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## Table of Contents

16-EDOis 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

## Intervals

16edo can be notated with conventional notation, including the staff, note names, relative notation, etc. in two ways. The first preserves the

melodicmeaning 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

harmonicmeaning 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)

Ratios*

major wider than minor

major narrower

than minor

Just

Names

Simplified

Septimal Whole-Tone

dim 4th

aug 4th

Supermajor 3rd

dim 5th

aug 5th

Dim 5th

dim 6th

Subminor 6th

Septimal Minor 7th

dim 8ve

aug 8ve

## 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).

## Selected just intervals by error

The following table shows how some prominent just intervals are represented in 16-EDO (ordered by absolute error).## 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".

dim 5th

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

## Rank two temperaments

List of 16et rank two temperaments by badnessper octave

MavilaDiminished[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 |.)

Armodue: Italian pages of theory for 16-tone (esadekaphonic) system, including compositions.Armodue Theory (4-line staff)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, 2011Sword, 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 AsleepIllegible 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