15edo: Difference between revisions

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This gives 15edo a whole new set of pitch symmetries and modes of limited transposition. Coupled with the lack of a [[5L 2s|5L 2s diatonic scale]] and of a standard tritone, this tuning can be disorienting at first. Nonetheless, 15edo is notable for being the next-smallest edo after 9edo, 12edo and 14edo that contains recognizable major and minor triads. Under a stricter definition excluding 9edo and 14edo, this is a property noted in the works of theorists like [[Ivor Darreg]] and [[Easley Blackwood]]. In addition, because the guitar can be tuned symmetrically, from E to e (6th to 1st strings) unlike the 12-tone system on guitars, the learning curve is very manageable. All chords look the same modulated anywhere, and minor arpeggios are vertically stacked, making them very easy to play. 15-tone may be a promising start for anyone interested in xenharmony, due to its manageable number of tones and for containing the relatively popular 5edo.

A possible analogue to the diatonic scale in 15edo is the [[Zarlino]] diatonic, which flattens one fifth to a large tritone in order to make all 7 notes distinct (and close to corresponding JI intervals, especially if you use the left-handed version). The fact that 15edo supports [[porcupine]] temperament is equivalent to the fact that both accidentals generally required to notate zarlino collapse to a single chromatic step. For a moment-of-symmetry scale, the [[1L 6s]] (onyx) and [[5L 5s]] (~~blackwood~~) scales are also an option.

A possible analogue to the diatonic scale in 15edo is the [[Zarlino]] diatonic, which flattens one fifth to a large tritone in order to make all 7 notes distinct (and close to corresponding JI intervals, especially if you use the left-handed version). The fact that 15edo supports [[porcupine]] temperament is equivalent to the fact that both accidentals generally required to notate zarlino collapse to a single chromatic step. For a moment-of-symmetry scale, the [[1L 6s]] (onyx) and [[5L 5s]] (pentawood) scales are also an option.

15edo is also the second-smallest edo (after [[10edo]]) that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]].

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

==Intervals==

== 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 [[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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{| class="wikitable center-all left-8"

|-

!Degree

! [[Degree]]

!~~Cents~~

! [[Cent]]s

!Approximate Ratios<ref group="note">{{rd|limit=11-limit}}</ref>

! [[Interval region]]

![[Solfege]] (porcupine-based)

! Approximate Ratios<ref group="note">{{sg|limit=11-limit}}</ref>

!Porcupine[7] (traditional)

! Audio

!Porcupine[8] (Greek)

|-

!Zarlino diatonic notation

| 0

| Unison (prime)

| 1/1

| [[File:piano_0_1edo.mp3]]

|-

| 1

| 80

| Minor second

| 25/24, 21/20, 16/15, 22/21

| [[File:piano_1_15edo.mp3]]

|-

| 2

| 160

| Submajor second

| 11/10, 12/11, 10/9

| [[File:piano_2_15edo.mp3]]

|-

| 3

| 240

| Supermajor second

| 8/7, 7/6, 9/8

| [[File:piano_1_5edo.mp3]]

|-

| 4

| 320

| Minor third

| 6/5, 11/9

| [[File:piano_4_15edo.mp3]]

|-

| 5

| 400

| Major third

| 5/4, 14/11

| [[File:piano_1_3edo.mp3]]

|-

| 6

| 480

| Perfect fourth/ subfourth

| 4/3, ''9/7'', 21/16

| [[File:piano_2_5edo.mp3]]

|-

| 7

| 560

| Narrow tritone

| 11/8, 7/5

| [[File:piano_7_15edo.mp3]]

|-

| 8

| 640

| Wide tritone

| 16/11, 10/7

| [[File:piano_8_15edo.mp3]]

|-

| 9

| 720

| Perfect fifth/ superfifth

| 3/2, ''14/9'', 32/21

| [[File:piano_3_5edo.mp3]]

|-

| 10

| 800

| Minor sixth

| 8/5, 11/7

| [[File:piano_2_3edo.mp3]]

|-

| 11

| 880

| Major sixth

| 5/3, 18/11

| [[File:piano_11_15edo.mp3]]

|-

| 12

| 960

| Subminor seventh

| 7/4, 12/7, 16/9

| [[File:piano_4_5edo.mp3]]

|-

| 13

| 1040

| Supraminor seventh

| 20/11, 11/6, 9/5

| [[File:piano_13_15edo.mp3]]

|-

| 14

| 1120

| Major seventh

| 48/25, 40/21, 15/8, 21/11

| [[File:piano_14_15edo.mp3]]

|-

| 15

| 1200

| Octave

| 2/1

|[[File:piano_1_1edo.mp3]]

|}

== 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.

