15edo

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15edo

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Prime factorization	3 × 5
Step size	80 ¢
Fifth	9\15 (720 ¢) (→ 3\5)
Semitones (A1:m2)	3:0 (240 ¢ : 0 ¢)
Consistency limit	7
Distinct consistency limit	5

English Wikipedia has an article on:

15 equal temperament

15 equal divisions of the octave (15edo), or 15(-tone) equal temperament (15tet, 15et) when viewed from a regular temperament perspective, is the tuning system derived by dividing the octave into 15 equally large steps.

Theory

Approximation of prime harmonics in 15edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.0	+18.0	+13.7	-8.8	+8.7	+39.5	-25.0	+22.5	+11.7	+10.4	-25.0
	Relative (%)	+0.0	+22.6	+17.1	-11.0	+10.9	+49.3	-31.2	+28.1	+14.7	+13.0	-31.3
Steps (reduced)		15 (0)	24 (9)	35 (5)	42 (12)	52 (7)	56 (11)	61 (1)	64 (4)	68 (8)	73 (13)	74 (14)

15edo can be thought of as three sets of 5edo which do not connect by fifths. The fifth at 720 cents is quite wide yet still usable 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 15edo returns to the octave if stacked five times which is radically different than a meantone system. Additionally, 15 being equal to 3 × 5 also implies that 15edo contains five sets of 3edo.

15edo 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 15edo would be best advised to avoid chords requiring those harmonics (or to at least treat them with sensitivity). 15edo 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 15edo 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 xenharmony, due to its manageable number of tones and for containing the relatively popular 5edo.

A recommended method to the notation of 15edo 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 (160¢) of 15edo while the "small quill" (80¢) is the chroma of 15edo. This produces a very consistent notation for both porcupine[8] and Blackwood[10] and seems to work much better than attempting to put 15edo into a seven nominal based framework.

• 
  Porcupine layout for 15edo
• 
  Hanson layout for 15edo

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	δ	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 15edo 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 absolute and relative notation:

…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…

…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…

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	vD# / vE / vF / vGb	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	vF# / vG / vAb	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	vG# / vA / vBb	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	vA# / vB / vC / vDb	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	vC# / vD / vEb	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. Alterations are always enclosed in parentheses, additions never are. An up or down immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13).

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 vF# A = Dv = "D 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 = D^m,7 = "D upminor, add seven", or D ^F A B = D^m,6 = "D upminor add-six"

0-5-9-12 = D vF# A C = Dv,7 = "D down add-seven", or D vF# A B = Dv,6 = "D down add-six"

0-6-9-12 = D F# A C = D7 = "D seven", or D F# A B = D6 = "D six"

0-5-9-14 = D vF# A vC# = DvM7 = "D downmajor seven"

0-4-9-13 = D ^F A ^C = D^m7 = "D upminor-seven", or D ^F A ^B = D^m6 = "D upminor-six"

For a more complete list, see Ups and Downs Notation #Chords and Chord Progressions.

Just approximation

15edo 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 15edo based on this temperament (and its 7- and 11-limit extensions), see Harmony in 15edo Blacksmith[10].

Selected just intervals

15-odd-limit mappings

The following table shows how 15-odd-limit intervals are represented in 15edo. Prime harmonics are in bold; inconsistent intervals are in italic.

Direct mapping (even if inconsistent)

Interval, complement	Error (abs, ¢)
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, 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

Selected 13-limit intervals

Notation

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

Blackwood Notation

• 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

• 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]

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], and in fact, Stephen Weigel has already done this with his pink Halberstadt keyboard.

Rank two temperaments (with horagrams for octave-periodic MOS)

• List of 15et rank two temperaments by badness
• List of edo-distinct 15et rank two temperaments

Periods per octave	Period	Generator	Temperaments	MOS Scales
1	15\15	1\15	Nautilus/valentine
1	15\15	2\15	Porcupine/opossum	1L 6s, 7L 1s
1	15\15	4\15	Hanson/keemun/orgone	3L 1s, 4L 3s, 4L 7s
1	15\15	7\15	Progress	2L 1s, 2L 3s, 2L 5s, 2L 7s 2L 9s, 2L 11s
3	5\15	1\15	Augmented/augene	3L 3s, 3L 6s, 3L 9s
3	5\15	2\15	Triforce	3L 3s, 6L 3s
5	3\15	1\15	Blackwood/blacksmith	5L 5s

