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== 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.
{| class="wikitable center-all left-8"
{| class="wikitable center-all left-8"
|-
|-
! Degree
! Degree
! Cents
! Cents
! Solfege<br>(porcupine-based)
! Approximate Ratios*
! [[Solfege]]<br>(porcupine-based)
! Porcupine[7]<br>(traditional)
! Porcupine[8]<br>(Greek)
! Porcupine[8]<br>(Greek)
! Blackwood<br>"guitar notation"
! Blackwood<br>"guitar notation"
! Porcupine[7]<br>(traditional)
! Blackwood<br>Decimal
! Blackwood<br>Decimal
! Approximate Ratios*
|-
|-
| 0
| 0
| 0
| 0
| 1/1
| do
| do
| G
| α
| α
| E
| E
| G
| 1
| 1
| 1/1
|-
|-
| 1
| 1
| 80
| 80
| 25/24, 21/20, 16/15
| di
| di
| G# / Abb
| α/ β\
| α/ β\
| E#
| E#
| G# / Abb
| 1# / 2b
| 1# / 2b
| 25/24, 21/20, 16/15
|-
|-
| 2
| 2
| 160
| 160
| 11/10, 12/11, 10/9
| ru
| ru
| Gx / Ab
| β
| β
| Gb
| Gb
| Gx / Ab
| 2
| 2
| 11/10, 12/11, 10/9
|-
|-
| 3
| 3
| 240
| 240
| 8/7, 7/6, 9/8
| re
| re
| A
| β/ χ\
| β/ χ\
| G
| G
| A
| 3
| 3
| 8/7, 7/6, 9/8
|-
|-
| 4
| 4
| 320
| 320
| 6/5, 11/9
| me
| me
| A# / Bb
| χ
| χ
| G#
| G#
| A# / Bb
| 3# / 4b
| 3# / 4b
| 6/5, 11/9
|-
|-
| 5
| 5
| 400
| 400
| 5/4, 14/11
| mi
| mi
| B
| χ/ δ\
| χ/ δ\
| Ab
| Ab
| B
| 4
| 4
| 5/4, 14/11
|-
|-
| 6
| 6
| 480
| 480
| 4/3, 9/7, 21/16
| fa
| fa
| B# / Cb
| δ
| δ
| A
| A
| B# / Cb
| 5
| 5
| 4/3, 9/7, 21/16
|-
|-
| 7
| 7
| 560
| 560
| 11/8, 7/5
| fu
| fu
| C
| δ/ ε\
| δ/ ε\
| A#
| A#
| C
| 5# / 6b
| 5# / 6b
| 11/8, 7/5
|-
|-
| 8
| 8
| 640
| 640
| 16/11, 10/7
| su
| su
| C# / Db
| ε
| ε
| Bb
| Bb
| C# / Db
| 6
| 6
| 16/11, 10/7
|-
|-
| 9
| 9
| 720
| 720
| 3/2, 14/9, 32/21
| sol
| sol
| D
| ε/ φ\
| ε/ φ\
| B
| B
| D
| 7
| 7
| 3/2, 14/9, 32/21
|-
|-
| 10
| 10
| 800
| 800
| 8/5, 11/7
| le
| le
| D# / Eb
| φ
| φ
| B#
| B#
| D# / Eb
| 7# / 8b
| 7# / 8b
| 8/5, 11/7
|-
|-
| 11
| 11
| 880
| 880
| 5/3, 18/11
| la
| la
| E
| φ/ γ\
| φ/ γ\
| Db
| Db
| E
| 8
| 8
| 5/3, 18/11
|-
|-
| 12
| 12
| 960
| 960
| 7/4, 12/7, 16/9
| ta
| ta
| E# / Fb
| γ
| γ
| D
| D
| E# / Fb
| 9
| 9
| 7/4, 12/7, 16/9
|-
|-
| 13
| 13
| 1040
| 1040
| 20/11, 11/6, 9/5
| tu
| tu
| F
| γ/ η\
| γ/ η\
| D#
| D#
| F
| 9# / 0b
| 9# / 0b
| 20/11, 11/6, 9/5
|-
|-
| 14
| 14
| 1120
| 1120
| 48/25, 40/21, 15/8
| ti
| ti
| F# /Gb
| η
| η
| Eb
| Eb
| F# /Gb
| 0
| 0
| 48/25, 40/21, 15/8
|-
|-
| 15
| 15
| 1200
| 1200
| 2/1
| do
| do
| G
| α
| α
| E
| E
| G
| 1
| 1
| 2/1
|}
|}
<nowiki>*</nowiki> based on treating 15edo as an 11-limit temperament; other approaches are possible
<nowiki>*</nowiki> based on treating 15edo as an 11-limit temperament; other approaches are possible