{| class="wikitable center-all left-8 mw-collapsible"

|+ style="font-size: 105%; white-space: nowrap;" | Overview of notation systems for 15edo

|-

! Degree

! Cents

! [[Solfege]] (porcupine-based)

! Porcupine[7] (traditional)

! Porcupine[8] (Greek)

! Zarlino diatonic notation

! Blackwood "guitar notation"

!Blackwood Decimal

! Blackwood Decimal

~~!Audio~~

|-

| 0

|0

| 0

~~|1/1~~

| do

|do

| D

|α

| α

|C

| C

|E

| E

|1

| 1

~~|[[File:piano_0_1edo.mp3]]~~

|-

|1

| 1

| 80

~~|25/24, 21/20, 16/15, 22/21~~

| di

|di

| D# / Eb

|D# / Eb

| α/ β\

|Db / C#

| Db / C#

| E#

|1# / 2b

| 1# / 2b

~~|[[File:piano_1_15edo.mp3]]~~

|-

|2

| 2

|160

| 160

~~|11/10, 12/11, 10/9~~

| ru

|ru

| E

|E

| β

|β

| D

|D

| Gb

|2

| 2

~~|[[File:piano_2_15edo.mp3]]~~

|-

|3

| 3

| 240

~~|8/7, 7/6, 9/8~~

| re

| E# / Fb

|β/ χ\

| β/ χ\

|D#

| D#

| G

|3

| 3

~~|[[File:piano_1_5edo.mp3]]~~

|-

|4

| 4

|320

| 320

~~|6/5, 11/9~~

| me

|me

| F

|F

| χ

|Eb

| Eb

|G#

| G#

|3# / 4b

| 3# / 4b

~~|[[File:piano_4_15edo.mp3]]~~

|-

|5

| 5

| 400

~~|5/4, 14/11~~

| mi

|mi

| F# / Gb

|F# / Gb

| χ/ δ\

|χ/ δ\

| E

|E

| Ab

|Ab

| 4

|4

~~|[[File:piano_1_3edo.mp3]]~~

|-

| 6

|480

| 480

~~|4/3, ''9/7'', 21/16~~

| fa

|fa

| G

|G

| δ

|δ

| F

|F

| A

|A

| 5

|5

~~|[[File:piano_2_5edo.mp3]]~~

|-

|7

| 7

|560

| 560

~~|11/8, 7/5~~

| fu

|fu

| G#

|G#

| δ/ ε\

|δ/ ε\

| F#

|F#

| A#

|A#

| 5# / 6b

|5# / 6b

~~|[[File:piano_7_15edo.mp3]]~~

|-

|8

| 8

|640

| 640

~~|16/11, 10/7~~

| su

|su

| Ab

|ε

| ε

|Gb

| Gb

|Bb

| Bb

|6

| 6

~~|[[File:piano_8_15edo.mp3]]~~

|-

|9

| 9

|720

| 720

~~|3/2, ''14/9'', 32/21~~

| sol

|sol

| A

|A

| ε/ φ\

|ε/ φ\

| G

|G

| B

|B

| 7

|7

~~|[[File:piano_3_5edo.mp3]]~~

|-

|10

| 10

|800

| 800

~~| 8/5, 11/7~~

| le

|le

| A# / Bb

|A# / Bb

| φ

|φ

| Ab / G#

|Ab / G#

| B#

|B#

| 7# / 8b

|7# / 8b

~~|[[File:piano_2_3edo.mp3]]~~

|-

|11

| 11

|880

| 880

~~|5/3, 18/11~~

| la

|la

| B

|B

| φ/ γ\

|φ/ γ\

| A

|A

| Db

|Db

| 8

|8

~~|[[File:piano_11_15edo.mp3]]~~

|-

|12

| 12

| 960

~~|7/4, 12/7, 16/9~~

| ta

|ta

| B# / Cb

|B# / Cb

| γ

|γ

| A# / Bbb

|A# / Bbb

| D

|D

| 9

~~|[[File:piano_4_5edo.mp3]]~~

|-

|13

| 13

| 1040

~~|20/11, 11/6, 9/5~~

| tu

|tu

| C

|C

| γ/ η\

|γ/ η\

| Bb

|Bb

| D#

|D#

| 9# / 0b

|9# / 0b

~~|[[File:piano_13_15edo.mp3]]~~

|-

|14

| 14

| 1120

~~|48/25, 40/21, 15/8, 21/11~~

| ti

|ti

| C# / Db

|C# / Db

| η

|η

| B

|B

| Eb

|Eb

| 0

|0

~~|[[File:piano_14_15edo.mp3]]~~

|-

| 15

|1200

| 1200

~~|2/1~~

| do

|do

| D

|D

| α

|C

| C

| E

| 1

~~|[[File:piano_1_1edo.mp3]]~~

|}

===Alternative interval names===

=== Alternative interval names ===

{| class="wikitable center-all"

|-

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

