- 1 Theory
- 2 Intervals
- 3 Notation
- 4 Rank two temperaments
- 5 Commas
- 6 Theory
- 7 Practical Theory / Books
- 8 Images
- 9 Compositions
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.
"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 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 and Blackwood and seems to work much better than attempting to put 15-edo into a seven nominal based framework.
|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|
- 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 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
|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.
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.
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)|
Patent val mapping
|Interval, complement||Error (abs., in cents)|
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.
- 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.
- See the main porcupine notation page.
Porcupine notation bases porcupine 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
|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 and 8 whites as porcupine, and in fact, Stephen Weigel has already done this with his pink Halberstadt keyboard.
Rank two temperaments
|Ratio||Monzo||Cents||Color Names||Name 1||Name 2||Name 3|
|256/243||| 8 -5 >||90.22||Sawa||Limma||Pythagorean Minor 2nd|
|250/243||| 1 -5 3 >||49.17||Triyo||Maximal Diesis||Porcupine Comma|
|128/125||| 7 0 -3 >||41.06||Trigu||Diesis||Augmented Comma|
|15625/15552||| -6 -5 6 >||8.11||Tribiyo||Kleisma||Semicomma Majeur|
|28/27||| 2 -3 0 1 >||62.96||Zo||Septimal Third Tone||Small Septimal Chroma|
|1029/1000||| -3 1 -3 3 >||49.49||Trizogu||Keega|
|49/48||| -4 -1 0 2 >||35.70||Zozo||Slendro Diesis|
|64/63||| 6 -2 0 -1 >||27.26||Ru||Septimal Comma||Archytas' Comma||Leipziger Komma|
|64827/64000||| -9 3 -3 4 >||22.23||Laquadzo-atrigu||Squalentine|
|875/864||| -5 -3 3 1 >||21.90||Zotriyo||Keema|
|126/125||| 1 2 -3 1 >||13.79||Zotrigu||Septimal Semicomma||Starling Comma|
|4000/3969||| 5 -4 3 -2 >||13.47||Rurutriyo||Octagar|
|1029/1024||| -10 1 0 3 >||8.43||Latrizo||Gamelisma|
|6144/6125||| 11 1 -3 -2 >||5.36||Saruru-atrigu||Porwell|
|250047/250000||| -4 6 -6 3 >||0.33||Trizogugu||Landscape Comma|
|100/99||| 2 -2 2 0 -1 >||17.40||Luyoyo||Ptolemisma|
|121/120||| -3 -1 -1 0 2 >||14.37||Lologu||Biyatisma|
|176/175||| 4 0 -2 -1 1 >||9.86||Lorugugu||Valinorsma|
|65536/65219||| 16 0 0 -2 -3 >||8.39||Satrilu-aruru||Orgonisma|
|385/384||| -7 -1 1 1 1 >||4.50||Lozoyo||Keenanisma|
|441/440||| -3 2 -1 2 -1 >||3.93||Luzozogu||Werckisma|
|4000/3993||| 5 -1 3 0 -3 >||3.03||Triluyo||Wizardharry|
|3025/3024||| -4 -3 2 -1 2 >||0.57||Loloruyoyo||Lehmerisma|
|91/90||| -1 -2 -1 1 0 1 >||19.13||Thozogu||Superleap|
|676/675||| 2 -3 -2 0 0 2 >||2.56||Bithogu||Parizeksma|
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.
Keep out of my Psyche by Stephen Weigel
Fugue a3 in 15ET by Aaron Andrew Hunt
Study for Kyle Gann by Aaron K. Johnson (12-out-of-15)
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.
PentadecafoniCoda (15et) by F.F.F. Fiale