15edo: Difference between revisions

<gallery widths=300px>

== Intervals ==

! Degree

! Cents

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

! [[Solfege]]<br>(porcupine-based)

! Porcupine[7]<br>(traditional)

! Porcupine[8]<br>(Greek)

! Blackwood<br>Decimal

! Audio

| 0

| 1/1

| do

| α

| E

| 1

| [[File:piano_0_1edo.mp3]]

| 1

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

| di

| D# / Eb

| 1# / 2b

| [[File:piano_1_15edo.mp3]]

| 2

| 160

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

| ru

| E

| β

| 2

| [[File:piano_2_15edo.mp3]]

| 3

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

| β/ χ\

| 3

| [[File:piano_1_5edo.mp3]]

| 4

| 320

| 6/5, 11/9

| me

| F

| G#

| 3# / 4b

| [[File:piano_4_15edo.mp3]]

| 5

| 5/4, 14/11

| mi

| F# / Gb

| χ/ δ\

| Ab

| 4

| [[File:piano_1_3edo.mp3]]

| 480

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

| fa

| G

| δ

| A

| 5

| [[File:piano_2_5edo.mp3]]

| 7

| 560

| 11/8, 7/5

| fu

| G#

| δ/ ε\

| A#

| 5# / 6b

| [[File:piano_7_15edo.mp3]]

| 8

| 640

| 16/11, 10/7

| su

| ε

| Bb

| 6

| [[File:piano_8_15edo.mp3]]

| 9

| 720

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

| sol

| A

| ε/ φ\

| B

| 7

| [[File:piano_3_5edo.mp3]]

| 10

| 800

| le

| A# / Bb

| φ

| B#

| 7# / 8b

| [[File:piano_2_3edo.mp3]]

| 11

| 880

| 5/3, 18/11

| la

| B

| φ/ γ\

| Db

| 8

| [[File:piano_11_15edo.mp3]]

| 12

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

| ta

| B# / Cb

| γ

| D

| [[File:piano_4_5edo.mp3]]

| 13

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

| tu

| C

| γ/ η\

| D#

| 9# / 0b

| [[File:piano_13_15edo.mp3]]

| 14

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

| ti

| C# / Db

| η

| Eb

| 0

| [[File:piano_14_15edo.mp3]]

| 1200

| 2/1

| do

| D

| [[File:piano_1_1edo.mp3]]

=== Alternate interval names ===

! step

! colspan="3" | [[Ups and downs notation|ups and downs]] notation ([[Enharmonic unisons in ups and downs notation|EUs]]: v³A1 and m2)<br>(partial list, e.g. M2/m3 is also A1 and d4)

