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**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.
**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 identical to that of 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
|| Degree || Cents ||= Approximate
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=Hexadecaphonic Octave Theory=  
=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".
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-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).
"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."


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.
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, 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="198" height="121"]]


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


=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"]]
[[image:http://ronsword.com/DSgoldsmith_piece.jpg width="1008" height="342"]]
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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:
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. 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, "Duple", or "Decave".
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)
[[Paul Erlich]] writes,


[[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...."
"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."  


0. 1/1 C or 1
**Mavila**
1. 75.00 cents C# Dbb or 1*
[5]: 5 2 5 2 2
2. 150.00 cents Cx Db or 2
[7]: 3 2 2 3 2 2 2
3. 225.00 cents D or 2*
[9]: 1 2 2 2 1 2 2 2 2
4. 300.00 cents D# Ebb or 3
**Diminished**
5. 375.00 cents Dx Eb or 3*
[8]: 1 3 1 3 1 3 1 3
6. 450.00 cents E Fb or 4
[12]: 1 1 2 1 1 2 1 1 2 1 1 2
7. 525.00 cents F or 5
**Magic**
8. 600.00 cents F# Gbb or 5*
[7]: 1 4 1 4 1 4 1
9. 675.00 cents Fx Gb or 6
[10]: 1 3 1 1 3 1 1 1 3 1
10. 750.00 cents G Abb or 6*
[13]: 1 1 2 1 1 1 2 1 1 1 2 1 1
11. 825.00 cents G# Ab or 7
**Cynder/Gorgo**
12. 900.00 cents A or 7*
[5]: 3 3 4 3 3
13. 975.00 cents A# Bbb or 8
[6]: 3 3 1 3 3 3
14. 1050.00 cents Ax Bb or 8*
[11]: 1 2 1 2 1 2 1 2 1 2 1
15. 1125.00 cents B Cb or 9
**Lemba**
16. 2/1 C or 1
[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...."


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=  
=Commas=  
16 EDO [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes [[val]] &lt; 16 25 37 45 55 59 |.)
16 EDO [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes [[val]] &lt; 16 25 37 45 55 59 |.)
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||= 441/440 ||&lt; | -3 2 -1 2 -1 &gt; ||&gt; 3.93 ||= Werckisma ||=  ||=  ||
||= 441/440 ||&lt; | -3 2 -1 2 -1 &gt; ||&gt; 3.93 ||= Werckisma ||=  ||=  ||
||= 3025/3024 ||&lt; | -4 -3 2 -1 2 &gt; ||&gt; 0.57 ||= Lehmerisma ||=  ||=  ||
||= 3025/3024 ||&lt; | -4 -3 2 -1 2 &gt; ||&gt; 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**=  
=**Armodue Theory (4-line staff)**=  


[[http://www.armodue.com/ricerche.htm|Armodue]]: Italian pages of theory for 16-tone (esadekaphonic) system, including compositions.
[[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..


Translations of parts of the Armodue pages can be found [[Armodue|here]] on this wiki..
=Books/Literature=


=External links=
Sword, Ronald. "Thesaurus of Melodic Patterns and Intervals for 16-Tones" IAAA Press, USA. First Ed: August, 2011
[[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. "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)
Sword, Ronald. "Esadekaphonic Scales for Guitar." IAAA Press, UK-USA. First Ed: April, 2009. (semi-diminished fourth tuning)
=External Links=
[[http://www.ronsword.com/16tonepianoproject.html]] "The 16-tone Piano Project"


