16edo: Difference between revisions

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<span style="display: block; text-align: right;">[[xenharmonie/16edo|Deutsch]] - [[16平均律|日本語]]
</span>

[[toc]]
**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= 

[[image:16edo wheel 01.png width="325" height="325"]]

=Intervals= 
|| Degree || Cents (coarse, fine)
DMS ||= Approximate
Ratios* || Interval Names
Just || Interval Names
Simplified ||
|| 0 || 0 ||= 1/1 || Unison || Unison ||
|| 1 || 75, 90
22º30' ||= 28/27, 27/26 || Subminor 2nd || Minor Second ||
|| 2 || 150, 180
45° ||= 12/11, 35/32 || Neutral 2nd || Major Second ||
|| 3 || 225, 270
67°30' ||= 8/7 || Supermajor 2nd,
Septimal Whole-Tone || Perfect Second ||
|| 4 || 300, 360
90° ||= 19/16, 32/27 || Minor 3rd || Minor Third ||
|| 5 || 375, 450
112°30' ||= 5/4, 26/21 || Major 3rd || Major Third ||
|| 6 || 450, 540
135° ||= 13/10, 35/27 || Sub-4th,
Supermajor 3rd || Minor Fourth ||
|| 7 || 525, 630
157°30' ||= 27/20, 52/35, 256/189 || Wide 4th || Major Fourth ||
|| 8 || 600, 720
180° ||= 7/5, 10/7 || Tritone || Augmented fourth/
Diminished fifth ||
|| 9 || 675, 810
202°30' ||= 40/27, 35/26, 189/128 || Narrow 5th || Minor Fifth ||
|| 10 || 750, 900
225° ||= 20/13, 54/35 || Super-5th,
Subminor 6th || Major Fifth ||
|| 11 || 825, 990
247°30' ||= 8/5, 21/13 || Minor 6th || Minor Sixth ||
|| 12 || 900, 1080
270° ||= 27/16, 32/19 || Major 6th || Major Sixth ||
|| 13 || 975, 1170
292°30' ||= 7/4 || Subminor 7th,
Septimal Minor 7th || Perfect Seventh ||
|| 14 || 1050, 1260
315° ||= 11/6, 64/35 || Neutral 7th || Minor Seventh ||
|| 15 || 1125, 1350
337°30' ||= 27/14, 52/27 || Supermajor 7th || Major Seventh ||
|| 16 || 1200, 1440
360° ||= 2/1 || Octave || Octave ||
*based on treating 16-EDO as a 2.5.7.13.19.27 subgroup temperament; other approaches are possible.

==Selected just intervals by error== 
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]]

=Hexadecaphonic Octave Theory= 

The scale supports the diminished temperament with its 1/4 octave period, though its generator size, equal to its step size of 75 cents, is smaller than ideal. Its very flat 3/2 of 675 cents supports Mavila temperament, where the mapping of major and minor is reversed. The temperament could be popular for its 150-cent "3/4-tone" equal division of the traditional 300-cent minor third.

16-EDO is also a tuning for the [[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**".

[[Easley Blackwood]] writes of 16-EDO:
"16 notes: This tuning is best thought of as a combination of four intertwined diminished seventh chords. Since 12-note tuning can be regarded as a combination of three diminished seventh chords, it is plain that the two tunings have elements in common. The most obvious difference in the way the two tunings sound and work is that triads in 16-note tuning, although recognizable, are too discordant to serve as the final harmony in cadences. Keys can still be established by successions of altered subdominant and dominant harmonies, however, and the Etude is based mainly upon this property. The fundamental consonant harmony employed is a minor triad with an added minor seventh."

Another interesting approach can include two interwoven [[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).
[[image:http://www.ronsword.com/161928%20copy.jpg width="316" height="179"]]

The interval between the 28th & 19th overtones, 28:19, measures approximately 671.3 cents, which is 3.7 cents away from 16edo's "narrow fifth". Another voicing for this chord is 14:16:19, which features 19:14 as the outer interval (528.7 cents just, 525.0 cents in 16edo). A perhaps more consonant open voicing is 7:16:19.


=Hexadecaphonic Notation= 

16-EDO notation can be easy utilizing Goldsmith's Circle of keys, nominals, and respective notation. The nominals for a 6 line staff can be switched for Wilson's Beta and Epsilon additions to A-G. Armodue of Italy uses a 4-line staff for 16-EDO.
[[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,
"Like the conventional 12-EDO diatonic and pentatonic (meantone) scales, these arise from tempering out a
unison vector from Fokker periodicity blocks. Only in 16-EDO, that unison vector is 135:128, instead of 81:80."

