16edo
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<span style="display: block; text-align: right;">Other languages: [[xenharmonie/16edo|Deutsch]] </span> [[toc|flat]] ---- **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. || Degree || Cents ||= Approximate Ratios* || Interval Name || || 0 || 0 ||= 1/1 || Unison || || 1 || 75 ||= 28/27, 27/26 || Subminor 2nd || || 2 || 150 ||= 12/11, 35/32 || Neutral 2nd || || 3 || 225 ||= 8/7 || Supermajor 2nd, Septimal Whole-Tone || || 4 || 300 ||= 19/16, 32/27 || Minor 3rd || || 5 || 375 ||= 5/4, 26/21 || Major 3rd || || 6 || 450 ||= 13/10, 35/27 || Sub-4th, Supermajor 3rd || || 7 || 525 ||= 27/20, 52/35, 256/189 || Wide 4th || || 8 || 600 ||= 7/5, 10/7 || Tritone || || 9 || 675 ||= 40/27, 35/26, 189/128 || Narrow 5th || || 10 || 750 ||= 20/13, 54/35 || Super-5th, Subminor 6th || || 11 || 825 ||= 8/5, 21/13 || Minor 6th || || 12 || 900 ||= 27/16, 32/19 || Major 6th || || 13 || 975 ||= 7/4 || Subminor 7th, Septimal Minor 7th || || 14 || 1050 ||= 11/6, 64/35 || Neutral 7th || || 15 || 1125 ||= 27/14, 52/27 || Supermajor 7th || || 16 || 1200 ||= 2/1 || Octave || *based on treating 16-EDO as a 2.5.7.13.19.27 subgroup temperament; other approaches are possible. [[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." =Rank two temperaments= [[List of 16et rank two temperaments by badness]] ||~ Periods per octave ||~ Generator ||~ Temperaments || || 1 || 1\16 ||= || || 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]] || || 4 || 1\16 ||= [[Diminished]] || || 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...." =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]] by [[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]]
Original HTML content:
<html><head><title>16edo</title></head><body><span style="display: block; text-align: right;">Other languages: <a class="wiki_link" href="http://xenharmonie.wikispaces.com/16edo">Deutsch</a><br />
</span><br />
<!-- ws:start:WikiTextTocRule:17:<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/> --><!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --><a href="#Hexadecaphonic Octave Theory">Hexadecaphonic Octave Theory</a><!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --> | <a href="#Hexadecaphonic Notation">Hexadecaphonic Notation</a><!-- ws:end:WikiTextTocRule:19 --><!-- ws:start:WikiTextTocRule:20: --> | <a href="#Rank two temperaments">Rank two temperaments</a><!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --> | <a href="#Commas">Commas</a><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --> | <a href="#Armodue Theory (4-line staff)">Armodue Theory (4-line staff)</a><!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextTocRule:23: --> | <a href="#Books/Literature">Books/Literature</a><!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --> | <a href="#Compositions">Compositions</a><!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: -->
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<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 />
<table class="wiki_table">
<tr>
<td>Degree<br />
</td>
<td>Cents<br />
</td>
<td style="text-align: center;">Approximate<br />
Ratios*<br />
</td>
<td>Interval Name<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>
</tr>
<tr>
<td>1<br />
</td>
<td>75<br />
</td>
<td style="text-align: center;">28/27, 27/26<br />
</td>
<td>Subminor 2nd<br />
</td>
</tr>
<tr>
<td>2<br />
</td>
<td>150<br />
</td>
<td style="text-align: center;">12/11, 35/32<br />
</td>
<td>Neutral 2nd<br />
</td>
</tr>
<tr>
<td>3<br />
</td>
<td>225<br />
</td>
<td style="text-align: center;">8/7<br />
</td>
<td>Supermajor 2nd,<br />
Septimal Whole-Tone<br />
</td>
</tr>
<tr>
<td>4<br />
</td>
<td>300<br />
</td>
<td style="text-align: center;">19/16, 32/27<br />
</td>
<td>Minor 3rd<br />
</td>
</tr>
<tr>
<td>5<br />
</td>
<td>375<br />
</td>
<td style="text-align: center;">5/4, 26/21<br />
</td>
<td>Major 3rd<br />
</td>
</tr>
<tr>
<td>6<br />
</td>
<td>450<br />
</td>
<td style="text-align: center;">13/10, 35/27<br />
</td>
<td>Sub-4th,<br />
Supermajor 3rd<br />
</td>
</tr>
<tr>
<td>7<br />
</td>
<td>525<br />
</td>
<td style="text-align: center;">27/20, 52/35, 256/189<br />
</td>
<td>Wide 4th<br />
</td>
</tr>
<tr>
<td>8<br />
</td>
<td>600<br />
</td>
<td style="text-align: center;">7/5, 10/7<br />
</td>
<td>Tritone<br />
</td>
</tr>
<tr>
<td>9<br />
</td>
<td>675<br />
</td>
<td style="text-align: center;">40/27, 35/26, 189/128<br />
</td>
<td>Narrow 5th<br />
</td>
</tr>
<tr>
<td>10<br />
</td>
<td>750<br />
</td>
<td style="text-align: center;">20/13, 54/35<br />
</td>
<td>Super-5th,<br />
Subminor 6th<br />
</td>
</tr>
<tr>
<td>11<br />
</td>
<td>825<br />
</td>
<td style="text-align: center;">8/5, 21/13<br />
</td>
<td>Minor 6th<br />
</td>
</tr>
<tr>
<td>12<br />
</td>
<td>900<br />
</td>
<td style="text-align: center;">27/16, 32/19<br />
</td>
<td>Major 6th<br />
</td>
</tr>
<tr>
<td>13<br />
</td>
<td>975<br />
</td>
<td style="text-align: center;">7/4<br />
</td>
<td>Subminor 7th,<br />
Septimal Minor 7th<br />
</td>
</tr>
<tr>
<td>14<br />
</td>
<td>1050<br />
</td>
<td style="text-align: center;">11/6, 64/35<br />
</td>
<td>Neutral 7th<br />
</td>
</tr>
<tr>
<td>15<br />
</td>
<td>1125<br />
</td>
<td style="text-align: center;">27/14, 52/27<br />
</td>
<td>Supermajor 7th<br />
</td>
</tr>