=Compositions=  
=Compositions=  
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[[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]]</pre></div>
[[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]]</pre></div>
<h4>Original HTML content:</h4>
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<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;16edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:14:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:14 --&gt;&lt;!-- ws:start:WikiTextTocRule:15: --&gt;&lt;a href="#Hexadecaphonic Octave Theory"&gt;Hexadecaphonic Octave Theory&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:15 --&gt;&lt;!-- ws:start:WikiTextTocRule:16: --&gt; | &lt;a href="#Hexadecaphonic Notation:"&gt;Hexadecaphonic Notation:&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:16 --&gt;&lt;!-- ws:start:WikiTextTocRule:17: --&gt; | &lt;a href="#Commas"&gt;Commas&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:17 --&gt;&lt;!-- ws:start:WikiTextTocRule:18: --&gt; | &lt;a href="#Armodue Theory (4-line staff)"&gt;Armodue Theory (4-line staff)&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:18 --&gt;&lt;!-- ws:start:WikiTextTocRule:19: --&gt; | &lt;a href="#Books/Literature"&gt;Books/Literature&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;!-- ws:start:WikiTextTocRule:20: --&gt; | &lt;a href="#External Links"&gt;External Links&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:20 --&gt;&lt;!-- ws:start:WikiTextTocRule:21: --&gt; | &lt;a href="#Compositions"&gt;Compositions&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:21 --&gt;&lt;!-- ws:start:WikiTextTocRule:22: --&gt;
&lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;hr /&gt;
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&lt;br /&gt;
&lt;br /&gt;
&lt;strong&gt;16edo&lt;/strong&gt; is the &lt;a class="wiki_link" href="/equal%20division%20of%20the%20octave"&gt;equal division of the octave&lt;/a&gt; into sixteen narrow chromatic semitones each of 75 &lt;a class="wiki_link" href="/cent"&gt;cent&lt;/a&gt;s exactly. It is not especially good at representing most low-integer musical intervals, but it has a &lt;a class="wiki_link" href="/7_4"&gt;7/4&lt;/a&gt; which is six cents sharp, and a &lt;a class="wiki_link" href="/5_4"&gt;5/4&lt;/a&gt; which is eleven cents flat. Four steps of it gives the 300 cent minor third interval identical to that of 12-edo, giving it four diminished seventh chords exactly like those of &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;, and a diminished triad on each scale step.&lt;br /&gt;
&lt;strong&gt;16-EDO&lt;/strong&gt; is the &lt;a class="wiki_link" href="/equal%20division%20of%20the%20octave"&gt;equal division of the octave&lt;/a&gt; into sixteen narrow chromatic semitones each of 75 &lt;a class="wiki_link" href="/cent"&gt;cent&lt;/a&gt;s exactly. It is not especially good at representing most low-integer musical intervals, but it has a &lt;a class="wiki_link" href="/7_4"&gt;7/4&lt;/a&gt; which is six cents sharp, and a &lt;a class="wiki_link" href="/5_4"&gt;5/4&lt;/a&gt; which is eleven cents flat. Four steps of it gives the 300 cent minor third interval identical to that of 12-EDO, giving it four diminished seventh chords exactly like those of &lt;a class="wiki_link" href="/12edo"&gt;12-EDO&lt;/a&gt;, and a diminished triad on each scale step.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;


