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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{interwiki
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| de = 16-EDO
: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-12-25 20:08:25 UTC</tt>.<br>
| en = 16edo
: The original revision id was <tt>602808826</tt>.<br>
| es = 16 EDO
: The revision comment was: <tt></tt><br>
| ja = 16平均律
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
}}
<h4>Original Wikitext content:</h4>
{{Infobox ET}}
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">&lt;span style="display: block; text-align: right;"&gt;[[xenharmonie/16edo|Deutsch]] - [[16平均律|日本語]]
{{ED intro}}
&lt;/span&gt;


[[toc]]
16edo's step size is sometimes called an '''eka''', a term proposed by [[Luca Attanasio]], from Sanskrit [[wikt:%E0%A4%8F%E0%A4%95#Sanskrit|एक]] (''éka'', "one", "unit"),<ref>[http://www.armodue.com/risorse.htm Armodue: le risorse di un nuovo sistema musicale]</ref> when used as an [[interval size unit]], especially in the context of [[Armodue]] theory.
**16-EDO** is the [[equal division of the octave]] into sixteen narrow chromatic semitones each of 75 [[cent]]s exactly. It is not especially good at representing most low-integer musical intervals, but it has a [[7_4|7/4]] which is six cents sharp, and a [[5_4|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 [[12edo|12-EDO]], and a diminished triad on each scale step.


=Images=  
== Theory ==
The [[3/2|perfect fifth]] of 16edo is 27 cents flat of 3/2, flatter than that of [[7edo]] so that it generates an [[2L 5s|antidiatonic]] instead of [[5L 2s|diatonic]] scale, but sharper than [[9edo]]'s fifth, to which it similarly retains the characteristic of being a fifth while being distinctly flat of 3/2. If the fifth is interpreted as 3/2, this befits a tuning of [[mavila]], the [[5-limit]] [[regular temperament|temperament]] that [[tempering out|tempers out]] [[135/128]], such that a stack of four fifths gives a [[6/5]] minor third instead of the familiar [[5/4]] major third as in [[meantone]]. A more accurate restriction is [[mabilic]], which discards the inaccurate mapping of 3 while keeping the fifth as a generator.


[[image:16edo wheel 01.png width="325" height="325"]]
This leads to some confusion in regards to interval names, as what would be major in diatonic now sounds minor; there are several ways to handle this (see in [[#Intervals]]).


=Intervals=
In general, 16edo tends to better approximate the differences between odd [[harmonic]]s than odd harmonics themselves, though it has a [[5/1|5th harmonic]] which is only 11 cents flat, and a [[7/1|7th harmonic]] which is only 6 cents sharp. As such, 16edo can be seen as an approach to tuning that takes advantage of the idea that simpler ratios can be functionally approximated with greater error (i.e. a 3/2 that's 25 cents flat is still recognizable, but a 5/4 that's 25 cents flat loses much of its identity and a 7/4 that's 25 cents flat is completely unrecognizable). In essence, 16edo's 3, 5, and 7 are backwards from 12edo's, with 7 being nearly perfect, 5 being decent, and 3 being distinctly out-of-tune.


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 the former is always higher pitched than the latter. 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.
In terms of higher primes, both 11 and 13 are approximated very flat, with the [[11/8]] not distinguished from [[4/3]], and [[13/8]] not distinguished from [[8/5]]. 16edo represents the no-9 no-15 [[25-odd-limit]] [[consistent]]ly, however.  


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, major is narrower than minor, etc. 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".
Four steps of 16edo gives the 300{{c}} minor third interval shared by [[12edo]] (and other multiples of [[4edo]]), which approximates [[6/5]], and thus tempers out 648/625, the [[diminished comma]]. This means that the familiar [[diminished seventh chord]] may be built on any scale step with four unique tetrads up to [[octave equivalence]]. The minor third is of course not distinguished from the septimal subminor third, [[7/6]], so [[36/35]] and moreover [[50/49]] are tempered out, making 16edo a possible tuning for [[diminished (temperament)|septimal diminished]]. Another possible interpretation for this interval is the 19th harmonic, [[19/16]].  


|| Degree || Cents ||= Approximate
16edo shares several similarities with 15edo. They both share mappings of [[8/7]], [[5/4]], and [[3/2]] in terms of edosteps – in fact, they are both [[valentine]] tunings, and thus [[slendric]] tunings. 16edo and 15edo also both have three types of seconds and two types of thirds (not including arto/tendo thirds). However, 15edo's fifth is sharp while 16's is flat.  
Ratios* || Interval
Names
Just ||= Interval
Names
Simplified ||||= Interval names,
major wider
than minor ||||= Interval names,
major narrower
than minor ||
|| 0 || 0 ||= 1/1 || Unison ||= unison ||= unison ||= D ||= unison ||= D ||
|| 1 || 75 ||= 28/27, 27/26 || Subminor 2nd ||= min 2nd ||= aug 1, dim 2nd ||= D#, Eb ||= dim 1, aug 2nd ||= Db, E# ||
|| 2 || 150 ||= 12/11, 35/32 || Neutral 2nd ||= maj 2nd ||= min 2nd ||= E ||= maj 2nd ||= E ||
|| 3 || 225 ||= 8/7 || Supermajor 2nd,
Septimal Whole-Tone ||= perf 2nd ||= maj 2nd ||= E# ||= min 2nd ||= Eb ||
|| 4 || 300 ||= 19/16, 32/27 || Minor 3rd ||= min 3rd ||= min 3rd ||= Fb ||= maj 3rd ||= F# ||
|| 5 || 375 ||= 5/4, 26/21 || Major 3rd ||= maj 3rd ||= maj 3rd ||= F ||= min 3rd ||= F ||
|| 6 || 450 ||= 13/10, 35/27 || Sub-4th,
Supermajor 3rd ||= min 4th ||= aug 3rd, dim 4th ||= F#, Gb ||= dim 3rd, aug 4th ||= Fb, G# ||
|| 7 || 525 ||= 27/20, 52/35, 256/189 || Wide 4th ||= maj 4th ||= perf 4th ||= G ||= perf 4th ||= G ||
|| 8 || 600 ||= 7/5, 10/7 || Tritone ||= aug 4th,
dim 5th ||= aug 4th, dim 5th ||= G#, Ab ||= dim 4th, aug 5th ||= Gb, A# ||
|| 9 || 675 ||= 40/27, 35/26, 189/128 || Narrow 5th ||= min 5th ||= perf 5th ||= A ||= perf 5th ||= A ||
|| 10 || 750 ||= 20/13, 54/35 || Super-5th,
Subminor 6th ||= maj 5th ||= aug 5th, dim 6th ||= A#, Bb ||= dim 5th, aug 6th ||= Ab, B# ||
|| 11 || 825 ||= 8/5, 21/13 || Minor 6th ||= min 6th ||= min 6th ||= B ||= maj 6th ||= B ||
|| 12 || 900 ||= 27/16, 32/19 || Major 6th ||= maj 6th ||= maj 6th ||= B# ||= min 6th ||= Bb ||
|| 13 || 975 ||= 7/4 || Subminor 7th,
Septimal Minor 7th ||= perf 7th ||= min 7th ||= Cb ||= maj 7th ||= C# ||
|| 14 || 1050 ||= 11/6, 64/35 || Neutral 7th ||= min 7th ||= maj 7th ||= C ||= min 7th ||= C ||
|| 15 || 1125 ||= 27/14, 52/27 || Supermajor 7th ||= maj 7th ||= aug 7th, dim 8ve ||= C# ||= dim 7th, aug 8ve ||= D# ||
|| 16 || 1200 ||= 2/1 || octave ||= octave ||= 8ve ||= D ||= 8ve ||= D ||
*based on treating 16-EDO as a 2.5.7.13.19.27 subgroup temperament; other approaches are possible.


==Selected just intervals by error==
16edo works as a tuning for [[extraclassical tonality]], due to its ultramajor third of 450 cents.  
The following table shows how [[Just-24|some prominent just intervals]] are represented in 16-EDO (ordered by absolute error).
|| **Interval, complement** || **Error (abs., in [[cent|cents]])** ||
||= [[12_11|12/11]], [[11_6|11/6]] ||= 0.637 ||
||= [[13_10|13/10]], [[20_13|20/13]] ||= 4.214 ||
||= [[8_7|8/7]], [[7_4|7/4]] ||= 6.174 ||
||= [[13_11|13/11]], [[22_13|22/13]] ||= 10.790 ||
||= [[5_4|5/4]], [[8_5|8/5]] ||= 11.314 ||
||= [[13_12|13/12]], [[24_13|24/13]] ||= 11.427 ||
||= [[15_11|15/11]], [[22_15|22/15]] ||= 11.951 ||
||= [[9_7|9/7]], [[14_9|14/9]] ||= 14.916 ||
||= [[11_10|11/10]], [[20_11|20/11]] ||= 15.004 ||
||= [[16_13|16/13]], [[13_8|13/8]] ||= 15.528 ||
||= [[6_5|6/5]], [[5_3|5/3]] ||= 15.641 ||
||= [[7_5|7/5]], [[10_7|10/7]] ||= 17.488 ||
||= [[9_8|9/8]], [[16_9|16/9]] ||= 21.090 ||
||= [[14_13|14/13]], [[13_7|13/7]] ||= 21.702 ||
||= [[15_13|15/13]], [[26_15|26/15]] ||= 22.741 ||
||= [[11_8|11/8]], [[16_11|16/11]] ||= 26.318 ||
||= [[4_3|4/3]], [[3_2|3/2]] ||= 26.955 ||
||= [[11_9|11/9]], [[18_11|18/11]] ||= 27.592 ||
||= [[15_14|15/14]], [[28_15|28/15]] ||= 30.557 ||
||= [[10_9|10/9]], [[9_5|9/5]] ||= 32.404 ||
||= [[14_11|14/11]], [[11_7|11/7]] ||= 32.492 ||
||= [[7_6|7/6]], [[12_7|12/7]] ||= 33.129 ||
||= [[18_13|18/13]], [[13_9|13/9]] ||= 36.618 ||
||= [[16_15|16/15]], [[15_8|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.
[[media type="custom" key="24810096"]]


[[file:16ed2-001.svg]]
=== Odd harmonics ===
{{Harmonics in equal|16}}


=Hexadecaphonic Octave Theory=  
=== Octave stretch ===
Having a flat tendency, 16et is best tuned with [[stretched octave]]s, which improve the accuracy of wide-voiced JI chords and [[rooted]] harmonics especially on inharmonic timbres such as bells and gamelans, with [[25edt]], [[41ed6]], and [[57ed12]] being good options.


