16edo
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[[toc|flat]] ---- **16edo** 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 identical to that of 12-edo, giving it four diminished seventh chords exactly like those of [[12edo]], 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. =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 "blown fifth" of 675 cents means it works as a mavila temperament tuning. For a 16-edo version of Indonesian music, four small steps of 225 cents and one large one of 300 cents gives a [[MOSScales|MOS]] version of the Slendro scale, and five small steps of 150 cents with two large ones of 225 steps a Pelog-like MOS. The temperament could be popular for its easy manageability of 150 cent intervals 3/4, 9/4 and 21/4-tones. The 25 cent difference in the steps can have a similar effect the [[scales of Olympos have]] with buried enharmonic genera. 16edo 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 16edo provides 5/16 octaves. For this, there are MOS of sizes 7, 10, and 13; these are shown below under "magic family of scales". 16-edo can be treated as four interwoven diminished seventh arpeggios, or as two interwoven [[8edo]] scales (narrow [[11-limit]] neutral seconds which, when stacked, produce traditional 300 cent minor third intervals). There are two minor seventh intervals, a harmonic seventh at step 13, a 7/4 ratio approximation, sharp by 6.174 cents, followed by an undecimal 11/6 ratio or neutral seventh. The septimal can be the 9/4th tone or septimal semi-diminished fourth (35/27 ratio) , semi-augmented fifth (54/35), harmonic seventh (7/4), and septimal whole tone 8/7. The undecimal intervals are the 3/4 tone or undecimal neutral second (12/11), and the 21/4th tone or undecimal neutral seventh (11/6). Another xenharmonic aspect of 16-tone is how the 11-limit whole tone scale, using the neutral second, interlocks with the diminished scale, similar to the augmented scale and whole tone relationship in 12-tone (the whole tone divides the major third in 12, in 16-its the minor third). In 16-tone, because of the 25 cent difference in the steps from 100 in 12-tone, a western "twelve tone ear" hears dissonance with more complexity and less familiarity than even [[24edo|24-tone]], yet within a more manageable number of tones and a strange familiarity - the diminished family - making 16-edo a truly xenharmonic system. 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. 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". Example on Goldsmith board: [[image:http://www.ronsword.com/161928%20copy.jpg width="158" height="92"]]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. [[image:http://ronsword.com/DSgoldsmith_piece.jpg width="1008" height="342"]] ---- In 16-edo diatonic scales are dissonant because of the 25 cent raised superfourth in conjunction with the 25 cent subtracted fifth / poor 3/2 approximation. The septimal semi diminished fourth can be more desirable. Perhaps using Moment of Symmetry Scales an alternative temperament families like the "Anti-Diatonic" Mavila (which reverses step sizes of diatonic), Diminished, Happy, Rice, Grumpy, Mosh, Magic, Lemba, Cynder, and Decatonic can be more interesting and suitable: Diminished family of scales (1 3 1 3 1 3 1 3, 1 1 2 1 1 2 1 1 2 1 1 2) Magic family of scales (1 4 1 4 1 4 1, 1 3 1 1 3 1 1 1 3 1, 1 1 2 1 1 1 2 1 1 1 2 1 1) Cynder family (3 3 4 3 3, 3 3 1 3 3 3, 1 2 1 2 1 2 1 2 1 2 1) Lemba family (3 2 3 3 2 3, 2 1 2 1 2 2 1 2 1 2) About Mavila Paul Erlich writes, "Like the conventional 12-tet 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." Mavila (1 2 2 2 1 2 2 2 2, 3 2 2 3 2 2 2, 5 2 5 2 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...." 0. 1/1 C or 1 1. 75.00 cents C# Dbb or 1* 2. 150.00 cents Cx Db or 2 3. 225.00 cents D or 2* 4. 300.00 cents D# Ebb or 3 5. 375.00 cents Dx Eb or 3* 6. 450.00 cents E Fb or 4 7. 525.00 cents F or 5 8. 600.00 cents F# Gbb or 5* 9. 675.00 cents Fx Gb or 6 10. 750.00 cents G Abb or 6* 11. 825.00 cents G# Ab or 7 12. 900.00 cents A or 7* 13. 975.00 cents A# Bbb or 8 14. 1050.00 cents Ax Bb or 8* 15. 1125.00 cents B Cb or 9 16. 2/1 C or 1 1 octave into 8 equal parts = 2 2 2 2 2 2 2 2 = 3/4 tone Neutral Second Progression 2 octaves into 8 equal parts = 4 4 4 4 4 4 4 4 = Classic Minor Third Progression 3 octaves into 8 equal parts = 6 6 6 6 6 6 6 6 = 9/4tone or Septimal semi-dim Fourth Progression 4 octaves into 8 equal parts = 8 8 8 8 8 8 8 8 = Tritone Progression 5 octaves into 8 equal parts = 10 10 10 10 10 10 10 10 = Septimal semi-aug Fifth Progression 6 octaves into 8 equal parts = 12 12 12 12 12 12 12 12 = Classic Sixth Progression 7 octaves into 8 equal parts = 14 14 14 14 14 14 14 14 = 21/4 tone or Neutral Seventh Progression 8 octaves into 8 equal parts = 16 16 16 16 16 16 16 16 = Octave Progression 9 octaves into 8 equal parts = 18 18 18 18 18 18 18 18 = Ninth Progression =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 ||= ||= || =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. =**Armodue theory**= [[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.. =External links= [[image:http://ronsword.com/images/ESG_sm.jpg width="120" height="161"]] 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.