<tr>
<td>16<br />
</td>
<td>1200<br />
</td>
<td style="text-align: center;">2/1<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 />
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<!-- ws:start:WikiTextHeadingRule:3:<h1> --><h1 id="toc0"><a name="Hexadecaphonic Octave Theory"></a><!-- ws:end:WikiTextHeadingRule:3 -->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 "3/4-tone" 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 "<strong>Magic family of scales</strong>".<br />
<br />
<a class="wiki_link" href="/Easley%20Blackwood">Easley Blackwood</a> writes of 16-EDO:<br />
"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."<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:564:<img src="http://www.ronsword.com/161928%20copy.jpg" alt="" title="" style="height: 179px; width: 316px;" /> --><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:564 --><br />
<br />
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.<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:5:<h1> --><h1 id="toc1"><a name="Hexadecaphonic Notation"></a><!-- ws:end:WikiTextHeadingRule:5 -->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:565:<img src="http://ronsword.com/DSgoldsmith_piece.jpg" alt="" title="" style="height: 342px; width: 1008px;" /> --><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:565 --><br />
<hr />
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.<br />
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. <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 "Nonatonic" MOS is adapted for 16-EDO, then perhaps it makes sense to refer to Octaves as 2/1, "<a class="wiki_link" href="/Decave">Decave</a>".<br />
<br />
<a class="wiki_link" href="/Paul%20Erlich">Paul Erlich</a> writes,<br />
"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."<br />
<br />
<!-- ws:start:WikiTextHeadingRule:7:<h1> --><h1 id="toc2"><a name="Rank two temperaments"></a><!-- ws:end:WikiTextHeadingRule:7 -->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;"><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><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><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:<img src="http://www.wikispaces.com/site/embedthumbnail/file-audio/MavilaAntidiatonic16edo.mp3?h=20&w=240" class="WikiMedia WikiMediaFile" id="wikitext@@media@@type=&quot;file&quot; key=&quot;MavilaAntidiatonic16edo.mp3&quot; width=&quot;240&quot; height=&quot;20&quot;" title="Local Media File"height="20" width="240"/> --><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:<img src="http://www.wikispaces.com/site/embedthumbnail/file-audio/MavilaSuperdiatonic16edo.mp3?h=20&w=240" class="WikiMedia WikiMediaFile" id="wikitext@@media@@type=&quot;file&quot; key=&quot;MavilaSuperdiatonic16edo.mp3&quot; width=&quot;240&quot; height=&quot;20&quot;" title="Local Media File"height="20" width="240"/> --><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: "The trouble (in 16-EDO) has ... to do with the fact that the distance between the major third<br />
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<br />
135/128 (the difference between 16/15 and 9/8) is tempered out...."<br />
<br />
<!-- ws:start:WikiTextHeadingRule:9:<h1> --><h1 id="toc3"><a name="Commas"></a><!-- ws:end:WikiTextHeadingRule:9 -->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> < 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 ><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 ><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 ><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 ><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 ><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 ><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 ><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 ><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 ><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 ><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 ><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 ><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 ><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 ><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 ><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 ><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 ><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:11:<h1> --><h1 id="toc4"><a name="Armodue Theory (4-line staff)"></a><!-- ws:end:WikiTextHeadingRule:11 --><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:13:<h1> --><h1 id="toc5"><a name="Books/Literature"></a><!-- ws:end:WikiTextHeadingRule:13 -->Books/Literature</h1>
Sword, Ronald. "Thesaurus of Melodic Patterns and Intervals for 16-Tones" IAAA Press, USA. First Ed: August, 2011<br />
Sword, Ronald. "Hexadecaphonic Scales for Guitar." IAAA Press, UK-USA. First Ed: Feb, 2010. (superfourth tuning)<br />
Sword, Ronald. "Esadekaphonic Scales for Guitar." IAAA Press, UK-USA. First Ed: April, 2009. (semi-diminished fourth tuning)<br />
<br />
<!-- ws:start:WikiTextHeadingRule:15:<h1> --><h1 id="toc6"><a name="Compositions"></a><!-- ws:end:WikiTextHeadingRule:15 -->Compositions</h1>
<a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=660895&songID=12370649&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> 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></body></html>