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&lt;br /&gt;
&lt;br /&gt;
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&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Hexadecaphonic Octave Theory"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Hexadecaphonic Octave Theory&lt;/h1&gt;
  The scale supports the diminished temperament with its 1/4 octave period, though its generator size, equal to its step size of 75 cents, is smaller than ideal. Its very flat &amp;quot;blown fifth&amp;quot; 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 &lt;a class="wiki_link" href="/MOSScales"&gt;MOS&lt;/a&gt; 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 &lt;a class="wiki_link" href="/scales%20of%20Olympos%20have"&gt;scales of Olympos have&lt;/a&gt; with buried enharmonic genera.&lt;br /&gt;
  &lt;br /&gt;
The scale supports the diminished temperament with its 1/4 octave period, though its generator size, equal to its step size of 75 cents, is smaller than ideal. Its very flat 3/2 of 675 cents supports Mavila temperament, where the mapping of major and minor is reversed. The temperament could be popular for its 150-cent &amp;quot;3/4-tone&amp;quot; equal division of the traditional 300-cent minor third.&lt;br /&gt;
16-EDO is also a tuning for the &lt;a class="wiki_link" href="/Jubilismic%20clan"&gt;no-threes 7-limit temperament tempering out 50/49&lt;/a&gt;. This has a flat major third as generator, for which 16-EDO provides 5/16 octaves. For this, there are MOS of sizes 7, 10, and 13; these are shown below under &amp;quot;&lt;strong&gt;Magic family of scales&lt;/strong&gt;&amp;quot;.&lt;br /&gt;
&lt;a class="wiki_link" href="/Easley%20Blackwood"&gt;Easley Blackwood&lt;/a&gt; writes of 16-EDO: &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
16edo is also a tuning for the &lt;a class="wiki_link" href="/Jubilismic%20clan"&gt;no-threes 7-limit temperament tempering out 50/49&lt;/a&gt;. This has a flat major third as generator, for which 16edo provides 5/16 octaves. For this, there are MOS of sizes 7, 10, and 13; these are shown below under &amp;quot;magic family of scales&amp;quot;.&lt;br /&gt;
&amp;quot;16 notes: This tuning is best thought of as a combination of four intertwined diminished seventh chords. Since 12-note tuning can be regarded as a combination of three diminished seventh chords, it is plain that the two tunings have elements in common. The most obvious difference in the way the two tunings sound and work is that triads in 16-note tuning, although recognizable, are too discordant to serve as the final harmony in cadences. Keys can still be established by successions of altered subdominant and dominant harmonies, however, and the Etude is based mainly upon this property. The fundamental consonant harmony employed is a minor triad with an added minor seventh.&amp;quot;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
16-edo can be treated as four interwoven diminished seventh arpeggios, or as two interwoven &lt;a class="wiki_link" href="/8edo"&gt;8edo&lt;/a&gt; scales (narrow &lt;a class="wiki_link" href="/11-limit"&gt;11-limit&lt;/a&gt; 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).&lt;br /&gt;
Another interesting approach can include two interwoven &lt;a class="wiki_link" href="/8edo"&gt;8&lt;/a&gt;-EDO scales (narrow 12/11 neutral seconds). There are two major 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 (which is mapped in 16's &lt;strong&gt;Mavila&lt;/strong&gt; as a major seventh).&lt;br /&gt;
If we take the 300-cent minor third as an approximation of the harmonic 19th (&lt;a class="wiki_link" href="/19_16"&gt;19/16&lt;/a&gt;, approximately 297.5 cents), that adds another overtone which can combine with the approximation of the harmonic seventh to form a 16:19:28 triad (pictured below). &lt;br /&gt;
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In 16-tone, because of the 25 cent difference in the steps from 100 in 12-tone, a western &amp;quot;twelve tone ear&amp;quot; hears dissonance with more complexity and less familiarity than even &lt;a class="wiki_link" href="/24edo"&gt;24-tone&lt;/a&gt;, yet within a more manageable number of tones and a strange familiarity - the diminished family - making 16-edo a truly xenharmonic system.&lt;br /&gt;
