15edo-a

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Introduction

15 Equal Divisions of the Octave (15edo) is a tuning which divides the octave into 15 equally spaced pitches. It can be thought of as three sets of 5edo connected to each other by intervals of 3edo.

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)."

It is notable for being the next-smallest EDO after 12edo that contains recognizable major and minor triads, a property noted in the works of theorists like Ivor Darreg and Easley Blackwood Jr. However, it also lacks a traditional fifth-generated diatonic scale, so unlike similarly-sized EDOs like 17edo and 19edo, it is not directly backwards-compatible with traditional Western music theory.

Intervals

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.

Degree Cents Extra-Diatonic Category
0 0 Unison
1 80 Subminor 2nd
2 160 Neutral 2nd
3 240 Supermajor 2nd
4 320 Minor 3rd
5 400 Major 3rd
6 480 Sub 4th
7 560 Narrow Tritone
8 640 Wide Tritone
9 720 Super 5th
10 800 Minor 6th
11 880 Major 6th
12 960 Subminor 7th
13 1040 Neutral 7th
14 1120 Supermajor 7th
15 1200 Octave

The fifth at 720 cents is quite wide, yet can still be functional as a perfect fifth by some standards. The perfect fifth of 15edo returns to the octave if stacked five times, which is radically different than what occurs in EDOs with fifths that are tuned closer to Just Intonation. This has a variety of ramifications for chord progressions based on functional harmony, because with a closed circle of five notes, the same interval can have multiple functions.

In the 15edo system, major thirds cannot be divided perfectly into two, while minor 3rds, 4ths, wide tritones, subminor 7ths, and supermajor 7ths can. Similarly, 4ths, 5ths, and subminor 7ths can all be divided into 3 equal parts, while minor 3rds, tritones, and major 6ths cannot. This gives 15edo a whole new set of pitch symmetries and modes of limited transposition.

Notation

Because 15edo's best fifth is the same as that of 5edo, the circle of fifths fully closes with only five notes, meaning the Pythagorean limma is tempered out. This slightly breaks traditional fifth-based notation, because E and F, as well as B and C, will merge into singular pitches. This also has the effect of making the Pythagorean apotome equal to the Pythagorean whole-tone, such that the sharp and flat accidentals will indicate raising or lowering by one step of 5edo, or 240 cents. There are two main approaches to dealing with this issue: the use of ups and downs in conjunction with the traditional A through G nominals based on the 5edo circle of fifths, and the use of Porcupine temperament instead of the circle of fifths as a notational basis.

Ups and Downs Notation

In ups and downs notation, which is fifth-generated, every 15edo note has at least three names.

step cents ups and downs relative notation

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

ups and downs

absolute notation

0 P1, m2 unison, min 2nd C# / D / Eb
1 80 ^1, ^m2 up-unison, upminor 2nd C#^ / D^ / Eb^
2 160 vM2 downmajor 2nd D#v / Ev / Fv / Gbv
3 240 M2, m3 major 2nd, minor 3rd D# / E / F / Gb
4 320 ^m3 upminor 3rd D#^ / E^ / F^ / Gb^
5 400 vM3 downmajor 3rd F#v / Gv / Abv
6 480 M3, P4, d5 major 3rd, perfect 4th, dim 5th F# / G / Ab
7 560 ^4, ^d5 up 4th, updim 5th F#^ / G^ / Ab^
8 640 vA4, v5 downaug 4th, down 5th G#v / Av / Bbv
9 720 A4, P5, m6 aug 4th, perfect 5th, minor 6th G# / A / Bb
10 800 ^5, ^m6 up 5th, upminor 6th G#^ / A^ / Bb^
11 880 vA5, vM6 downaug 5th, downmajor 6th A#v / Bv / Cv / Dbv
12 960 M6, m7 major 6th, minor 7th A# / B / C / Db
13 1040 ^m7 upminor 7th A#^ / B^ / C^ / Db^
14 1120 vM7, v8 downmajor 7th, down octave C#v / Dv / Ebv
15 1200 M7, P8 major 7th, octave C# / D / Eb

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.

Porcupine[7] Notation

See the main porcupine notation page.

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...

porcupine

relative notation

porcupine

absolute notation

Solfege

(porcupine-based)

unison D do
aug unison, dim 2nd D# / Eb di
perfect 2nd E ru
aug 2nd, dim 3rd E# / Fb re
minor 3rd F me
major 3rd, dim 4th F# / Gb mi
aug 3rd, minor 4th Fx / G fa
major 4th, dim 5th G# / Abb fu
aug 4th, minor 5th Gx / Ab su
major 5th, dim 6th A / Bbb sol
aug 5th, minor 6th A# / Bb le
major 6th B la
aug 6th, dim 7th B# / Cb ta
perfect 7th C tu
aug 7th, dim 8ve C# / Db ti
8ve D do

Porcupine[8] Notation:

An alternative porcupine notation is based on the 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.

