15edo: Difference between revisions

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A recommended method to the notation of 15-edo by some is a system based on porcupine[8] in which eight nominals form the base diatonic scale. In this sense, the "quill" is the name given to the two step interval (160c) of 15-edo while the "small quill" (80c) is the chroma of 15-edo. This produces a very consistent notation for both porcupine[8] and Blackwood[10] and seems to work much better than attempting to put 15-edo into a seven nominal based framework.

~~=Images=~~

=Intervals=

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*based on treating 15-EDO 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 relative ~~and absolute~~ notation:

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:

...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...

~~...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...~~

{| class="wikitable"

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| style="text-align:center;" | ^1, ^m2

| style="text-align:center;" | up-unison, upminor 2nd

| style="text-align:center;" | C#^ / D^ / Eb^

| style="text-align:center;" | ^C# / ^D / ^Eb

| style="text-align:center;" | aug unison, dim 2nd

| style="text-align:center;" | D# / Eb

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| style="text-align:center;" | vM2

| style="text-align:center;" | downmajor 2nd

| style="text-align:center;" | D#v / Ev / Fv / ~~Gbv~~

| style="text-align:center;" | vD# / vE / vF / vGb

| style="text-align:center;" | perfect 2nd

| style="text-align:center;" | E

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| style="text-align:center;" | ^m3

| style="text-align:center;" | upminor 3rd

| style="text-align:center;" | D#^ / E^ / F^ / Gb^

| style="text-align:center;" | ^D# / ^E / ^F / ^Gb

| style="text-align:center;" | minor 3rd

| style="text-align:center;" | F

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| style="text-align:center;" | vM3

| style="text-align:center;" | downmajor 3rd

| style="text-align:center;" | F#v / Gv / ~~Abv~~

| style="text-align:center;" | vF# / vG / vAb

| style="text-align:center;" | major 3rd, dim 4th

| style="text-align:center;" | F# / Gb

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| style="text-align:center;" | ^4, ^d5

| style="text-align:center;" | up 4th, updim 5th

| style="text-align:center;" | F#^ / G^ / Ab^

| style="text-align:center;" | ^F# / ^G / ^Ab

| style="text-align:center;" | major 4th, dim 5th

| style="text-align:center;" | G# / Abb

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| style="text-align:center;" | vA4, v5

| style="text-align:center;" | downaug 4th, down 5th

| style="text-align:center;" | G#v / Av / ~~Bbv~~

| style="text-align:center;" | vG# / vA / vBb

| style="text-align:center;" | aug 4th, minor 5th

| style="text-align:center;" | Gx / Ab

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| style="text-align:center;" | ^5, ^m6

| style="text-align:center;" | up 5th, upminor 6th

| style="text-align:center;" | G#^ / A^ / Bb^

| style="text-align:center;" | ^G# / ^A / ^Bb

| style="text-align:center;" | aug 5th, minor 6th

| style="text-align:center;" | A# / Bb

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| style="text-align:center;" | vA5, vM6

| style="text-align:center;" | downaug 5th, downmajor 6th

| style="text-align:center;" | A#v / Bv / Cv / ~~Dbv~~

| style="text-align:center;" | vA# / vB / vC / vDb

| style="text-align:center;" | major 6th

| style="text-align:center;" | B

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| style="text-align:center;" | ^m7

| style="text-align:center;" | upminor 7th

| style="text-align:center;" | A#^ / B^ / C^ / Db^

| style="text-align:center;" | ^A# / ^B / ^C / ^Db

| style="text-align:center;" | perfect 7th

| style="text-align:center;" | C

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| style="text-align:center;" | vM7, v8

| style="text-align:center;" | downmajor 7th, down octave

| style="text-align:center;" | C#v / Dv / ~~Ebv~~

| style="text-align:center;" | vC# / vD / vEb

| style="text-align:center;" | aug 7th, dim 8ve

| style="text-align:center;" | C# / Db

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0-3-9 = D E A = D2 = "D sus 2", or D F A = Dm = "D minor" (approximate 6:7:9)

0-4-9 = D F^ A = D.^m = "D upminor" (approximate 10:12:15)

0-4-9 = D ^F A = D^m = "D upminor" (approximate 10:12:15)

0-5-9 = D F#v A = ~~D.v~~ = "D ~~dot~~ down" or "D downmajor" (approximate 4:5:6)

