15edo

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15 tone keyboard.png

Summary

15 Equal or 15 EDO is a tuning which divides the octave into 15 equally spaced pitches. It can be thought of as three sets of 5-EDO which do not connect by fifths. The fifth at 720 cents is quite wide yet still useable as a perfect fifth. Some would describe the fifth as more shimmery and pungent than anything closer to a just 3/2. The perfect fifth of 15 EDO returns to the octave if stacked five times which is radically different than a meantone system.

From Wikipedia:

"In music, 15 equal temperament, called 15-TET, 15-EDO, or 15-ET, is the tempered scale derived by dividing the octave into 15 equal steps. Each step represents a frequency ratio of 2^(1/15), or 80 cents. Because 15 factors into 3 times 5, it can be seen as being made up of three scales of 5 equal divisions of the octave (or five scales of 3edo)."

15-edo can be seen as a 7-limit temperament because of its ability to approximate some septimal intervals, but it also contains some fairly obvious approximations to 11-limit intervals, so it can reasonably be described as an 11-limit temperament; however, due to its rather distant approximation of the 3rd harmonic (and therefore the 9th harmonic as well), those seeking to approximate JI with 15-edo would be best advised to avoid chords requiring those harmonics (or to at least treat them with sensitivity). 15-edo is also notable for being the smallest edo with recognizable, distinct representations of 5-odd limit intervals (3/2, 5/4, 6/5, and their octave inverses) that has a positive syntonic comma.

In the 15-edo system, major thirds cannot be divided perfectly into two, and coupled with the lack of a standard tritone, this tuning at first can be disorienting. However, 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 superior harmony and xenharmony, a manageable number of tones, and the sonic fingerprint of multiples of 5-edo.

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

15edo wheel.png15edo wheel 02.png15edo wheel 03.png

Intervals

Degree Cents Solfege

(porcupine-based)

Porcupine[8]

(Greek)

Blackwood

"guitar notation"

Porcupine[7]

(traditional)

Blackwood

Decimal

Approximate Ratios*
0 0 do α E G 1 1/1
1 80 di α/ β\ E# G# / Abb 1# / 2b 25/24, 21/20, 16/15
2 160 ru β Gb Gx / Ab 2 11/10, 12/11, 10/9
3 240 re β/ χ\ G A 3 8/7, 7/6, 9/8
4 320 me χ G# A# / Bb 3# / 4b 6/5, 11/9
5 400 mi χ/ δ\ Ab B 4 5/4, 14/11
6 480 fa ð A B# / Cb 5 4/3, 9/7, 21/16
7 560 fu δ/ ε\ A# C 5# / 6b 11/8, 7/5
8 640 su ε Bb C# / Db 6 16/11, 10/7
9 720 sol ε/ φ\ B D 7 3/2, 14/9, 32/21
10 800 le φ B# D# / Eb 7# / 8b 8/5, 11/7
11 880 la φ/ γ\ Db E 8 5/3, 18/11
12 960 ta γ D E# / Fb 9 7/4, 12/7, 16/9
13 1040 tu γ/ η\ D# F 9# / 0b 20/11, 11/6, 9/5
14 1120 ti η Eb F# /Gb 0 48/25, 40/21, 15/8
15 1200 do α E G 1 2/1
  • based on treating 15-EDO as an 11-limit temperament; other approaches are possible

In ups and downs notation, which is fifth-generated, every 15edo note has at least three names. 15edo can also be notated using the 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:

...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...

step cents ups and downs relative notation

(partial list, e.g. M2 is also A1 and d4)

ups and downs

absolute notation

porcupine

relative notation

porcupine

absolute notation

0 P1, m2 unison, min 2nd C# / D / Eb unison D
1 80 ^1, ^m2 up-unison, upminor 2nd C#^ / D^ / Eb^ aug unison, dim 2nd D# / Eb
2 160 vM2 downmajor 2nd D#v / Ev / Fv / Gbv perfect 2nd E
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
5 400 vM3 downmajor 3rd F#v / Gv / Abv major 3rd, dim 4th F# / Gb
6 480 M3, P4, d5 major 3rd, perfect 4th, dim 5th F# / G / Ab aug 3rd, minor 4th Fx / G
7 560 ^4, ^d5 up 4th, updim 5th F#^ / G^ / Ab^ major 4th, dim 5th G# / Abb
8 640 vA4, v5 downaug 4th, down 5th G#v / Av / Bbv 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 ^5, ^m6 up 5th, upminor 6th G#^ / A^ / Bb^ aug 5th, minor 6th A# / Bb
11 880 vA5, vM6 downaug 5th, downmajor 6th A#v / Bv / Cv / Dbv major 6th B
12 960 M6, m7 major 6th, minor 7th A# / B / C / Db aug 6th, dim 7th B# / Cb
13 1040 ^m7 upminor 7th A#^ / B^ / C^ / Db^ perfect 7th C
14 1120 vM7, v8 downmajor 7th, down octave C#v / Dv / Ebv aug 7th, dim 8ve C# / Db
15 1200 M7, P8 major 7th, octave C# / D / Eb 8ve D

All 15edo chords can be named using ups and downs. Because many intervals have several names, many chords do too.