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.
=== Subsets and supersets ===
Since 16 factors into primes as 2<sup>4</sup>, 16edo has subset edos {{EDOs| 2, 4, and 8 }}.


16-EDO is also a tuning for the [[Jubilismic clan|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**".
=== Composition theory ===
* [[User:VectorGraphics/16edo theory|Vector's approach]]
* [[Armodue harmony]]


[[Easley Blackwood]] writes of 16-EDO:
{{Todo|inline=1| expand }}
"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 [[8edo|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|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).
== Intervals ==
[[image:http://www.ronsword.com/161928%20copy.jpg width="316" height="179"]]
{{Mavila}}


The interval between the 28th &amp; 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.
Alternatively, one can use Armodue nine-nominal notation.


{| class="wikitable center-all"
|-
! rowspan="2" | Degree
! rowspan="2" | [[Cent]]s
! rowspan="2" | Approximate<br>ratios*
! colspan="6" | Names
|-
! colspan="2" | Antidiatonic
! colspan="2" | Diatonic
! Just
! Simplified
|-
| 0
| 0
| 1/1
| unison
| D
| unison
| D
| unison
| unison
|-
| 1
| 75
| 28/27, 27/26
| aug 1, dim 2nd
| D♯, E♭
| dim 1, aug 2nd
| D♭, 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
| E♭
| supermajor 2nd,<br>septimal whole-tone
| perf 2nd
|-
| 4
| 300
| 19/16, 32/27
| minor 3rd
| F♭
| 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,<br>dim 4th
| F♯, G♭
| dim 3rd,<br>aug 4th
| F♭, G♯
| sub-4th,<br>supermajor 3rd
| min 4th
|-
| 7
| 525
| 19/14, 27/20, 35/26, 256/189
| perfect 4th
| G
| perfect 4th
| G
| wide 4th
| maj 4th
|-
| 8
| 600
| 7/5, 10/7
| aug 4th,<br>dim 5th
| G♯, A♭
| dim 4th,<br>aug 5th
| G♭, A♯
| tritone
| aug 4th,<br>dim 5th
|-
| 9
| 675
| 28/19, 40/27, 52/35, 189/128
| perfect 5th
| A
| perfect 5th
| A
| narrow 5th
| min 5th
|-
| 10
| 750
| 20/13, 54/35
| aug 5th,<br>dim 6th
| A♯, B♭
| dim 5th,<br>aug 6th
| A♭, B♯
| super-5th,<br>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
| B♭
| major 6th
| maj 6th
|-
| 13
| 975
| 7/4
| minor 7th
| C♭
| major 7th
| C♯
| subminor 7th,<br>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,<br>dim 8ve
| C♯, D♭
| dim 7th,<br>aug 8ve
| C♭, D♯
| supermajor 7th
| maj 7th
|-
| 16
| 1200
| 2/1
| 8ve
| D
| 8ve
| D
| octave
| octave
|}
<nowiki />* Based on treating 16edo as a 2.5.7.13.19.27 subgroup temperament; other approaches are possible.


=Hexadecaphonic Notation=  
== Notation ==
16edo notation can be easy utilizing [[Goldsmith's Circle]] of keys, nominals, and respective notation{{clarify}}. The nominals for a 6 line staff can be switched for [[Erv Wilson]]'s Beta and Epsilon additions to A–G. The Armodue model uses a 4-line staff for 16edo.


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.
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 ♯/♭ 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 theory|Armodue notation]] of 16edo "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 the octave ([[2/1]]) as the "[[decave]]". This is identical to the KISS notation for this scale when using numbers.
[[image:http://ronsword.com/DSgoldsmith_piece.jpg width="1008" height="342"]]
----
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. [[23edo|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,
{| class="wikitable center-all"
"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."
! Degree
! Cents
! colspan="2" | Mavila[9] notation
|-
| 0
| 0
| unison
| 1
|-
| 1
| 75
| aug unison, minor 2nd
| 1♯, 2♭
|-
| 2
| 150
| major 2nd
| 2
|-
| 3
| 225
| aug 2nd, minor 3rd
| 2♯, 3♭
|-
| 4
| 300
| major 3rd, dim 4th
| 3, 4𝄫
|-
| 5
| 375
| minor 4th
| 4♭
|-
| 6
| 450
| major 4th,<br>dim 5th
| 4, 5♭
|-
| 7
| 525
| aug 4th, minor 5th
| 4♯, 5
|-
| 8
| 600
| aug 5th, dim 6th
| 5♯, 6♭
|-
| 9
| 675
| perfect 6th, dim 7th
| 6, 7𝄫
|-
| 10
| 750
| aug 6th, minor 7th
| 6♯, 7♭
|-
| 11
| 825
| major 7th
| 7
|-
| 12
| 900
| aug 7th, minor 8th
| 7♯, 8♭
|-
| 13
| 975
| major 8th, dim 9th
| 8, 9𝄫
|-
| 14
| 1050
| minor 9th
| 9
|-
| 15
| 1125
| major 9th, dim 10ve
| 9♯, 1♭
|-
| 16
| 1200
| 10ve (Decave)
| 1
|}


==16 Tone Piano Layout Based on the Anti-diatonic Scale==  
=== Sagittal notation ===
[[image:16-EDO-PIano-Diagram.png width="748" height="293"]]
This notation uses the same sagittal sequence as [[21edo #Sagittal notation|21edo]].
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=
<imagemap>
[[List of 16et rank two temperaments by badness]]
File:16-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 471 0 631 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 471 106 [[Fractional_3-limit_notation#Bad-fifths_limma-fraction_notation | limma-fraction notation]]
default [[File:16-EDO_Sagittal.svg]]
</imagemap>


||~ Periods
=== Armodue notation (4-line staff) ===
per octave ||~ Generator ||~ Temperaments ||
[http://www.armodue.com/ricerche.htm Armodue]: Pierpaolo Beretta's website for his Armodue theory for 16edo (esadekaphonic), including compositions.
|| 1 || 1\16 ||= [[Valentine]], [[Slurpee|slurpee]] ||
 
|| 1 || 3\16 ||= [[Gorgo]] ||
For resources on the Armodue theory, see the [[Armodue]] on this wiki
|| 1 || 5\16 ||= Messed-up [[Magic|magic]]/muggles ||
 
|| 1 || 7\16 ||= [[Mavila]]/armodue ||
== Chord names ==
|| 2 || 1\16 ||= [[Bipelog]] ||
16edo chords can be named using ups and downs. Using diatonic interval names, chord names bear little relationship to the sound: a minor chord (spelled {{dash|A, C, E|med}}) sounds like [[4:5:6]], the classical major triad, and a major chord (spelled {{dash|C, E, G|med}}) sounds like [[10:12:15]], a classical minor triad! Instead, using antidiatonic names, the chord names will match the sound&mdash;but finding the name from the spelling follows the rules of antidiatonic rather than diatonic interval arithmetic.
|| 2 || 3\16 ||= [[Lemba]], [[Astrology|astrology]] ||
 
|| 4 || 1\16 ||= [[Diminished]]/demolished ||
{| class="wikitable center-all"
|| 8 || 1\16 ||=   ||
|-
! rowspan="2" | Chord
! rowspan="2" | JI ratios
! colspan="6" | Name
|-
! colspan="3" | Diatonic
! colspan="3" | Antidiatonic
|-
| {{dash|0, 5, 9|med}}
| 4:5:6
| D F A
| Dm
| D minor
| D F A
| D
| D major
|-
| {{dash|0, 4, 9|med}}
| 10:12:15
| D F♯ A
| D
| D major
| D F♭ A
| Dm
| D minor
|-
| {{dash|0, 4, 8|med}}
| 5:6:7
| D F♯ A♯
| Daug
| D augmented
| D F♭ A♭
| Ddim
| D diminished
|-
| {{dash|0, 5, 10|med}}
|
| D F A♭
| Ddim
| D diminished
| D F A♯
| Daug
| D augmented
|-
| {{dash|0, 5, 9, 13|med}}
| 4:5:6:7
| D F A C♯
| Dm(M7)
| D minor-major
| D F A C♭
| D7
| D seven
|-
| {{dash|0, 5, 9, 12|med}}
|
| D F A Bb
| Dm(♭6)
| D minor flat-six
| D F A B♯
| D6
| D six
|-
| {{dash|0, 5, 9, 14|med}}
|
| D F A C
| Dm7
| D minor seven
| D F A C
| DM7
| D major seven
|-
| {{dash|0, 4, 9, 13|med}}
|  
| D F♯ A C♯
| DM7
| D major seven
| D F♭ A C♭
| 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 {{dash|6, 1, 3, 5, 7, 9, 11, 13}}). See [[Ups and downs notation #Chords and chord progressions]] for more examples.
 
Using antidiatonic names, if you're used to diatonic interval arithmetic, you can do antidiatonic interval arithmetic by following the simple guideline that qualities are '''reversed''' from standard diatonic. As in, just as adding two major seconds gives you a major third in 12edo, adding two minor seconds gives a minor third in 16edo.
 
That is, reversing means exchanging major for minor, aug for dim, and sharp for flat. Perfect and natural are unaffected.
 