0-5-9 = D vF# A = Dv = "D down" or "D downmajor" (approximate 4:5:6)

0-6-9 = D G A = D4, or D F# A = D = "D" or "D major" (approximate 14:18:21)

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0-3-9-12 = D F A C = Dm7 = "D minor seven", or D F A B = Dm6 = "D minor six"

0-4-9-12 = D F^ A C = Dm7(^3) = "D minor seven up-three", or D F^ A B = Dm6(^3) = "D minor six up-three"

0-4-9-12 = D ^F A C = Dm7(^3) = "D minor seven up-three", or D ^F A B = Dm6(^3) = "D minor six up-three"

0-5-9-12 = D F#v A C = D7(v3) = "D seven down-three", or D F#v A B = D6(v3) = "D six down-three"

0-5-9-12 = D vF# A C = D7(v3) = "D seven down-three", or D vF# A B = D6(v3) = "D six down-three"

0-6-9-12 = D F# A C = D7 = "D seven", or D F# A B = D6 = "D six"

0-5-9-14 = D F#v A C#v = ~~D.vM7~~ = "D downmajor seven"

0-5-9-14 = D vF# A vC# = DvM7 = "D downmajor seven"

0-4-9-13 = D F^ A C^ = D.^m7 = "D ~~dot up minor~~-seven", or D F^ A B^ = D.^m6 = "D ~~dot up minor~~-six"

0-4-9-13 = D ^F A ^C = D^m7 = "D upminor-seven", or D ^F A ^B = D^m6 = "D upminor-six"

For a more complete list, see [[~~Ups_and_Downs_Notation~~#Chord ~~names in other EDOs~~|Ups and Downs Notation - Chord ~~names in other EDOs~~]].

For a more complete list, see [[Ups and Downs Notation#Chords and Chord Progressions|Ups and Downs Notation - Chords and Chord Progressions]].

15-EDO offers some minor improvements over 12-TET in ratios of 5 (particularly in 6/5 and 5/3), and has a much better approximation to the 7th and 11th harmonics, but its approximation to the 3rd harmonic is rather off. However, the particular way in which this approximation is off is as much a feature as it is a bug, for it allows the construction of a 5L5s MOS scale wherein every note of the scale can serve as a root for a 7-limit otonal or utonal tetrad, as well as either a 5-limit major or minor 7th chord. This is known as Blackwood temperament, named after Easley Blackwood, Jr., who is the first to document its existence. It has also been written on extensively by [[IgliashonJones|Igliashon Jones]] in the paper "[http://www.cityoftheasleep.com/etc/5nEDOs.pdf Five is Not an Odd Number]". For an in-depth treatment of harmony in 15-edo based on this temperament (and its 7- and 11-limit extensions), see [[Harmony_in_15edo_Blacksmith|Harmony in 15edo Blacksmith[10]]].

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{| class="wikitable"

|-

! | ~~Rational~~

! | [[Ratios|Ratio]]

! | Monzo

! | [[Monzo]]

! | ~~Size (~~Cents)

! | [[Cent|Cents]]

![[Color notation/Temperament Names|Color Names]]

! | Name 1

! | Name 2

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| |<nowiki> | 8 -5 </nowiki>>

| style="text-align:right;" | 90.22

| style="text-align:center;" |Sawa

| style="text-align:center;" | Limma

| style="text-align:center;" | Pythagorean Minor 2nd

| style="text-align:center;" |

|-

~~| style="text-align:center;" | 28/27~~

~~| |<nowiki> | 2 -3 0 1 </nowiki>>~~

~~| style="text-align:right;" | 62.96~~

~~| style="text-align:center;" | Septimal Third Tone~~

~~| style="text-align:center;" | Small Septimal Chroma~~

~~| |~~

|-

| style="text-align:center;" | 250/243

| |<nowiki> | 1 -5 3 </nowiki>>

| style="text-align:right;" | 49.17

| style="text-align:center;" |Triyo

| style="text-align:center;" | Maximal Diesis

| style="text-align:center;" | Porcupine Comma

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| |<nowiki> | 7 0 -3 </nowiki>>

| style="text-align:right;" | 41.06

| style="text-align:center;" |Trigu

| style="text-align:center;" | Diesis

| style="text-align:center;" | Augmented Comma

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| |<nowiki> | -6 -5 6 </nowiki>>