The interval between the 28th &amp;amp; 19th overtones, 28:19, measures approximately 671.3 cents, which is 3.7 cents away from 16edo's &amp;quot;narrow fifth&amp;quot;. Another voicing for this chord is 14:16:19, which features 19:14 as the outer interval (528.7 cents just, 525.0 cents in 16edo). A perhaps more consonant open voicing is 7:16:19.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
If we take the 300-cent minor third as an approximation of the harmonic 19th (&lt;a class="wiki_link" href="/19_16"&gt;19/16&lt;/a&gt;, approximately 297.5 cents), that adds another overtone which can combine with the approximation of the harmonic seventh to form a 16:19:28 triad. The interval between the 28th &amp;amp; 19th overtones, 28:19, measures approximately 671.3 cents, which is 3.7 cents away from 16edo's &amp;quot;narrow fifth&amp;quot;. Example on Goldsmith board: &lt;!-- ws:start:WikiTextRemoteImageRule:458:&amp;lt;img src=&amp;quot;http://www.ronsword.com/161928%20copy.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 92px; width: 158px;&amp;quot; /&amp;gt; --&gt;&lt;img src="http://www.ronsword.com/161928%20copy.jpg" alt="external image 161928%20copy.jpg" title="external image 161928%20copy.jpg" style="height: 92px; width: 158px;" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:458 --&gt;Another voicing for this chord is 14:16:19, which features 19:14 as the outer interval (528.7 cents just, 525.0 cents in 16edo). A perhaps more consonant open voicing is 7:16:19.&lt;br /&gt;
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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&lt;br /&gt;
additions to A-G. Armodue of Italy uses a 4-line staff for 16-EDO.&lt;br /&gt;
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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 &amp;quot;Anti-Diatonic&amp;quot; Mavila (which reverses step sizes of diatonic), Diminished, Happy, Rice, Grumpy, Mosh, Magic, Lemba, Cynder, and Decatonic can be more interesting and suitable:&lt;br /&gt;
In 16-EDO diatonic scales are dissonant and &amp;quot;shimmery&amp;quot; because of the 25 cent raised superfourth in conjunction with the 25 cent subtracted fifth / poor 3/2 approximation. Scales like the Harmonic Minor scale in 16-EDO require 4 step sizes. &lt;br /&gt;
Moment of Symmetry Scales like Mavila [7] (or &amp;quot;Inverse/Anti-Diatonic&amp;quot; which reverses step sizes of diatonic from LLsLLLs to ssLsssL in the heptatonic variation) can work as an alternative to the traditional diatonic scale. The 6-line 16-EDO &amp;quot;Mavila-[9] Staff&amp;quot; does just this, and places the arrangement (222122221) on nine white &amp;quot;natural&amp;quot; keys of the 16-EDO Keyboard. 23-EDO also works with the Mavila-[9] 6-line Staff, notated as 1/3 tones of 16-EDO. If the 9-note &amp;quot;Nonatonic&amp;quot; MOS is adapted for 16-EDO, then perhaps it makes sense to refer to Octaves as 2/1, &amp;quot;Duple&amp;quot;, or &amp;quot;Decave&amp;quot;.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Diminished family of scales (1 3 1 3 1 3 1 3, 1 1 2 1 1 2 1 1 2 1 1 2)&lt;br /&gt;
&lt;a class="wiki_link" href="/Paul%20Erlich"&gt;Paul Erlich&lt;/a&gt; writes,&lt;br /&gt;
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)&lt;br /&gt;
Cynder family (3 3 4 3 3, 3 3 1 3 3 3, 1 2 1 2 1 2 1 2 1 2 1)&lt;br /&gt;
Lemba family (3 2 3 3 2 3, 2 1 2 1 2 2 1 2 1 2)&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
About Mavila Paul Erlich writes, &amp;quot;Like the conventional 12-tet diatonic and pentatonic&lt;br /&gt;
&amp;quot;Like the conventional 12-EDO diatonic and pentatonic (meantone) scales, these arise from tempering out a &lt;br /&gt;
(meantone) scales, these arise from tempering out a unison vector from Fokker periodicity&lt;br /&gt;
unison vector from Fokker periodicity blocks.Only in 16-EDO, that unison vector is 135:128, instead of 81:80.&amp;quot; &lt;br /&gt;