Examples can be found at the bottom of the page.
 
== Approximation to JI ==
=== Selected just intervals by error ===
{{Q-odd-limit intervals|16}}
 
It's worth noting that the 525{{c}} 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.
 
[[File:16ed2-001.svg|alt=alt : Your browser has no SVG support.]]
 
[[:File:16ed2-001.svg|16ed2-001.svg]]
 
== 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{{c}}, is smaller than ideal. Its very flat 3/2 of 675{{c}} [[support]]s Mavila temperament, where the mapping of major and minor is reversed. The temperament could be popular for its 150{{c}} "3/4-tone" equal division of the traditional 300{{c}} minor third.
 
16edo is also a tuning for the [[Jubilismic clan|no-threes 7-limit temperament tempering out 50/49]]. This has a period of a half-octave (600{{c}}), 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 [http://x31eq.com/cgi-bin/uv.cgi?uvs=%5B-19%2C7%2C1%3E&limit=2_5_7 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{{c}}, and take the 300{{c}} minor third as an approximation of the harmonic 19th ([[19/16]], approximately 297.5{{c}}), that can combine with the approximation of the harmonic seventh to form a 16:19:28 triad .
 
The interval between the 28th &amp; 19th harmonics, 28:19, measures approximately 671.3{{c}}, which is 3.7{{c}} 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{{c}} just, 525.0{{c}} in 16edo). A perhaps more consonant open voicing is 7:16:19
 
== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|edo=16}}
 
=== Commas ===
16et [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes [[val]] {{val| 16 25 37 45 55 59 }}.)
 
{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime<br>limit]]
! [[Ratio]]<ref group=note>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 5
| [[135/128]]
| {{monzo| -7 3 1 }}
| 92.18
| Layobi
| Mavila comma, major chroma
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, major diesis
|-
| 5
| [[3125/3072]]
| {{monzo| -10 -1 5 }}
| 29.61
| Laquinyo
| Magic comma
|-
| 5
| [[6115295232/6103515625|(20 digits)]]
| {{monzo| 23 6 -14 }}
| 3.34
| Sasepbiru
| [[Vishnuzma]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quartertone
|-
| 7
| [[525/512]]
| {{monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64827/64000]]
| {{monzo| -9 3 -3 4 }}
| 22.23
| Laquadzo-atrigu
| Squalentine comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[1029/1024]]
| {{monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[6144/6125]]
| {{monzo| 11 1 -3 -2 }}
| 5.36
| Sarurutrigu
| Porwell comma
|-
| 11
| [[121/120]]
| {{monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[385/384]]
| {{monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[3025/3024]]
| {{monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|}
 
=== Rank-2 temperaments ===
* [[List of 16et rank two temperaments by badness]]
 
{| class="wikitable center-1 center-2"
|+ Table of temperaments by generator
|-
! Periods<br>per 8ve
! 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 (temperament)|Diminished]]/[[demolished]]
|-
| 8
| 1\16
| [[Semidim]]
|}
 
== Scales ==
* {{Main|List of MOS scales in {{PAGENAME}}}}
Important mosses include:
* [[magic]] anti-diatonic 3L4s 1414141 (5\16, 1\1)
* [[magic]] superdiatonic 3L7s 1311311311 (5\16, 1\1)
* [[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)
 
 
'''Mavila'''
 
{| class="wikitable"
|-
| [5]:
| 5 2 5 2 2
|
|-
| [7]:
| 3 2 2 3 2 2 2
|[[File:MavilaAntidiatonic16edo.mp3]]
|-
| [9]:
| 1 2 2 2 1 2 2 2 2
|[[File:MavilaSuperdiatonic16edo.mp3]]
|}
See also [[Mavila Temperament Modal Harmony]].
 
'''Diminished'''
 
{| class="wikitable"
|-
| [8]:
| 1 3 1 3 1 3 1 3
|[[File:htgt16edo.mp3]]
|-
| [12]:
| 1 1 2 1 1 2 1 1 2 1 1 2
|
|}
 
'''Magic'''


**Mavila**
|| [5]: || 5 2 5 2 2 ||  ||
|| [7]: || 3 2 2 3 2 2 2 || [[media type="file" key="MavilaAntidiatonic16edo.mp3" width="240" height="20"]] ||
|| [9]: || 1 2 2 2 1 2 2 2 2 || [[media type="file" key="MavilaSuperdiatonic16edo.mp3" width="240" height="20"]] ||
**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
[7]: 1 4 1 4 1 4 1
[10]: 1 3 1 1 3 1 1 1 3 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
[13]: 1 1 2 1 1 1 2 1 1 1 2 1 1
**Cynder/Gorgo**
 
'''Cynder/Gorgo'''
 
[5]: 3 3 4 3 3
[5]: 3 3 4 3 3
[6]: 3 3 1 3 3 3
[6]: 3 3 1 3 3 3
[11]: 1 2 1 2 1 2 1 2 1 2 1
[11]: 1 2 1 2 1 2 1 2 1 2 1
**Lemba**
 
'''Lemba/Astrology'''
 
[4]: 3 5 3 5
 
[6]: 3 2 3 3 2 3
[6]: 3 2 3 3 2 3
[10]: 2 1 2 1 2 2 1 2 1 2
[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
== Metallic harmony ==
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
In 16edo, triadic harmony can be based on on heptatonic sevenths (or seconds) rather than thirds. For instance, 16edo approximates 7/4 well enough to use
135/128 (the difference between 16/15 and 9/8) is tempered out...."
 
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{{c}}). Stacking these two intervals reaches 2025{{c}}, or a minor 6th plus an octave. Thus the out-of-tune 675{{c}} 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, {{nowrap|{{dash|0, 975, 2025{{c}}}}}}, and a large one, {{nowrap|{{dash|0, 1050, 2025{{c}}}}}}. 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 {{nowrap|{{dash|0, 975, 1950{{c}}}}}}, and a wide symmetrical triad at {{nowrap|{{dash|0, 1050, 2100{{c}}}}}}. 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{{c}}. In Mavila[9], hard and soft triads cease to share a triad class, as 975{{c}} is a major 8th, while 1050{{c}} 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, according to antidiatonic notation.
 
[[File:16-EDO-PIano-Diagram.png|alt=16-EDO-PIano-Diagram.png|748x293px|16-EDO-PIano-Diagram.png]]
 
'''Interleaved edos'''
 
A visualization of 16edo being two interleaved copies of [[8edo]] and four interleaved copies of [[4edo]].
 
[[File:16edo_wheel_01.png|alt=16edo wheel 01.png|325x325px|16edo wheel 01.png]]
 
=== Lumatone mapping ===
 
See: [[Lumatone mapping for 16edo]]
 
== Interval arithmetic examples ==
These examples show the correspondence between interval arithmetic using diatonic and antidiatonic notation.
{| class="wikitable" style="text-align: center;"
! colspan="2" |Diatonic (i.e. 12edo)
! colspan="2" |Antidiatonic (i.e. 16edo)
|-
! Question
! Result
! Question
! Result
|-
| M2 + M2
| aug3
| m2 + m2
| dim3
|-
| D to F♯
| aug3
| D to F♭
| dim3
|-
| D to F
| M3
| D to F
| m3
|-
| E♭ + m3
| Gbb
| E♯ + M3
| G♯♯
|-
| E♭ + P5
| B♭
| E♯ + P5
| B♯
|-
| A minor chord
| A C♭ E
| A major chord
| A C♯ E
|-
| E♭ major chord
| E♭ G♭ D♭
| E♯ minor chord
| E♯ G♯ B♯
|-
| Gm7 = G + m3 + P5 + m7
| G B D F♭
| G + M3 + P5 + M7
| G B D F♯
|-
| A♭7aug = A♭ + M3 + A5 + m7
| A♭ C♭ E Gbb
| A♯ + m3 + d5 + M7
| A♯ C♯ E G♯♯
|-
| what chord is D F A♯?
| D + M3 + A5 = Daug
| D F A♭
| D + m3 + d5
|-
| what chord is C E G♭ B♭?
| C + m3 + d5 + d7 = Cdim7
| C E G♯ B♯
| C + M3 + A5 + A7
|-
| C major scale = C + M2 + M3<br>+ P4 + P5 + M6 + M7 + P8
| C D♯ E♯ F<br>G A♯ B♯ C
| C + m2 + m3 + P4<br>+ P5 + m6 + m7 + P8
| C D♭ E♭ F<br>G A♭ B♭ C
|-
| C minor scale = C + M2 + m3<br>+ P4 + P5 + m6 + m7 + P8
| C D♯ E F<br>G A B C
| C + m2 + M3 + P4<br>+ P5 + M6 + M7 + P8
| C D♭ E F<br>G A B C
|-
| what scale is A B♯ C♭ D<br>E F G♭ A?
| A + M2 + m3 + P4<br>+ P5 + M6 + m7 = A dorian
| A B♭ C♯ D<br>E F G♯ A
| A + m2 + M3 + P4<br>+ P5 + m6 + M7
|}
 
== Music ==
{{Catrel| 16edo tracks }}
 
; [[Abnormality]]
* [https://www.youtube.com/watch?v=zao6E8GdQh0 ''it's not not opposite day''] (2023)
* [https://www.youtube.com/watch?v=1pa3dztk8o0 ''nightfall''] (2024)
 
; [[Beheld]]
* [https://www.youtube.com/watch?v=kzPeVB2mncc ''Nebulous vibe'']
 
; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/huckleberry-regional-preserve ''Huckleberry Regional Preserve'']
* [https://cityoftheasleep.bandcamp.com/track/illegible-red-ink ''Illegible Red Ink'']
 
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/IfVvjoRqqNk ''16edo jam''] (2025)
* [https://www.youtube.com/watch?v=cUgbkkIvy0g ''Waltz in 16edo''] (2025)
 
; [[E8 Heterotic]]
* [https://youtu.be/a8Jgb_XIj7c "Hexed"]