==16 Tone Piano Layout Based on the Anti-diatonic Scale== 
[[image:16-EDO-PIano-Diagram.png width="748" height="293"]]
This Layout places Mavila[7] on the white keys and Mavila[9] on the white keys. As you can see, flats are higher than naturals and sharps are lower.

=Rank two temperaments= 
[[List of 16et rank two temperaments by badness]]

||~ Periods
per octave ||~ Generator ||~ Temperaments ||
|| 1 || 1\16 ||= [[Valentine]], [[Slurpee|slurpee]] ||
|| 1 || 3\16 ||= [[Gorgo]] ||
|| 1 || 5\16 ||= Messed-up [[Magic|magic]]/muggles ||
|| 1 || 7\16 ||= [[Mavila]]/armodue ||
|| 2 || 1\16 ||= [[Bipelog]] ||
|| 2 || 3\16 ||= [[Lemba]], [[Astrology|astrology]] ||
|| 4 || 1\16 ||= [[Diminished]]/demolished ||
|| 8 || 1\16 ||=   ||

**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
[10]: 1 3 1 1 3 1 1 1 3 1
[13]: 1 1 2 1 1 1 2 1 1 1 2 1 1
**Cynder/Gorgo**
[5]: 3 3 4 3 3
[6]: 3 3 1 3 3 3
[11]: 1 2 1 2 1 2 1 2 1 2 1
**Lemba**
[6]: 3 2 3 3 2 3
[10]: 2 1 2 1 2 2 1 2 1 2

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

==Harmonizing Mavila/Armodue in 16 EDO== 

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

===MOS that support metallic harmony=== 
Mavila 7 contains two hard triads on degrees 1 and 4 and two soft triads on degrees 2 and 6. The other three chords
are wide symmetrical triads 0-1050-1050. Mavila 9 introduced two more soft triads while the Wilson scale introduces two more hard triads.

See [[Metallic Harmony]].


=Commas= 
16 EDO [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes [[val]] < 16 25 37 45 55 59 |.)
||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 135/128 ||< | -7 3 1 > ||> 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||= 648/625 ||< | 3 4 -4 > ||> 62.57 ||= Major Diesis ||= Diminished Comma ||=   ||
||= 3125/3072 ||< | -10 -1 5 > ||> 29.61 ||= Small Diesis ||= Magic Comma ||=   ||
||= 1212717/1210381 ||< | 23 6 -14 > ||> 3.34 ||= Vishnuzma ||= Semisuper ||=   ||
||= 36/35 ||< | 2 2 -1 -1 > ||> 48.77 ||= Septimal Quarter Tone ||=   ||=   ||
||= 525/512 ||< | -9 1 2 1 > ||> 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||=   ||
||= 50/49 ||< | 1 0 2 -2 > ||> 34.98 ||= Tritonic Diesis ||= Jubilisma ||=   ||
||= 64827/64000 ||< | -9 3 -3 4 > ||> 22.23 ||= Squalentine ||=   ||=   ||
||= 3125/3087 ||< | 0 -2 5 -3 > ||> 21.18 ||= Gariboh ||=   ||=   ||
||= 126/125 ||< | 1 2 -3 1 > ||> 13.79 ||= Septimal Semicomma ||= Starling Comma ||=   ||
||= 1029/1024 ||< | -10 1 0 3 > ||> 8.43 ||= Gamelisma ||=   ||=   ||
||= 6144/6125 ||< | 11 1 -3 -2 > ||> 5.36 ||= Porwell ||=   ||=   ||
||= 121/120 ||< | -3 -1 -1 0 2 > ||> 14.37 ||= Biyatisma ||=   ||=   ||
||= 176/175 ||< | 4 0 -2 -1 1 > ||> 9.86 ||= Valinorsma ||=   ||=   ||
||= 385/384 ||< | -7 -1 1 1 1 > ||> 4.50 ||= Keenanisma ||=   ||=   ||
||= 441/440 ||< | -3 2 -1 2 -1 > ||> 3.93 ||= Werckisma ||=   ||=   ||
||= 3025/3024 ||< | -4 -3 2 -1 2 > ||> 0.57 ||= Lehmerisma ||=   ||=   ||

=**Armodue Theory (4-line staff)**= 
[[http://www.armodue.com/ricerche.htm|Armodue]]: Italian pages of theory for 16-tone (esadekaphonic) system, including compositions.
Translations of parts of the Armodue pages can be found [[Armodue|here]] on this wiki.