~~==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.~~

=== Notations generated by the fifth ===

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) ====

15edo 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:

==== Kite's ups and downs notation (heptatonic) ====

15edo can be notated with [[Kite's ups and downs notation|Kite's 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).

==== Sagittal notation (heptatonic) ====

This notation uses the same sagittal sequence as edos [[22edo #Sagittal notation|22]] and [[29edo #Sagittal notation|29]], is a subset of the notation for [[30edo #Sagittal notation|30edo]], and is a superset of the notation for [[5edo #Sagittal notation|5edo]].

==== "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

|^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

|^1

|v2

|P2

|^2

|v3

|P3

|^3

|v4

|P4

|^4

|v5

|P5

|^5

|v6

|P6

|}

~~==== Sagittal notation (heptatonic)====~~

This notation uses the same sagittal sequence as EDOs [[22edo#Sagittal notation|22]] and [[29edo#Sagittal notation|29]], is a subset of the notation for [[30edo#Sagittal notation|30-EDO]], and is a superset of the notation for [[5edo#Sagittal notation|5-EDO]].

~~{{Sagittal chart|}}~~

==== 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 ===

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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.

====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.

{| class="wikitable"

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===Rank-2 temperaments===

=== Rank-2 temperaments ===

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

* [[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]].

== Scales==

== Scales ==

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[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

===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]]

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

Line 1,124:

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*[[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 12.04.45 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 ]]

==Diagrams==

== Diagrams ==

=== 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.]]

===Lumatone ===

== Instruments ==

=== Halberstadt-Inspired Keyboards ===

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]]''

==Music==

== 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 ==

; [http://micro.soonlabel.com/15-ET/ XA 15-ET Directory]

; [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===

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

Line 1,156:

Line 1,211:

*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.)''

~~==Notes==~~

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

[[Category:15edo| ]]