==Harmonizing Mavila/Armodue in 16 EDO==
; [[Fabrizio Fiale]]
* [https://www.soundclick.com/music/songInfo.cfm?songID=12370649 ''Prenestyna Highway'']
* [https://www.soundclick.com/music/songInfo.cfm?songID=7715803 ''Palestrina Morta, fantasia quasi una sonata'']
* [https://soundcloud.com/fff-fiale/in-sospensione-neutra ''In Sospensione Neutra'']


Because 16 edo doesn't approximate 3/2 well at all, triadic harmony based on thirds isn't a good option.
; [[Aaron Andrew Hunt]]
However triadic harmony can be based on on sevenths rather than thirds. For instance, 16 edo approximates 7/4 well enough to use
* [https://soundcloud.com/uz1kt3k/fuga-a3-in-16et ''Fuga a3 in 16ET'']
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===
; [[Last Sacrament]]
Mavila 7 contains two hard triads on degrees 1 and 4 and two soft triads on degrees 2 and 6. The other three chords
* [http://lastsacrament.bandcamp.com/album/enantiodromia ''Enantiodromia''] (album) (from 2013)
are wide symmetrical triads 0-1050-1050. Mavila 9 introduces two more soft triads while the Wilson scale introduces two more hard triads.
* [https://lastsacrament.bandcamp.com/album/maniacal-meditations-ep ''Maniacal Meditations''] (EP) (2013 EP)


See [[Metallic Harmony]].
; [[William Lynch]]
* [[:File:Mavila_Jazz_Rhodes_1.mp3|''Mavila Jazz Groove'']]
* [[:File:mavila4.mp3|''Cold, Dark Night for a Dance'']]


; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=vIWxP_C0aUM ''Mavila Fugue'']
* [https://www.youtube.com/watch?v=KYkmT46oGhw ''Canon at the Semitone on The Mother's Malison Theme'', for Cor Anglais and Violin] ([https://www.youtube.com/watch?v=I6BUauD8EaE for Organ])
* [https://www.youtube.com/watch?v=P7LUSRd1kMg ''Canon on Twinkle Twinkle Little Star'', for Organ] (2023) ([https://www.youtube.com/watch?v=QHJYyqge_JQ for Baroque Oboe and Viola])
* [https://www.youtube.com/shorts/I4-URAGgQMQ ''Baroque Micropiece in 16edo''] (2024)


=Commas=
; [[Herman Miller]]
16 EDO [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes [[val]] &lt; 16 25 37 45 55 59 |.)
* [http://www.io.com/%7Ehmiller/midi/16tet.mid ''Etude in 16-tone equal tuning'']{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/16tet.mp3 play]{{dead link}} ([http://soonlabel.com/xenharmonic/archives/2604 organ version]{{dead link}})
||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 135/128 ||&lt; | -7 3 1 &gt; ||&gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||= 648/625 ||&lt; | 3 4 -4 &gt; ||&gt; 62.57 ||= Major Diesis ||= Diminished Comma ||=  ||
||= 3125/3072 ||&lt; | -10 -1 5 &gt; ||&gt; 29.61 ||= Small Diesis ||= Magic Comma ||=  ||
||=  ||&lt; | 23 6 -14 &gt; ||&gt; 3.34 ||= Vishnuzma ||= Semisuper ||=  ||
||= 36/35 ||&lt; | 2 2 -1 -1 &gt; ||&gt; 48.77 ||= Septimal Quarter Tone ||=  ||=  ||
||= 525/512 ||&lt; | -9 1 2 1 &gt; ||&gt; 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||=  ||
||= 50/49 ||&lt; | 1 0 2 -2 &gt; ||&gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||=  ||
||= 64827/64000 ||&lt; | -9 3 -3 4 &gt; ||&gt; 22.23 ||= Squalentine ||=  ||=  ||
||= 3125/3087 ||&lt; | 0 -2 5 -3 &gt; ||&gt; 21.18 ||= Gariboh ||=  ||=  ||
||= 126/125 ||&lt; | 1 2 -3 1 &gt; ||&gt; 13.79 ||= Septimal Semicomma ||= Starling Comma ||=  ||
||= 1029/1024 ||&lt; | -10 1 0 3 &gt; ||&gt; 8.43 ||= Gamelisma ||=  ||=  ||
||= 6144/6125 ||&lt; | 11 1 -3 -2 &gt; ||&gt; 5.36 ||= Porwell ||=  ||=  ||
||= 121/120 ||&lt; | -3 -1 -1 0 2 &gt; ||&gt; 14.37 ||= Biyatisma ||=  ||=  ||
||= 176/175 ||&lt; | 4 0 -2 -1 1 &gt; ||&gt; 9.86 ||= Valinorsma ||=  ||=  ||
||= 385/384 ||&lt; | -7 -1 1 1 1 &gt; ||&gt; 4.50 ||= Keenanisma ||=  ||=  ||
||= 441/440 ||&lt; | -3 2 -1 2 -1 &gt; ||&gt; 3.93 ||= Werckisma ||=  ||=  ||
||= 3025/3024 ||&lt; | -4 -3 2 -1 2 &gt; ||&gt; 0.57 ||= Lehmerisma ||=  ||=  ||


=**Armodue Theory (4-line staff)**=  
; [[Nae Ayy]]
[[http://www.armodue.com/ricerche.htm|Armodue]]: Italian pages of theory for 16-tone (esadekaphonic) system, including compositions.
* [https://www.youtube.com/watch?v=H74psBvdeT4 ''Rambling'']
Translations of parts of the Armodue pages can be found [[Armodue|here]] on this wiki.
* [https://www.youtube.com/watch?v=OAhV8ol2Hbw ''a n g e r y'']
* [https://www.youtube.com/watch?v=-MboZelse90 ''Maundering'']


=Books/Literature=
; [[NullPointerException Music]]
Sword, Ronald. "Thesaurus of Melodic Patterns and Intervals for 16-Tones" IAAA Press, USA. First Ed: August, 2011
* [https://www.youtube.com/watch?v=LXsZIbT6wpM ''Edolian - Seventhic''] (2020)
Sword, Ronald. "Hexadecaphonic Scales for Guitar." IAAA Press, UK-USA. First Ed: Feb, 2010. (superfourth tuning)
* [https://www.youtube.com/watch?v=UrQPr7V9feA ''Finality''] (2021)
Sword, Ronald. "Esadekaphonic Scales for Guitar." IAAA Press, UK-USA. First Ed: April, 2009. (semi-diminished fourth tuning)


