16edo

From Xenharmonic Wiki
Jump to: navigation, search

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).

Best direct mapping, even if inconsistent

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

Patent val mapping

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
11/10, 20/11 15.004
16/13, 13/8 15.528
6/5, 5/3 15.641
7/5, 10/7 17.488
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
14/11, 11/7 32.492
7/6, 12/7 33.129
16/15, 15/8 38.269
18/13, 13/9 38.382
10/9, 9/5 42.596
15/14, 28/15 44.443
9/8, 16/9 53.910
9/7, 14/9 60.084

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.

alt : Your browser has no SVG support.

16ed2-001.svg

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

See also Mavila Temperament Modal Harmony.

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

The Foggy Road from Pasadena 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