# 16edo

(Redirected from 16-edo)
Jump to navigation Jump to search
 ← 15edo 16edo 17edo →
Prime factorization 24
Step size 75¢
Fifth 9\16 (675¢)
Semitones (A1:m2) -1:3 (-75¢ : 225¢)
Dual sharp fifth 10\16 (750¢) (→5\8)
Dual flat fifth 9\16 (675¢)
Dual major 2nd 3\16 (225¢)
Consistency limit 7
Distinct consistency limit 5

16 equal divisions of the octave (abbreviated 16edo or 16ed2), also called 16-tone equal temperament (16tet) or 16 equal temperament (16et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 16 equal parts of exactly 75 ¢ each. Each step represents a frequency ratio of 21/16, or the 16th root of 2.

16edo's step size is sometimes called an eka, a term proposed by Luca Attanasio, from Sanskrit एक (éka, "one", "unit"),[1] when used as an interval size unit, especially in the context of Armodue theory.

## Theory

16edo is not especially good at representing most low-odd-limit 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 12edo, giving it four diminished seventh chords exactly like those of 12edo, and a diminished triad on each scale step.

### Odd harmonics

Approximation of odd harmonics in 16edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -27.0 -11.3 +6.2 +21.1 -26.3 -15.5 +36.7 -30.0 +2.5 -20.8 -28.3
Relative (%) -35.9 -15.1 +8.2 +28.1 -35.1 -20.7 +49.0 -39.9 +3.3 -27.7 -37.7
Steps
(reduced)
25
(9)
37
(5)
45
(13)
51
(3)
55
(7)
59
(11)
63
(15)
65
(1)
68
(4)
70
(6)
72
(8)

## 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 - this shouldn't be surprising as conventional interval arithmetic is designed for meantone/(super)pythagorean systems and 16edo is neither - e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G is not 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. This approach may seem bizarre at first. However, it carries over the way interval arithmetic and chord names work from diatonic notation. Furthermore, conventional 12edo music can be directly translated to 16edo "on the fly".

Alternatively, one can use Armodue nine-nominal notation; see Armodue theory

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 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, 16/13, 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 19/14, 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 28/19, 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, 13/8, 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 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 16edo as a 2.5.7.13.19.27 subgroup temperament; other approaches are possible.

## Approximation to JI

### Selected just intervals by error

The following tables show how 15-odd-limit intervals are represented in 16edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 16edo (direct approximation, even if inconsistent)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
11/6, 12/11 0.637 0.8
13/10, 20/13 4.214 5.6
7/4, 8/7 6.174 8.2
13/11, 22/13 10.790 14.4
5/4, 8/5 11.314 15.1
13/12, 24/13 11.427 15.2
15/11, 22/15 11.951 15.9
9/7, 14/9 14.916 19.9
11/10, 20/11 15.004 20.0
13/8, 16/13 15.528 20.7
5/3, 6/5 15.641 20.9
7/5, 10/7 17.488 23.3
9/8, 16/9 21.090 28.1
13/7, 14/13 21.702 28.9
15/13, 26/15 22.741 30.3
11/8, 16/11 26.318 35.1
3/2, 4/3 26.955 35.9
11/9, 18/11 27.592 36.8
15/14, 28/15 30.557 40.7
9/5, 10/9 32.404 43.2
11/7, 14/11 32.492 43.3
7/6, 12/7 33.129 44.2
13/9, 18/13 36.618 48.8
15/8, 16/15 36.731 49.0
15-odd-limit intervals in 16edo (patent val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
11/6, 12/11 0.637 0.8
13/10, 20/13 4.214 5.6
7/4, 8/7 6.174 8.2
13/11, 22/13 10.790 14.4
5/4, 8/5 11.314 15.1
13/12, 24/13 11.427 15.2
15/11, 22/15 11.951 15.9
11/10, 20/11 15.004 20.0
13/8, 16/13 15.528 20.7
5/3, 6/5 15.641 20.9
7/5, 10/7 17.488 23.3
13/7, 14/13 21.702 28.9
15/13, 26/15 22.741 30.3
11/8, 16/11 26.318 35.1
3/2, 4/3 26.955 35.9
11/9, 18/11 27.592 36.8
11/7, 14/11 32.492 43.3
7/6, 12/7 33.129 44.2
15/8, 16/15 38.269 51.0
13/9, 18/13 38.382 51.2
9/5, 10/9 42.596 56.8
15/14, 28/15 44.443 59.3
9/8, 16/9 53.910 71.9
9/7, 14/9 60.084 80.1

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.

## Notation

16edo 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 16edo.

Mos 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 16edo keyboard. If the 9-note "Enneatonic" MOS is adopted as a notational basis for 16edo, 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

### Armodue theory (4-line staff)

Armodue: Pierpaolo Beretta's website for his "Armodue" theory for 16edo (esadekaphonic), including compositions.

For translations of parts of the Armodue pages see the Armodue on this wiki

### Chord names

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

## 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.

16edo 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 16edo 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 (16edo supports both, but is not a very accurate tuning of either).

16edo 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 Jr writes of 16edo:

"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 can combine with the approximation of the harmonic seventh to form a 16:19:28 triad .

