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.

First implemented by R.M.A Kusumadinata (Sundra) somewhere between the 1930s and the 1960s and Augusto Novaro in 1951, 15edo saw it's most notable early use in 1991 by Easley Blackwood Jr. in his Opus 28, Twelve Microtonal Etudes for Electronic Music Media and Opus 33, Suite for Guitar in 15-note Equal Tuning.

Along with the Blackwood Decatonic, a 10-note scale named after Blackwood for it's use is his Opus 28, and his ensuing discussion of it in his accompanying paper, Modes and Chord Progressions in Equal Tunings, published in the same year, 15edo is commonly associated with Porcupine Temperament, who's 7 and 8 note MOS scales are reasonably popular around these parts.

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.

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.

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.


Degree	Cents	Extra-Diatonic Category	Represented Ratios
0	0	Unison	1/1
1	80	Subminor 2nd	25/24, 21/20, 16/15
2	160	Neutral 2nd	11/10, 12/11, 10/9
3	240	Supermajor 2nd	8/7, 7/6, 9/8
4	320	Minor 3rd	6/5, 11/9
5	400	Major 3rd	5/4, 14/11
6	480	Perfect / Sub 4th	4/3, 9/7, 21/16
7	560	Narrow Tritone	11/8, 7/5
8	640	Wide Tritone	16/11, 10/7
9	720	Perfect / Super 5th	3/2, 14/9, 32/21
10	800	Minor 6th	8/5, 11/7
11	880	Major 6th	5/3, 18/11
12	960	Subminor 7th	7/4, 12/7, 16/9
13	1040	Neutral 7th	20/11, 11/6, 9/5
14	1120	Supermajor 7th	48/25, 40/21, 15/8
15	1200	Octave	2/1

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. Using the 11-limit mapping from the table below leads to the intervals of 15edo representing 11-limit ratios shown in the able above.

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

The following diagram shows how closely some prominent JI intervals are approximated in 15edo, using their best direct mappings, even if they lead to inconsistencies.

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

Scales

Some scales commonly used in 15edo, written in a common mode, in steps of 15edo:

MOS Scales

Augmented[6]: 414141

Porcupine[7]: 3222222

Hanson/Keemun/Orgone[7]: 1313133

Porcupine[8]: 21222222

Augmented[9]: 311311311

Blackwood[10]: 2121212121 (Blackwood Decatonic)

Augmented[12]: 211121112111

Other Scales

Zarlino/Ptolemy diatonic, "just" major, Ma grama - 3213231

inverse of Zarlino/Ptolemy diatonic, natural minor - 3123132

tetrachordal major, Sa grama - 3213321

inverse of tetrachordal major, "just"/tetrachordal minor - 3123123

Porcupine bright major #7 - Porcupine[7] 6|0 #7 - 3222231

Porcupine bright major #6 #7 - Porcupine[7] 6|0 #7 - 3222321

Porcupine bright minor #2 - Porcupine[7] 4|2 #2 3132222 (mode of bright major #7)

Porcupine dark minor #2 - Porcupine[7] 3|3 #2 3123222 (inverse of bright major #6 #7)

Porcupine bright harmonic 11th mode - Porcupine[7] 6|0 b7 3222213

"just" harmonic minor - 3123141

"just" harmonic major - 3213141

"just" melodic minor ascending - 3123231

"just" double harmonic - 1413141

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	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].

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