|7\15 MOS using 1L 1s, 2L 1s, 2L 3s, 2L 5s, 2L 7s, 2L 9s, 2L 11s ]]

[[File:15edo_wheel.png|alt=15edo wheel.png|225x225px|15edo wheel.png]] [[File:15edo_wheel_02.png|alt=15edo wheel 02.png|250x250px|15edo wheel 02.png]] [[File:15edo_wheel_03.png|alt=15edo wheel 03.png|220x220px|15edo wheel 03.png]]

[[File:Porcupine keyboard major triad shapes.png|none|thumb|500x500px|Major chord shapes for a Porcupine keyboard in 15edo, using Blackwood logic (i.e. native fifth notation) for the letters.]]

''See [[Lumatone mapping for 15edo]]''

{{Main| 15edo/Music }}

{{Catrel|15edo tracks}}

; [http://micro.soonlabel.com/15-ET/ XA 15-ET Directory]{{dead link}}

=== Theory===

* Carson, Brent. [http://web.archive.org/web/20121025054304/http://home.comcast.net/~brentishere/15noteequaltempermenttutorial.html Fifteen Note Equal Temperment]

*InTeAS. ''[https://web.archive.org/web/20110713044141/http://www.inteas.com/Penta01.htm The Pentadecaphonic System]'' (2001, archived)

*[[Sword, Ron]]. [http://www.metatonalmusic.com/books.html Pentadecaphonic Scales for Guitar: Scales, Chord-Scales, Notation, and Theory for Fifteen Equal Divisions of the Octave]''. (A large repository of all known scales and temperament families in the 15edo system. 300+ examples with chord-scale progressions.)''

[[Category:15edo| ]] <!-- main article -->