=Compositions=
; [[Jean-Pierre Poulin]]
[[@http://www.soundclick.com/bands/page_songInfo.cfm?bandID=660895&amp;songID=12370649&amp;showPlayer=true|Prenestyna Highway]] by [[@http://fiale.tk|Fabrizio Fulvio Fausto Fiale]]
* [http://www.jeanpierrepoulin.com/mp3/Armodue78.mp3 ''Armodue78'']
[[@http://lastsacrament.bandcamp.com/album/enantiodromia|Enantiodromia (album)]] by Last Sacrament
[[@http://www.cdbaby.com/cd/aeternamusic|Tribute to Armodue]] by Aeterna
[[http://www.io.com/%7Ehmiller/midi/16tet.mid|Etude in 16-tone equal tuning]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/16tet.mp3|play]] ([[@http://soonlabel.com/xenharmonic/archives/2604|organ version]]) by [[Herman Miller|Herman Miller ]]
[[http://www.ronsword.com/sounds/Ron%20Sword%20-%2016-tone%20acoustic%20improvisation.mp3|16-tone steel string acoustic diddle]] by Ron Sword
[[http://www.jeanpierrepoulin.com/mp3/Armodue78.mp3|Armodue78]] by [[@Jean-Pierre Poulin]]
[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Fiale/ffffiale+palestrinamortafantasiaquasiunasonata.mp3|Palestrina Morta, fantasia quasi una sonata]] by [[@http://fiale.tk|Fabrizio Fulvio Fausto Fiale]]
[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/05%20-%205.%2016%20octave.mp3|Comets Over Flatland 5]] by [[Randy Winchester]]
[[http://micro.soonlabel.com/16-ET/20120527-16-malathion.mp3|Malathion]] by [[http://chrisvaisvil.com/?p=2358#comments|Chris Vaisvil ]]
[[http://micro.soonlabel.com/16-ET/20130216_16edo_vesta.mp3|Being of Vesta]] by [[http://chrisvaisvil.com/?p=3061|Chris Vaisvil]]
[[http://micro.soonlabel.com/simultaneous-tunings/20130607_thin_ice_christiane.mp3|Thin Ice]] by [[Chris Vaisvil]] ; [[http://chrisvaisvil.com/thin-ice-for-alto-female-choir-harp-and-percussion-in-adaptive-ji-16-edo-and-8-edo/|information on the composition]]
[[file:Mavila Jazz Rhodes 1.mp3|Mavila Jazz Groove]] by [[https://soundcloud.com/pianofreak96|William Lynch]]
[[file:mavila4.mp3|Cold, Dark Night for a Dance ]]by [[https://soundcloud.com/pianofreak96|William Lynch]]
[[https://soundcloud.com/fff-fiale/in-sospensione-neutra|In Sospensione Neutra by Fabrizio Fulvio Fiale]]</pre></div>
<h4>Original HTML content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;16edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;span style="display: block; text-align: right;"&gt;&lt;a class="wiki_link" href="http://xenharmonie.wikispaces.com/16edo"&gt;Deutsch&lt;/a&gt; - &lt;a class="wiki_link" href="/16%E5%B9%B3%E5%9D%87%E5%BE%8B"&gt;日本語&lt;/a&gt;&lt;br /&gt;
&lt;/span&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextTocRule:29:&amp;lt;img id=&amp;quot;wikitext@@toc@@normal&amp;quot; class=&amp;quot;WikiMedia WikiMediaToc&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/normal?w=225&amp;amp;h=100&amp;quot;/&amp;gt; --&gt;&lt;div id="toc"&gt;&lt;h1 class="nopad"&gt;Table of Contents&lt;/h1&gt;&lt;!-- ws:end:WikiTextTocRule:29 --&gt;&lt;!-- ws:start:WikiTextTocRule:30: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Images"&gt;Images&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:30 --&gt;&lt;!-- ws:start:WikiTextTocRule:31: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Intervals"&gt;Intervals&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:31 --&gt;&lt;!-- ws:start:WikiTextTocRule:32: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Intervals-Selected just intervals by error"&gt;Selected just intervals by error&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:32 --&gt;&lt;!-- ws:start:WikiTextTocRule:33: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Hexadecaphonic Octave Theory"&gt;Hexadecaphonic Octave Theory&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:33 --&gt;&lt;!-- ws:start:WikiTextTocRule:34: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Hexadecaphonic Notation"&gt;Hexadecaphonic Notation&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:34 --&gt;&lt;!-- ws:start:WikiTextTocRule:35: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Hexadecaphonic Notation-16 Tone Piano Layout Based on the Anti-diatonic Scale"&gt;16 Tone Piano Layout Based on the Anti-diatonic Scale&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:35 --&gt;&lt;!-- ws:start:WikiTextTocRule:36: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Rank two temperaments"&gt;Rank two temperaments&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:36 --&gt;&lt;!-- ws:start:WikiTextTocRule:37: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Rank two temperaments-Harmonizing Mavila/Armodue in 16 EDO"&gt;Harmonizing Mavila/Armodue in 16 EDO&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:37 --&gt;&lt;!-- ws:start:WikiTextTocRule:38: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Rank two temperaments-Harmonizing Mavila/Armodue in 16 EDO-MOS that support metallic harmony"&gt;MOS that support metallic harmony&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:38 --&gt;&lt;!-- ws:start:WikiTextTocRule:39: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Commas"&gt;Commas&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:39 --&gt;&lt;!-- ws:start:WikiTextTocRule:40: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Armodue Theory (4-line staff)"&gt;Armodue Theory (4-line staff)&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:40 --&gt;&lt;!-- ws:start:WikiTextTocRule:41: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Books/Literature"&gt;Books/Literature&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:41 --&gt;&lt;!-- ws:start:WikiTextTocRule:42: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Compositions"&gt;Compositions&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:42 --&gt;&lt;!-- ws:start:WikiTextTocRule:43: --&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:43 --&gt;&lt;strong&gt;16-EDO&lt;/strong&gt; is the &lt;a class="wiki_link" href="/equal%20division%20of%20the%20octave"&gt;equal division of the octave&lt;/a&gt; into sixteen narrow chromatic semitones each of 75 &lt;a class="wiki_link" href="/cent"&gt;cent&lt;/a&gt;s exactly. It is not especially good at representing most low-integer musical intervals, but it has a &lt;a class="wiki_link" href="/7_4"&gt;7/4&lt;/a&gt; which is six cents sharp, and a &lt;a class="wiki_link" href="/5_4"&gt;5/4&lt;/a&gt; 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 &lt;a class="wiki_link" href="/12edo"&gt;12-EDO&lt;/a&gt;, and a diminished triad on each scale step.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:3:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Images"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:3 --&gt;Images&lt;/h1&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextLocalImageRule:909:&amp;lt;img src=&amp;quot;/file/view/16edo%20wheel%2001.png/516947132/325x325/16edo%20wheel%2001.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 325px; width: 325px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/16edo%20wheel%2001.png/516947132/325x325/16edo%20wheel%2001.png" alt="16edo wheel 01.png" title="16edo wheel 01.png" style="height: 325px; width: 325px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:909 --&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:5:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Intervals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:5 --&gt;Intervals&lt;/h1&gt;
&lt;br /&gt;
16edo can be notated with conventional notation, including the staff, note names, relative notation, etc. in two ways. The first preserves the &lt;u&gt;melodic&lt;/u&gt; meaning of sharp/flat, major/minor and aug/dim, in that the former is always higher pitched than the latter. 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.&lt;br /&gt;
&lt;br /&gt;
The second approach is to preserve the &lt;u&gt;harmonic&lt;/u&gt; 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, major is narrower than minor, etc. 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 &amp;quot;on the fly&amp;quot;.&lt;br /&gt;
&lt;br /&gt;


; [[Ron Sword]]
* [https://soundcloud.com/ron-sword/mavila-fog ''The Foggy Road from Pasadena'']{{dead link}}


&lt;table class="wiki_table"&gt;
; [[Chris Vaisvil]]
    &lt;tr&gt;
* [http://micro.soonlabel.com/16-ET/20120527-16-malathion.mp3 ''Malathion''] - [http://chrisvaisvil.com/?p=2358 details]
        &lt;td&gt;Degree&lt;br /&gt;
* [http://micro.soonlabel.com/16-ET/20130216_16edo_vesta.mp3 ''Being of Vesta''] - [http://chrisvaisvil.com/?p=3061 details]
&lt;/td&gt;
* [http://micro.soonlabel.com/simultaneous-tunings/20130607_thin_ice_christiane.mp3 ''Thin Ice''] - [http://chrisvaisvil.com/?p=3354 details]
        &lt;td&gt;Cents&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Approximate&lt;br /&gt;
Ratios*&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Interval &lt;br /&gt;
Names&lt;br /&gt;
Just&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Interval&lt;br /&gt;
Names&lt;br /&gt;
Simplified&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2" style="text-align: center;"&gt;Interval names,&lt;br /&gt;
major wider &lt;br /&gt;
than minor&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2" style="text-align: center;"&gt;Interval names, &lt;br /&gt;
major narrower&lt;br /&gt;
than minor&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;0&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;1/1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Unison&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;unison&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;unison&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;unison&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;75&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;28/27, 27/26&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Subminor 2nd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;min 2nd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;aug 1, dim 2nd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;D#, Eb&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;dim 1, aug 2nd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Db, E#&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;150&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;12/11, 35/32&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Neutral 2nd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;maj 2nd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;min 2nd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;E&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;maj 2nd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;E&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;225&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;8/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Supermajor 2nd,&lt;br /&gt;
Septimal Whole-Tone&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;perf 2nd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;maj 2nd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;E#&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;min 2nd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Eb&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;300&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;19/16, 32/27&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Minor 3rd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;min 3rd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;min 3rd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Fb&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;maj 3rd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;F#&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;375&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;5/4, 26/21&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Major 3rd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;maj 3rd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;maj 3rd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;F&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;min 3rd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;F&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;450&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;13/10, 35/27&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Sub-4th,&lt;br /&gt;
Supermajor 3rd&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;min 4th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;aug 3rd, dim 4th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;F#, Gb&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;dim 3rd, aug 4th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Fb, G#&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;525&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;27/20, 52/35, 256/189&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Wide 4th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;maj 4th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;perf 4th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;G&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;perf 4th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;G&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;600&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;7/5, 10/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Tritone&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;aug 4th,&lt;br /&gt;
dim 5th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;aug 4th, dim 5th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;G#, Ab&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;dim 4th, aug 5th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Gb, A#&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;675&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;40/27, 35/26, 189/128&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Narrow 5th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;min 5th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;perf 5th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;A&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;perf 5th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;A&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;10&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;750&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;20/13, 54/35&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Super-5th,&lt;br /&gt;
Subminor 6th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;maj 5th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;aug 5th, dim 6th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;A#, Bb&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;dim 5th, aug 6th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Ab, B#&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;825&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;8/5, 21/13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Minor 6th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;min 6th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;min 6th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;B&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;maj 6th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;B&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;12&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;900&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;27/16, 32/19&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Major 6th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;maj 6th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;maj 6th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;B#&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;min 6th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Bb&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;975&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;7/4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Subminor 7th,&lt;br /&gt;
Septimal Minor 7th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;perf 7th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;min 7th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Cb&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;maj 7th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;14&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1050&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;11/6, 64/35&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Neutral 7th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;min 7th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;maj 7th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;min 7th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;15&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1125&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;27/14, 52/27&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Supermajor 7th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;maj 7th&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;aug 7th, dim 8ve&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;dim 7th, aug 8ve&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;D#&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;16&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1200&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;2/1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;octave&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;octave&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;8ve&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;8ve&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


*based on treating 16-EDO as a 2.5.7.13.19.27 subgroup temperament; other approaches are possible.&lt;br /&gt;
; [[Stephen Weigel]]
&lt;br /&gt;
* [https://www.youtube.com/watch?v=0t7ZmlmrE0Q ''Shot Fades the Sum Of'']
&lt;!-- ws:start:WikiTextHeadingRule:7:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Intervals-Selected just intervals by error"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:7 --&gt;Selected just intervals by error&lt;/h2&gt;
* [https://www.youtube.com/watch?v=2y01AlgOPvk ''When the Saints go Marching'']
The following table shows how &lt;a class="wiki_link" href="/Just-24"&gt;some prominent just intervals&lt;/a&gt; are represented in 16-EDO (ordered by absolute error).&lt;br /&gt;


; [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/05%20-%205.%2016%20octave.mp3 Comets Over Flatland 5]{{dead link}}