In [[Ups and downs notation#Summary of EDO notation-"Pentatonic" EDOs|ups and downs notation]], which is fifth-generated, every 15edo note has at least three names. 15edo can also be notated using the [[Ups and downs notation #Natural Generators|natural generator]], which is not the 9\15 5th but the 2\15 2nd. For 15edo, this is also known as porcupine notation. The 15edo porcupine genchain in both absolute and relative notation:
=== Alternate interval names ===
 
…Fx - Gx - A# - B# - C# - D# - E# - F# - G# --- A --- B --- C -- D --- E --- F --- G -- Ab -- Bb - Cb - Db - Eb - Fb - Gb - Abb - Bbb…
 
…A3 - A4 - A5 - A6 - A7 - A1 - A2 - M3 - M4 - M5 - M6 - P7 - P1 - P2 - m3 - m4 - m5 - m6 - d7 -- d8 - d2 - d3 -- d4 - d5 -- d6…
 
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.
{| class="wikitable center-all"
{| class="wikitable center-all"
|-
|-
Line 330: Line 325:
| D
| D
|}
|}
The 15edo porcupine genchain in both absolute and relative notation:
* …Fx - Gx - A# - B# - C# - D# - E# - F# - G# --- A --- B --- C -- D --- E --- F --- G -- Ab -- Bb - Cb - Db - Eb - Fb - Gb - Abb - Bbb…
* …A3 - A4 - A5 - A6 - A7 - A1 - A2 - M3 - M4 - M5 - M6 - P7 - P1 - P2 - m3 - m4 - m5 - m6 - d7 -- d8 - d2 - d3 -- d4 - d5 -- d6…


All 15edo chords can be named using ups and downs. Because many intervals have several names, many chords do too. Alterations are always enclosed in parentheses, additions never are. An up or down immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13).
All 15edo chords can be named using ups and downs. Because many intervals have several names, many chords do too. Alterations are always enclosed in parentheses, additions never are. An up or down immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13).
Line 1,008: Line 1,008:


; [[Jonathan Rabson]]
; [[Jonathan Rabson]]
* [http://www.youtube.com/watch?v=kQECU5ecCd4 Portrait of insects with 15-tone equal tempered guitar music]
* [https://www.youtube.com/watch?v=kQECU5ecCd4 Portrait of insects with 15-tone equal tempered guitar music]


; [[Carlo Serafini]]
; [[Carlo Serafini]]
* [http://youtu.be/6TyQ9kDm2fk ''Ode For Ada''] ([http://www.seraph.it/blog_files/5f07ebbdf8b4ba8ef067ed3d00a38357-158.html blog entry])
* [https://youtu.be/6TyQ9kDm2fk ''Ode For Ada''] ([http://www.seraph.it/blog_files/5f07ebbdf8b4ba8ef067ed3d00a38357-158.html blog entry])
* [http://www.seraph.it/dep/det/Sahara.mp3 ''Sahara''] ([http://www.seraph.it/blog_files/152ffe3684d7a11a91e5bc6c92a5bc9a-271.html blog entry])
* [http://www.seraph.it/dep/det/Sahara.mp3 ''Sahara''] ([http://www.seraph.it/blog_files/152ffe3684d7a11a91e5bc6c92a5bc9a-271.html blog entry])


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