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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".

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 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."

Another interesting approach can include two interwoven 8-EDO scales (narrow 12/11 neutral seconds). There are two major seventh intervals, a harmonic seventh at step 13\16, a 7/4 ratio approximation, sharp by 6.174 cents, followed by an undecimal 11/6 ratio or neutral seventh (which is mapped in 16's Mavila as a major seventh). If we 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 (pictured below).
external image 161928%20copy.jpg

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.
external image DSgoldsmith_piece.jpg

In 16-EDO diatonic scales are dissonant and "shimmery" because of the 25 cent raised superfourth in conjunction with the 25 cent subtracted fifth / poor 3/2 approximation. Scales like the Harmonic Minor scale in 16-EDO require 4 step sizes.
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. The 6-line 16-EDO "Mavila-[9] Staff" does just this, and places the arrangement (222122221) on nine white "natural" keys of the 16-EDO keyboard. 23-EDO also works with the Mavila-[9] 6-line staff, notated as 1/3 tones of 16-EDO. If the 9-note "Nonatonic" MOS is adapted for 16-EDO, then perhaps it makes sense to refer to Octaves as 2/1, "Decave".

Paul Erlich writes,
"Like the conventional 12-EDO diatonic and pentatonic (meantone) scales, these arise from tempering out a
unison vector from Fokker periodicity blocks. Only in 16-EDO, that unison vector is 135:128, instead of 81:80."

16 Tone Piano Layout Based on the 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.

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
[6]: 3 2 3 3 2 3
[10]: 2 1 2 1 2 2 1 2 1 2

Igliashon Jones writes: "The trouble (in 16-EDO) has ... to do with the fact that the distance between the major third
and the "fourth" is the same as the distance between the "fourth" and the "fifth" (i.e. near a 12/11)...This mean(s) that
135/128 (the difference between 16/15 and 9/8) is tempered out...."

Harmonizing Mavila/Armodue in 16 EDO


Because 16 edo doesn't approximate 3/2 well at all, triadic harmony based on thirds isn't a good option.
However triadic harmony can be based on on sevenths rather than thirds. For instance, 16 edo approximates 7/4 well enough to use
it in place of the usual 3/2. A triad can then be constructed by adding another seventh on top producing two possibilities for asymmetric
sevenths triads. a small one: 0-975-1050 called hard and a large one: 0-1050-975 called soft. In addition two other symmetrical triads
0-975-975 and 0-1050-1050 are also obvious possible chords.
Their characteristic metallic sound has earned them the name "Metallic triads".

MOS that support metallic harmony

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 0-1050-1050. Mavila 9 introduces two more soft triads while the Wilson scale introduces two more hard 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

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