@@ Line 20: / Line 20: @@
 This gives 15edo a whole new set of pitch symmetries and modes of limited transposition. Coupled with the lack of a [[5L 2s|5L 2s diatonic scale]] and of a standard tritone, this tuning can be disorienting at first. Nonetheless, 15edo is notable for being the next-smallest edo after 9edo, 12edo and 14edo that contains recognizable major and minor triads. Under a stricter definition excluding 9edo and 14edo, this is a property noted in the works of theorists like [[Ivor Darreg]] and [[Easley Blackwood]]. In addition, because the guitar can be tuned symmetrically, from E to e (6th to 1st strings) unlike the 12-tone system on guitars, the learning curve is very manageable. All chords look the same modulated anywhere, and minor arpeggios are vertically stacked, making them very easy to play. 15-tone may be a promising start for anyone interested in xenharmony, due to its manageable number of tones and for containing the relatively popular 5edo.
-A possible analogue to the diatonic scale in 15edo is the [[Zarlino]] diatonic, which flattens one fifth to a large tritone in order to make all 7 notes distinct (and close to corresponding JI intervals, especially if you use the left-handed version). The fact that 15edo supports [[porcupine]] temperament is equivalent to the fact that both accidentals generally required to notate zarlino collapse to a single chromatic step. For a moment-of-symmetry scale, the [[1L&nbsp;6s]] (onyx) and [[5L&nbsp;5s]] (blackwood) scales are also an option.
+A possible analogue to the diatonic scale in 15edo is the [[Zarlino]] diatonic, which flattens one fifth to a large tritone in order to make all 7 notes distinct (and close to corresponding JI intervals, especially if you use the left-handed version). The fact that 15edo supports [[porcupine]] temperament is equivalent to the fact that both accidentals generally required to notate zarlino collapse to a single chromatic step. For a moment-of-symmetry scale, the [[1L&nbsp;6s]] (onyx) and [[5L&nbsp;5s]] (pentawood) scales are also an option.
 edo is also the second-smallest edo (after [[10edo]]) that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]].
@@ Line 33: / Line 33: @@
 * [[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}}
 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]].
@@ Line 39: / Line 39: @@
 {| class="wikitable center-all left-8"
 |-
-!Degree
+! [[Degree]]
-!Cents
+! [[Cent]]s
-!Approximate Ratios<ref group="note">{{rd|limit=11-limit}}</ref>
+! [[Interval region]]
-![[Solfege]]<br>(porcupine-based)
+! Approximate Ratios<ref group="note">{{sg|limit=11-limit}}</ref>
-!Porcupine[7]<br>(traditional)
+! Audio
-!Porcupine[8]<br>(Greek)
+|-
-!Zarlino diatonic notation
+| 0
+| 0
+| Unison<br>(prime)
+| 1/1
+| [[File:piano_0_1edo.mp3]]
+|-
+| 1
+| 80
+| Minor second
+| 25/24, 21/20, 16/15, 22/21
+| [[File:piano_1_15edo.mp3]]
+|-
+| 2
+| 160
+| Submajor second
+| 11/10, 12/11, 10/9
+| [[File:piano_2_15edo.mp3]]
+|-
+| 3
+| 240
+| Supermajor second
+| 8/7, 7/6, 9/8
+| [[File:piano_1_5edo.mp3]]
+|-
+| 4
+| 320
+| Minor third
+| 6/5, 11/9
+| [[File:piano_4_15edo.mp3]]
+|-
+| 5
+| 400
+| Major third
+| 5/4, 14/11
+| [[File:piano_1_3edo.mp3]]
+|-
+| 6
+| 480
+| Perfect fourth/<br>subfourth