=Books/Literature= 
Sword, Ronald. "Thesaurus of Melodic Patterns and Intervals for 16-Tones" IAAA Press, USA. First Ed: August, 2011
Sword, Ronald. "Hexadecaphonic Scales for Guitar." IAAA Press, UK-USA. First Ed: Feb, 2010. (superfourth tuning)
Sword, Ronald. "Esadekaphonic Scales for Guitar." IAAA Press, UK-USA. First Ed: April, 2009. (semi-diminished fourth tuning)

=Compositions= 
[[@http://www.soundclick.com/bands/page_songInfo.cfm?bandID=660895&songID=12370649&showPlayer=true|Prenestyna Highway]] by [[@http://fiale.tk|Fabrizio Fulvio Fausto Fiale]]
[[@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]]

Original HTML content:

<html><head><title>16edo</title></head><body><span style="display: block; text-align: right;"><a class="wiki_link" href="http://xenharmonie.wikispaces.com/16edo">Deutsch</a> - <a class="wiki_link" href="/16%E5%B9%B3%E5%9D%87%E5%BE%8B">日本語</a><br />
</span><br />
<br />
<!-- ws:start:WikiTextTocRule:29:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --><div style="margin-left: 1em;"><a href="#Images">Images</a></div>
<!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --><div style="margin-left: 1em;"><a href="#Intervals">Intervals</a></div>
<!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --><div style="margin-left: 2em;"><a href="#Intervals-Selected just intervals by error">Selected just intervals by error</a></div>
<!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --><div style="margin-left: 1em;"><a href="#Hexadecaphonic Octave Theory">Hexadecaphonic Octave Theory</a></div>
<!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --><div style="margin-left: 1em;"><a href="#Hexadecaphonic Notation">Hexadecaphonic Notation</a></div>
<!-- ws:end:WikiTextTocRule:34 --><!-- ws:start:WikiTextTocRule:35: --><div style="margin-left: 2em;"><a href="#Hexadecaphonic Notation-16 Tone Piano Layout Based on the Anti-diatonic Scale">16 Tone Piano Layout Based on the Anti-diatonic Scale</a></div>
<!-- ws:end:WikiTextTocRule:35 --><!-- ws:start:WikiTextTocRule:36: --><div style="margin-left: 1em;"><a href="#Rank two temperaments">Rank two temperaments</a></div>
<!-- ws:end:WikiTextTocRule:36 --><!-- ws:start:WikiTextTocRule:37: --><div style="margin-left: 2em;"><a href="#Rank two temperaments-Harmonizing Mavila/Armodue in 16 EDO">Harmonizing Mavila/Armodue in 16 EDO</a></div>
<!-- ws:end:WikiTextTocRule:37 --><!-- ws:start:WikiTextTocRule:38: --><div style="margin-left: 3em;"><a href="#Rank two temperaments-Harmonizing Mavila/Armodue in 16 EDO-MOS that support metallic harmony">MOS that support metallic harmony</a></div>
<!-- ws:end:WikiTextTocRule:38 --><!-- ws:start:WikiTextTocRule:39: --><div style="margin-left: 1em;"><a href="#Commas">Commas</a></div>
<!-- ws:end:WikiTextTocRule:39 --><!-- ws:start:WikiTextTocRule:40: --><div style="margin-left: 1em;"><a href="#Armodue Theory (4-line staff)">Armodue Theory (4-line staff)</a></div>
<!-- ws:end:WikiTextTocRule:40 --><!-- ws:start:WikiTextTocRule:41: --><div style="margin-left: 1em;"><a href="#Books/Literature">Books/Literature</a></div>
<!-- ws:end:WikiTextTocRule:41 --><!-- ws:start:WikiTextTocRule:42: --><div style="margin-left: 1em;"><a href="#Compositions">Compositions</a></div>
<!-- ws:end:WikiTextTocRule:42 --><!-- ws:start:WikiTextTocRule:43: --></div>
<!-- ws:end:WikiTextTocRule:43 --><strong>16-EDO</strong> is the <a class="wiki_link" href="/equal%20division%20of%20the%20octave">equal division of the octave</a> into sixteen narrow chromatic semitones each of 75 <a class="wiki_link" href="/cent">cent</a>s exactly. It is not especially good at representing most low-integer musical intervals, but it has a <a class="wiki_link" href="/7_4">7/4</a> which is six cents sharp, and a <a class="wiki_link" href="/5_4">5/4</a> 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 <a class="wiki_link" href="/12edo">12-EDO</a>, and a diminished triad on each scale step.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:3:&lt;h1&gt; --><h1 id="toc0"><a name="Images"></a><!-- ws:end:WikiTextHeadingRule:3 -->Images</h1>
 <br />
<!-- ws:start:WikiTextLocalImageRule:769:&lt;img src=&quot;/file/view/16edo%20wheel%2001.png/516947132/325x325/16edo%20wheel%2001.png&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 325px; width: 325px;&quot; /&gt; --><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;" /><!-- ws:end:WikiTextLocalImageRule:769 --><br />
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<!-- ws:start:WikiTextHeadingRule:5:&lt;h1&gt; --><h1 id="toc1"><a name="Intervals"></a><!-- ws:end:WikiTextHeadingRule:5 -->Intervals</h1>
 

