15edo - Revision history

FloraC: /* Notations generated by the fifth */ SZG notation and misc. cleanup

2026-05-12T11:41:20Z

Notations generated by the fifth: SZG notation and misc. cleanup

@@ Line 472: / Line 472: @@
 In these notations, the nominals form a circle of perfect fifths. The other notes are notated using accidentals that raise or lower by one edostep.
-==== Ups and downs notation (heptatonic) ====
+==== Stein–Zimmermann–Gould notation (heptatonic) ====
-edo can be notated with [[ups and downs]], spoken as up, dup, downsharp, sharp, upsharp etc. and down, dud, upflat etc. Note that downsharp is equivalent to dup (double-up) and upflat is equivalent to dud (double-down).{{Sharpness-sharp3a}}[[Alternative symbols for ups and downs notation]] uses sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]:{{Sharpness-sharp3}}
+[[Stein–Zimmermann–Gould notation]] uses sharps and flats with arrows:
 ==== "Eef" notation (pentatonic) ====
 [[Kite Giedraitis]] proposes pentatonic (as opposed to heptatonic) note names that omit B and merge E and F into a new letter "eef" that rhymes with "leaf". Eef, like E, is a 5th above A. Eef, like F, is a 4th above C. Eef is written like an E, but with the bottom horizontal line going not right but left from the vertical line. Eef can be typed as ꘙ (unicode A619) or ⊧ (unicode 22A7) or 𐐆 (unicode 10406). The circle of 5ths is C G D A ꘙ C. All intervals are either perfect, upperfect or downperfect (never major or minor). This is similar to heptatonic interval names in 7edo, 14edo, 21edo, etc.
 {| class="wikitable"
-|C
+| C || ^C || vD || D || ^D || vꘙ || ꘙ || ^ꘙ || vG || G || ^G || vA || A || ^A || vC || C
 |-
-|P1
+| P1 || ^1 || v2 || P2 || ^2 || v3 || P3 || ^3 || v4 || P4 || ^4 || v5 || P5 || ^5 || v6 || P6
 |}
 ==== Blackwood guitar notation ====
-On a 15edo guitar, because the "perfect fourth" comes from 5edo, all of the open strings can be tuned a perfect fourth apart and still span exactly two octaves. If one starts the [[circle of fourths]] on B — B-E-A-D-G-(B) — then the open strings of the guitar can be notated as usual (E-A-D-G-B-E). However, because the circle of fourths closes at five, and does not continue to circulate through the other 10 notes of 15edo, it is necessary to use accidentals to notate intervals on the other two chains of 5edo. This notation is not particularly ideal as a basis for a staff notation (as it requires all non-5edo chords to be notated with accidentals). It is nevertheless useful because it reflects an intuitive approach to 15edo on the guitar, since 5edo provides a useful set of 3-limit landmarks (or "perfect fourths" and "perfect fifths") that can be used to navigate the fretboard. It's especially convenient for writing chord charts, where the funky accidental-laden spellings can be more or less ignored.
+On a 15edo guitar, because the "perfect fourth" comes from 5edo, all of the open strings can be tuned a perfect fourth apart and still span exactly two octaves. If one starts the [[circle of fourths]] on B—B–E–A–D–G-(B)—then the open strings of the guitar can be notated as usual (E–A–D–G–B–E). However, because the circle of fourths closes at five, and does not continue to circulate through the other 10 notes of 15edo, it is necessary to use accidentals to notate intervals on the other two chains of 5edo. This notation is not particularly ideal as a basis for a staff notation (as it requires all non-5edo chords to be notated with accidentals). It is nevertheless useful because it reflects an intuitive approach to 15edo on the guitar, since 5edo provides a useful set of 3-limit landmarks (or "perfect fourths" and "perfect fifths") that can be used to navigate the fretboard. It's especially convenient for writing chord charts, where the funky accidental-laden spellings can be more or less ignored.
 === Blackwood decatonic notation ===

BudjarnLambeth: /* Instruments */ add link

2026-04-11T10:25:53Z

Instruments: add link

← Older revision		Revision as of 10:25, 11 April 2026
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	=== Lumatone ===		=== Lumatone ===
	''See [[Lumatone mapping for 15edo]]''		''See [[Lumatone mapping for 15edo]]''

			== Guitars ==
			[[Benjamin Strange]] built a 15edo Hello Kitty guitar and documented the process on his blog: https://www.strangeguitarworks.com/benjamins-fender-hello-kitty-microtonal-strat-of-doom/

