- 1 Theory
- 2 Intervals
- 3 Chord Names
- 4 Selected just intervals by error
- 5 Hexadecaphonic Octave Theory
- 6 Hexadecaphonic Notation
- 7 16 Tone Piano Layout Based on the Mavila/"Anti-diatonic" Scale
- 8 Rank two temperaments
- 9 Metallic Harmony in 16 EDO
- 10 Commas
- 11 Armodue Theory (4-line staff)
- 12 Images
- 13 Books/Literature
- 14 Compositions
16-EDO is the equal division of the octave into sixteen narrow chromatic semitones each of 75 cents exactly. It is not especially good at representing most low-integer musical intervals, but it has a 7/4 which is only six cents sharp, and a 5/4 which is only eleven cents flat. Four steps of it gives the 300 cent minor third interval, the same of that 12-EDO, giving it four diminished seventh chords exactly like those of 12-EDO, and a diminished triad on each scale step.
16edo can be notated with conventional notation, including the staff, note names, relative notation, etc. in two ways. The first preserves the melodic meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5. (But see below in "Chord Names".)
The second approach is to preserve the harmonic meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 16edo "on the fly".
Alternatively, one can use Armodue nine-nominal notation; see below
major wider than minor
|1||75||96 (6016)||28/27, 27/26||aug 1, dim 2nd||D#, Eb||dim 1, aug 2nd||Db, E#||Subminor 2nd||Min 2nd|
|2||150||192 (C016)||35/32||minor 2nd||E||major 2nd||E||Neutral 2nd||Maj 2nd|
|3||225||288 (12016)||8/7||major 2nd||E#||minor 2nd||Eb||Supermajor 2nd,
|4||300||384 (18016)||19/16, 32/27||minor 3rd||Fb||major 3rd||F#||Minor 3rd||Min 3rd|
|5||375||480 (1E016)||5/4, 16/13, 26/21||major 3rd||F||minor 3rd||F||Major 3rd||Maj 3rd|
|6||450||576 (24016)||13/10, 35/27||aug 3rd
|F#, Gb||dim 3rd,
|7||525||672 (2A016)||19/14, 27/20, 52/35, 256/189||perfect 4th||G||perfect 4th||G||Wide 4th||Maj 4th|
|8||600||768 (30016)||7/5, 10/7||aug 4th
|G#, Ab||dim 4th,
|Gb, A#||Tritone||Aug 4th,
|9||675||864 (36016)||28/19, 40/27, 35/26, 189/128||perfect 5th||A||perfect 5th||A||Narrow 5th||Min 5th|
|10||750||960 (3C016)||20/13, 54/35||aug 5th
|A#, Bb||dim 5th, aug 6th||Ab, B#||Super-5th,
|11||825||1056 (42016)||8/5, 13/8, 21/13||minor 6th||B||major 6th||B||Minor 6th||Min 6th|
|12||900||1152 (48016)||27/16, 32/19||major 6th||B#||minor 6th||Bb||Major 6th||Maj 6th|
|13||975||1248 (4E016)||7/4||minor 7th||Cb||major 7th||C#||Subminor 7th,
Septimal Minor 7th
|14||1050||1344 (54016)||64/35||major 7th||C||minor 7th||C||Neutral 7th||Min 7th|
|15||1125||1440 (5A016)||27/14, 52/27||aug 7th
|C#, Db||dim 7th,
|Cb, D#||Supermajor 7th||Maj 7th|
- based on treating 16-EDO as a 188.8.131.52.19.27 subgroup temperament; other approaches are possible.
16edo chords can be named using ups and downs. Using harmonic interval names, the names are easy to find, but they bear little relationship to the sound. 4:5:6 is a minor chord and 10:12:15 is a major chord! Using melodic names, the chord names will match the sound, but finding the name is much more complicated (see below).
|chord||JI ratios||harmonic name||melodic name|
|0-5-9||4:5:6||D F A||Dm||D minor||D F A||D||D major|
|0-4-9||10:12:15||D F# A||D||D major||D Fb A||Dm||D minor|
|0-4-8||5:6:7||D F# A#||Daug||D augmented||D Fb Ab||Ddim||D diminished|
|0-5-10||D F Ab||Ddim||D diminished||D F A#||Daug||D augmented|
|0-5-9-13||4:5:6:7||D F A C#||Dm(M7)||D minor-major||D F A Cb||D7||D seven|
|0-5-9-12||D F A Bb||Dm(b6)||D minor flat-six||D F A B#||D6||D six|
|0-5-9-14||D F A C||Dm7||D minor seven||D F A C||DM7||D major seven|
|0-4-9-13||D F# A C#||DM7||D major seven||D Fb A Cb||DM7||D minor seven|
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). See Ups and Downs Notation - Chords and Chord Progressions for more examples.