<table class="wiki_table">
    <tr>
        <td>Degree<br />
</td>
        <td>Cents (coarse, fine)<br />
DMS<br />
</td>
        <td style="text-align: center;">Approximate<br />
Ratios*<br />
</td>
        <td>Interval Names<br />
Just<br />
</td>
        <td>Interval Names<br />
Simplified<br />
</td>
    </tr>
    <tr>
        <td>0<br />
</td>
        <td>0<br />
</td>
        <td style="text-align: center;">1/1<br />
</td>
        <td>Unison<br />
</td>
        <td>Unison<br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>75, 90<br />
22º30'<br />
</td>
        <td style="text-align: center;">28/27, 27/26<br />
</td>
        <td>Subminor 2nd<br />
</td>
        <td>Minor Second<br />
</td>
    </tr>
    <tr>
        <td>2<br />
</td>
        <td>150, 180<br />
45°<br />
</td>
        <td style="text-align: center;">12/11, 35/32<br />
</td>
        <td>Neutral 2nd<br />
</td>
        <td>Major Second<br />
</td>
    </tr>
    <tr>
        <td>3<br />
</td>
        <td>225, 270<br />
67°30'<br />
</td>
        <td style="text-align: center;">8/7<br />
</td>
        <td>Supermajor 2nd,<br />
Septimal Whole-Tone<br />
</td>
        <td>Perfect Second<br />
</td>
    </tr>
    <tr>
        <td>4<br />
</td>
        <td>300, 360<br />
90°<br />
</td>
        <td style="text-align: center;">19/16, 32/27<br />
</td>
        <td>Minor 3rd<br />
</td>
        <td>Minor Third<br />
</td>
    </tr>
    <tr>
        <td>5<br />
</td>
        <td>375, 450<br />
112°30'<br />
</td>
        <td style="text-align: center;">5/4, 26/21<br />
</td>
        <td>Major 3rd<br />
</td>
        <td>Major Third<br />
</td>
    </tr>
    <tr>
        <td>6<br />
</td>
        <td>450, 540<br />
135°<br />
</td>
        <td style="text-align: center;">13/10, 35/27<br />
</td>
        <td>Sub-4th,<br />
Supermajor 3rd<br />
</td>
        <td>Minor Fourth<br />
</td>
    </tr>
    <tr>
        <td>7<br />
</td>
        <td>525, 630<br />
157°30'<br />
</td>
        <td style="text-align: center;">27/20, 52/35, 256/189<br />
</td>
        <td>Wide 4th<br />
</td>
        <td>Major Fourth<br />
</td>
    </tr>
    <tr>
        <td>8<br />
</td>
        <td>600, 720<br />
180°<br />
</td>
        <td style="text-align: center;">7/5, 10/7<br />
</td>
        <td>Tritone<br />
</td>
        <td>Augmented fourth/<br />
Diminished fifth<br />
</td>
    </tr>
    <tr>
        <td>9<br />
</td>
        <td>675, 810<br />
202°30'<br />
</td>
        <td style="text-align: center;">40/27, 35/26, 189/128<br />
</td>
        <td>Narrow 5th<br />
</td>
        <td>Minor Fifth<br />
</td>
    </tr>
    <tr>
        <td>10<br />
</td>
        <td>750, 900<br />
225°<br />
</td>
        <td style="text-align: center;">20/13, 54/35<br />
</td>
        <td>Super-5th,<br />
Subminor 6th<br />
</td>
        <td>Major Fifth<br />
</td>
    </tr>
    <tr>
        <td>11<br />
</td>
        <td>825, 990<br />
247°30'<br />
</td>
        <td style="text-align: center;">8/5, 21/13<br />
</td>
        <td>Minor 6th<br />
</td>
        <td>Minor Sixth<br />
</td>
    </tr>
    <tr>
        <td>12<br />
</td>
        <td>900, 1080<br />
270°<br />
</td>
        <td style="text-align: center;">27/16, 32/19<br />
</td>
        <td>Major 6th<br />
</td>
        <td>Major Sixth<br />
</td>
    </tr>
    <tr>
        <td>13<br />
</td>
        <td>975, 1170<br />
292°30'<br />
</td>
        <td style="text-align: center;">7/4<br />
</td>
        <td>Subminor 7th,<br />
Septimal Minor 7th<br />
</td>
        <td>Perfect Seventh<br />
</td>
    </tr>
    <tr>
        <td>14<br />
</td>
        <td>1050, 1260<br />
315°<br />
</td>
        <td style="text-align: center;">11/6, 64/35<br />
</td>
        <td>Neutral 7th<br />
</td>
        <td>Minor Seventh<br />
</td>
    </tr>
    <tr>
        <td>15<br />
</td>
        <td>1125, 1350<br />
337°30'<br />
</td>
        <td style="text-align: center;">27/14, 52/27<br />
</td>
        <td>Supermajor 7th<br />
</td>
        <td>Major Seventh<br />
</td>
    </tr>
    <tr>
        <td>16<br />
</td>
        <td>1200, 1440<br />
360°<br />
</td>
        <td style="text-align: center;">2/1<br />
</td>
        <td>Octave<br />
</td>
        <td>Octave<br />
</td>
    </tr>
</table>