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|Ron Sword - 16-tone steel string acoustic diddle]] [[http://www.jeanpierrepoulin.com/mp3/Armodue78.mp3|Armodue78]] by [[@http://www.jeanpierrepoulin.com/|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]]
Original HTML content:
<html><head><title>16edo</title></head><body><!-- ws:start:WikiTextTocRule:12:<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/> --><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: --><a href="#Hexadecaphonic Octave Theory">Hexadecaphonic Octave Theory</a><!-- ws:end:WikiTextTocRule:13 --><!-- ws:start:WikiTextTocRule:14: --> | <a href="#Commas">Commas</a><!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --> | <a href="#Hexadecaphonic Notation:">Hexadecaphonic Notation:</a><!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --> | <a href="#Armodue theory">Armodue theory</a><!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --> | <a href="#External links">External links</a><!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --> | <a href="#Compositions">Compositions</a><!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: -->
<!-- ws:end:WikiTextTocRule:19 --><hr />
<br />
<strong>16edo</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 identical to that of 12-edo, giving it four diminished seventh chords exactly like those of <a class="wiki_link" href="/12edo">12edo</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 />
<!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Hexadecaphonic Octave Theory"></a><!-- ws:end:WikiTextHeadingRule:0 -->Hexadecaphonic Octave Theory</h1>
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 "blown fifth" of 675 cents means it works as a mavila temperament tuning. For a 16-edo version of Indonesian music, four small steps of 225 cents and one large one of 300 cents gives a <a class="wiki_link" href="/MOSScales">MOS</a> version of the Slendro scale, and five small steps of 150 cents with two large ones of 225 steps a Pelog-like MOS. The temperament could be popular for its easy manageability of 150 cent intervals 3/4, 9/4 and 21/4-tones. The 25 cent difference in the steps can have a similar effect the <a class="wiki_link" href="/scales%20of%20Olympos%20have">scales of Olympos have</a> with buried enharmonic genera.<br />
<br />
16edo 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 16edo provides 5/16 octaves. For this, there are MOS of sizes 7, 10, and 13; these are shown below under "magic family of scales".<br />
<br />
16-edo can be treated as four interwoven diminished seventh arpeggios, or as two interwoven <a class="wiki_link" href="/8edo">8edo</a> scales (narrow <a class="wiki_link" href="/11-limit">11-limit</a> neutral seconds which, when stacked, produce traditional 300 cent minor third intervals). There are two minor seventh intervals, a harmonic seventh at step 13, a 7/4 ratio approximation, sharp by 6.174 cents, followed by an undecimal 11/6 ratio or neutral seventh. The septimal can be the 9/4th tone or septimal semi-diminished fourth (35/27 ratio) , semi-augmented fifth (54/35), harmonic seventh (7/4), and septimal whole tone 8/7. The undecimal intervals are the 3/4 tone or undecimal neutral second (12/11), and the 21/4th tone or undecimal neutral seventh (11/6). Another xenharmonic aspect of 16-tone is how the 11-limit whole tone scale, using the neutral second, interlocks with the diminished scale, similar to the augmented scale and whole tone relationship in 12-tone (the whole tone divides the major third in 12, in 16-its the minor third).<br />
<br />
In 16-tone, because of the 25 cent difference in the steps from 100 in 12-tone, a western "twelve tone ear" hears dissonance with more complexity and less familiarity than even <a class="wiki_link" href="/24edo">24-tone</a>, yet within a more manageable number of tones and a strange familiarity - the diminished family - making 16-edo a truly xenharmonic system.<br />
<br />
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. 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". Example on Goldsmith board: <!-- ws:start:WikiTextRemoteImageRule:458:<img src="http://www.ronsword.com/161928%20copy.jpg" alt="" title="" style="height: 92px; width: 158px;" /> --><img src="http://www.ronsword.com/161928%20copy.jpg" alt="external image 161928%20copy.jpg" title="external image 161928%20copy.jpg" style="height: 92px; width: 158px;" /><!-- ws:end:WikiTextRemoteImageRule:458 -->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 />