blocks. Only in 16-EDO, that unison vector is 135:128, instead of 81:80.&amp;quot;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Mavila (1 2 2 2 1 2 2 2 2, 3 2 2 3 2 2 2, 5 2 5 2 2)&lt;br /&gt;
&lt;strong&gt;Mavila&lt;/strong&gt;&lt;br /&gt;
[5]: 5 2 5 2 2&lt;br /&gt;
[7]: 3 2 2 3 2 2 2&lt;br /&gt;
[9]: 1 2 2 2 1 2 2 2 2&lt;br /&gt;
&lt;strong&gt;Diminished&lt;/strong&gt;&lt;br /&gt;
[8]: 1 3 1 3 1 3 1 3 &lt;br /&gt;
[12]: 1 1 2 1 1 2 1 1 2 1 1 2&lt;br /&gt;
&lt;strong&gt;Magic&lt;/strong&gt;&lt;br /&gt;
[7]: 1 4 1 4 1 4 1 &lt;br /&gt;
[10]: 1 3 1 1 3 1 1 1 3 1 &lt;br /&gt;
[13]: 1 1 2 1 1 1 2 1 1 1 2 1 1&lt;br /&gt;
&lt;strong&gt;Cynder/Gorgo&lt;/strong&gt; &lt;br /&gt;
[5]: 3 3 4 3 3&lt;br /&gt;
[6]: 3 3 1 3 3 3 &lt;br /&gt;
[11]: 1 2 1 2 1 2 1 2 1 2 1&lt;br /&gt;
&lt;strong&gt;Lemba&lt;/strong&gt;&lt;br /&gt;
[6]: 3 2 3 3 2 3 &lt;br /&gt;
[10]: 2 1 2 1 2 2 1 2 1 2&lt;br /&gt;
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&lt;a class="wiki_link" href="/Igliashon%20Jones"&gt;Igliashon Jones&lt;/a&gt; writes, &amp;quot;The trouble (in 16-EDO) has ... to do with the fact that the distance between the major third and the &amp;quot;fourth&amp;quot; is the same as the distance between the &amp;quot;fourth&amp;quot; and the &amp;quot;fifth&amp;quot; (i.e. near a 12/11)...This mean(s) that 135/128 (the difference between 16/15 and 9/8) is tempered out....&amp;quot;&lt;br /&gt;
&lt;a class="wiki_link" href="/Igliashon%20Jones"&gt;Igliashon Jones&lt;/a&gt; writes, &amp;quot;The trouble (in 16-EDO) has ... to do with the fact that the distance between the major third &lt;br /&gt;
and the &amp;quot;fourth&amp;quot; is the same as the distance between the &amp;quot;fourth&amp;quot; and the &amp;quot;fifth&amp;quot; (i.e. near a 12/11)...This mean(s) that &lt;br /&gt;
135/128 (the difference between 16/15 and 9/8) is tempered out....&amp;quot;&lt;br /&gt;
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0. 1/1 C or 1&lt;br /&gt;
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1. 75.00 cents C# Dbb or 1*&lt;br /&gt;
2. 150.00 cents Cx Db or 2&lt;br /&gt;
3. 225.00 cents D or 2*&lt;br /&gt;
4. 300.00 cents D# Ebb or 3&lt;br /&gt;
5. 375.00 cents Dx Eb or 3*&lt;br /&gt;
6. 450.00 cents E Fb or 4&lt;br /&gt;
7. 525.00 cents F or 5&lt;br /&gt;
8. 600.00 cents F# Gbb or 5*&lt;br /&gt;
9. 675.00 cents Fx Gb or 6&lt;br /&gt;
10. 750.00 cents G Abb or 6*&lt;br /&gt;
11. 825.00 cents G# Ab or 7&lt;br /&gt;
12. 900.00 cents A or 7*&lt;br /&gt;
13. 975.00 cents A# Bbb or 8&lt;br /&gt;
14. 1050.00 cents Ax Bb or 8*&lt;br /&gt;
15. 1125.00 cents B Cb or 9&lt;br /&gt;
16. 2/1 C or 1&lt;br /&gt;
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1 octave into 8 equal parts = 2 2 2 2 2 2 2 2 = 3/4 tone Neutral Second Progression&lt;br /&gt;
2 octaves into 8 equal parts = 4 4 4 4 4 4 4 4 = Classic Minor Third Progression&lt;br /&gt;
3 octaves into 8 equal parts = 6 6 6 6 6 6 6 6 = 9/4tone or Septimal semi-dim Fourth Progression&lt;br /&gt;
4 octaves into 8 equal parts = 8 8 8 8 8 8 8 8 = Tritone Progression&lt;br /&gt;
5 octaves into 8 equal parts = 10 10 10 10 10 10 10 10 = Septimal semi-aug Fifth Progression&lt;br /&gt;
6 octaves into 8 equal parts = 12 12 12 12 12 12 12 12 = Classic Sixth Progression&lt;br /&gt;
7 octaves into 8 equal parts = 14 14 14 14 14 14 14 14 = 21/4 tone or Neutral Seventh Progression&lt;br /&gt;
8 octaves into 8 equal parts = 16 16 16 16 16 16 16 16 = Octave Progression&lt;br /&gt;
9 octaves into 8 equal parts = 18 18 18 18 18 18 18 18 = Ninth Progression&lt;br /&gt;
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  16 EDO &lt;a class="wiki_link" href="/tempering%20out"&gt;tempers out&lt;/a&gt; the following &lt;a class="wiki_link" href="/comma"&gt;comma&lt;/a&gt;s. (Note: This assumes &lt;a class="wiki_link" href="/val"&gt;val&lt;/a&gt; &amp;lt; 16 25 37 45 55 59 |.)&lt;br /&gt;
  16 EDO &lt;a class="wiki_link" href="/tempering%20out"&gt;tempers out&lt;/a&gt; the following &lt;a class="wiki_link" href="/comma"&gt;comma&lt;/a&gt;s. (Note: This assumes &lt;a class="wiki_link" href="/val"&gt;val&lt;/a&gt; &amp;lt; 16 25 37 45 55 59 |.)&lt;br /&gt;