| style="text-align:right;" | 8.11

| style="text-align:center;" |Tribiyo

| style="text-align:center;" | Kleisma

| style="text-align:center;" | Semicomma Majeur

| style="text-align:center;" |

|-

| style="text-align:center;" | 28/27

| |<nowiki> | 2 -3 0 1 </nowiki>>

| style="text-align:right;" | 62.96

| style="text-align:center;" |Zo

| style="text-align:center;" | Septimal Third Tone

| style="text-align:center;" | Small Septimal Chroma

| |

|-

| style="text-align:center;" | 1029/1000

| |<nowiki> | -3 1 -3 3 </nowiki>>

| style="text-align:right;" | 49.49

| style="text-align:center;" |Trizogu

| style="text-align:center;" | Keega

| style="text-align:center;" |

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| |<nowiki> | -4 -1 0 2 </nowiki>>

| style="text-align:right;" | 35.70

| style="text-align:center;" |Zozo

| style="text-align:center;" | Slendro Diesis

| style="text-align:center;" |

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| |<nowiki> | 6 -2 0 -1 </nowiki>>

| style="text-align:right;" | 27.26

| style="text-align:center;" |Ru

| style="text-align:center;" | Septimal Comma

| style="text-align:center;" | Archytas' Comma

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| |<nowiki> | -9 3 -3 4 </nowiki>>

| style="text-align:right;" | 22.23

| style="text-align:center;" |Laquadzo-atrigu

| style="text-align:center;" | Squalentine

| style="text-align:center;" |

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| |<nowiki> | -5 -3 3 1 </nowiki>>

| style="text-align:right;" | 21.90

| style="text-align:center;" |Zotriyo

| style="text-align:center;" | Keema

| style="text-align:center;" |

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| |<nowiki> | 1 2 -3 1 </nowiki>>

| style="text-align:right;" | 13.79

| style="text-align:center;" |Zotrigu

| style="text-align:center;" | Septimal Semicomma

| style="text-align:center;" | Starling Comma

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| |<nowiki> | 5 -4 3 -2 </nowiki>>

| style="text-align:right;" | 13.47

| style="text-align:center;" |Rurutriyo

| style="text-align:center;" | Octagar

| style="text-align:center;" |

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| |<nowiki> | -10 1 0 3 </nowiki>>

| style="text-align:right;" | 8.43

| style="text-align:center;" |Latrizo

| style="text-align:center;" | Gamelisma

| style="text-align:center;" |

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| |<nowiki> | 11 1 -3 -2 </nowiki>>

| style="text-align:right;" | 5.36

| style="text-align:center;" |Saruru-atrigu

| style="text-align:center;" | Porwell

| style="text-align:center;" |

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| |<nowiki> | -4 6 -6 3 </nowiki>>

| style="text-align:right;" | 0.33

| style="text-align:center;" |Trizogugu

| style="text-align:center;" | Landscape Comma

| style="text-align:center;" |

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| |<nowiki> | 2 -2 2 0 -1 </nowiki>>

| style="text-align:right;" | 17.40

| style="text-align:center;" |Luyoyo

| style="text-align:center;" | Ptolemisma

| style="text-align:center;" |

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| |<nowiki> | -3 -1 -1 0 2 </nowiki>>

| style="text-align:right;" | 14.37

| style="text-align:center;" |Lologu

| style="text-align:center;" | Biyatisma

| style="text-align:center;" |

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| |<nowiki> | 4 0 -2 -1 1 </nowiki>>

| style="text-align:right;" | 9.86

| style="text-align:center;" |Lorugugu

| style="text-align:center;" | Valinorsma

| style="text-align:center;" |

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| |<nowiki> | 16 0 0 -2 -3 </nowiki>>

| style="text-align:right;" | 8.39

| style="text-align:center;" |Satrilu-aruru

| style="text-align:center;" | Orgonisma

| style="text-align:center;" |

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| |<nowiki> | -7 -1 1 1 1 </nowiki>>

| style="text-align:right;" | 4.50

| style="text-align:center;" |Lozoyo

| style="text-align:center;" | Keenanisma

| style="text-align:center;" |

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| |<nowiki> | -3 2 -1 2 -1 </nowiki>>

| style="text-align:right;" | 3.93

| style="text-align:center;" |Luzozogu

| style="text-align:center;" | Werckisma

| style="text-align:center;" |

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| |<nowiki> | 5 -1 3 0 -3 </nowiki>>