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)

0-5-9 = D F#v A = D.v = "D dot 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)

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"

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

0-6-9-12 = 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"

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

For a more complete list, see Ups and Downs Notation - Chord names in other EDOs.

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 Igliashon Jones in the paper "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[10].

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15ed2-001.svg

Selected just intervals by error

The following table shows how some prominent just intervals are represented in 15edo (ordered by absolute error).

Best direct mapping, even if inconsistent

Interval, complement Error (abs., in cents)
18/13, 13/9 3.382
6/5, 5/3 4.359
11/10, 20/11 5.004
15/13, 26/15 7.741
11/8, 16/11 8.682
8/7, 7/4 8.826
12/11, 11/6 9.363
5/4, 8/5 13.686
14/11, 11/7 17.508
4/3, 3/2 18.045
13/12, 24/13 21.427
10/9, 9/5 22.404
7/5, 10/7 22.512
15/11, 22/15 23.049
13/10, 20/13 25.786
7/6, 12/7 26.871
11/9, 18/11 27.408
13/11, 22/13 30.790
14/13, 13/7 31.702
16/15, 15/8 31.731
9/7, 14/9 35.084
9/8, 16/9 36.090
15/14, 28/15 39.443
16/13, 13/8 39.472


Patent val mapping

Interval, complement Error (abs., in cents)
18/13, 13/9 3.382
6/5, 5/3 4.359
11/10, 20/11 5.004
15/13, 26/15 7.741
11/8, 16/11 8.682
8/7, 7/4 8.826
12/11, 11/6 9.363
5/4, 8/5 13.686
14/11, 11/7 17.508
4/3, 3/2 18.045
13/12, 24/13 21.427
10/9, 9/5 22.404
7/5, 10/7 22.512
15/11, 22/15 23.049
13/10, 20/13 25.786
7/6, 12/7 26.871
11/9, 18/11 27.408
13/11, 22/13 30.790
16/15, 15/8 31.731
9/8, 16/9 36.090
16/13, 13/8 39.472
15/14, 28/15 40.557
9/7, 14/9 44.916
14/13, 13/7 48.298

Notation

There are a variety of other ways to notate 15-edo, and the choice of notation depends heavily which rank-2 temperament or MOS scale one wishes to treat as being the "main focus" of 15-edo composition.

Blackwood Notation

  • Decimal Version: Using the nominals 1-0 (with 0 representing "10"), one of the three chains of 5-edo 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 15-edo guitar, because the "perfect fourth" comes from 5-edo, 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 15-edo, it is necessary to use accidentals to notate intervals on the other two chains of 5-edo. 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 15-edo on the guitar, since 5-edo 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

Porcupine notation bases porcupine[8] LLLLLLLs scale using eight nominals α β χ δ ε φ γ η. Others have proposed ABCDEFGHA but conflicts with european notation have caused many to reject this approach. Thus greek letters can be used in place with a close resemblance to the spelling of ABCDEFGHA.

Interval names for Porcupine[8]:

Cents Interval Name Note names
80 Half Quill α - β\
160 Quill α - β
240 Small Diquill α - χ\
320 Large Diquill α - χ
400 Small Triquill α - δ\
480 Large Triquill α - δ
560 Small Fourquill α - ε\
640 Large Fourquill 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 Large Sevenquill α - η
1200 Octoquill α - α

A regular keyboard can be designed using this system placing 7 black keys as porcupine[7] and 8 whites as porcupine[8].

