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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{interwiki
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| de = 16-EDO
: This revision was by author [[User:guest|guest]] and made on <tt>2010-08-16 02:04:52 UTC</tt>.<br>
| en = 16edo
: The original revision id was <tt>156694117</tt>.<br>
| es = 16 EDO
: The revision comment was: <tt></tt><br>
| ja = 16平均律
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
}}
<h4>Original Wikitext content:</h4>
{{Infobox ET}}
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">[[toc|flat]]
{{ED intro}}
----


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


[[user:Andrew_Heathwaite|1281203319]] adds: If we take the 300-cent minor third as an approximation of the harmonic 19th (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".
== Theory ==
The [[3/2|perfect fifth]] of 16edo is 27 cents flat of 3/2, flatter than that of [[7edo]] so that it generates an [[2L 5s|antidiatonic]] instead of [[5L 2s|diatonic]] scale, but sharper than [[9edo]]'s fifth, to which it similarly retains the characteristic of being a fifth while being distinctly flat of 3/2. If the fifth is interpreted as 3/2, this befits a tuning of [[mavila]], the [[5-limit]] [[regular temperament|temperament]] that [[tempering out|tempers out]] [[135/128]], such that a stack of four fifths gives a [[6/5]] minor third instead of the familiar [[5/4]] major third as in [[meantone]]. A more accurate restriction is [[mabilic]], which discards the inaccurate mapping of 3 while keeping the fifth as a generator.  


=Hexadecaphonic Octave Theory=
This leads to some confusion in regards to interval names, as what would be major in diatonic now sounds minor; there are several ways to handle this (see in [[#Intervals]]).  
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.


16-edo can be treated as four interwoven diminished seventh arpeggios, or as two interwoven 8-edo scales (narrow 11-limit whole tones 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, off by 3.5879 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 general, 16edo tends to better approximate the differences between odd [[harmonic]]s than odd harmonics themselves, though it has a [[5/1|5th harmonic]] which is only 11 cents flat, and a [[7/1|7th harmonic]] which is only 6 cents sharp. As such, 16edo can be seen as an approach to tuning that takes advantage of the idea that simpler ratios can be functionally approximated with greater error (i.e. a 3/2 that's 25 cents flat is still recognizable, but a 5/4 that's 25 cents flat loses much of its identity and a 7/4 that's 25 cents flat is completely unrecognizable). In essence, 16edo's 3, 5, and 7 are backwards from 12edo's, with 7 being nearly perfect, 5 being decent, and 3 being distinctly out-of-tune.  


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 24-tone, yet within a more manageable number of tones and a strange familiarity - the diminished family
In terms of higher primes, both 11 and 13 are approximated very flat, with the [[11/8]] not distinguished from [[4/3]], and [[13/8]] not distinguished from [[8/5]]. 16edo represents the no-9 no-15 [[25-odd-limit]] [[consistent]]ly, however.
- making 16-edo is a truly xenharmonic system.
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)
Four steps of 16edo gives the 300{{c}} minor third interval shared by [[12edo]] (and other multiples of [[4edo]]), which approximates [[6/5]], and thus tempers out 648/625, the [[diminished comma]]. This means that the familiar [[diminished seventh chord]] may be built on any scale step with four unique tetrads up to [[octave equivalence]]. The minor third is of course not distinguished from the septimal subminor third, [[7/6]], so [[36/35]] and moreover [[50/49]] are tempered out, making 16edo a possible tuning for [[diminished (temperament)|septimal diminished]]. Another possible interpretation for this interval is the 19th harmonic, [[19/16]].
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
16edo shares several similarities with 15edo. They both share mappings of [[8/7]], [[5/4]], and [[3/2]] in terms of edosteps – in fact, they are both [[valentine]] tunings, and thus [[slendric]] tunings. 16edo and 15edo also both have three types of seconds and two types of thirds (not including arto/tendo thirds). However, 15edo's fifth is sharp while 16's is flat.  
(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)
16edo works as a tuning for [[extraclassical tonality]], due to its ultramajor third of 450 cents.


