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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 ===
Retrieved from "https://en.xen.wiki/w/15edo"