&lt;table class="wiki_table"&gt;
; [[Woyten]]
    &lt;tr&gt;
* [https://www.youtube.com/watch?v=LLgClI8pyNw ''Don't Take Five''] (2021)
        &lt;td&gt;&lt;strong&gt;Interval, complement&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;strong&gt;Error (abs., in &lt;a class="wiki_link" href="/cent"&gt;cents&lt;/a&gt;)&lt;/strong&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/12_11"&gt;12/11&lt;/a&gt;, &lt;a class="wiki_link" href="/11_6"&gt;11/6&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;0.637&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/13_10"&gt;13/10&lt;/a&gt;, &lt;a class="wiki_link" href="/20_13"&gt;20/13&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;4.214&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/8_7"&gt;8/7&lt;/a&gt;, &lt;a class="wiki_link" href="/7_4"&gt;7/4&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;6.174&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/13_11"&gt;13/11&lt;/a&gt;, &lt;a class="wiki_link" href="/22_13"&gt;22/13&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;10.790&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/5_4"&gt;5/4&lt;/a&gt;, &lt;a class="wiki_link" href="/8_5"&gt;8/5&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;11.314&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/13_12"&gt;13/12&lt;/a&gt;, &lt;a class="wiki_link" href="/24_13"&gt;24/13&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;11.427&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/15_11"&gt;15/11&lt;/a&gt;, &lt;a class="wiki_link" href="/22_15"&gt;22/15&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;11.951&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/9_7"&gt;9/7&lt;/a&gt;, &lt;a class="wiki_link" href="/14_9"&gt;14/9&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;14.916&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/11_10"&gt;11/10&lt;/a&gt;, &lt;a class="wiki_link" href="/20_11"&gt;20/11&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;15.004&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/16_13"&gt;16/13&lt;/a&gt;, &lt;a class="wiki_link" href="/13_8"&gt;13/8&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;15.528&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/6_5"&gt;6/5&lt;/a&gt;, &lt;a class="wiki_link" href="/5_3"&gt;5/3&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;15.641&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/7_5"&gt;7/5&lt;/a&gt;, &lt;a class="wiki_link" href="/10_7"&gt;10/7&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;17.488&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/9_8"&gt;9/8&lt;/a&gt;, &lt;a class="wiki_link" href="/16_9"&gt;16/9&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;21.090&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/14_13"&gt;14/13&lt;/a&gt;, &lt;a class="wiki_link" href="/13_7"&gt;13/7&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;21.702&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/15_13"&gt;15/13&lt;/a&gt;, &lt;a class="wiki_link" href="/26_15"&gt;26/15&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;22.741&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/11_8"&gt;11/8&lt;/a&gt;, &lt;a class="wiki_link" href="/16_11"&gt;16/11&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;26.318&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/4_3"&gt;4/3&lt;/a&gt;, &lt;a class="wiki_link" href="/3_2"&gt;3/2&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;26.955&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/11_9"&gt;11/9&lt;/a&gt;, &lt;a class="wiki_link" href="/18_11"&gt;18/11&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;27.592&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/15_14"&gt;15/14&lt;/a&gt;, &lt;a class="wiki_link" href="/28_15"&gt;28/15&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;30.557&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/10_9"&gt;10/9&lt;/a&gt;, &lt;a class="wiki_link" href="/9_5"&gt;9/5&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;32.404&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/14_11"&gt;14/11&lt;/a&gt;, &lt;a class="wiki_link" href="/11_7"&gt;11/7&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;32.492&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/7_6"&gt;7/6&lt;/a&gt;, &lt;a class="wiki_link" href="/12_7"&gt;12/7&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;33.129&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/18_13"&gt;18/13&lt;/a&gt;, &lt;a class="wiki_link" href="/13_9"&gt;13/9&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;36.618&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/16_15"&gt;16/15&lt;/a&gt;, &lt;a class="wiki_link" href="/15_8"&gt;15/8&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;36.731&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


It's worth noting that the 525-cent interval is almost exactly halfway in between 4/3 and 11/8, making it EXTREMELY discordant.&lt;br /&gt;
; [[Xotla]]
&lt;!-- ws:start:WikiTextMediaRule:0:&amp;lt;img src=&amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/24810096?h=0&amp;amp;w=0&amp;quot; class=&amp;quot;WikiMedia WikiMediaCustom&amp;quot; id=&amp;quot;wikitext@@media@@type=&amp;amp;quot;custom&amp;amp;quot; key=&amp;amp;quot;24810096&amp;amp;quot;&amp;quot; title=&amp;quot;Custom Media&amp;quot;/&amp;gt; --&gt;&lt;object id="example" type="image/svg+xml" data="http://xenharmonic.wikispaces.com/file/view/16ed2-001.svg"&gt;alt : Your browser has no SVG support.&lt;/object&gt;&lt;!-- ws:end:WikiTextMediaRule:0 --&gt;&lt;br /&gt;
* "Robotic Dialogue" from ''Microtones & Garden Gnomes'' (2017) [https://xotla.bandcamp.com/track/robotic-dialogue-16edo Bandcamp] | [https://youtu.be/sFxny2JNGpo?si=8MKPuIMCR_Xx1DTi YouTube]
&lt;br /&gt;
* "Cognitive Climate" from Science Fraction (2022) [https://open.spotify.com/track/52v382I0OUotQjHo0pPoXs Spotify] | [https://xotla.bandcamp.com/track/cognitive-climate-16edo Bandcamp] | [https://youtu.be/dNBDG4wymN8?si=XGbpNkRp3qUo0Xgb YouTube]
&lt;!-- ws:start:WikiTextFileRule:913:&amp;lt;img src=&amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/16ed2-001.svg?h=52&amp;amp;w=320&amp;quot; class=&amp;quot;WikiFile&amp;quot; id=&amp;quot;wikitext@@file@@16ed2-001.svg&amp;quot; title=&amp;quot;File: 16ed2-001.svg&amp;quot; width=&amp;quot;320&amp;quot; height=&amp;quot;52&amp;quot; /&amp;gt; --&gt;&lt;div class="objectEmbed"&gt;&lt;a href="/file/view/16ed2-001.svg/480989378/16ed2-001.svg" onclick="ws.common.trackFileLink('/file/view/16ed2-001.svg/480989378/16ed2-001.svg');"&gt;&lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="16ed2-001.svg" /&gt;&lt;/a&gt;&lt;div&gt;&lt;a href="/file/view/16ed2-001.svg/480989378/16ed2-001.svg" onclick="ws.common.trackFileLink('/file/view/16ed2-001.svg/480989378/16ed2-001.svg');" class="filename" title="16ed2-001.svg"&gt;16ed2-001.svg&lt;/a&gt;&lt;br /&gt;&lt;ul&gt;&lt;li&gt;&lt;a href="/file/detail/16ed2-001.svg"&gt;Details&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a href="/file/view/16ed2-001.svg/480989378/16ed2-001.svg"&gt;Download&lt;/a&gt;&lt;/li&gt;&lt;li style="color: #666"&gt;18 KB&lt;/li&gt;&lt;/ul&gt;&lt;/div&gt;&lt;/div&gt;&lt;!-- ws:end:WikiTextFileRule:913 --&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:9:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Hexadecaphonic Octave Theory"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:9 --&gt;Hexadecaphonic Octave Theory&lt;/h1&gt;
&lt;br /&gt;
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 &amp;quot;3/4-tone&amp;quot; equal division of the traditional 300-cent minor third.&lt;br /&gt;
&lt;br /&gt;
16-EDO is also a tuning for the &lt;a class="wiki_link" href="/Jubilismic%20clan"&gt;no-threes 7-limit temperament tempering out 50/49&lt;/a&gt;. 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 &amp;quot;&lt;strong&gt;Magic family of scales&lt;/strong&gt;&amp;quot;.&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/Easley%20Blackwood"&gt;Easley Blackwood&lt;/a&gt; writes of 16-EDO:&lt;br /&gt;
&amp;quot;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.&amp;quot;&lt;br /&gt;
&lt;br /&gt;
Another interesting approach can include two interwoven &lt;a class="wiki_link" href="/8edo"&gt;8-EDO&lt;/a&gt; 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 &lt;strong&gt;Mavila&lt;/strong&gt; as a major seventh). If we take the 300-cent minor third as an approximation of the harmonic 19th (&lt;a class="wiki_link" href="/19_16"&gt;19/16&lt;/a&gt;, 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).&lt;br /&gt;
&lt;!-- ws:start:WikiTextRemoteImageRule:911:&amp;lt;img src=&amp;quot;http://www.ronsword.com/161928%20copy.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 179px; width: 316px;&amp;quot; /&amp;gt; --&gt;&lt;img src="http://www.ronsword.com/161928%20copy.jpg" alt="external image 161928%20copy.jpg" title="external image 161928%20copy.jpg" style="height: 179px; width: 316px;" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:911 --&gt;&lt;br /&gt;
&lt;br /&gt;
The interval between the 28th &amp;amp; 19th overtones, 28:19, measures approximately 671.3 cents, which is 3.7 cents away from 16edo's &amp;quot;narrow fifth&amp;quot;. 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.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:11:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Hexadecaphonic Notation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:11 --&gt;Hexadecaphonic Notation&lt;/h1&gt;
&lt;br /&gt;
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.&lt;br /&gt;
&lt;!-- ws:start:WikiTextRemoteImageRule:912:&amp;lt;img src=&amp;quot;http://ronsword.com/DSgoldsmith_piece.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 342px; width: 1008px;&amp;quot; /&amp;gt; --&gt;&lt;img src="http://ronsword.com/DSgoldsmith_piece.jpg" alt="external image DSgoldsmith_piece.jpg" title="external image DSgoldsmith_piece.jpg" style="height: 342px; width: 1008px;" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:912 --&gt;&lt;br /&gt;
&lt;hr /&gt;
In 16-EDO diatonic scales are dissonant and &amp;quot;shimmery&amp;quot; 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.&lt;br /&gt;
Moment of Symmetry Scales like Mavila [7] (or &amp;quot;Inverse/Anti-Diatonic&amp;quot; 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 &amp;quot;Mavila-[9] Staff&amp;quot; does just this, and places the arrangement (222122221) on nine white &amp;quot;natural&amp;quot; keys of the 16-EDO keyboard. &lt;a class="wiki_link" href="/23edo"&gt;23-EDO&lt;/a&gt; also works with the Mavila-[9] 6-line staff, notated as 1/3 tones of 16-EDO. If the 9-note &amp;quot;Nonatonic&amp;quot; MOS is adapted for 16-EDO, then perhaps it makes sense to refer to Octaves as 2/1, &amp;quot;&lt;a class="wiki_link" href="/Decave"&gt;Decave&lt;/a&gt;&amp;quot;.&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/Paul%20Erlich"&gt;Paul Erlich&lt;/a&gt; writes,&lt;br /&gt;
&amp;quot;Like the conventional 12-EDO diatonic and pentatonic (meantone) scales, these arise from tempering out a&lt;br /&gt;
unison vector from Fokker periodicity blocks. Only in 16-EDO, that unison vector is 135:128, instead of 81:80.&amp;quot;&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:13:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Hexadecaphonic Notation-16 Tone Piano Layout Based on the Anti-diatonic Scale"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:13 --&gt;16 Tone Piano Layout Based on the Anti-diatonic Scale&lt;/h2&gt;
&lt;!-- ws:start:WikiTextLocalImageRule:910:&amp;lt;img src=&amp;quot;/file/view/16-EDO-PIano-Diagram.png/599092760/748x293/16-EDO-PIano-Diagram.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 293px; width: 748px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/16-EDO-PIano-Diagram.png/599092760/748x293/16-EDO-PIano-Diagram.png" alt="16-EDO-PIano-Diagram.png" title="16-EDO-PIano-Diagram.png" style="height: 293px; width: 748px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:910 --&gt;&lt;br /&gt;
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.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:15:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc6"&gt;&lt;a name="Rank two temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:15 --&gt;Rank two temperaments&lt;/h1&gt;
&lt;a class="wiki_link" href="/List%20of%2016et%20rank%20two%20temperaments%20by%20badness"&gt;List of 16et rank two temperaments by badness&lt;/a&gt;&lt;br /&gt;
&lt;br /&gt;