| style="text-align:right;" | 3.03

| style="text-align:center;" |Triluyo

| style="text-align:center;" | Wizardharry

| style="text-align:center;" |

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| |<nowiki> | -4 -3 2 -1 2 </nowiki>>

| style="text-align:right;" | 0.57

| style="text-align:center;" |Loloruyoyo

| style="text-align:center;" | Lehmerisma

| style="text-align:center;" |

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| |<nowiki> | -1 -2 -1 1 0 1 </nowiki>>

| style="text-align:right;" | 19.13

| style="text-align:center;" |Thozogu

| style="text-align:center;" | Superleap

| style="text-align:center;" |

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| |<nowiki> | 2 -3 -2 0 0 2 </nowiki>>

| style="text-align:right;" | 2.56

| style="text-align:center;" |Bithogu

| style="text-align:center;" | Parizeksma

| style="text-align:center;" |

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=Practical Theory / Books=

[http://www.swordguitars.com Sword, Ronald. "Pendecaphonic Scales for Guitar" IAAA Press, UK-USA. First Ed: June 2009.] - A large repository of all known scales and temperament families in the 15-edo system. 300+ examples /w chord-scale progressions.

=Images=

=Compositions=

@@ Line 22: / Line 22: @@
 A recommended method to the notation of 15-edo by some is a system based on porcupine[8] in which eight nominals form the base diatonic scale. In this sense, the "quill" is the name given to the two step interval (160c) of 15-edo while the "small quill" (80c) is the chroma of 15-edo. This produces a very consistent notation for both porcupine[8] and Blackwood[10] and seems to work much better than attempting to put 15-edo into a seven nominal based framework.
-=Images=
-[[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]]
 =Intervals=
@@ Line 196: / Line 192: @@
 *based on treating 15-EDO 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 relative and absolute notation:
+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:
+...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...
-...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...
 {| class="wikitable"
@@ Line 231: / Line 227: @@
 | style="text-align:center;" | ^1, ^m2
 | style="text-align:center;" | up-unison, upminor 2nd
-| style="text-align:center;" | C#^ / D^ / Eb^
+| style="text-align:center;" | ^C# / ^D / ^Eb
 | style="text-align:center;" | aug unison, dim 2nd
 | style="text-align:center;" | D# / Eb
@@ Line 239: / Line 235: @@
 | style="text-align:center;" | vM2
 | style="text-align:center;" | downmajor 2nd
-| style="text-align:center;" | D#v / Ev / Fv / Gbv
+| style="text-align:center;" | vD# / vE / vF / vGb
 | style="text-align:center;" | perfect 2nd
 | style="text-align:center;" | E
@@ Line 255: / Line 251: @@
 | style="text-align:center;" | ^m3
 | style="text-align:center;" | upminor 3rd
-| style="text-align:center;" | D#^ / E^ / F^ / Gb^
+| style="text-align:center;" | ^D# / ^E / ^F / ^Gb
 | style="text-align:center;" | minor 3rd
 | style="text-align:center;" | F
@@ Line 263: / Line 259: @@
 | style="text-align:center;" | vM3
 | style="text-align:center;" | downmajor 3rd
-| style="text-align:center;" | F#v / Gv / Abv
+| style="text-align:center;" | vF# / vG / vAb
 | style="text-align:center;" | major 3rd, dim 4th
 | style="text-align:center;" | F# / Gb
@@ Line 279: / Line 275: @@
 | style="text-align:center;" | ^4, ^d5
 | style="text-align:center;" | up 4th, updim 5th
-| style="text-align:center;" | F#^ / G^ / Ab^
+| style="text-align:center;" | ^F# / ^G / ^Ab