Using melodic names, interval arithmetic is done using a simple trick: first reverse everything, then perform normal arithmetic, then reverse everything again. Reversing means exchanging major for minor, aug for dim, and sharp for flat. Perfect and natural are unaffected. Examples:
|initial question||reverse everything||do the math||reverse again|
|M2 + M2||m2 + m2||dim3||aug3|
|D to F#||D to Fb||dim3||aug3|
|D to F||D to F||m3||M3|
|Eb + m3||E# + M3||G##||Gbb|
|Eb + P5||E# + P5||B#||Bb|
|A minor chord||A major chord||A C# E||A Cb E|
|Eb major chord||E# minor chord||E# G# B#||Eb Gb Db|
|Gm7 = G + m3 + P5 + m7||G + M3 + P5 + M7||G B D F#||G B D Fb|
|Ab7aug = Ab + M3 + A5 + m7||A# + m3 + d5 + M7||A# C# E G##||Ab Cb E Gbb|
|what chord is D F A#?||D F Ab||D + m3 + d5||D + M3 + A5 = Daug|
|what chord is C E Gb Bb?||C E G# B#||C + M3 + A5 + A7||C + m3 + d5 + d7 = Cdim7|
|C major scale = C + M2 + M3
+ P4 + P5 + M6 + M7 + P8
|C + m2 + m3 + P4
+ P5 + m6 + m7 + P8
|C Db Eb F
G Ab Bb C
|C D# E# F
G A# B# C
|C minor scale = C + M2 + m3
+ P4 + P5 + m6 + m7 + P8
|C + m2 + M3 + P4
+ P5 + M6 + M7 + P8
|C Db E F
G A B C
|C D# E F
G A B C
|what scale is A B# Cb D
E F Gb A?
|A Bb C# D
E F G# A
|A + m2 + M3 + P4
+ P5 + m6 + M7
|A + M2 + m3 + P4
+ P5 + M6 + m7 = A dorian
Selected just intervals by error
The following table shows how some prominent just intervals are represented in 16-EDO (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)|
It's worth noting that the 525-cent interval is almost exactly halfway in between 4/3 and 11/8, making it very discordant, although playing this in the context of a larger chord, and with specialized timbres, can make this less noticeable.
Hexadecaphonic Octave Theory
The scale supports the diminished temperament with its 1/4 octave period, though its generator size, equal to its step size of 75 cents, is smaller than ideal. Its very flat 3/2 of 675 cents supports Mavila temperament, where the mapping of major and minor is reversed. The temperament could be popular for its 150-cent "3/4-tone" equal division of the traditional 300-cent minor third.
16-EDO is also a tuning for the no-threes 7-limit temperament tempering out 50/49. This has a period of a half-octave (600¢), and a generator of a flat septimal major 2nd, for which 16-EDO uses 3\16. For this, there are MOS scales of sizes 4, 6, and 10; extending this temperament to the full 7-limit can produce either Lemba or Astrology (16-EDO supports both, but is not a very accurate tuning of either).
16-EDO is also a tuning for the no-threes 7-limit temperament tempering out 546875:524288, which has a flat major third as generator, for which 16-EDO provides 5\16 octaves. For this, there are MOS of sizes 7, 10, and 13; these are shown below under "Magic family of scales".