; [[User:Nick_Vuci|Nick Vuci]]
* [https://en.xen.wiki/images/4/44/NickVuci-20220206-16edo-Prelude.mp3 ''Prelude'']
* [https://en.xen.wiki/images/9/9a/NickVuci-20231102-16edo-SofterForJ.mp3 ''Softer for J'']
* [https://en.xen.wiki/images/4/48/NickVuci-20220306-16edo-Invention.mp3 ''2-Part Invention'']
* [https://en.xen.wiki/w/User:Nick_Vuci#Modal_Studies ''Mavila Modal Studies'']
* [https://en.xen.wiki/images/c/c6/NV-20210526-16NEJI128-SerialismDubstepSketch.mp3 ''EDM based on a tone row'']


&lt;table class="wiki_table"&gt;
; [[Zewen Senpai]]
    &lt;tr&gt;
* [https://www.youtube.com/watch?v=QOzBGd64Pi4 ''Simple Ambient Study No. 1'']
        &lt;th&gt;Periods&lt;br /&gt;
per octave&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Generator&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Temperaments&lt;br /&gt;
&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1\16&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/Valentine"&gt;Valentine&lt;/a&gt;, &lt;a class="wiki_link" href="/Slurpee"&gt;slurpee&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3\16&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/Gorgo"&gt;Gorgo&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5\16&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Messed-up &lt;a class="wiki_link" href="/Magic"&gt;magic&lt;/a&gt;/muggles&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7\16&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/Mavila"&gt;Mavila&lt;/a&gt;/armodue&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1\16&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/Bipelog"&gt;Bipelog&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3\16&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/Lemba"&gt;Lemba&lt;/a&gt;, &lt;a class="wiki_link" href="/Astrology"&gt;astrology&lt;/a&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1\16&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/Diminished"&gt;Diminished&lt;/a&gt;/demolished&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1\16&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;br /&gt;
== Notes ==
&lt;strong&gt;Mavila&lt;/strong&gt;&lt;br /&gt;
<references group=note/>


== See also ==
* [[57ed12]] - octave stretched version of 16edo; 57ed12 improves 3.5.11.13.17 but damages 2.7


&lt;table class="wiki_table"&gt;
=== Approaches ===
    &lt;tr&gt;
* [[User:VectorGraphics/16edo theory|Vector's approach]]
        &lt;td&gt;[5]:&lt;br /&gt;
* [[Armodue theory]]
&lt;/td&gt;
** [[Armodue armonia]]
        &lt;td&gt;5 2 5 2 2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;[7]:&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3 2 2 3 2 2 2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;!-- ws:start:WikiTextMediaRule:1:&amp;lt;img src=&amp;quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/MavilaAntidiatonic16edo.mp3?h=20&amp;amp;w=240&amp;quot; class=&amp;quot;WikiMedia WikiMediaFile&amp;quot; id=&amp;quot;wikitext@@media@@type=&amp;amp;quot;file&amp;amp;quot; key=&amp;amp;quot;MavilaAntidiatonic16edo.mp3&amp;amp;quot; width=&amp;amp;quot;240&amp;amp;quot; height=&amp;amp;quot;20&amp;amp;quot;&amp;quot; title=&amp;quot;Local Media File&amp;quot;height=&amp;quot;20&amp;quot; width=&amp;quot;240&amp;quot;/&amp;gt; --&gt;&lt;embed src="/s/mediaplayer.swf" pluginspage="http://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" quality="high" width="240" height="20" wmode="transparent" flashvars="file=http%253A%252F%252Fxenharmonic.wikispaces.com%252Ffile%252Fview%252FMavilaAntidiatonic16edo.mp3?file_extension=mp3&amp;autostart=false&amp;repeat=false&amp;showdigits=true&amp;showfsbutton=false&amp;width=240&amp;height=20"&gt;&lt;/embed&gt;&lt;!-- ws:end:WikiTextMediaRule:1 --&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;[9]:&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1 2 2 2 1 2 2 2 2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;!-- ws:start:WikiTextMediaRule:2:&amp;lt;img src=&amp;quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/MavilaSuperdiatonic16edo.mp3?h=20&amp;amp;w=240&amp;quot; class=&amp;quot;WikiMedia WikiMediaFile&amp;quot; id=&amp;quot;wikitext@@media@@type=&amp;amp;quot;file&amp;amp;quot; key=&amp;amp;quot;MavilaSuperdiatonic16edo.mp3&amp;amp;quot; width=&amp;amp;quot;240&amp;amp;quot; height=&amp;amp;quot;20&amp;amp;quot;&amp;quot; title=&amp;quot;Local Media File&amp;quot;height=&amp;quot;20&amp;quot; width=&amp;quot;240&amp;quot;/&amp;gt; --&gt;&lt;embed src="/s/mediaplayer.swf" pluginspage="http://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" quality="high" width="240" height="20" wmode="transparent" flashvars="file=http%253A%252F%252Fxenharmonic.wikispaces.com%252Ffile%252Fview%252FMavilaSuperdiatonic16edo.mp3?file_extension=mp3&amp;autostart=false&amp;repeat=false&amp;showdigits=true&amp;showfsbutton=false&amp;width=240&amp;height=20"&gt;&lt;/embed&gt;&lt;!-- ws:end:WikiTextMediaRule:2 --&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;strong&gt;Diminished&lt;/strong&gt;&lt;br /&gt;
== References ==
[8]: 1 3 1 3 1 3 1 3&lt;br /&gt;
<references />
[12]: 1 1 2 1 1 2 1 1 2 1 1 2&lt;br /&gt;
&lt;strong&gt;Magic&lt;/strong&gt;&lt;br /&gt;
[7]: 1 4 1 4 1 4 1&lt;br /&gt;
[10]: 1 3 1 1 3 1 1 1 3 1&lt;br /&gt;
[13]: 1 1 2 1 1 1 2 1 1 1 2 1 1&lt;br /&gt;
&lt;strong&gt;Cynder/Gorgo&lt;/strong&gt;&lt;br /&gt;
[5]: 3 3 4 3 3&lt;br /&gt;
[6]: 3 3 1 3 3 3&lt;br /&gt;
[11]: 1 2 1 2 1 2 1 2 1 2 1&lt;br /&gt;
&lt;strong&gt;Lemba&lt;/strong&gt;&lt;br /&gt;
[6]: 3 2 3 3 2 3&lt;br /&gt;
[10]: 2 1 2 1 2 2 1 2 1 2&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/Igliashon%20Jones"&gt;Igliashon Jones&lt;/a&gt; writes: &amp;quot;The trouble (in 16-EDO) has ... to do with the fact that the distance between the major third&lt;br /&gt;
and the &amp;quot;fourth&amp;quot; is the same as the distance between the &amp;quot;fourth&amp;quot; and the &amp;quot;fifth&amp;quot; (i.e. near a 12/11)...This mean(s) that&lt;br /&gt;
135/128 (the difference between 16/15 and 9/8) is tempered out....&amp;quot;&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:17:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="Rank two temperaments-Harmonizing Mavila/Armodue in 16 EDO"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:17 --&gt;Harmonizing Mavila/Armodue in 16 EDO&lt;/h2&gt;
&lt;br /&gt;
Because 16 edo doesn't approximate 3/2 well at all, triadic harmony based on thirds isn't a good option.&lt;br /&gt;
However triadic harmony can be based on on sevenths rather than thirds. For instance, 16 edo approximates 7/4 well enough to use&lt;br /&gt;
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&lt;br /&gt;
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&lt;br /&gt;
0-975-975 and 0-1050-1050 are also obvious possible chords.&lt;br /&gt;
Their characteristic metallic sound has earned them the name &amp;quot;Metallic triads&amp;quot;.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:19:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc8"&gt;&lt;a name="Rank two temperaments-Harmonizing Mavila/Armodue in 16 EDO-MOS that support metallic harmony"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:19 --&gt;MOS that support metallic harmony&lt;/h3&gt;
Mavila 7 contains two hard triads on degrees 1 and 4 and two soft triads on degrees 2 and 6. The other three chords&lt;br /&gt;
are wide symmetrical triads 0-1050-1050. Mavila 9 introduces two more soft triads while the Wilson scale introduces two more hard triads.&lt;br /&gt;
&lt;br /&gt;
See &lt;a class="wiki_link" href="/Metallic%20Harmony"&gt;Metallic Harmony&lt;/a&gt;.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:21:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc9"&gt;&lt;a name="Commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:21 --&gt;Commas&lt;/h1&gt;
16 EDO &lt;a class="wiki_link" href="/tempering%20out"&gt;tempers out&lt;/a&gt; the following &lt;a class="wiki_link" href="/comma"&gt;comma&lt;/a&gt;s. (Note: This assumes &lt;a class="wiki_link" href="/val"&gt;val&lt;/a&gt; &amp;lt; 16 25 37 45 55 59 |.)&lt;br /&gt;