[[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...."
=== Odd harmonics ===
{{Harmonics in equal|16}}


0. 1/1 C or 1
=== Octave stretch ===
1. 75.00 cents C# Dbb or 1*
Having a flat tendency, 16et is best tuned with [[stretched octave]]s, which improve the accuracy of wide-voiced JI chords and [[rooted]] harmonics especially on inharmonic timbres such as bells and gamelans, with [[25edt]], [[41ed6]], and [[57ed12]] being good options.
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


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


1 octave into 8 equal parts = 2 2 2 2 2 2 2 2 = 3/4 tone Neutral Second Progression
=== Composition theory ===
2 octaves into 8 equal parts = 4 4 4 4 4 4 4 4 = Classic Minor Third Progression
* [[User:VectorGraphics/16edo theory|Vector's approach]]
3 octaves into 8 equal parts = 6 6 6 6 6 6 6 6 = 9/4tone or Septimal semi-dim Fourth Progression
* [[Armodue harmony]]
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


{{Todo|inline=1| expand }}


== Intervals ==
{{Mavila}}


=External theory links=
Alternatively, one can use Armodue nine-nominal notation.


[[http://www.armodue.com/ricerche.htm|Armodue]]: Italian pages of theory for 16-tone (esadekaphonic) system, including compositions - translation, anyone?
{| class="wikitable center-all"
|-
! rowspan="2" | Degree
! rowspan="2" | [[Cent]]s
! rowspan="2" | Approximate<br>ratios*
! colspan="6" | Names
|-
! colspan="2" | Antidiatonic
! colspan="2" | Diatonic
! Just
! Simplified
|-
| 0
| 0
| 1/1
| unison
| D
| unison
| D
| unison
| unison
|-
| 1
| 75
| 28/27, 27/26
| aug 1, dim 2nd
| D♯, E♭
| dim 1, aug 2nd
| D♭, E♯
| subminor 2nd
| min 2nd
|-
| 2
| 150
| 35/32
| minor 2nd
| E
| major 2nd
| E
| neutral 2nd
| maj 2nd
|-
| 3
| 225
| 8/7
| major 2nd
| E♯
| minor 2nd
| E♭
| supermajor 2nd,<br>septimal whole-tone
| perf 2nd
|-
| 4
| 300
| 19/16, 32/27
| minor 3rd
| F♭
| major 3rd
| F♯
| minor 3rd
| min 3rd
|-
| 5
| 375
| 5/4, 16/13, 26/21
| major 3rd
| F
| minor 3rd
| F
| major 3rd
| maj 3rd
|-
| 6
| 450
| 13/10, 35/27
| aug 3rd,<br>dim 4th
| F♯, G♭
| dim 3rd,<br>aug 4th
| F♭, G♯
| sub-4th,<br>supermajor 3rd
| min 4th
|-
| 7
| 525
| 19/14, 27/20, 35/26, 256/189
| perfect 4th
| G
| perfect 4th
| G
| wide 4th
| maj 4th
|-
| 8
| 600
| 7/5, 10/7
| aug 4th,<br>dim 5th
| G♯, A♭
| dim 4th,<br>aug 5th
| G♭, A♯
| tritone
| aug 4th,<br>dim 5th
|-
| 9
| 675
| 28/19, 40/27, 52/35, 189/128
| perfect 5th
| A
| perfect 5th
| A
| narrow 5th
| min 5th
|-
| 10
| 750
| 20/13, 54/35
| aug 5th,<br>dim 6th
| A♯, B♭
| dim 5th,<br>aug 6th
| A♭, B♯
| super-5th,<br>subminor 6th
| maj 5th
|-
| 11
| 825
| 8/5, 13/8, 21/13
| minor 6th
| B
| major 6th
| B
| minor 6th
| min 6th
|-
| 12
| 900
| 27/16, 32/19
| major 6th
| B♯
| minor 6th
| B♭
| major 6th
| maj 6th
|-
| 13
| 975
| 7/4
| minor 7th
| C♭
| major 7th
| C♯
| subminor 7th,<br>septimal minor 7th
| perf 7th
|-
| 14
| 1050
| 64/35
| major 7th
| C
| minor 7th
| C
| neutral 7th
| min 7th
|-
| 15
| 1125
| 27/14, 52/27
| aug 7th,<br>dim 8ve
| C♯, D♭
| dim 7th,<br>aug 8ve
| C♭, D♯
| supermajor 7th
| maj 7th
|-
| 16
| 1200
| 2/1
| 8ve
| D
| 8ve
| D
| octave
| octave
|}
<nowiki />* Based on treating 16edo as a 2.5.7.13.19.27 subgroup temperament; other approaches are possible.


