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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 six cents sharp, and a 5/4 which is 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.

Images


16edo wheel 01.png

Intervals


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.

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)

Degree
Cents
Approximate
Ratios*
Melodic names,
major wider than minor
Harmonic names,
major narrower
than minor
Interval Names
Just
Interval
Names
Simplified
0
0
1/1
unison
D
unison
D
Unison
Unison
1
75
28/27, 27/26
aug 1, dim 2nd
D#, Eb
dim 1, aug 2nd
Db, E#
Subminor 2nd
Min 2nd
2
150
12/11, 35/32
minor 2nd
E
major 2nd
E
Neutral 2nd
Maj 2nd
3
225
8/7
major 2nd
E#
minor 2nd
Eb
Supermajor 2nd,
Septimal Whole-Tone
Perf 2nd
4
300
19/16, 32/27
minor 3rd
Fb
major 3rd
F#
Minor 3rd
Min 3rd
5
375
5/4, 26/21
major 3rd
F
minor 3rd
F
Major 3rd
Maj 3rd
6
450
13/10, 35/27
aug 3rd
dim 4th
F#, Gb
dim 3rd,
aug 4th
Fb, G#
Sub-4th,
Supermajor 3rd
Min 4th
7
525
27/20, 52/35, 256/189
perfect 4th
G
perfect 4th
G
Wide 4th
Maj 4th
8
600
7/5, 10/7
aug 4th
dim 5th
G#, Ab
dim 4th,
aug 5th
Gb, A#
Tritone
Aug 4th,
Dim 5th
9
675
40/27, 35/26, 189/128
perfect 5th
A
perfect 5th
A
Narrow 5th
Min 5th
10
750
20/13, 54/35
aug 5th
dim 6th
A#, Bb
dim 5th, aug 6th
Ab, B#
Super-5th,
Subminor 6th
Maj 5th
11
825
8/5, 21/13
minor 6th
B
major 6th
B
Minor 6th
Min 6th
12
900
27/16, 32/19
major 6th
B#
minor 6th
Bb
Major 6th
Maj 6th
13
975
7/4
minor 7th
Cb
major 7th
C#
Subminor 7th,
Septimal Minor 7th
Perf 7th
14
1050
11/6, 64/35
major 7th
C
minor 7th
C
Neutral 7th
Min 7th
15
1125
27/14, 52/27
aug 7th
dim 8ve
C#, Db
dim 7th,
aug 8ve
Cb, D#
Supermajor 7th
Maj 7th
16
1200
2/1
8ve
D
8ve
D
Octave
Octave
*based on treating 16-EDO as a 2.5.7.13.19.27 subgroup temperament; other approaches are possible.

Chord names


16edo chords can be named using us 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. First change sharps to flats, then find every interval from the root, then exchange major for minor and aug for dim, then name the chord from the intervals. (See xenharmonic.wikispaces.com/Ups+and+Downs+Notation#Other%20EDOs).

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

Selected just intervals by error

The following table shows how some prominent just intervals are represented in 16-EDO (ordered by absolute error).
Interval, complement
Error (abs., in cents)
12/11, 11/6
0.637
13/10, 20/13
4.214
8/7, 7/4
6.174
13/11, 22/13
10.790
5/4, 8/5
11.314
13/12, 24/13
11.427
15/11, 22/15
11.951
9/7, 14/9
14.916
11/10, 20/11
15.004
16/13, 13/8
15.528
6/5, 5/3
15.641
7/5, 10/7
17.488
9/8, 16/9
21.090
14/13, 13/7
21.702
15/13, 26/15
22.741
11/8, 16/11
26.318
4/3, 3/2
26.955
11/9, 18/11
27.592
15/14, 28/15
30.557
10/9, 9/5
32.404
14/11, 11/7
32.492
7/6, 12/7
33.129
18/13, 13/9
36.618
16/15, 15/8
36.731
It's worth noting that the 525-cent interval is almost exactly halfway in between 4/3 and 11/8, making it EXTREMELY discordant.
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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.

Hexadecaphonic Notation


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. Armodue of Italy uses a 4-line staff for 16-EDO.

Moment of Symmetry Scales like Mavila [7] (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[9] MOS, for a sort of "hyper-diatonic" scale of 7 large steps and 2 small steps. Armodue notation of 16-EDO "Mavila-[9] 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".