The interval between the 28th & 19th harmonics, 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

## Regular temperament properties

### Uniform maps

13-limit uniform maps between 15.5 and 16.5
Min. size Max. size Wart notation Map
15.5000 15.5387 16ccdeeffff 16 25 36 44 54 57]
15.5387 15.7197 16ccdeeff 16 25 36 44 54 58]
15.7197 15.7540 16deeff 16 25 37 44 54 58]
15.7540 15.8089 16dff 16 25 37 44 55 58]
15.8089 15.8512 16d 16 25 37 44 55 59]
15.8512 16.0431 16 16 25 37 45 55 59]
16.0431 16.0792 16e 16 25 37 45 56 59]
16.0792 16.0887 16ef 16 25 37 45 56 60]
16.0887 16.1504 16bef 16 26 37 45 56 60]
16.1504 16.2074 16bcef 16 26 38 45 56 60]
16.2074 16.3322 16bcddef 16 26 38 46 56 60]
16.3322 16.3494 16bcddeeef 16 26 38 46 57 60]
16.3494 16.5000 16bcddeeefff 16 26 38 46 57 61]

### Commas

16et tempers out the following commas. (Note: This assumes val 16 25 37 45 55 59].)

Prime
Limit
Ratio[note 1] Monzo Cents Color name Name
5 135/128 [-7 3 1 92.18 Layobi Major chroma
5 648/625 [3 4 -4 62.57 Quadgu Major diesis
5 3125/3072 [-10 -1 5 29.61 Laquinyo Magic comma
5 (20 digits) [23 6 -14 3.34 Sasepbiru Vishnuzma
7 36/35 [2 2 -1 -1 48.77 Rugu Septimal quartertone
7 525/512 [-9 1 2 1 43.41 Lazoyoyo Avicennma
7 50/49 [1 0 2 -2 34.98 Biruyo Jubilisma
7 64827/64000 [-9 3 -3 4 22.23 Laquadzo-atrigu Squalentine
7 3125/3087 [0 -2 5 -3 21.18 Triru-aquinyo Gariboh
7 126/125 [1 2 -3 1 13.79 Zotrigu Starling comma
7 1029/1024 [-10 1 0 3 8.43 Latrizo Gamelisma
7 6144/6125 [11 1 -3 -2 5.36 Sarurutrigu Porwell comma
11 121/120 [-3 -1 -1 0 2 14.37 Lologu Biyatisma
11 176/175 [4 0 -2 -1 1 9.86 Lorugugu Valinorsma
11 385/384 [-7 -1 1 1 1 4.50 Lozoyo Keenanisma
11 441/440 [-3 2 -1 2 -1 3.93 Luzozogu Werckisma
11 3025/3024 [-4 -3 2 -1 2 0.57 Loloruyoyo Lehmerisma

### Rank-2 temperaments

Important mosses include:

• magic anti-diatonic 3L4s 1414141 (5\16, 1\1)
• magic superdiatonic 3L7s 1311311311 (5\16, 1\1)
• Pathological magic chromatic 11121121112 3L10s (5\16, 1\1)
• mavila anti-diatonic 2L5s 2223223 (9\16, 1\1)
• mavila superdiatonic 7L2s 222212221 (9\16, 1\1)
• gorgo 5L1s 333331 (3\16, 1\1)
• lemba 4L2s 332332 (3\16, 1\2)
• Pathological 1L 12s 4 1 1 1 1 1 1 1 1 1 1 1 (1\16, 1\1)
• Pathological 1L 13s 3 1 1 1 1 1 1 1 1 1 1 1 1 1 (1\16, 1\1)
• Pathological 2L 12s 2 1 1 1 1 1 1 2 1 1 1 1 1 1 (1\16, 1\2)
• Pathological 1L 14s 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 (1\16, 1\1)

Temperaments listed by generator size:

Periods
per octave
Generator Temperaments
1 1\16 Valentine, slurpee
1 3\16 Gorgo
1 5\16 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 https://en.xen.wiki/w/File:MavilaAntidiatonic16edo.mp3 [9]: 1 2 2 2 1 2 2 2 2 https://en.xen.wiki/w/File:MavilaSuperdiatonic16edo.mp3

See also Mavila Temperament Modal Harmony.

Diminished

 [8]: 1 3 1 3 1 3 1 3 https://en.xen.wiki/w/File:Htgt16edo.mp3 [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

Because 16edo does not approximate 3/2 well at all, triadic harmony based on heptatonic thirds is not 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[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.

## Diagrams

16-tone piano layout based on the mavila[7]/antidiatonic scale

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".

Un-annotated diagram

Please explain this image.

Lumatone mapping

## Music

See also: Category:16edo tracks
Abnormality
Beheld
City of the Asleep
E8 Heterotic
Fabrizio Fiale
Aaron Andrew Hunt
Last Sacrament
William Lynch
Claudi Meneghin
Herman Miller
Nae Ayy
NullPointerException Music
Jean-Pierre Poulin
Ron Sword
Chris Vaisvil
Stephen Weigel
Randy Winchester
Woyten
Xotla
Nick Vuci
Zewen Senpai

## Notes

1. Ratios longer than 10 digits are presented by placeholders with informative hints