2\15 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 7L 1s

4\15 MOS using 1L 1s, 1L 2s, 3L 1s, 4L 3s, 4L 7s

7\15 MOS using 1L 1s, 2L 1s, 2L 3s, 2L 5s, 2L 7s, 2L 9s, 2L 11s

Commas

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

Prime Limit	Ratio	Monzo	Cents	Color name	Name(s)
3	256/243	[8 -5⟩	90.22	Sawa	Limma, Pythagorean minor 2nd
5	250/243	[1 -5 3⟩	49.17	Triyo	Maximal diesis, Porcupine comma
5	128/125	[7 0 -3⟩	41.06	Trigu	Diesis, Augmented comma
5	15625/15552	[-6 -5 6⟩	8.11	Tribiyo	Kleisma, semicomma majeur
7	28/27	[2 -3 0 1⟩	62.96	Zo	Septimal third tone, Small septimal chroma
7	1029/1000	[-3 1 -3 3⟩	49.49	Trizogu	Keega
7	49/48	[-4 -1 0 2⟩	35.70	Zozo	Slendro diesis
7	64/63	[6 -2 0 -1⟩	27.26	Ru	Septimal comma, Archytas' comma, Leipziger Komma
7	64827/64000	[-9 3 -3 4⟩	22.23	Laquadzo-atrigu	Squalentine
7	875/864	[-5 -3 3 1⟩	21.90	Zotriyo	Keema
7	126/125	[1 2 -3 1⟩	13.79	Zotrigu	Septimal semicomma, Starling comma
7	4000/3969	[5 -4 3 -2⟩	13.47	Rurutriyo	Octagar
7	1029/1024	[-10 1 0 3⟩	8.43	Latrizo	Gamelisma
7	6144/6125	[11 1 -3 -2⟩	5.36	Saruru-atrigu	Porwell
7	(12 digits)	[-4 6 -6 3⟩	0.33	Trizogugu	Landscape comma
11	100/99	[2 -2 2 0 -1⟩	17.40	Luyoyo	Ptolemisma
11	121/120	[-3 -1 -1 0 2⟩	14.37	Lologu	Biyatisma
11	176/175	[4 0 -2 -1 1⟩	9.86	Lorugugu	Valinorsma
11	65536/65219	[16 0 0 -2 -3⟩	8.39	Satrilu-aruru	Orgonisma
11	385/384	[-7 -1 1 1 1⟩	4.50	Lozoyo	Keenanisma
11	441/440	[-3 2 -1 2 -1⟩	3.93	Luzozogu	Werckisma
11	4000/3993	[5 -1 3 0 -3⟩	3.03	Triluyo	Wizardharry
11	3025/3024	[-4 -3 2 -1 2⟩	0.57	Loloruyoyo	Lehmerisma
13	91/90	[-1 -2 -1 1 0 1⟩	19.13	Thozogu	Superleap
13	676/675	[2 -3 -2 0 0 2⟩	2.56	Bithogu	Parizeksma

Books

Theory

• The 15-Tone Scale System by Ivor Darreg (Mirrored at http://www.webcitation.org/5xZyzKBEW. Originally at [1], now broken)
• The Pentadecaphonic System (Originally at [2], now broken)
• 15-EDO Tutorial by Brent Carson (Originally at [3], now broken)

Guitar

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 15edo system. 300+ examples /w chord-scale progressions.

Keyboard

Major chord shapes for a Porcupine keyboard in 15edo, using Blackwood logic (i.e. native fifth notation) for the letters.

Images

Music

See also: Category:15edo tracks

• XA 15-ET Directory
• WEIGEL FAMILY CHRISTMAS (xenharmonic chocolate), an album of xenharmonic Christmas covers played by Stephen Weigel, many are in 15edo
• Keep out of my Psyche by Stephen Weigel
• Mizarian Porcupine Overture play^{[dead link]} by Herman Miller (porcupine chord progressions; this is the song that 'porcupine' is named after)
• Study for Bells^{[dead link]} by Daniel Thompson (Jan. 2007)
• Hyperimprovisation 3.3 play^{[dead link]} by Jacob Barton (2003)
• OFOIOB play^{[dead link]} by Jacob Barton
• 15 Tone ET Improvisationn^{[dead link]} by Norbert Oldani
• Elegy in 15ET^{[dead link]} by Aaron Andrew Hunt
• Fugue a3 in 15ET^{[dead link]} by Aaron Andrew Hunt
• Comets Over Flatland 12^{[dead link]} by Randy Winchester
• Comets Over Flatland 13^{[dead link]} by Randy Winchester
• Comets Over Flatland 16^{[dead link]} by Randy Winchester
• Study for Kyle Gann by Aaron Krister Johnson (12-out-of-15)
• Four Ballet Scenes^{[dead link]} by Rick McGowan
• "Gently Playing With Miller's Porcupine" by Chris Vaisvil - details (uses Miller's Porcupine-7 mode 2 2 2 3 2 2 2)
• After Dark on the Pedway by Chris Vaisvil - details
• 15 Sandles by Chris Vaisvil - details, scordatura midi file
• Improv on 15 EDO by Chris Vaisvil - scordatura midi file, scordatura PDF score
• 15 edo Trolls by Chris Vaisvil - details
• Through the Fire of the Sun (15 edo rock band) by Chris Vaisvil - details
• 2-2-1-2-2-1-2-2-1 mode of 15 edo by Chris Vaisvil - details
• Ode For Ada by Carlo Serafini (blog entry)
• Sahara 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.ogg Sonic experiment in 15. Somewhat familiar tonality.
• Portrait of insects with 15-tone equal tempered guitar music by Jonathan Rabson
• PentadecafoniCoda (15et) by F.F.F. Fiale
• Cuckoo-Rag Fugue^{[dead link]} by Claudi Meneghin
• 15-penny jingle^{[dead link]} by Claudi Meneghin
• Tocada in 15edo^{[dead link]} by Claudi Meneghin
• A Broken Stern (2012) by Andrew J Milne on SoundCloud
• Canada Geese by Diamond Doll (punk song)
• Mask On by Diamond Doll (punk song)
• 15 Somethings' Alien Disco by Jutomi
• March of the Lichens by Jutomi
• The Wish To Wish The Wish by Userminusone
• "Semantics" by Brendan Byrnes (from his 2020 album "Realism")