Rank two temperaments

List of 15et rank two temperaments by badness

List of edo-distinct 15et rank two temperaments

Periods

per octave

Period Generator Temperaments
1 15\15 1\15 Nautilus/valentine
1 15\15 2\15 Porcupine/opossum
1 15\15 4\15 Hanson/keemun/orgone
1 15\15 7\15 Progress
3 5\15 1\15 Augmented/augene
3 5\15 2\15 Triforce
5 3\15 1\15 Blackwood/blacksmith

Commas

15 EDO tempers out the following commas. (Note: This assumes the val < 15 24 35 42 52 56 |.)

Rational Monzo Size (Cents) Name 1 Name 2 Name 3
256/243 | 8 -5 > 90.22 Limma Pythagorean Minor 2nd
28/27 | 2 -3 0 1 > 62.96 Septimal Third Tone Small Septimal Chroma
250/243 | 1 -5 3 > 49.17 Maximal Diesis Porcupine Comma
128/125 | 7 0 -3 > 41.06 Diesis Augmented Comma
15625/15552 | -6 -5 6 > 8.11 Kleisma Semicomma Majeur
1029/1000 | -3 1 -3 3 > 49.49 Keega
49/48 | -4 -1 0 2 > 35.70 Slendro Diesis
64/63 | 6 -2 0 -1 > 27.26 Septimal Comma Archytas' Comma Leipziger Komma
64827/64000 | -9 3 -3 4 > 22.23 Squalentine
875/864 | -5 -3 3 1 > 21.90 Keema
126/125 | 1 2 -3 1 > 13.79 Septimal Semicomma Starling Comma
4000/3969 | 5 -4 3 -2 > 13.47 Octagar
1029/1024 | -10 1 0 3 > 8.43 Gamelisma
6144/6125 | 11 1 -3 -2 > 5.36 Porwell
250047/250000 | -4 6 -6 3 > 0.33 Landscape Comma
100/99 | 2 -2 2 0 -1 > 17.40 Ptolemisma
121/120 | -3 -1 -1 0 2 > 14.37 Biyatisma
176/175 | 4 0 -2 -1 1 > 9.86 Valinorsma
65536/65219 | 16 0 0 -2 -3 > 8.39 Orgonisma
385/384 | -7 -1 1 1 1 > 4.50 Keenanisma
441/440 | -3 2 -1 2 -1 > 3.93 Werckisma
4000/3993 | 5 -1 3 0 -3 > 3.03 Wizardharry
3025/3024 | -4 -3 2 -1 2 > 0.57 Lehmerisma
91/90 | -1 -2 -1 1 0 1 > 19.13 Superleap
676/675 | 2 -3 -2 0 0 2 > 2.56 Parizeksma

Theory

The 15-Tone Scale System by Ivor Darreg (Originally at [1], now broken)

The Pentadecaphonic System (Originally at [2], now broken)

15-EDO Tutorial by Brent Carson (Originally at [3], now broken)

Practical Theory / Books

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.

Compositions

XA 15-ET Directory

Mizarian Porcupine Overture play by Herman Miller (Herman Miller) (porcupine chord progressions)

Study for Bells by Daniel Thompson (Daniel Thompson) (Jan. 2007)

Hyperimprovisation 3.3 play by Jacob Barton (2003)

OFOIOB play by Jacob Barton

15 Tone ET Improvisationn by Norbert Oldani

Elegy in 15ET by Aaron Andrew Hunt

Fugue a3 in 15ET by Aaron Andrew Hunt

Comets Over Flatland 12 by Randy Winchester

Comets Over Flatland 13 by Randy Winchester

Comets Over Flatland 16 by Randy Winchester

Study for Kyle Gann by Aaron K. Johnson (12-out-of-15)

Rick McGowan: Four Ballet Scenes

"Gently Playing With Miller's Porcupine" by Chris Vaisvil (uses Miller's Porcupine-7 mode 2 2 2 3 2 2 2)

After Dark on the Pedway by Chris Vaisvil

15 Sandles by Chris Vaisvil scordatura midi file

Improv on 15 EDO by Chris Vaisvil scordatura midi file and scordatura PDF score

15 edo Trolls by Chris Vaisvil - details

Through the Fire of the Sun (15 edo rock band) by Chris Vaisvil

2-2-1-2-2-1-2-2-1 mode of 15 edo play by Chris Vaisvil

Ode For Ada by Carlo Serafini (blog entry)

Suite in 15-Note Equal Tuning, opus 33 by Easley Blackwood (as well as one of the Twelve Microtonal Etudes, opus 28)

15edo-Chords.ogg Some nice sounds I found in 15 EDO

happenstance15.oggSonic experiment in 15. Somewhat familiar tonality.

Portrait of insects with 15-tone equal tempered guitar music

PentadecafoniCoda (15et) by F.F.F. Fiale

Cuckoo-Rag Fugue by Claudi Meneghin

15-penny jingle, by Claudi Meneghin

Tocada in 15edo, by Claudi Meneghin

A Broken Stern (2012) by Andrew J Milne on SoundCloud