Summary[edit]

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[edit]

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

Intervals[edit]

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	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].

File:15ed2-001.svg

15ed2-001.svg

Selected just intervals by error[edit]

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

Best direct mapping, even if inconsistent[edit]

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[edit]

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[edit]

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[edit]

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[edit]

See the main porcupine notation page.

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]:[edit]

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[edit]

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[edit]

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