 | style="text-align:center;" | major 4th, dim 5th
 | style="text-align:center;" | G# / Abb
@@ Line 287: / Line 283: @@
 | style="text-align:center;" | vA4, v5
 | style="text-align:center;" | downaug 4th, down 5th
-| style="text-align:center;" | G#v / Av / Bbv
+| style="text-align:center;" | vG# / vA / vBb
 | style="text-align:center;" | aug 4th, minor 5th
 | style="text-align:center;" | Gx / Ab
@@ Line 303: / Line 299: @@
 | style="text-align:center;" | ^5, ^m6
 | style="text-align:center;" | up 5th, upminor 6th
-| style="text-align:center;" | G#^ / A^ / Bb^
+| style="text-align:center;" | ^G# / ^A / ^Bb
 | style="text-align:center;" | aug 5th, minor 6th
 | style="text-align:center;" | A# / Bb
@@ Line 311: / Line 307: @@
 | style="text-align:center;" | vA5, vM6
 | style="text-align:center;" | downaug 5th, downmajor 6th
-| style="text-align:center;" | A#v / Bv / Cv / Dbv
+| style="text-align:center;" | vA# / vB / vC / vDb
 | style="text-align:center;" | major 6th
 | style="text-align:center;" | B
@@ Line 327: / Line 323: @@
 | style="text-align:center;" | ^m7
 | style="text-align:center;" | upminor 7th
-| style="text-align:center;" | A#^ / B^ / C^ / Db^
+| style="text-align:center;" | ^A# / ^B / ^C / ^Db
 | style="text-align:center;" | perfect 7th
 | style="text-align:center;" | C
@@ Line 335: / Line 331: @@
 | style="text-align:center;" | vM7, v8
 | style="text-align:center;" | downmajor 7th, down octave
-| style="text-align:center;" | C#v / Dv / Ebv
+| style="text-align:center;" | vC# / vD / vEb
 | style="text-align:center;" | aug 7th, dim 8ve
 | style="text-align:center;" | C# / Db
@@ Line 351: / Line 347: @@
 -3-9 = D E A = D2 = "D sus 2", or D F A = Dm = "D minor" (approximate 6:7:9)
--4-9 = D F^ A = D.^m = "D upminor" (approximate 10:12:15)
+-4-9 = D ^F A = D^m = "D upminor" (approximate 10:12:15)
--5-9 = D F#v A = D.v = "D dot down" or "D downmajor" (approximate 4:5:6)
+-5-9 = D vF# A = Dv = "D down" or "D downmajor" (approximate 4:5:6)
 -6-9 = D G A = D4, or D F# A = D = "D" or "D major" (approximate 14:18:21)
@@ Line 359: / Line 355: @@
 -3-9-12 = D F A C = Dm7 = "D minor seven", or D F A B = Dm6 = "D minor six"
--4-9-12 = D F^ A C = Dm7(^3) = "D minor seven up-three", or D F^ A B = Dm6(^3) = "D minor six up-three"
+-4-9-12 = D ^F A C = Dm7(^3) = "D minor seven up-three", or D ^F A B = Dm6(^3) = "D minor six up-three"
--5-9-12 = D F#v A C = D7(v3) = "D seven down-three", or D F#v A B = D6(v3) = "D six down-three"
+-5-9-12 = D vF# A C = D7(v3) = "D seven down-three", or D vF# A B = D6(v3) = "D six down-three"
 -6-9-12 = D F# A C = D7 = "D seven", or D F# A B = D6 = "D six"
--5-9-14 = D F#v A C#v = D.vM7 = "D downmajor seven"
+-5-9-14 = D vF# A vC# = DvM7 = "D downmajor seven"
--4-9-13 = D F^ A C^ = D.^m7 = "D dot up minor-seven", or D F^ A B^ = D.^m6 = "D dot up minor-six"
+-4-9-13 = D ^F A ^C = D^m7 = "D upminor-seven", or D ^F A ^B = D^m6 = "D upminor-six"
-For a more complete list, see [[Ups_and_Downs_Notation#Chord names in other EDOs|Ups and Downs Notation - Chord names in other EDOs]].
+For a more complete list, see [[Ups and Downs Notation#Chords and Chord Progressions|Ups and Downs Notation - Chords and Chord Progressions]].