| 0

| 0

| unison, min 2nd

| C# / D / Eb

| unison

| D

| 1

| 80

| ^1, ^m2

| up-unison, upminor 2nd

| ^C# / ^D / ^Eb

| aug unison, dim 2nd

| 2

| 160

| vM2

| vD# / vE / vF / vGb

| 3

| 240

| M2, m3

| major 2nd, minor 3rd

| D# / E / F / Gb

| aug 2nd, dim 3rd

| E# / Fb

| 4

| 320

| ^m3

| upminor 3rd

| ^D# / ^E / ^F / ^Gb

| minor 3rd

| F

| vM3

| downmajor 3rd

| major 3rd, dim 4th

| F# / Gb

| 6

| 480

| major 3rd, perfect 4th, dim 5th

| aug 3rd, minor 4th

| Fx / G

| 7

| up 4th, updim 5th

| ^F# / ^G / ^Ab

| major 4th, dim 5th

| G# / Abb

| 8

| 640

| vA4, v5

| downaug 4th, down 5th

| aug 4th, minor 5th

| Gx / Ab

| 9

| 720

| A4, P5, m6

| aug 4th, perfect 5th, minor 6th

| G# / A / Bb

| major 5th, dim 6th

| A / Bbb

| 10

| 800

| up 5th, upminor 6th

| aug 5th, minor 6th

| A# / Bb

| 11

| vA5, vM6

| downaug 5th, downmajor 6th

| vA# / vB / vC / vDb

| major 6th

| 12

| 960

| M6, m7

| major 6th, minor 7th

| A# / B / C / Db

| aug 6th, dim 7th

| 1040

| ^m7

| upminor 7th

| perfect 7th

| C

| 14

| downmajor 7th, down octave

| vC# / vD / vEb

| aug 7th, dim 8ve

| C# / Db

| 15

| 1200

| major 7th, octave

| C# / D / Eb

| 8ve

| D

* …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…

== Notation ==

=== Ups and downs notation ===

=== Sagittal notation ===

=== Blackwood Notation (Pentatonic) ===

| C

| ^C

| vD

| D

| ^D

| vꘙ

| ꘙ

| vG

| G

| ^G

| A

| ^A

| vC

| C

| P1

| ^1

| v2

| P2

| ^2

| v3

| P3

| ^3

| v4

| P4

| ^4

| v5

| P5

| v6

| P6

=== Blackwood Notation (Decatonic) ===

* '''Decimal Version:''' Using the nominals 1-0 (with 0 representing "10"), one of the three chains of 5edo is represented by the odd numbers, the second by the even numbers, and the third by numbers with accidentals (either odd numbers with sharps, or even numbers with flats).

* '''Guitar Version:''' 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.  

=== Porcupine Notation (Heptatonic) ===

! Cents

| 0

| D

| Augmented Unison / Minor Second

| Major Second

| E

| Augmented Second / Diminished Third

| E# / Fb

| F

| 400

| F# / Gb

| 480

| 560

| Augmented Fourth

| G#

| 640

| Diminished Fifth

| Ab

| 720

| Perfect Fifth

| A# / Bb

| 880

| 960

| Augmented Sixth / Diminished Seventh

| B# / Cb

| 1040

| Minor Seventh

| C

| 1120

| Major Seventh / Diminished Octave

| C# / Db

| Octave

=== Porcupine Notation (Octatonic) ===

! Cents

! Interval Name

| 0

| Zeroquill

| α - α

| 80

| Small Quill / Half Quill

| α - β\

| 160

| Quill

| α - β

| 240

| Small Diquill

| α - χ\

| 320

| Large Diquill

| α - χ

| 400

| Small Triquill

| α - δ\

| 480

| α - δ

| 560

| Small Fourquill

| α - ε\

| 640

| α - ε

| 720

| Small Fivequill

| α - φ\

| 800

| Large Fivequill

| α - φ

| 880

| Small Sixquill

| α - γ\

| 960

| Large Sixquill

| α - γ

| 1040

| Small Sevenquill

| α - η\

| 1120

| 1200

| Octoquill

| α - α

== Approximation to JI ==

=== 15-odd-limit interval mappings ===

=== Zeta peak index ===

! colspan="3" | Tuning

! colspan="2" | Closest edo

! colspan="2" | Integer limit

! ZPI

! Steps per octave

! Step size (cents)

! Height

! Integral

! Gap

! Edo

! Octave (cents)

! Consistent

! Distinct

| [[47zpi]]

| 79.7157343943591

| 5.050324

| 1.104057

| 14.918297

| 15edo

| 1195.73601591539

| 7

== Regular temperament properties ==

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal<br>8ve stretch (¢)

! colspan="2" | Tuning error

! [[TE error|Absolute]] (¢)

| 2.3.5

| {{mapping| 15 24 35 }}

| 4.63

| 5.81

| 2.3.5.7

| 28/27, 49/48, 126/125

| {{mapping| 15 24 35 42 }}

| −3.55

| 5.56

| 6.97

| 2.3.5.7.11

| 28/27, 49/48, 55/54, 77/75

| {{mapping| 15 24 35 42 52 }}

| −3.34

| 4.99

=== Errors by subgroup ===

|+ style="font-size: 105%; white-space: nowrap;" | Errors by subgroup

! Subgroup

! Mapping

! Adjusted error (¢)

| 2.3

| {{mapping| 15 24 }}

| 8.979801

| 2.5

| 6.826357

| 2.7

| {{mapping| 15 42 }}

| 4.418738

| 2.11

| {{mapping| 15 52 }}

| 4.336492

| 2.3.5

| {{mapping| 15 24 35 }}

| 10.742841

| 2.3.7

| {{mapping| 15 24 42 }}

| 17.481581

| 2.3.11

| {{mapping| 15 24 52 }}

| 16.831238

| 2.5.7

| {{mapping| 15 35 42 }}

| 10.509269

| 2.5.11

| {{mapping| 15 35 52 }}

| 2.7.11

| {{mapping| 15 42 52 }}

| 2.3.5.7

| {{mapping| 15 24 35 42 }}

| 15.603114

| 2.3.5.11

| {{mapping| 15 24 35 52 }}

| 14.693746

| 2.3.7.11

| {{mapping| 15 24 42 52 }}

| 18.660367

| {{mapping| 15 35 42 52 }}

| 11.462127

| 2.3.5.7.11

| {{mapping| 15 24 35 42 52 }}

| 17.258371

=== Uniform maps ===

=== Rank-2 temperaments ===

* [[List of edo-distinct 15et rank two temperaments]]