<!-- ws:start:WikiTextRemoteImageRule:459:<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:459 --><br />
<hr />
In 16-edo diatonic scales are dissonant because of the 25 cent raised superfourth in conjunction with the 25 cent subtracted fifth / poor 3/2 approximation. The septimal semi diminished fourth can be more desirable. Perhaps using Moment of Symmetry Scales an alternative temperament families like the "Anti-Diatonic" Mavila (which reverses step sizes of diatonic), Diminished, Happy, Rice, Grumpy, Mosh, Magic, Lemba, Cynder, and Decatonic can be more interesting and suitable:<br />
<br />
Diminished family of scales (1 3 1 3 1 3 1 3, 1 1 2 1 1 2 1 1 2 1 1 2)<br />
Magic family of scales (1 4 1 4 1 4 1, 1 3 1 1 3 1 1 1 3 1, 1 1 2 1 1 1 2 1 1 1 2 1 1)<br />
Cynder family (3 3 4 3 3, 3 3 1 3 3 3, 1 2 1 2 1 2 1 2 1 2 1)<br />
Lemba family (3 2 3 3 2 3, 2 1 2 1 2 2 1 2 1 2)<br />
<br />
About Mavila Paul Erlich writes, "Like the conventional 12-tet diatonic and pentatonic<br />
(meantone) scales, these arise from tempering out a unison vector from Fokker periodicity<br />
blocks. Only in 16-EDO, that unison vector is 135:128, instead of 81:80."<br />
<br />
Mavila (1 2 2 2 1 2 2 2 2, 3 2 2 3 2 2 2, 5 2 5 2 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 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...."<br />
<br />
0. 1/1 C or 1<br />
1. 75.00 cents C# Dbb or 1*<br />
2. 150.00 cents Cx Db or 2<br />
3. 225.00 cents D or 2*<br />
4. 300.00 cents D# Ebb or 3<br />
5. 375.00 cents Dx Eb or 3*<br />
6. 450.00 cents E Fb or 4<br />
7. 525.00 cents F or 5<br />
8. 600.00 cents F# Gbb or 5*<br />
9. 675.00 cents Fx Gb or 6<br />
10. 750.00 cents G Abb or 6*<br />
11. 825.00 cents G# Ab or 7<br />
12. 900.00 cents A or 7*<br />
13. 975.00 cents A# Bbb or 8<br />
14. 1050.00 cents Ax Bb or 8*<br />
15. 1125.00 cents B Cb or 9<br />
16. 2/1 C or 1<br />
<br />
<br />
1 octave into 8 equal parts = 2 2 2 2 2 2 2 2 = 3/4 tone Neutral Second Progression<br />
2 octaves into 8 equal parts = 4 4 4 4 4 4 4 4 = Classic Minor Third Progression<br />
3 octaves into 8 equal parts = 6 6 6 6 6 6 6 6 = 9/4tone or Septimal semi-dim Fourth Progression<br />
4 octaves into 8 equal parts = 8 8 8 8 8 8 8 8 = Tritone Progression<br />
5 octaves into 8 equal parts = 10 10 10 10 10 10 10 10 = Septimal semi-aug Fifth Progression<br />
6 octaves into 8 equal parts = 12 12 12 12 12 12 12 12 = Classic Sixth Progression<br />
7 octaves into 8 equal parts = 14 14 14 14 14 14 14 14 = 21/4 tone or Neutral Seventh Progression<br />
8 octaves into 8 equal parts = 16 16 16 16 16 16 16 16 = Octave Progression<br />
9 octaves into 8 equal parts = 18 18 18 18 18 18 18 18 = Ninth Progression<br />
<!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><a name="Commas"></a><!-- ws:end:WikiTextHeadingRule:2 -->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>
<!-- ws:start:WikiTextHeadingRule:4:<h1> --><h1 id="toc2"><a name="Hexadecaphonic Notation:"></a><!-- ws:end:WikiTextHeadingRule:4 -->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<br />
additions to A-G. Armodue of Italy uses a 4-line staff for 16-EDO.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:<h1> --><h1 id="toc3"><a name="Armodue theory"></a><!-- ws:end:WikiTextHeadingRule:6 --><strong>Armodue theory</strong></h1>
<br />
<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 />
<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:8:<h1> --><h1 id="toc4"><a name="External links"></a><!-- ws:end:WikiTextHeadingRule:8 -->External links</h1>
<!-- ws:start:WikiTextRemoteImageRule:460:<img src="http://ronsword.com/images/ESG_sm.jpg" alt="" title="" style="height: 161px; width: 120px;" /> --><img src="http://ronsword.com/images/ESG_sm.jpg" alt="external image ESG_sm.jpg" title="external image ESG_sm.jpg" style="height: 161px; width: 120px;" /><!-- ws:end:WikiTextRemoteImageRule:460 --><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:10:<h1> --><h1 id="toc5"><a name="Compositions"></a><!-- ws:end:WikiTextHeadingRule:10 -->Compositions</h1>
<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 Herman Miller<br />
<a class="wiki_link_ext" href="http://www.ronsword.com/sounds/Ron%20Sword%20-%2016-tone%20acoustic%20improvisation.mp3" rel="nofollow">Ron Sword - 16-tone steel string acoustic diddle</a><br />
<a class="wiki_link_ext" href="http://www.jeanpierrepoulin.com/mp3/Armodue78.mp3" rel="nofollow">Armodue78</a> by <a class="wiki_link_ext" href="http://www.jeanpierrepoulin.com/" rel="nofollow" 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></body></html>