*based on treating 16-EDO as a 2.5.7.13.19.27 subgroup temperament; other approaches are possible.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:7:&lt;h2&gt; --><h2 id="toc2"><a name="Intervals-Selected just intervals by error"></a><!-- ws:end:WikiTextHeadingRule:7 -->Selected just intervals by error</h2>
 The following table shows how <a class="wiki_link" href="/Just-24">some prominent just intervals</a> are represented in 16-EDO (ordered by absolute error).<br />


<table class="wiki_table">
    <tr>
        <td><strong>Interval, complement</strong><br />
</td>
        <td><strong>Error (abs., in <a class="wiki_link" href="/cent">cents</a>)</strong><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><a class="wiki_link" href="/12_11">12/11</a>, <a class="wiki_link" href="/11_6">11/6</a><br />
</td>
        <td style="text-align: center;">0.637<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><a class="wiki_link" href="/13_10">13/10</a>, <a class="wiki_link" href="/20_13">20/13</a><br />
</td>
        <td style="text-align: center;">4.214<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><a class="wiki_link" href="/8_7">8/7</a>, <a class="wiki_link" href="/7_4">7/4</a><br />
</td>
        <td style="text-align: center;">6.174<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><a class="wiki_link" href="/13_11">13/11</a>, <a class="wiki_link" href="/22_13">22/13</a><br />
</td>
        <td style="text-align: center;">10.790<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><a class="wiki_link" href="/5_4">5/4</a>, <a class="wiki_link" href="/8_5">8/5</a><br />
</td>
        <td style="text-align: center;">11.314<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><a class="wiki_link" href="/13_12">13/12</a>, <a class="wiki_link" href="/24_13">24/13</a><br />
</td>
        <td style="text-align: center;">11.427<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><a class="wiki_link" href="/15_11">15/11</a>, <a class="wiki_link" href="/22_15">22/15</a><br />
</td>
        <td style="text-align: center;">11.951<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><a class="wiki_link" href="/9_7">9/7</a>, <a class="wiki_link" href="/14_9">14/9</a><br />
</td>
        <td style="text-align: center;">14.916<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><a class="wiki_link" href="/11_10">11/10</a>, <a class="wiki_link" href="/20_11">20/11</a><br />
</td>
        <td style="text-align: center;">15.004<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><a class="wiki_link" href="/16_13">16/13</a>, <a class="wiki_link" href="/13_8">13/8</a><br />
</td>
        <td style="text-align: center;">15.528<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><a class="wiki_link" href="/6_5">6/5</a>, <a class="wiki_link" href="/5_3">5/3</a><br />
</td>
        <td style="text-align: center;">15.641<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><a class="wiki_link" href="/7_5">7/5</a>, <a class="wiki_link" href="/10_7">10/7</a><br />
</td>
        <td style="text-align: center;">17.488<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><a class="wiki_link" href="/9_8">9/8</a>, <a class="wiki_link" href="/16_9">16/9</a><br />
</td>
        <td style="text-align: center;">21.090<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><a class="wiki_link" href="/14_13">14/13</a>, <a class="wiki_link" href="/13_7">13/7</a><br />
</td>
        <td style="text-align: center;">21.702<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><a class="wiki_link" href="/15_13">15/13</a>, <a class="wiki_link" href="/26_15">26/15</a><br />
</td>
        <td style="text-align: center;">22.741<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><a class="wiki_link" href="/11_8">11/8</a>, <a class="wiki_link" href="/16_11">16/11</a><br />
</td>
        <td style="text-align: center;">26.318<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><a class="wiki_link" href="/4_3">4/3</a>, <a class="wiki_link" href="/3_2">3/2</a><br />
</td>
        <td style="text-align: center;">26.955<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><a class="wiki_link" href="/11_9">11/9</a>, <a class="wiki_link" href="/18_11">18/11</a><br />
</td>
        <td style="text-align: center;">27.592<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><a class="wiki_link" href="/15_14">15/14</a>, <a class="wiki_link" href="/28_15">28/15</a><br />
</td>
        <td style="text-align: center;">30.557<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><a class="wiki_link" href="/10_9">10/9</a>, <a class="wiki_link" href="/9_5">9/5</a><br />
</td>
        <td style="text-align: center;">32.404<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><a class="wiki_link" href="/14_11">14/11</a>, <a class="wiki_link" href="/11_7">11/7</a><br />
</td>
        <td style="text-align: center;">32.492<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><a class="wiki_link" href="/7_6">7/6</a>, <a class="wiki_link" href="/12_7">12/7</a><br />
</td>
        <td style="text-align: center;">33.129<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><a class="wiki_link" href="/18_13">18/13</a>, <a class="wiki_link" href="/13_9">13/9</a><br />
</td>
        <td style="text-align: center;">36.618<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><a class="wiki_link" href="/16_15">16/15</a>, <a class="wiki_link" href="/15_8">15/8</a><br />
</td>
        <td style="text-align: center;">36.731<br />
</td>
    </tr>
</table>