	== Music ==		== Music ==

Overthink: /* Intervals */ I think this looks better

2026-03-21T01:25:46Z

Intervals: I think this looks better

← Older revision		Revision as of 01:25, 21 March 2026
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	\| 6		\| 6
	\| 480		\| 480
	\| Perfect fourth /<br>~~Subfourth~~		\| Perfect fourth/<br>subfourth
	\| 4/3, ''9/7'', 21/16		\| 4/3, ''9/7'', 21/16
	\| [[File:piano_2_5edo.mp3]]		\| [[File:piano_2_5edo.mp3]]
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	\| 9		\| 9
	\| 720		\| 720
	\| Perfect fifth /<br>~~Superfifth~~		\| Perfect fifth/<br>superfifth
	\| 3/2, ''14/9'', 32/21		\| 3/2, ''14/9'', 32/21
	\| [[File:piano_3_5edo.mp3]]		\| [[File:piano_3_5edo.mp3]]

Overthink: Split interval table

2026-03-21T01:24:56Z

Split interval table

Show changes

Lucius Chiaraviglio: Move notes to be under the corresponding tables; consolidate instruments into Instruments section; fix section headings

2026-01-22T10:39:13Z

Move notes to be under the corresponding tables; consolidate instruments into Instruments section; fix section headings

← Older revision		Revision as of 10:39, 22 January 2026
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	* [[Metallic harmony]] - harmony involving stacking sevenths instead of thirds; 15edo is one of the systems it is intended for.		* [[Metallic harmony]] - harmony involving stacking sevenths instead of thirds; 15edo is one of the systems it is intended for.

	==Intervals==		== Intervals ==
	{{See also\|15edo-interval names}}		{{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]].		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]].
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	\|[[File:piano_1_1edo.mp3]]		\|[[File:piano_1_1edo.mp3]]
	\|}		\|}
			<references group="note" />

	===Alternative interval names===		=== Alternative interval names ===
	{\| class="wikitable center-all"		{\| class="wikitable center-all"
	\|-		\|-
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	For a more complete list, see [[Ups and downs notation#Chords and Chord Progressions]].		For a more complete list, see [[Ups and downs notation#Chords and Chord Progressions]].

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

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	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.		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 (heptatonic) ====		==== Porcupine notation (heptatonic) ====
	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.		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.
	{\| class="wikitable"		{\| class="wikitable"
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	{{Uniform map\|edo=15}}		{{Uniform map\|edo=15}}

	===Rank-2 temperaments===		=== Rank-2 temperaments ===
	* [[List of 15et rank two temperaments by badness]]		* [[List of 15et rank two temperaments by badness]]
	* [[List of edo-distinct 15et rank two temperaments]]		* [[List of edo-distinct 15et rank two temperaments]]
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	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]].		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]].

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

	=== MOS scales===		=== MOS scales ===
	* Augene[6] [[3L 3s]] (period = 5\15, gen = 1\15): 4 1 4 1 4 1		* 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		*Augene[9] [[3L 6s]] (period = 5\15, gen = 1\15): 3 1 1 3 1 1 3 1 1
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	Blackwood decatonic, major mode, in 15edo		Blackwood decatonic, major mode, in 15edo

	===Other scales===		=== Other scales ===
	*[[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]]		*[[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		*inverse of [[Zarlino]]/Ptolemy diatonic, natural minor (Porcupine[7] 3\|3 #2 b6): 3 1 2 3 1 3 2
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	*[[5- to 10-tone scales in 47zpi]] (slightly stretched 15edo)		*[[5- to 10-tone scales in 47zpi]] (slightly stretched 15edo)

	=== Horagrams===		=== Horagrams ===
	[[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 12.04.03 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 12.04.45 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: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 ]]

	==Diagrams==		== Diagrams ==
	[[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: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]]

	=== Keyboard===		=== Porcupine (Halberstadt-Inspired) Keyboard Chord Shapes ===
			The following chord shapes use the porcupine layout for 15edo (quasi-Halberstadt) (see [[#Halberstadt-Inspired_Keyboards\|Halberstadt-Inspired Keyboards]] below).
	[[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.]]		[[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.]]

	===Lumatone ===		== Instruments ==
			=== Halberstadt-Inspired Keyboards ===
			<gallery widths="300px">
			File:15_tone_keyboard.png\|Porcupine layout for 15edo
			File:Screen Shot 2020-04-23 at 11.59.17 PM.png\|Hanson layout for 15edo
			File:15edo kb3.png\|Zarlino layout for 15edo
			</gallery>

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

	==Music==		== Music ==
	{{Main\| 15edo/Music }}		{{Main\| 15edo/Music }}
	{{Catrel\|15edo tracks}}		{{Catrel\|15edo tracks}}
	; [http://micro.soonlabel.com/15-ET/ XA 15-ET Directory]{{dead link}}		; [http://micro.soonlabel.com/15-ET/ XA 15-ET Directory]{{dead link}}