Easley Blackwood writes of 16-EDO:
"16 notes: This tuning is best thought of as a combination of four intertwined diminished seventh chords. Since 12-note tuning can be regarded as a combination of three diminished seventh chords, it is plain that the two tunings have elements in common. The most obvious difference in the way the two tunings sound and work is that triads in 16-note tuning, although recognizable, are too discordant to serve as the final harmony in cadences. Keys can still be established by successions of altered subdominant and dominant harmonies, however, and the Etude is based mainly upon this property. The fundamental consonant harmony employed is a minor triad with an added minor seventh."
From a harmonic series perspective, if we take 13\16 as a 7/4 ratio approximation, sharp by 6.174 cents, and take the 300-cent minor third as an approximation of the harmonic 19th (19/16, approximately 297.5 cents), that adds another overtone which can combine with the approximation of the harmonic seventh to form a 16:19:28 triad .
The interval between the 28th & 19th overtones, 28:19, measures approximately 671.3 cents, which is 3.7 cents away from 16edo's "narrow fifth". Another voicing for this chord is 14:16:19, which features 19:14 as the outer interval (528.7 cents just, 525.0 cents in 16edo). A perhaps more consonant open voicing is 7:16:19.
16-EDO notation can be easy utilizing Goldsmith's Circle of keys, nominals, and respective notation. The nominals for a 6 line staff can be switched for Wilson's Beta and Epsilon additions to A-G. The Armodue model uses a 4-line staff for 16-EDO.
Moment of Symmetry Scales like Mavila  (or "Inverse/Anti-Diatonic" which reverses step sizes of diatonic from LLsLLLs to ssLsssL in the heptatonic variation) can work as an alternative to the traditional diatonic scale, while maintaining conventional A-G #/b notation as described above. Alternatively, one can utilize the Mavila MOS, for a sort of "hyper-diatonic" scale of 7 large steps and 2 small steps. Armodue notation of 16-EDO "Mavila- Staff" does just this, and places the arrangement (222122221) on nine white "natural" keys of the 16-EDO keyboard. If the 9-note "Enneatonic" MOS is adopted as a notational basis for 16-EDO, then we need an entirely different set of interval classes than any of the heptatonic classes described above; perhaps it even makes sense to refer to Octaves as 2/1, "Decave".
|1||75||aug unison, minor 2nd||1#, 2b|
|3||225||aug 2nd, minor 3rd||2#, 3b|
|4||300||major 3rd, dim 4th||3, 4bb|
|7||525||aug 4th, minor 5th||4#, 5|
|8||600||aug 5th, dim 6th||5#, 6b|
|9||675||perfect 6th, dim 7th||6, 7bb|
|10||750||aug 6th, minor 7th||6#, 7b|
|12||900||aug 7th, minor 8th||7#, 8b|
|13||975||major 8th, dim 9th||8, 9bb|
|15||1125||major 9th, dim 10ve||9#, 1b|
16 Tone Piano Layout Based on the Mavila/"Anti-diatonic" Scale
This Layout places Mavila on the black keys and Mavila on the white keys. As you can see, flats are higher than naturals and sharps are lower, as per the "harmonic notation" above. Simply swap sharps with flats for "melodic notation".
Rank two temperaments
Temperaments listed by generator size:
|:||5 2 5 2 2|
|:||3 2 2 3 2 2 2|
|:||1 2 2 2 1 2 2 2 2|
See also Mavila Temperament Modal Harmony.
|:||1 3 1 3 1 3 1 3|
|:||1 1 2 1 1 2 1 1 2 1 1 2|
: 1 4 1 4 1 4 1
: 1 3 1 1 3 1 1 1 3 1
: 1 1 2 1 1 1 2 1 1 1 2 1 1
: 3 3 4 3 3
: 3 3 1 3 3 3
: 1 2 1 2 1 2 1 2 1 2 1
: 3 5 3 5
: 3 2 3 3 2 3
: 2 1 2 1 2 2 1 2 1 2
Metallic Harmony in 16 EDO
Because 16edo doesn't approximate 3/2 well at all, triadic harmony based on heptatonic thirds isn't a great option for typical harmonic timbres.
However, triadic harmony can be based on on heptatonic sevenths (or seconds) rather than thirds. For instance, 16edo approximates 7/4 well enough to use
it in place of the usual 3/2, and in Mavila this 7/4 approximation shares an interval class with a well-approximated 11/6 (at 1050 cents). Stacking these two intervals reaches 2025¢, or a minor 6th plus an octave. Thus the out-of-tune 675¢ interval is bypassed, and all the dyads in the triad are consonant.