It's worth noting that the 525-cent interval is almost exactly halfway in between 4/3 and 11/8, making it EXTREMELY discordant.<br />
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<br />
<!-- ws:start:WikiTextHeadingRule:9:&lt;h1&gt; --><h1 id="toc3"><a name="Hexadecaphonic Octave Theory"></a><!-- ws:end:WikiTextHeadingRule:9 -->Hexadecaphonic Octave Theory</h1>
 <br />
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 &quot;3/4-tone&quot; equal division of the traditional 300-cent minor third.<br />
<br />
16-EDO is also a tuning for the <a class="wiki_link" href="/Jubilismic%20clan">no-threes 7-limit temperament tempering out 50/49</a>. 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 &quot;<strong>Magic family of scales</strong>&quot;.<br />
<br />
<a class="wiki_link" href="/Easley%20Blackwood">Easley Blackwood</a> writes of 16-EDO:<br />
&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.&quot;<br />
<br />
Another interesting approach can include two interwoven <a class="wiki_link" href="/8edo">8-EDO</a> 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 <strong>Mavila</strong> as a major seventh). If we take the 300-cent minor third as an approximation of the harmonic 19th (<a class="wiki_link" href="/19_16">19/16</a>, 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).<br />
<!-- ws:start:WikiTextRemoteImageRule:771:&lt;img src=&quot;http://www.ronsword.com/161928%20copy.jpg&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 179px; width: 316px;&quot; /&gt; --><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;" /><!-- ws:end:WikiTextRemoteImageRule:771 --><br />
<br />
The interval between the 28th &amp; 19th overtones, 28:19, measures approximately 671.3 cents, which is 3.7 cents away from 16edo's &quot;narrow fifth&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.<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:11:&lt;h1&gt; --><h1 id="toc4"><a name="Hexadecaphonic Notation"></a><!-- ws:end:WikiTextHeadingRule:11 -->Hexadecaphonic Notation</h1>
 <br />
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.<br />
<!-- ws:start:WikiTextRemoteImageRule:772:&lt;img src=&quot;http://ronsword.com/DSgoldsmith_piece.jpg&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 342px; width: 1008px;&quot; /&gt; --><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;" /><!-- ws:end:WikiTextRemoteImageRule:772 --><br />
<hr />
In 16-EDO diatonic scales are dissonant and &quot;shimmery&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.<br />
Moment of Symmetry Scales like Mavila [7] (or &quot;Inverse/Anti-Diatonic&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 &quot;Mavila-[9] Staff&quot; does just this, and places the arrangement (222122221) on nine white &quot;natural&quot; keys of the 16-EDO keyboard. <a class="wiki_link" href="/23edo">23-EDO</a> also works with the Mavila-[9] 6-line staff, notated as 1/3 tones of 16-EDO. If the 9-note &quot;Nonatonic&quot; MOS is adapted for 16-EDO, then perhaps it makes sense to refer to Octaves as 2/1, &quot;<a class="wiki_link" href="/Decave">Decave</a>&quot;.<br />
<br />
<a class="wiki_link" href="/Paul%20Erlich">Paul Erlich</a> writes,<br />
&quot;Like the conventional 12-EDO diatonic and pentatonic (meantone) scales, these arise from tempering out a<br />
unison vector from Fokker periodicity blocks. Only in 16-EDO, that unison vector is 135:128, instead of 81:80.&quot;<br />
<br />
<!-- ws:start:WikiTextHeadingRule:13:&lt;h2&gt; --><h2 id="toc5"><a name="Hexadecaphonic Notation-16 Tone Piano Layout Based on the Anti-diatonic Scale"></a><!-- ws:end:WikiTextHeadingRule:13 -->16 Tone Piano Layout Based on the Anti-diatonic Scale</h2>
 <!-- ws:start:WikiTextLocalImageRule:770:&lt;img src=&quot;/file/view/16-EDO-PIano-Diagram.png/599092760/748x293/16-EDO-PIano-Diagram.png&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 293px; width: 748px;&quot; /&gt; --><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;" /><!-- ws:end:WikiTextLocalImageRule:770 --><br />
This Layout places Mavila[7] on the white keys and Mavila[9] on the white keys. As you can see, flats are higher than naturals and sharps are lower.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:15:&lt;h1&gt; --><h1 id="toc6"><a name="Rank two temperaments"></a><!-- ws:end:WikiTextHeadingRule:15 -->Rank two temperaments</h1>
 <a class="wiki_link" href="/List%20of%2016et%20rank%20two%20temperaments%20by%20badness">List of 16et rank two temperaments by badness</a><br />
<br />