+| 4/3, ''9/7'', 21/16
+| [[File:piano_2_5edo.mp3]]
+|-
+| 7
+| 560
+| Narrow tritone
+| 11/8, 7/5
+| [[File:piano_7_15edo.mp3]]
+|-
+| 8
+| 640
+| Wide tritone
+| 16/11, 10/7
+| [[File:piano_8_15edo.mp3]]
+|-
+| 9
+| 720
+| Perfect fifth/<br>superfifth
+| 3/2, ''14/9'', 32/21
+| [[File:piano_3_5edo.mp3]]
+|-
+| 10
+| 800
+| Minor sixth
+| 8/5, 11/7
+| [[File:piano_2_3edo.mp3]]
+|-
+| 11
+| 880
+| Major sixth
+| 5/3, 18/11
+| [[File:piano_11_15edo.mp3]]
+|-
+| 12
+| 960
+| Subminor seventh
+| 7/4, 12/7, 16/9
+| [[File:piano_4_5edo.mp3]]
+|-
+| 13
+| 1040
+| Supraminor seventh
+| 20/11, 11/6, 9/5
+| [[File:piano_13_15edo.mp3]]
+|-
+| 14
+| 1120
+| Major seventh
+| 48/25, 40/21, 15/8, 21/11
+| [[File:piano_14_15edo.mp3]]
+|-
+| 15
+| 1200
+| Octave
+| 2/1
+|[[File:piano_1_1edo.mp3]]
+|}
+<references group="note" />
+== 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.
+{| class="wikitable center-all left-8 mw-collapsible"
+|+ style="font-size: 105%; white-space: nowrap;" | Overview of notation systems for 15edo
+|-
+! Degree
+! Cents
+! [[Solfege]]<br>(porcupine-based)
+! Porcupine[7]<br>(traditional)
+! Porcupine[8]<br>(Greek)
+! Zarlino diatonic notation
 ! Blackwood<br>"guitar notation"
-!Blackwood<br>Decimal
+! Blackwood<br>Decimal
-!Audio
 |-
 | 0
-|0
+| 0
-|1/1
+| do
-|do
 | D
-|α
+| α
-|C
+| C
-|E
+| E
-|1
+| 1
-|[[File:piano_0_1edo.mp3]]
 |-
-|1
+| 1
 | 80
-|25/24, 21/20, 16/15, 22/21
+| di
-|di
+| D# / Eb
-|D# / Eb
 | α/ β\
-|Db / C#
+| Db / C#
 | E#
-|1# / 2b
+| 1# / 2b
-|[[File:piano_1_15edo.mp3]]
 |-
-|2
+| 2
-|160
+| 160
-|11/10, 12/11, 10/9
+| ru
-|ru
+| E
-|E
+| β
-|β
+| D
-|D
 | Gb
-|2
+| 2
-|[[File:piano_2_15edo.mp3]]
 |-
-|3
+| 3
 | 240
-|8/7, 7/6, 9/8
 | re
 | E# / Fb
-|β/ χ\
+| β/ χ\
-|D#
+| D#
 | G
-|3
+| 3
-|[[File:piano_1_5edo.mp3]]
 |-
-|4
+| 4
-|320
+| 320
-|6/5, 11/9
+| me
-|me
+| F
-|F
 | χ
-|Eb
+| Eb
-|G#
+| G#
-|3# / 4b
+| 3# / 4b
-|[[File:piano_4_15edo.mp3]]
 |-
-|5
+| 5
 | 400
-|5/4, 14/11
+| mi
-|mi
+| F# / Gb
-|F# / Gb
+| χ/ δ\
-|χ/ δ\
+| E
-|E
+| Ab
-|Ab
+| 4
-|4
-|[[File:piano_1_3edo.mp3]]
 |-
 | 6
-|480
+| 480
-|4/3, ''9/7'', 21/16
+| fa
-|fa
+| G
-|G
+| δ
-|δ
+| F
-|F
+| A
-|A
+| 5
-|5
-|[[File:piano_2_5edo.mp3]]
 |-
-|7
+| 7
-|560
+| 560
-|11/8, 7/5
+| fu
-|fu
+| G#
-|G#
+| δ/ ε\
-|δ/ ε\
+| F#
-|F#
+| A#
-|A#
+| 5# / 6b
-|5# / 6b
-|[[File:piano_7_15edo.mp3]]
 |-
-|8
+| 8
-|640
+| 640
-|16/11, 10/7
+| su
-|su
 | Ab
-|ε
+| ε
-|Gb
+| Gb
-|Bb
+| Bb
-|6
+| 6
-|[[File:piano_8_15edo.mp3]]
 |-
-|9
+| 9
-|720
+| 720
-|3/2, ''14/9'', 32/21
+| sol
-|sol
+| A
-|A
+| ε/ φ\
-|ε/ φ\
+| G
-|G
+| B
-|B
+| 7
-|7
-|[[File:piano_3_5edo.mp3]]
 |-
-|10
+| 10
-|800
+| 800
-| 8/5, 11/7
+| le
-|le
+| A# / Bb
-|A# / Bb
+| φ
-|φ
+| Ab / G#
-|Ab / G#
+| B#
-|B#
+| 7# / 8b
-|7# / 8b
-|[[File:piano_2_3edo.mp3]]
 |-
-|11
+| 11
-|880
+| 880
-|5/3, 18/11
+| la
-|la
+| B
-|B
+| φ/ γ\
-|φ/ γ\
+| A
-|A
+| Db
-|Db
+| 8
-|8
-|[[File:piano_11_15edo.mp3]]
 |-
-|12
+| 12
 | 960
-|7/4, 12/7, 16/9
+| ta
-|ta
+| B# / Cb
-|B# / Cb
+| γ
-|γ
+| A# / Bbb
-|A# / Bbb
+| D
-|D
 | 9
-|[[File:piano_4_5edo.mp3]]
 |-
-|13
+| 13
 | 1040
-|20/11, 11/6, 9/5
+| tu
-|tu
+| C
-|C
+| γ/ η\
-|γ/ η\
+| Bb
-|Bb
+| D#
-|D#
+| 9# / 0b
-|9# / 0b
-|[[File:piano_13_15edo.mp3]]
 |-
-|14
+| 14
 | 1120
-|48/25, 40/21, 15/8, 21/11
+| ti
-|ti
+| C# / Db