 -EDO offers some minor improvements over 12-TET in ratios of 5 (particularly in 6/5 and 5/3), and has a much better approximation to the 7th and 11th harmonics, but its approximation to the 3rd harmonic is rather off. However, the particular way in which this approximation is off is as much a feature as it is a bug, for it allows the construction of a 5L5s MOS scale wherein every note of the scale can serve as a root for a 7-limit otonal or utonal tetrad, as well as either a 5-limit major or minor 7th chord. This is known as Blackwood temperament, named after Easley Blackwood, Jr., who is the first to document its existence. It has also been written on extensively by [[IgliashonJones|Igliashon Jones]] in the paper "[http://www.cityoftheasleep.com/etc/5nEDOs.pdf Five is Not an Odd Number]". For an in-depth treatment of harmony in 15-edo based on this temperament (and its 7- and 11-limit extensions), see [[Harmony_in_15edo_Blacksmith|Harmony in 15edo Blacksmith[10]]].
@@ Line 676: / Line 672: @@
 {| class="wikitable"
 |-
-! | Rational
+! | [[Ratios|Ratio]]
-! | Monzo
+! | [[Monzo]]
-! | Size (Cents)
+! | [[Cent|Cents]]
+![[Color notation/Temperament Names|Color Names]]
 ! | Name 1
 ! | Name 2
@@ Line 686: / Line 683: @@
 | |<nowiki> | 8 -5 </nowiki>&gt;
 | style="text-align:right;" | 90.22
+| style="text-align:center;" |Sawa
 | style="text-align:center;" | Limma
 | style="text-align:center;" | Pythagorean Minor 2nd
 | style="text-align:center;" |
-|-
-| style="text-align:center;" | 28/27
-| |<nowiki> | 2 -3 0 1 </nowiki>&gt;
-| style="text-align:right;" | 62.96
-| style="text-align:center;" | Septimal Third Tone
-| style="text-align:center;" | Small Septimal Chroma
-| |
 |-
 | style="text-align:center;" | 250/243
 | |<nowiki> | 1 -5 3 </nowiki>&gt;
 | style="text-align:right;" | 49.17
+| style="text-align:center;" |Triyo
 | style="text-align:center;" | Maximal Diesis
 | style="text-align:center;" | Porcupine Comma
@@ Line 707: / Line 699: @@
 | |<nowiki> | 7 0 -3 </nowiki>&gt;
 | style="text-align:right;" | 41.06
+| style="text-align:center;" |Trigu
 | style="text-align:center;" | Diesis
 | style="text-align:center;" | Augmented Comma
@@ Line 714: / Line 707: @@
 | |<nowiki> | -6 -5 6 </nowiki>&gt;
 | style="text-align:right;" | 8.11
+| style="text-align:center;" |Tribiyo
 | style="text-align:center;" | Kleisma
 | style="text-align:center;" | Semicomma Majeur
 | style="text-align:center;" |
+|-
+| style="text-align:center;" | 28/27
+| |<nowiki> | 2 -3 0 1 </nowiki>&gt;
+| style="text-align:right;" | 62.96
+| style="text-align:center;" |Zo
+| style="text-align:center;" | Septimal Third Tone
+| style="text-align:center;" | Small Septimal Chroma
+| |
 |-
 | style="text-align:center;" | 1029/1000
 | |<nowiki> | -3 1 -3 3 </nowiki>&gt;
 | style="text-align:right;" | 49.49
+| style="text-align:center;" |Trizogu
 | style="text-align:center;" | Keega
 | style="text-align:center;" |
@@ Line 728: / Line 731: @@
 | |<nowiki> | -4 -1 0 2 </nowiki>&gt;
 | style="text-align:right;" | 35.70
+| style="text-align:center;" |Zozo
 | style="text-align:center;" | Slendro Diesis
 | style="text-align:center;" |
@@ Line 735: / Line 739: @@
 | |<nowiki> | 6 -2 0 -1 </nowiki>&gt;
 | style="text-align:right;" | 27.26
+| style="text-align:center;" |Ru
 | style="text-align:center;" | Septimal Comma
 | style="text-align:center;" | Archytas' Comma
@@ Line 742: / Line 747: @@
 | |<nowiki> | -9 3 -3 4 </nowiki>&gt;
 | style="text-align:right;" | 22.23
+| style="text-align:center;" |Laquadzo-atrigu
 | style="text-align:center;" | Squalentine
 | style="text-align:center;" |
@@ Line 749: / Line 755: @@
 | |<nowiki> | -5 -3 3 1 </nowiki>&gt;
 | style="text-align:right;" | 21.90
+| style="text-align:center;" |Zotriyo
 | style="text-align:center;" | Keema
 | style="text-align:center;" |
@@ Line 756: / Line 763: @@
 | |<nowiki> | 1 2 -3 1 </nowiki>&gt;
 | style="text-align:right;" | 13.79
+| style="text-align:center;" |Zotrigu