	~~==Other info==~~		== Further reading ==
	~~==== Keyboard layouts ====~~
	~~<gallery widths="300px">~~
	~~File:15_tone_keyboard.png\|Porcupine layout for 15edo~~
	~~File:Screen Shot 2020-04-23 at 11.59.17 PM.png\|Hanson layout for 15edo~~
	~~File:15edo kb3.png\|Zarlino layout for 15edo~~
	~~</gallery>~~
	==Further reading==
	=== Theory===		=== Theory===
	* Carson, Brent. [http://web.archive.org/web/20121025054304/http://home.comcast.net/~brentishere/15noteequaltempermenttutorial.html Fifteen Note Equal Temperment]		* Carson, Brent. [http://web.archive.org/web/20121025054304/http://home.comcast.net/~brentishere/15noteequaltempermenttutorial.html Fifteen Note Equal Temperment]
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	*InTeAS. ''[https://web.archive.org/web/20110713044141/http://www.inteas.com/Penta01.htm The Pentadecaphonic System]'' (2001, archived)		*InTeAS. ''[https://web.archive.org/web/20110713044141/http://www.inteas.com/Penta01.htm The Pentadecaphonic System]'' (2001, archived)

	===Guitar===		=== Guitar ===
	*[[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.)''		*[[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.)''

	~~==Notes==~~
	~~<references group="note" />~~

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

Lucius Chiaraviglio: /* Music */ XA 15-ET Directory: dead link

2026-01-22T06:57:54Z

Music: XA 15-ET Directory: dead link

← Older revision		Revision as of 06:57, 22 January 2026
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	{{Main\| 15edo/Music }}		{{Main\| 15edo/Music }}
	{{Catrel\|15edo tracks}}		{{Catrel\|15edo tracks}}
	; [http://micro.soonlabel.com/15-ET/ XA 15-ET Directory]		; [http://micro.soonlabel.com/15-ET/ XA 15-ET Directory]{{dead link}}

	==Other info==		==Other info==

Overthink: Undo revision 222143 by Overthink (talk) accidentally had edit conflict with myself (probably double clicked save changes)

2026-01-22T04:21:09Z

Undo revision 222143 by Overthink (talk) accidentally had edit conflict with myself (probably double clicked save changes)

← Older revision		Revision as of 04:21, 22 January 2026
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	!Degree		!Degree
	!Cents		!Cents
	!Approximate Ratios<ref group="note">{{rd\|limit=11-limit}}</ref>		!Approximate Ratios<ref group="note">{{sg\|limit=11-limit}}</ref>
	![[Solfege]]<br>(porcupine-based)		![[Solfege]]<br>(porcupine-based)
	!Porcupine[7]<br>(traditional)		!Porcupine[7]<br>(traditional)

Overthink: /* Intervals */ fix note

2026-01-22T04:20:12Z

Intervals: fix note

← Older revision		Revision as of 04:20, 22 January 2026
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	!Degree		!Degree
	!Cents		!Cents
	!Approximate Ratios<ref group="note">{{sg\|limit=11-limit}}</ref>		!Approximate Ratios<ref group="note">{{rd\|limit=11-limit}}</ref>
	![[Solfege]]<br>(porcupine-based)		![[Solfege]]<br>(porcupine-based)
	!Porcupine[7]<br>(traditional)		!Porcupine[7]<br>(traditional)

Overthink: /* Intervals */ fix note

2026-01-22T04:19:37Z

Intervals: fix note

← Older revision		Revision as of 04:19, 22 January 2026
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	!Degree		!Degree
	!Cents		!Cents
	!Approximate Ratios<ref group="note">{{rd\|limit=11-limit}}</ref>		!Approximate Ratios<ref group="note">{{sg\|limit=11-limit}}</ref>
	![[Solfege]]<br>(porcupine-based)		![[Solfege]]<br>(porcupine-based)
	!Porcupine[7]<br>(traditional)		!Porcupine[7]<br>(traditional)

Overthink: /* Intervals */mark inconsistent 14/9 and 9/7

2025-12-20T23:02:36Z

Intervals: mark inconsistent 14/9 and 9/7

← Older revision		Revision as of 23:02, 20 December 2025
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	\| 6		\| 6
	\|480		\|480
	\|4/3, 9/7, 21/16		\|4/3, ''9/7'', 21/16
	\|fa		\|fa
	\|G		\|G
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	\|9		\|9
	\|720		\|720
	\|3/2, 14/9, 32/21		\|3/2, ''14/9'', 32/21
	\|sol		\|sol
	\|A		\|A
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	For a more complete list, see [[Ups and downs notation#Chords and Chord Progressions]].		For a more complete list, see [[Ups and downs notation#Chords and Chord Progressions]].

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