-|C# / Db
+| η
-|η
+| B
-|B
+| Eb
-|Eb
+| 0
-|0
-|[[File:piano_14_15edo.mp3]]
 |-
 | 15
-|1200
+| 1200
-|2/1
+| do
-|do
+| D
-|D
 | α
-|C
+| C
 | E
 | 1
-|[[File:piano_1_1edo.mp3]]
 |}
-===Alternative interval names===
+=== Alternative interval names ===
 {| class="wikitable center-all"
 |-
@@ Line 392: / Line 468: @@
 For a more complete list, see [[Ups and downs notation#Chords and Chord Progressions]].
-==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.
 === Notations generated by the fifth ===
 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:
+{{Sharpness-sharp3-szg}}
+==== Kite's ups and downs notation (heptatonic) ====
+edo can be notated with [[Kite's ups and downs notation|Kite's 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}}
+==== Sagittal notation (heptatonic) ====
+This notation uses the same sagittal sequence as edos [[22edo #Sagittal notation|22]] and [[29edo #Sagittal notation|29]], is a subset of the notation for [[30edo #Sagittal notation|30edo]], and is a superset of the notation for [[5edo #Sagittal notation|5edo]].
+{{Sagittal chart|}}
 ==== "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
-|^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
-|^1
-|v2
-|P2
-|^2
-|v3
-|P3
-|^3
-|v4
-|P4
-|^4
-|v5
-|P5
-|^5
-|v6
-|P6
 |}
-==== Sagittal notation (heptatonic)====
-This notation uses the same sagittal sequence as EDOs [[22edo#Sagittal notation|22]] and [[29edo#Sagittal notation|29]], is a subset of the notation for [[30edo#Sagittal notation|30-EDO]], and is a superset of the notation for [[5edo#Sagittal notation|5-EDO]].
-{{Sagittal chart|}}
 ==== 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 ===
@@ Line 455: / Line 505: @@
 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.
 {| class="wikitable"
@@ Line 822: / Line 872: @@
 {{Uniform map|edo=15}}
-===Rank-2 temperaments===
+=== Rank-2 temperaments ===
 * [[List of 15et rank two temperaments by badness]]
 * [[List of edo-distinct 15et rank two temperaments]]
@@ Line 1,080: / Line 1,130: @@
 edo'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:
-=== MOS scales===
+=== MOS scales ===
 * 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
@@ Line 1,098: / Line 1,148: @@
 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]]
 *inverse of [[Zarlino]]/Ptolemy diatonic, natural minor (Porcupine[7] 3|3 #2 b6): 3 1 2 3 1 3 2
@@ Line 1,124: / Line 1,174: @@
 *[[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 12.04.45 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 ]]
-==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]]
-=== 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.]]
-===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]]''
-==Music==
+== 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 ==
 {{Main| 15edo/Music }}
 {{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==
+== 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===
 * Carson, Brent. [http://web.archive.org/web/20121025054304/http://home.comcast.net/~brentishere/15noteequaltempermenttutorial.html Fifteen Note Equal Temperment]
@@ Line 1,156: / Line 1,211: @@
 *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.)''
-==Notes==
-<references group="note" />
 [[Category:15edo| ]] <!-- main article -->