Cents Interval Name Porcupine[8]

(Greek)

0 Unison α
80 Half Quill α/ β\
160 Quill β
240 Small Diquill β/ χ\
320 Large Diquill χ
400 Small Triquill χ/ δ\
480 Large Triquill ð
560 Small Fourquill δ/ ε\
640 Large Fourquill ε
720 Small Fivequill ε/ φ\
800 Large Fivequill φ
880 Small Sixquill φ/ γ\
960 Large Sixquill γ
1040 Small Sevenquill γ/ η\
1120 Large Sevenquill η
1200 Octoquill α

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

15edo as a Regular Temperament

15edo may be treated as a regular temperament of 5-, 7-, and 11-limit JI, or as a 2.5.7.11 subgroup temperament. While it does significant damage to the ratios of 3, it offers significant improvement over 12edo in approximating ratios of 5, 7, and 11. As a 5-limit temperament, it is 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.

15ed2-001.svg

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 some ratios of 7 and 11, but its approximation to most ratios of 3 and 9 are rather off. Its mappings and error for various 11-limit subgroups is shown in the table below.

15edo Subgroup Errors
Subgroup Mapping Adjusted Error (cents)
2.3 <15 24] 8.979801
2.5 <15 35] 6.826357
2.7 <15 42] 4.418738
2.11 <15 52] 4.336492
2.3.5 <15 24 35] 10.742841
2.3.7 <15 24 42] 17.481581
2.3.11 <15 24 52] 16.831238
2.5.7 <15 35 42] 10.509269
2.5.11 <15 35 52] 8.335693
2.7.11 <15 42 52] 8.002641
2.3.5.7 <15 24 35 42] 15.603114
2.3.5.11 <15 24 35 52] 14.693746
2.3.7.11 <15 24 42 52] 18.660367
2.5.7.11 <15 35 42 52] 11.462127
2.3.5.7.11 <15 24 35 42 52] 17.258371

Using the 11-limit mapping from the table above leads to the intervals of 15edo representing 11-limit ratios as follows:

Degree Cents Represented Ratios
0 0 1/1
1 80 25/24, 21/20, 16/15
2 160 11/10, 12/11, 10/9
3 240 8/7, 7/6, 9/8
4 320 6/5, 11/9
5 400 5/4, 14/11
6 480 4/3, 9/7, 21/16
7 560 11/8, 7/5
8 640 16/11, 10/7
9 720 3/2, 14/9, 32/21
10 800 8/5, 11/7
11 880 5/3, 18/11
12 960 7/4, 12/7, 16/9
13 1040 20/11, 11/6, 9/5
14 1120 48/25, 40/21, 15/8
15 1200 2/1

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 Patent val mapping
Interval, complement Error (abs., in cents) Interval, complement Error (abs., in cents)
18/13, 13/9 3.382 18/13, 13/9 3.382
6/5, 5/3 4.359 6/5, 5/3 4.359
11/10, 20/11 5.004 11/10, 20/11 5.004
15/13, 26/15 7.741 15/13, 26/15 7.741
11/8, 16/11 8.682 11/8, 16/11 8.682
8/7, 7/4 8.826 8/7, 7/4 8.826
12/11, 11/6 9.363 12/11, 11/6 9.363
5/4, 8/5 13.686 5/4, 8/5 13.686
14/11, 11/7 17.508 14/11, 11/7 17.508
4/3, 3/2 18.045 4/3, 3/2 18.045
13/12, 24/13 21.427 13/12, 24/13 21.427
10/9, 9/5 22.404 10/9, 9/5 22.404
7/5, 10/7 22.512 7/5, 10/7 22.512
15/11, 22/15 23.049 15/11, 22/15 23.049
13/10, 20/13 25.786 13/10, 20/13 25.786
7/6, 12/7 26.871 7/6, 12/7 26.871
11/9, 18/11 27.408 11/9, 18/11 27.408
13/11, 22/13 30.790 13/11, 22/13 30.790
14/13, 13/7 31.702 16/15, 15/8 31.731
16/15, 15/8 31.731 9/8, 16/9 36.090
9/7, 14/9 35.084 16/13, 13/8 39.472
9/8, 16/9 36.090 15/14, 28/15 40.557
15/14, 28/15 39.443 9/7, 14/9 44.916
16/13, 13/8 39.472 14/13, 13/7 48.298

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 13-limit 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.

Musical Examples

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

Images

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

15 tone keyboard.png