! Generator

! Temperaments

! Mos scales

| 1\15

| 21/20

| [[Nautilus]]<br>[[Valentine]]

|  

| 1

| 2\15

| 11/10

| [[Porcupine]] / [[opossum]]

| [[1L 6s]], [[7L 1s]]

| 1

| 4\15

| 6/5<br>77/64

| [[Cata]] / [[keemun]] / [[catalan]]<br>[[Orgone]] / [[superkleismic]]

| 1

| 7\15

| 7/5

| [[Progress]]<br>[[Parakangaroo]]

| [[2L 1s]], [[2L 3s]], [[2L 5s]], [[2L 7s]] [[2L 9s]], [[2L 11s]]

| [[3L 3s]], [[3L 6s]], [[3L 9s]]

| 3

| 2\15

| 7/6

| [[Triforce]]

| [[3L 3s]], [[6L 3s]]

| 5

| 16/15

| [[Blackwood]] / [[blacksmith]]

=== Commas ===

! [[Monzo]]

! [[Cent]]s

! [[Color name]]

! Name

| 3

| [[256/243]]

| 90.225

| Sawa

| Blackwood comma, Pythagorean limma

| 5

| <abbr title="254803968/244140625">(18 digits)</abbr>

| {{monzo| 20 5 -12 }}

| 74.01

| Saquadtrigu

| 5

| [[250/243]]

| {{monzo| 1 -5 3 }}

| Triyo

| Porcupine comma, maximal

| 5

| [[128/125]]

| {{monzo| 7 0 -3 }}

| 41.059

| Trigu

| 5

| [[15625/15552]]

| {{monzo| -6 -5 6 }}

| 8.107

| Tribiyo

| Kleisma

| 7

| [[28/27]]

| {{monzo| 2 -3 0 1 }}

| 62.961

| Zo

| 7

| [[1029/1000]]

| {{monzo| -3 1 -3 3 }}

| 49.492

| Keega

| 7

| [[49/48]]

| {{monzo| -4 -1 0 2 }}

| 35.697

| Zozo

| Semaphoresma, Slendro diesis

| 7

| [[64/63]]

| {{monzo| 6 -2 0 -1 }}

| 27.264

| Ru

| Archytas' comma, septimal comma

| 7

| {{monzo| -9 3 -3 4 }}

| 22.227

| Laquadzo-atrigu

| Squalentine comma

| 7

| [[875/864]]

| {{monzo| -5 -3 3 1 }}

| 21.902

| Zotriyo

| Keema

| 7

| [[126/125]]

| {{monzo| 1 2 -3 1 }}

| 7

| [[4000/3969]]

| {{monzo| 5 -4 3 -2 }}

| 13.469

| Octagar comma

| 7

| {{monzo| -10 1 0 3 }}

| 8.433

| Latrizo

| Gamelisma

| 7

| [[6144/6125]]

| {{monzo|  11 1 -3 -2 }}

| 5.362

| Saruru-atrigu

| Porwell comma

| 7

| <abbr title="250047/250000">(12 digits)</abbr>

| {{monzo| -4 6 -6 3 }}

| 0.325

| [[Landscape comma]]

| 11

| [[100/99]]

| {{monzo| 2 -2 2 0 -1 }}

| Luyoyo

| Ptolemisma

| [[121/120]]

| {{monzo| -3 -1 -1 0 2 }}

| 14.367

| Biyatisma

| 11

| [[176/175]]

| {{monzo| 4 0 -2 -1 1 }}

| Lorugugu

| 11

| [[65536/65219]]

| {{monzo| 16 0 0 -2 -3 }}

| 8.394

| Satrilu-aruru

| Orgonisma

| [[385/384]]

| {{monzo| -7 -1 1 1 1 }}

| Lozoyo

| 11

| [[441/440]]

| 3.930

| Luzozogu

| Werckisma

| 11

| [[4000/3993]]

| {{monzo| 5 -1 3 0 -3 }}

| 3.032

| Triluyo

| 11

| [[3025/3024]]