Degree
Cents
Mavila[9] Notation
0
0
unison
1
1
75
aug unison, minor 2nd
1#, 2b
2
150
major 2nd
2
3
225
aug 2nd, minor 3rd
2#, 3b
4
300
major 3rd, dim 4th
3, 4bb
5
375
minor 4th
4b
6
450
major 4th,
dim 5th
4, 5b
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
11
825
major 7th
7
12
900
aug 7th, minor 8th
7#, 8b
13
975
major 8th, dim 9th
8, 9bb
14
1050
minor 9th
9
15
1125
major 9th, dim 10ve
9#, 1b
16
1200
10ve (Decave)
1

16 Tone Piano Layout Based on the Mavila[7]/"Anti-diatonic" Scale

16-EDO-PIano-Diagram.png
This Layout places Mavila[7] on the black keys and Mavila[9] 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

List of 16et rank two temperaments by badness

Periods
per octave
Generator
Temperaments
1
1\16
Valentine, slurpee
1
3\16
Gorgo
1
5\16
Messed-up magic/muggles
1
7\16
Mavila/armodue
2
1\16
Bipelog
2
3\16
Lemba, astrology
4
1\16
Diminished/demolished
8
1\16


Mavila
[5]:
5 2 5 2 2

[7]:
3 2 2 3 2 2 2

[9]:
1 2 2 2 1 2 2 2 2

Diminished
[8]: 1 3 1 3 1 3 1 3
[12]: 1 1 2 1 1 2 1 1 2 1 1 2
Magic
[7]: 1 4 1 4 1 4 1
[10]: 1 3 1 1 3 1 1 1 3 1
[13]: 1 1 2 1 1 1 2 1 1 1 2 1 1
Cynder/Gorgo
[5]: 3 3 4 3 3
[6]: 3 3 1 3 3 3
[11]: 1 2 1 2 1 2 1 2 1 2 1
Lemba/Astrology
[4]: 3 5 3 5
[6]: 3 2 3 3 2 3
[10]: 2 1 2 1 2 2 1 2 1 2

Metallic Harmony in 16 EDO


Because 16 edo 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, 16 edo approximates 7/4 well enough to use
it in place of the usual 3/2, and in Mavila[7] 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[7] 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[7] 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[9], 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[6], which gives two each of the soft, hard, and narrow symmetric triads.

See Metallic Harmony.

Commas

16 EDO tempers out the following commas. (Note: This assumes val < 16 25 37 45 55 59 |.)
Comma
Monzo
Value (Cents)
Name 1
Name 2
Name 3
135/128
| -7 3 1 >
92.18
Major Chroma
Major Limma
Pelogic Comma
648/625
| 3 4 -4 >
62.57
Major Diesis
Diminished Comma

3125/3072
| -10 -1 5 >
29.61
Small Diesis
Magic Comma


| 23 6 -14 >
3.34
Vishnuzma
Semisuper

36/35
| 2 2 -1 -1 >
48.77
Septimal Quarter Tone


525/512
| -9 1 2 1 >
43.41
Avicennma
Avicenna's Enharmonic Diesis

50/49
| 1 0 2 -2 >
34.98
Tritonic Diesis
Jubilisma

64827/64000
| -9 3 -3 4 >
22.23
Squalentine


3125/3087
| 0 -2 5 -3 >
21.18
Gariboh


126/125
| 1 2 -3 1 >
13.79
Septimal Semicomma
Starling Comma

1029/1024
| -10 1 0 3 >
8.43
Gamelisma


6144/6125
| 11 1 -3 -2 >
5.36
Porwell


121/120
| -3 -1 -1 0 2 >
14.37
Biyatisma


176/175
| 4 0 -2 -1 1 >
9.86
Valinorsma


385/384
| -7 -1 1 1 1 >
4.50
Keenanisma


441/440
| -3 2 -1 2 -1 >
3.93
Werckisma


3025/3024
| -4 -3 2 -1 2 >
0.57
Lehmerisma



Armodue Theory (4-line staff)

Armodue: Italian pages of theory for 16-tone (esadekaphonic) system, including compositions.
Translations of parts of the Armodue pages can be found here on this wiki.

Books/Literature

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)

Compositions

Huckleberry Regional Preserve by City of the Asleep
Illegible Red Ink by City of the Asleep
Prenestyna Highway by Fabrizio Fulvio Fausto Fiale
Enantiodromia (album) by Last Sacrament
Tribute to Armodue by Aeterna
Etude in 16-tone equal tuning play (organ version) by Herman Miller
16-tone steel string acoustic diddle by Ron Sword
Armodue78 by Jean-Pierre Poulin
Palestrina Morta, fantasia quasi una sonata by Fabrizio Fulvio Fausto Fiale
Comets Over Flatland 5 by Randy Winchester
Malathion by Chris Vaisvil
Being of Vesta by Chris Vaisvil
Thin Ice by Chris Vaisvil ; information on the composition
Mavila Jazz Groove by William Lynch
Cold, Dark Night for a Dance by William Lynch
In Sospensione Neutra by Fabrizio Fulvio Fiale
546875:524288