[[image:http://ronsword.com/images/ESG_sm.jpg width="120" height="161"]]
== Notation ==
Sword, Ronald. "Hexadecaphonic Scales for Guitar." IAAA Press, UK-USA. First Ed: Feb, 2010. (superfourth tuning).
16edo notation can be easy utilizing [[Goldsmith's Circle]] of keys, nominals, and respective notation{{clarify}}. The nominals for a 6 line staff can be switched for [[Erv Wilson]]'s Beta and Epsilon additions to A–G. The Armodue model uses a 4-line staff for 16edo.
Sword, Ronald. "Esadekaphonic Scales for Guitar." IAAA Press, UK-USA. First Ed: April, 2009. (semi-diminished fourth tuning)


=Compositions=
Mos scales like Mavila[7] (or "inverse/anti-diatonic" which reverses step sizes of diatonic from LLsLLLs to ssLsssL in the heptatonic variation) can work as an alternative to the traditional diatonic scale, while maintaining conventional A–G ♯/♭ notation as described above. Alternatively, one can utilize the Mavila[9] mos, for a sort of "hyper-diatonic" scale of 7 large steps and 2 small steps. [[Armodue theory|Armodue notation]] of 16edo "Mavila[9] Staff" does just this, and places the arrangement (222122221) on nine white "natural" keys of the 16edo keyboard. If the 9-note (enneatonic) mos is adopted as a notational basis for 16edo, then we need an entirely different set of interval classes than any of the heptatonic classes described above; perhaps it even makes sense to refer to the octave ([[2/1]]) as the "[[decave]]". This is identical to the KISS notation for this scale when using numbers.


[[http://www.io.com/%7Ehmiller/midi/16tet.mid|Etude in 16-tone equal tuning]] by Herman Miller
{| class="wikitable center-all"
[[http://www.jeanpierrepoulin.com/mp3/Armodue78.mp3|Armodue78]] by [[@http://www.jeanpierrepoulin.com/|Jean-Pierre Poulin]]
|-
! Degree
! Cents
! colspan="2" | Mavila[9] notation
|-
| 0
| 0
| unison
| 1
|-
| 1
| 75
| aug unison, minor 2nd
| 1♯, 2♭
|-
| 2
| 150
| major 2nd
| 2
|-
| 3
| 225
| aug 2nd, minor 3rd
| 2♯, 3♭
|-
| 4
| 300
| major 3rd, dim 4th
| 3, 4𝄫
|-
| 5
| 375
| minor 4th
| 4♭
|-
| 6
| 450
| major 4th,<br>dim 5th
| 4, 5♭
|-
| 7
| 525
| aug 4th, minor 5th
| 4♯, 5
|-
| 8
| 600
| aug 5th, dim 6th
| 5♯, 6♭
|-
| 9
| 675
| perfect 6th, dim 7th
| 6, 7𝄫
|-
| 10
| 750
| aug 6th, minor 7th
| 6♯, 7♭
|-
| 11
| 825
| major 7th
| 7
|-
| 12
| 900
| aug 7th, minor 8th
| 7♯, 8♭
|-
| 13
| 975
| major 8th, dim 9th
| 8, 9𝄫
|-
| 14
| 1050
| minor 9th
| 9
|-
| 15
| 1125
| major 9th, dim 10ve
| 9♯, 1♭
|-
| 16
| 1200
| 10ve (Decave)
| 1
|}