| {{monzo| -4 -3 2 -1 2 }}

| 0.572

| Loloruyoyo

| 13

| [[91/90]]

| {{monzo| -1 -2 -1 1 0 1 }}

| 19.130

| Superleap comma, biome comma

| 13

| [[676/675]]

| {{monzo| 2 -3 -2 0 0 2 }}

| 2.563

| Bithogu

| Island comma

<references/>

== Scales ==

=== MOS scales ===

* Augene[9] [[3L 6s]] (period = 5\15, gen = 1\15): 3 1 1 3 1 1 3 1 1

* Augene[12] [[3L 9s]] (period = 5\15, gen = 1\15): 2 1 1 1 2 1 1 1 2 1 1 1

* Triforce[6] [[3L 3s]] (period = 5\15, gen = 2\15): 3 2 3 2 3 2

* Triforce[9] [[6L 3s]] (period = 5\15, gen = 2\15): 2 1 2 2 1 2 2 1 2

* Porcupine[7] [[1L 6s]] (gen = 2\15): 3 2 2 2 2 2 2

* Porcupine[8] [[7L 1s]] (gen = 2\15): 2 1 2 2 2 2 2 2

* Hanson/Keemun/Orgone[7] [[4L 3s]] (gen = 4\15): 1 3 1 3 1 3 3

* Hanson/Keemun/Orgone[11] [[4L 7s]] (gen = 4\15): 1 2 1 1 2 1 1 2 1 2 1

* Blackwood[10] [[5L 5s]] (period = 3\15, gen = 1\15): 2 1 2 1 2 1 2 1 2 1 (Blackwood Decatonic)

=== Other scales ===

* [[Zarlino]]/Ptolemy diatonic, "just" major (Porcupine[7] 6|0 b4 #7): 3 2 1 3 2 3 1

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

* tetrachordal major: 3 2 1 3 3 2 1

* inverse of tetrachordal major, tetrachordal minor: 3 1 2 3 1 2 3

* "just"/[[The Pinetone System#Pinetone pentatonic|Pinetone major pentatonic]] (subset of Porcupine[7]): 3 2 4 2 4

* Porcupine bright major #7 (Porcupine harmonic major) - Porcupine[7] 6|0 #7: 3 2 2 2 2 3 1

* Porcupine bright major #6 #7 (Porcupine melodic major) - Porcupine[7] 6|0 #7: 3 2 2 2 3 2 1

* Porcupine bright minor #2 (Porcupine harmonic minor) - Porcupine[7] 4|2 #2: 3 1 3 2 2 2 2 (mode of bright major #7)

* Porcupine dark minor #2 (Porcupine melodic minor) - Porcupine[7] 3|3 #2: 3 1 2 3 2 2 2 (inverse of bright major #6 #7)

* Porcupine bright harmonic 11th mode - Porcupine[7] 6|0 b7: 3 2 2 2 2 1 3

* "just" harmonic minor: 3 1 2 3 1 4 1

* "just" harmonic major: 3 2 1 3 1 4 1

* Marvel double harmonic nonatonic (Augene[9] [[4M]]): 1 3 1 1 3 1 3 1 1, 1 1 3 1 3 1 1 3 1

* Marvel double harmonic decatonic: 1 3 1 1 2 1 1 3 1 1

* enharmonic trichord octave species: 1 5 3 1 5 , 5 1 3 5 1

* chromatic tetrachord octave species: 1 1 4 3 1 1 4, 4 1 1 3 4 1 1, 1 4 1 3 1 4 1

* [[5- to 10-tone scales in 47zpi]] (slightly stretched 15edo)

=== Horagrams ===

== Diagrams ==

=== Keyboard ===

=== Lumatone ===

== Music ==

== See also ==

* [[User:Unque/15edo Composition Theory|Unque's approach]]

* [[Metallic harmony]]: 15edo is one of the systems it is intended for.

== Further reading ==

=== Theory ===

* [[Darreg, Ivor]]. ''[http://tonalsoft.com/sonic-arts/darreg/dar35.htm 15-Tone Scale System]''. 1991. (Originally at [http://sonic-arts.org/darreg/dar35.htm], now broken)

* InTeAS. ''[https://web.archive.org/web/20110713044141/http://www.inteas.com/Penta01.htm The Pentadecaphonic System]'' (2001, archived)

=== 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 ==