<table class="wiki_table">
    <tr>
        <th>Periods<br />
per octave<br />
</th>
        <th>Generator<br />
</th>
        <th>Temperaments<br />
</th>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>1\16<br />
</td>
        <td style="text-align: center;"><a class="wiki_link" href="/Valentine">Valentine</a>, <a class="wiki_link" href="/Slurpee">slurpee</a><br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>3\16<br />
</td>
        <td style="text-align: center;"><a class="wiki_link" href="/Gorgo">Gorgo</a><br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>5\16<br />
</td>
        <td style="text-align: center;">Messed-up <a class="wiki_link" href="/Magic">magic</a>/muggles<br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>7\16<br />
</td>
        <td style="text-align: center;"><a class="wiki_link" href="/Mavila">Mavila</a>/armodue<br />
</td>
    </tr>
    <tr>
        <td>2<br />
</td>
        <td>1\16<br />
</td>
        <td style="text-align: center;"><a class="wiki_link" href="/Bipelog">Bipelog</a><br />
</td>
    </tr>
    <tr>
        <td>2<br />
</td>
        <td>3\16<br />
</td>
        <td style="text-align: center;"><a class="wiki_link" href="/Lemba">Lemba</a>, <a class="wiki_link" href="/Astrology">astrology</a><br />
</td>
    </tr>
    <tr>
        <td>4<br />
</td>
        <td>1\16<br />
</td>
        <td style="text-align: center;"><a class="wiki_link" href="/Diminished">Diminished</a>/demolished<br />
</td>
    </tr>
    <tr>
        <td>8<br />
</td>
        <td>1\16<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
</table>

<br />
<strong>Mavila</strong><br />


<table class="wiki_table">
    <tr>
        <td>[5]:<br />
</td>
        <td>5 2 5 2 2<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>[7]:<br />
</td>
        <td>3 2 2 3 2 2 2<br />
</td>
        <td><!-- ws:start:WikiTextMediaRule:1:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/MavilaAntidiatonic16edo.mp3?h=20&amp;w=240&quot; class=&quot;WikiMedia WikiMediaFile&quot; id=&quot;wikitext@@media@@type=&amp;quot;file&amp;quot; key=&amp;quot;MavilaAntidiatonic16edo.mp3&amp;quot; width=&amp;quot;240&amp;quot; height=&amp;quot;20&amp;quot;&quot; title=&quot;Local Media File&quot;height=&quot;20&quot; width=&quot;240&quot;/&gt; --><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&autostart=false&repeat=false&showdigits=true&showfsbutton=false&width=240&height=20"></embed><!-- ws:end:WikiTextMediaRule:1 --><br />
</td>
    </tr>
    <tr>
        <td>[9]:<br />
</td>
        <td>1 2 2 2 1 2 2 2 2<br />
</td>
        <td><!-- ws:start:WikiTextMediaRule:2:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/MavilaSuperdiatonic16edo.mp3?h=20&amp;w=240&quot; class=&quot;WikiMedia WikiMediaFile&quot; id=&quot;wikitext@@media@@type=&amp;quot;file&amp;quot; key=&amp;quot;MavilaSuperdiatonic16edo.mp3&amp;quot; width=&amp;quot;240&amp;quot; height=&amp;quot;20&amp;quot;&quot; title=&quot;Local Media File&quot;height=&quot;20&quot; width=&quot;240&quot;/&gt; --><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&autostart=false&repeat=false&showdigits=true&showfsbutton=false&width=240&height=20"></embed><!-- ws:end:WikiTextMediaRule:2 --><br />
</td>
    </tr>
</table>

