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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.
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]].
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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{| class="wikitable center-all left-8"
{| class="wikitable center-all left-8"
|-
|-
!Degree
! [[Degree]]
!Cents
! [[Cent]]s
!Approximate Ratios<ref group="note">{{sg|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>"guitar notation"
!Blackwood<br>Decimal
! Blackwood<br>Decimal
!Audio
|-
|-
| 0
| 0
|0
| 0
|1/1
| do
|do
| D
| D
|α
| α
|C
| C
|E
| E
|1
| 1
|[[File:piano_0_1edo.mp3]]
|-
|-
|1
| 1
| 80
| 80
|25/24, 21/20, 16/15, 22/21
| di
|di
| D# / Eb
|D# / Eb
| α/ β\
| α/ β\
|Db / C#
| Db / C#
| E#
| 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
| Gb
|2
| 2
|[[File:piano_2_15edo.mp3]]
|-
|-
|3
| 3
| 240
| 240
|8/7, 7/6, 9/8
| re
| re
| E# / Fb
| E# / Fb
|β/ χ\
| β/ χ\
|D#
| D#
| G
| G
|3
| 3
|[[File:piano_1_5edo.mp3]]
|-
|-
|4
| 4
|320
| 320
|6/5, 11/9
| me
|me
| F
|F
| χ
| χ
|Eb
| Eb
|G#
| G#
|3# / 4b
| 3# / 4b
|[[File:piano_4_15edo.mp3]]
|-
|-
|5
| 5
| 400
| 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
| 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
| 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
| 960
|7/4, 12/7, 16/9
| ta
|ta
| B# / Cb
|B# / Cb
| γ
|γ
| A# / Bbb
|A# / Bbb
| D
|D
| 9
| 9
|[[File:piano_4_5edo.mp3]]
|-
|-
|13
| 13
| 1040
| 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
| 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
| 15
|1200
| 1200
|2/1
| do
|do
| D
|D
| α
| α
|C
| C
| E
| E
| 1
| 1
|[[File:piano_1_1edo.mp3]]
|}
|}
<references group="note" />


=== Alternative interval names ===
=== Alternative interval names ===
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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 ==
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 ===
=== 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.
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:
{{Sharpness-sharp3-szg}}
 
==== 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).
{{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) ====
==== "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.  
[[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"
{| 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 ====
==== 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 ===
=== Blackwood decatonic notation ===
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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 ==
Retrieved from "https://en.xen.wiki/w/15edo"