== Further reading ==
* [[Sword, Ron]]. ''[https://ronsword.bigcartel.com/product/esadekaphonic-scales-for-guitar Hexadecaphonic Scales for Guitar: A Microtonal Guitar Method Book, for Theory, Scales, and Information on the Sixteen Equal Division Octave System]''. 2009. (semi-diminished fourth tuning)
* Sword, Ron. ''[http://www.metatonalmusic.com/books.html Hexadecaphonic Scales for Guitar: Theory, Scales and Information on the Sixteen Equal Division Octave system]''. 2010? (superfourth tuning)
* Sword, Ron. "Thesaurus of Melodic Patterns and Intervals for 16-Tones" IAAA Press, USA. First Ed: August, 2011{{citation needed}}


&lt;table class="wiki_table"&gt;
[[Category:Teentuning]]
    &lt;tr&gt;
[[Category:Listen]]
        &lt;th&gt;Comma&lt;br /&gt;
[[Category:Mavila]]
&lt;/th&gt;
[[Category:Guitar]]
        &lt;th&gt;Monzo&lt;br /&gt;
[[Category:Pages with internal sound examples]]
&lt;/th&gt;
        &lt;th&gt;Value (Cents)&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Name 1&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Name 2&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Name 3&lt;br /&gt;
&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;135/128&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;| -7 3 1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;92.18&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Major Chroma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Major Limma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Pelogic Comma&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;648/625&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;| 3 4 -4 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;62.57&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Major Diesis&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Diminished Comma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;3125/3072&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;| -10 -1 5 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;29.61&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Small Diesis&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Magic Comma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;| 23 6 -14 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;3.34&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Vishnuzma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Semisuper&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;36/35&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;| 2 2 -1 -1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;48.77&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Septimal Quarter Tone&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;525/512&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;| -9 1 2 1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;43.41&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Avicennma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Avicenna's Enharmonic Diesis&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;50/49&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;| 1 0 2 -2 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;34.98&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Tritonic Diesis&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Jubilisma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;64827/64000&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;| -9 3 -3 4 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;22.23&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Squalentine&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;3125/3087&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;| 0 -2 5 -3 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;21.18&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Gariboh&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;126/125&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;| 1 2 -3 1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;13.79&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Septimal Semicomma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Starling Comma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;1029/1024&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;| -10 1 0 3 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;8.43&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Gamelisma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;6144/6125&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;| 11 1 -3 -2 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;5.36&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Porwell&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;121/120&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;| -3 -1 -1 0 2 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;14.37&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Biyatisma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;176/175&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;| 4 0 -2 -1 1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;9.86&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Valinorsma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;385/384&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;| -7 -1 1 1 1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;4.50&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Keenanisma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;441/440&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;| -3 2 -1 2 -1 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;3.93&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Werckisma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;3025/3024&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: left;"&gt;| -4 -3 2 -1 2 &amp;gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: right;"&gt;0.57&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Lehmerisma&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;br /&gt;
{{Todo|cleanup}}
&lt;!-- ws:start:WikiTextHeadingRule:23:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc10"&gt;&lt;a name="Armodue Theory (4-line staff)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:23 --&gt;&lt;strong&gt;Armodue Theory (4-line staff)&lt;/strong&gt;&lt;/h1&gt;
&lt;a class="wiki_link_ext" href="http://www.armodue.com/ricerche.htm" rel="nofollow"&gt;Armodue&lt;/a&gt;: Italian pages of theory for 16-tone (esadekaphonic) system, including compositions.&lt;br /&gt;
Translations of parts of the Armodue pages can be found &lt;a class="wiki_link" href="/Armodue"&gt;here&lt;/a&gt; on this wiki.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:25:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc11"&gt;&lt;a name="Books/Literature"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:25 --&gt;Books/Literature&lt;/h1&gt;
Sword, Ronald. &amp;quot;Thesaurus of Melodic Patterns and Intervals for 16-Tones&amp;quot; IAAA Press, USA. First Ed: August, 2011&lt;br /&gt;
Sword, Ronald. &amp;quot;Hexadecaphonic Scales for Guitar.&amp;quot; IAAA Press, UK-USA. First Ed: Feb, 2010. (superfourth tuning)&lt;br /&gt;
Sword, Ronald. &amp;quot;Esadekaphonic Scales for Guitar.&amp;quot; IAAA Press, UK-USA. First Ed: April, 2009. (semi-diminished fourth tuning)&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:27:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc12"&gt;&lt;a name="Compositions"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:27 --&gt;Compositions&lt;/h1&gt;
&lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=660895&amp;amp;songID=12370649&amp;amp;showPlayer=true" rel="nofollow" target="_blank"&gt;Prenestyna Highway&lt;/a&gt; by &lt;a class="wiki_link_ext" href="http://fiale.tk" rel="nofollow" target="_blank"&gt;Fabrizio Fulvio Fausto Fiale&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link_ext" href="http://lastsacrament.bandcamp.com/album/enantiodromia" rel="nofollow" target="_blank"&gt;Enantiodromia (album)&lt;/a&gt; by Last Sacrament&lt;br /&gt;
&lt;a class="wiki_link_ext" href="http://www.cdbaby.com/cd/aeternamusic" rel="nofollow" target="_blank"&gt;Tribute to Armodue&lt;/a&gt; by Aeterna&lt;br /&gt;
&lt;a class="wiki_link_ext" href="http://www.io.com/%7Ehmiller/midi/16tet.mid" rel="nofollow"&gt;Etude in 16-tone equal tuning&lt;/a&gt; &lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/16tet.mp3" rel="nofollow"&gt;play&lt;/a&gt; (&lt;a class="wiki_link_ext" href="http://soonlabel.com/xenharmonic/archives/2604" rel="nofollow" target="_blank"&gt;organ version&lt;/a&gt;) by &lt;a class="wiki_link" href="/Herman%20Miller"&gt;Herman Miller &lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link_ext" href="http://www.ronsword.com/sounds/Ron%20Sword%20-%2016-tone%20acoustic%20improvisation.mp3" rel="nofollow"&gt;16-tone steel string acoustic diddle&lt;/a&gt; by Ron Sword&lt;br /&gt;
&lt;a class="wiki_link_ext" href="http://www.jeanpierrepoulin.com/mp3/Armodue78.mp3" rel="nofollow"&gt;Armodue78&lt;/a&gt; by &lt;a class="wiki_link" href="/Jean-Pierre%20Poulin" target="_blank"&gt;Jean-Pierre Poulin&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Fiale/ffffiale+palestrinamortafantasiaquasiunasonata.mp3" rel="nofollow"&gt;Palestrina Morta, fantasia quasi una sonata&lt;/a&gt; by &lt;a class="wiki_link_ext" href="http://fiale.tk" rel="nofollow" target="_blank"&gt;Fabrizio Fulvio Fausto Fiale&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/05%20-%205.%2016%20octave.mp3" rel="nofollow"&gt;Comets Over Flatland 5&lt;/a&gt; by &lt;a class="wiki_link" href="/Randy%20Winchester"&gt;Randy Winchester&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/16-ET/20120527-16-malathion.mp3" rel="nofollow"&gt;Malathion&lt;/a&gt; by &lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=2358#comments" rel="nofollow"&gt;Chris Vaisvil &lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/16-ET/20130216_16edo_vesta.mp3" rel="nofollow"&gt;Being of Vesta&lt;/a&gt; by &lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=3061" rel="nofollow"&gt;Chris Vaisvil&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/simultaneous-tunings/20130607_thin_ice_christiane.mp3" rel="nofollow"&gt;Thin Ice&lt;/a&gt; by &lt;a class="wiki_link" href="/Chris%20Vaisvil"&gt;Chris Vaisvil&lt;/a&gt; ; &lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/thin-ice-for-alto-female-choir-harp-and-percussion-in-adaptive-ji-16-edo-and-8-edo/" rel="nofollow"&gt;information on the composition&lt;/a&gt;&lt;br /&gt;
&lt;a href="/file/view/Mavila%20Jazz%20Rhodes%201.mp3/533097734/Mavila%20Jazz%20Rhodes%201.mp3" onclick="ws.common.trackFileLink('/file/view/Mavila%20Jazz%20Rhodes%201.mp3/533097734/Mavila%20Jazz%20Rhodes%201.mp3');"&gt;Mavila Jazz Groove&lt;/a&gt; by &lt;a class="wiki_link_ext" href="https://soundcloud.com/pianofreak96" rel="nofollow"&gt;William Lynch&lt;/a&gt;&lt;br /&gt;
&lt;a href="/file/view/mavila4.mp3/533097942/mavila4.mp3" onclick="ws.common.trackFileLink('/file/view/mavila4.mp3/533097942/mavila4.mp3');"&gt;Cold, Dark Night for a Dance &lt;/a&gt;by &lt;a class="wiki_link_ext" href="https://soundcloud.com/pianofreak96" rel="nofollow"&gt;William Lynch&lt;/a&gt;&lt;br /&gt;
&lt;a class="wiki_link_ext" href="https://soundcloud.com/fff-fiale/in-sospensione-neutra" rel="nofollow"&gt;In Sospensione Neutra by Fabrizio Fulvio Fiale&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
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