Line 647: Line 640:
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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&lt;br /&gt;
additions to A-G. Armodue of Italy uses a 4-line staff for 16-EDO.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
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&lt;a class="wiki_link_ext" href="http://www.armodue.com/ricerche.htm" rel="nofollow"&gt;Armodue&lt;/a&gt;: Italian pages of theory for 16-tone (esadekaphonic) system, including compositions.&lt;br /&gt;
&lt;a class="wiki_link_ext" href="http://www.armodue.com/ricerche.htm" rel="nofollow"&gt;Armodue&lt;/a&gt;: Italian pages of theory for 16-tone (esadekaphonic) system, including compositions.&lt;br /&gt;
&lt;br /&gt;
Translations of parts of the Armodue pages can be found &lt;a class="wiki_link" href="/Armodue"&gt;here&lt;/a&gt; on this wiki..&lt;br /&gt;
Translations of parts of the Armodue pages can be found &lt;a class="wiki_link" href="/Armodue"&gt;here&lt;/a&gt; on this wiki..&lt;br /&gt;
&lt;br /&gt;
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Sword, Ronald. &amp;quot;Hexadecaphonic Scales for Guitar.&amp;quot; IAAA Press, UK-USA. First Ed: Feb, 2010. (superfourth tuning).&lt;br /&gt;
Sword, Ronald. &amp;quot;Thesaurus of Melodic Patterns and Intervals for 16-Tones&amp;quot; IAAA Press, USA. First Ed: August, 2011&lt;br /&gt;
Sword, Ronald. &amp;quot;Hexadecaphonic Scales for Guitar.&amp;quot; IAAA Press, UK-USA. First Ed: Feb, 2010. (superfourth tuning)&lt;br /&gt;
Sword, Ronald. &amp;quot;Esadekaphonic Scales for Guitar.&amp;quot; IAAA Press, UK-USA. First Ed: April, 2009. (semi-diminished fourth tuning)&lt;br /&gt;
Sword, Ronald. &amp;quot;Esadekaphonic Scales for Guitar.&amp;quot; IAAA Press, UK-USA. First Ed: April, 2009. (semi-diminished fourth tuning)&lt;br /&gt;
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&lt;a class="wiki_link_ext" href="http://www.ronsword.com/16tonepianoproject.html" rel="nofollow"&gt;http://www.ronsword.com/16tonepianoproject.html&lt;/a&gt; &amp;quot;The 16-tone Piano Project&amp;quot;&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc6"&gt;&lt;a name="Compositions"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Compositions&lt;/h1&gt;
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  &lt;br /&gt;
&lt;a class="wiki_link_ext" href="http://www.io.com/%7Ehmiller/midi/16tet.mid" rel="nofollow"&gt;Etude in 16-tone equal tuning&lt;/a&gt; &lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/16tet.mp3" rel="nofollow"&gt;play&lt;/a&gt; by Herman Miller&lt;br /&gt;
&lt;a class="wiki_link_ext" href="http://www.io.com/%7Ehmiller/midi/16tet.mid" rel="nofollow"&gt;Etude in 16-tone equal tuning&lt;/a&gt; &lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/16tet.mp3" rel="nofollow"&gt;play&lt;/a&gt; by Herman Miller&lt;br /&gt;
Retrieved from "https://en.xen.wiki/w/16edo"