 | style="text-align:center;" | Septimal Semicomma
 | style="text-align:center;" | Starling Comma
@@ Line 763: / Line 771: @@
 | |<nowiki> | 5 -4 3 -2 </nowiki>&gt;
 | style="text-align:right;" | 13.47
+| style="text-align:center;" |Rurutriyo
 | style="text-align:center;" | Octagar
 | style="text-align:center;" |
@@ Line 770: / Line 779: @@
 | |<nowiki> | -10 1 0 3 </nowiki>&gt;
 | style="text-align:right;" | 8.43
+| style="text-align:center;" |Latrizo
 | style="text-align:center;" | Gamelisma
 | style="text-align:center;" |
@@ Line 777: / Line 787: @@
 | |<nowiki> | 11 1 -3 -2 </nowiki>&gt;
 | style="text-align:right;" | 5.36
+| style="text-align:center;" |Saruru-atrigu
 | style="text-align:center;" | Porwell
 | style="text-align:center;" |
@@ Line 784: / Line 795: @@
 | |<nowiki> | -4 6 -6 3 </nowiki>&gt;
 | style="text-align:right;" | 0.33
+| style="text-align:center;" |Trizogugu
 | style="text-align:center;" | Landscape Comma
 | style="text-align:center;" |
@@ Line 791: / Line 803: @@
 | |<nowiki> | 2 -2 2 0 -1 </nowiki>&gt;
 | style="text-align:right;" | 17.40
+| style="text-align:center;" |Luyoyo
 | style="text-align:center;" | Ptolemisma
 | style="text-align:center;" |
@@ Line 798: / Line 811: @@
 | |<nowiki> | -3 -1 -1 0 2 </nowiki>&gt;
 | style="text-align:right;" | 14.37
+| style="text-align:center;" |Lologu
 | style="text-align:center;" | Biyatisma
 | style="text-align:center;" |
@@ Line 805: / Line 819: @@
 | |<nowiki> | 4 0 -2 -1 1 </nowiki>&gt;
 | style="text-align:right;" | 9.86
+| style="text-align:center;" |Lorugugu
 | style="text-align:center;" | Valinorsma
 | style="text-align:center;" |
@@ Line 812: / Line 827: @@
 | |<nowiki> | 16 0 0 -2 -3 </nowiki>&gt;
 | style="text-align:right;" | 8.39
+| style="text-align:center;" |Satrilu-aruru
 | style="text-align:center;" | Orgonisma
 | style="text-align:center;" |
@@ Line 819: / Line 835: @@
 | |<nowiki> | -7 -1 1 1 1 </nowiki>&gt;
 | style="text-align:right;" | 4.50
+| style="text-align:center;" |Lozoyo
 | style="text-align:center;" | Keenanisma
 | style="text-align:center;" |
@@ Line 826: / Line 843: @@
 | |<nowiki> | -3 2 -1 2 -1 </nowiki>&gt;
 | style="text-align:right;" | 3.93
+| style="text-align:center;" |Luzozogu
 | style="text-align:center;" | Werckisma
 | style="text-align:center;" |
@@ Line 833: / Line 851: @@
 | |<nowiki> | 5 -1 3 0 -3 </nowiki>&gt;
 | style="text-align:right;" | 3.03
+| style="text-align:center;" |Triluyo
 | style="text-align:center;" | Wizardharry
 | style="text-align:center;" |
@@ Line 840: / Line 859: @@
 | |<nowiki> | -4 -3 2 -1 2 </nowiki>&gt;
 | style="text-align:right;" | 0.57
+| style="text-align:center;" |Loloruyoyo
 | style="text-align:center;" | Lehmerisma
 | style="text-align:center;" |
@@ Line 847: / Line 867: @@
 | |<nowiki> | -1 -2 -1 1 0 1 </nowiki>&gt;
 | style="text-align:right;" | 19.13
+| style="text-align:center;" |Thozogu
 | style="text-align:center;" | Superleap
 | style="text-align:center;" |
@@ Line 854: / Line 875: @@
 | |<nowiki> | 2 -3 -2 0 0 2 </nowiki>&gt;
 | style="text-align:right;" | 2.56
+| style="text-align:center;" |Bithogu
 | style="text-align:center;" | Parizeksma
 | style="text-align:center;" |
@@ Line 868: / Line 890: @@
 =Practical Theory / Books=
 [http://www.swordguitars.com Sword, Ronald. "Pendecaphonic Scales for Guitar" IAAA Press, UK-USA. First Ed: June 2009.] - A large repository of all known scales and temperament families in the 15-edo system. 300+ examples /w chord-scale progressions.
+=Images=
+[[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]]
 =Compositions=