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


<table class="wiki_table">
    <tr>
        <th>Comma<br />
</th>
        <th>Monzo<br />
</th>
        <th>Value (Cents)<br />
</th>
        <th>Name 1<br />
</th>
        <th>Name 2<br />
</th>
        <th>Name 3<br />
</th>
    </tr>
    <tr>
        <td style="text-align: center;">135/128<br />
</td>
        <td style="text-align: left;">| -7 3 1 &gt;<br />
</td>
        <td style="text-align: right;">92.18<br />
</td>
        <td style="text-align: center;">Major Chroma<br />
</td>
        <td style="text-align: center;">Major Limma<br />
</td>
        <td style="text-align: center;">Pelogic Comma<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">648/625<br />
</td>
        <td style="text-align: left;">| 3 4 -4 &gt;<br />
</td>
        <td style="text-align: right;">62.57<br />
</td>
        <td style="text-align: center;">Major Diesis<br />
</td>
        <td style="text-align: center;">Diminished Comma<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">3125/3072<br />
</td>
        <td style="text-align: left;">| -10 -1 5 &gt;<br />
</td>
        <td style="text-align: right;">29.61<br />
</td>
        <td style="text-align: center;">Small Diesis<br />
</td>
        <td style="text-align: center;">Magic Comma<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">1212717/1210381<br />
</td>
        <td style="text-align: left;">| 23 6 -14 &gt;<br />
</td>
        <td style="text-align: right;">3.34<br />
</td>
        <td style="text-align: center;">Vishnuzma<br />
</td>
        <td style="text-align: center;">Semisuper<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">36/35<br />
</td>
        <td style="text-align: left;">| 2 2 -1 -1 &gt;<br />
</td>
        <td style="text-align: right;">48.77<br />
</td>
        <td style="text-align: center;">Septimal Quarter Tone<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">525/512<br />
</td>
        <td style="text-align: left;">| -9 1 2 1 &gt;<br />
</td>
        <td style="text-align: right;">43.41<br />
</td>
        <td style="text-align: center;">Avicennma<br />
</td>
        <td style="text-align: center;">Avicenna's Enharmonic Diesis<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">50/49<br />
</td>
        <td style="text-align: left;">| 1 0 2 -2 &gt;<br />
</td>
        <td style="text-align: right;">34.98<br />
</td>
        <td style="text-align: center;">Tritonic Diesis<br />
</td>
        <td style="text-align: center;">Jubilisma<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">64827/64000<br />
</td>
        <td style="text-align: left;">| -9 3 -3 4 &gt;<br />
</td>
        <td style="text-align: right;">22.23<br />
</td>
        <td style="text-align: center;">Squalentine<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">3125/3087<br />
</td>
        <td style="text-align: left;">| 0 -2 5 -3 &gt;<br />
</td>
        <td style="text-align: right;">21.18<br />
</td>
        <td style="text-align: center;">Gariboh<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">126/125<br />
</td>
        <td style="text-align: left;">| 1 2 -3 1 &gt;<br />
</td>
        <td style="text-align: right;">13.79<br />
</td>
        <td style="text-align: center;">Septimal Semicomma<br />
</td>
        <td style="text-align: center;">Starling Comma<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">1029/1024<br />
</td>
        <td style="text-align: left;">| -10 1 0 3 &gt;<br />
</td>
        <td style="text-align: right;">8.43<br />
</td>
        <td style="text-align: center;">Gamelisma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">6144/6125<br />
</td>
        <td style="text-align: left;">| 11 1 -3 -2 &gt;<br />
</td>
        <td style="text-align: right;">5.36<br />
</td>
        <td style="text-align: center;">Porwell<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">121/120<br />
</td>
        <td style="text-align: left;">| -3 -1 -1 0 2 &gt;<br />
</td>
        <td style="text-align: right;">14.37<br />
</td>
        <td style="text-align: center;">Biyatisma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">176/175<br />
</td>
        <td style="text-align: left;">| 4 0 -2 -1 1 &gt;<br />
</td>
        <td style="text-align: right;">9.86<br />
</td>
        <td style="text-align: center;">Valinorsma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">385/384<br />
</td>
        <td style="text-align: left;">| -7 -1 1 1 1 &gt;<br />
</td>
        <td style="text-align: right;">4.50<br />
</td>
        <td style="text-align: center;">Keenanisma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">441/440<br />
</td>
        <td style="text-align: left;">| -3 2 -1 2 -1 &gt;<br />
</td>
        <td style="text-align: right;">3.93<br />
</td>
        <td style="text-align: center;">Werckisma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">3025/3024<br />
</td>
        <td style="text-align: left;">| -4 -3 2 -1 2 &gt;<br />
</td>
        <td style="text-align: right;">0.57<br />
</td>
        <td style="text-align: center;">Lehmerisma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
</table>

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

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