Depending on whether the Mavila major 7th or minor 7th is used, one of two triads is produced: a small one, 0-975-2025¢, and a large one, 0-1050-2025¢. William Lynch, a major proponent of this style of harmony, calls these two triads "hard" and "soft", respectively. In addition, two other "symmetrical" triads are also obvious possible chords: a narrow symmetrical triad at 0-975-1950¢, and a wide symmetrical triad at 0-1050-2100¢. These are sort of analogous to "diminished" and "augmented" triads. The characteristic buzzy/metallic sound of these seventh-based triads inspired William Lynch to call them "Metallic triads".
MOS scales supporting Metallic Harmony in 16edo
The ssLsssL mode of Mavila contains two hard triads on degrees 1 and 4 and two soft triads on degrees 2 and 6. The other three chords are wide symmetrical triads 0-1050-2025¢. In Mavila, hard and soft triads cease to share a triad class, as 975¢ is a major 8th, while 1050¢ is a minor 9th; the triads may still be used, but parallel harmonic motion will function differently.
Another possible MOS scales for this approach would be Lemba, which gives two each of the soft, hard, and narrow symmetric triads.
See Metallic Harmony.
|Ratio||Monzo||Cents||Color Name||Name 1||Name 2||Name 3|
|135/128||| -7 3 1 >||92.18||Layobi||Major Chroma||Major Limma||Pelogic Comma|
|648/625||| 3 4 -4 >||62.57||Quadgu||Major Diesis||Diminished Comma|
|3125/3072||| -10 -1 5 >||29.61||Laquinyo||Small Diesis||Magic Comma|
|36/35||| 2 2 -1 -1 >||48.77||Rugu||Septimal Quarter Tone|
|525/512||| -9 1 2 1 >||43.41||Lazoyoyo||Avicennma||Avicenna's Enharmonic Diesis|
|50/49||| 1 0 2 -2 >||34.98||Biruyo||Tritonic Diesis||Jubilisma|
|64827/64000||| -9 3 -3 4 >||22.23||Laquadzo-atrigu||Squalentine|
|3125/3087||| 0 -2 5 -3 >||21.18||Triru-aquinyo||Gariboh|
|126/125||| 1 2 -3 1 >||13.79||Zotrigu||Septimal Semicomma||Starling Comma|
|1029/1024||| -10 1 0 3 >||8.43||Latrizo||Gamelisma|
|6144/6125||| 11 1 -3 -2 >||5.36||Sarurutrigu||Porwell|
|| 23 6 -14 >||3.34||Sasepbiru||Vishnuzma||Semisuper|
|121/120||| -3 -1 -1 0 2 >||14.37||Lologu||Biyatisma|
|176/175||| 4 0 -2 -1 1 >||9.86||Lorugugu||Valinorsma|
|385/384||| -7 -1 1 1 1 >||4.50||Lozoyo||Keenanisma|
|441/440||| -3 2 -1 2 -1 >||3.93||Luzozogu||Werckisma|
|3025/3024||| -4 -3 2 -1 2 >||0.57||Loloruyoyo||Lehmerisma|
Armodue Theory (4-line staff)
Armodue: Pierpaolo Beretta's website for his "Armodue" theory for 16edo (esadekaphonic), including compositions.
Translations of parts of the Armodue pages can be found here on this wiki.
Sword, Ronald. "Thesaurus of Melodic Patterns and Intervals for 16-Tones" IAAA Press, USA. First Ed: August, 2011
Sword, Ronald. "Hexadecaphonic Scales for Guitar." IAAA Press, UK-USA. First Ed: Feb, 2010. (superfourth tuning)
Sword, Ronald. "Esadekaphonic Scales for Guitar." IAAA Press, UK-USA. First Ed: April, 2009. (semi-diminished fourth tuning)
Huckleberry Regional Preserve by City of the Asleep
Illegible Red Ink by City of the Asleep
Enantiodromia (album) by Last Sacrament
Tribute to Armodue by Aeterna
The Foggy Road from Pasadena by Ron Sword