[[@http://ronsword.com/sounds/16chordscale_improv.mp3|Chord-scale Improvisation in 16-tet]] by Ron Sword
=== Sagittal notation ===
[[@http://www.ronsword.com/sounds/ron_sword_16_improv.mp3|Chromatic 16-tet Improvisation]] by Ron Sword
This notation uses the same sagittal sequence as [[21edo #Sagittal notation|21edo]].
[[@http://www.ronsword.com/sounds/Ron%20Sword%20-%2016-tone%20acoustic%20improvisation.mp3|16-tet Acoustic Improvisation]] by Ron Sword
 
[[@http://www.ronsword.com/sounds/ronsword_miracle528_part3.mp3|16-tet Magic Drone]] by Ron Sword</pre></div>
<imagemap>
<h4>Original HTML content:</h4>
File:16-EDO_Sagittal.svg
<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:6:&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:6 --&gt;&lt;!-- ws:start:WikiTextTocRule:7: --&gt;&lt;a href="#Hexadecaphonic Octave Theory"&gt;Hexadecaphonic Octave Theory&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:7 --&gt;&lt;!-- ws:start:WikiTextTocRule:8: --&gt; | &lt;a href="#External theory links"&gt;External theory links&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:8 --&gt;&lt;!-- ws:start:WikiTextTocRule:9: --&gt; | &lt;a href="#Compositions"&gt;Compositions&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:9 --&gt;&lt;!-- ws:start:WikiTextTocRule:10: --&gt;
desc none
&lt;!-- ws:end:WikiTextTocRule:10 --&gt;&lt;hr /&gt;
rect 80 0 300 50 [[Sagittal_notation]]
&lt;br /&gt;
rect 471 0 631 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
16-edo equal temperament is the division of the octave into sixteen narrow chromatic semitones each of 75 cents exactly. It is not especially good at representing most low-integer musical intervals, but it has a 7/4 which is six cents sharp, and a 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 12-edo, and a diminished triad on each scale step.&lt;br /&gt;
rect 20 80 471 106 [[Fractional_3-limit_notation#Bad-fifths_limma-fraction_notation | limma-fraction notation]]
&lt;br /&gt;
default [[File:16-EDO_Sagittal.svg]]
&lt;!-- ws:start:WikiTextUserlinkRule:00:[[user:Andrew_Heathwaite|1281203319]] --&gt;&lt;span class="membersnap"&gt;- &lt;a class="userLink" href="http://www.wikispaces.com/user/view/Andrew_Heathwaite" style="outline: none;"&gt;&lt;img src="http://www.wikispaces.com/user/pic/Andrew_Heathwaite-lg.jpg" width="16" height="16" alt="Andrew_Heathwaite" class="userPicture" /&gt;&lt;/a&gt; &lt;a class="userLink" href="http://www.wikispaces.com/user/view/Andrew_Heathwaite" style="outline: none;"&gt;Andrew_Heathwaite&lt;/a&gt; &lt;small&gt;Aug 7, 2010&lt;/small&gt;&lt;/span&gt;&lt;!-- ws:end:WikiTextUserlinkRule:00 --&gt; adds: If we take the 300-cent minor third as an approximation of the harmonic 19th (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;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;.&lt;br /&gt;
</imagemap>
&lt;br /&gt;
 
&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;
=== Armodue notation (4-line staff) ===
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;
[http://www.armodue.com/ricerche.htm Armodue]: Pierpaolo Beretta's website for his Armodue theory for 16edo (esadekaphonic), including compositions.
&lt;br /&gt;
 
16-edo can be treated as four interwoven diminished seventh arpeggios, or as two interwoven 8-edo scales (narrow 11-limit whole tones 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, off by 3.5879 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;
For resources on the Armodue theory, see the [[Armodue]] on this wiki
&lt;br /&gt;
 
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 24-tone, yet within a more manageable number of tones and a strange familiarity - the diminished family&lt;br /&gt;
== Chord names ==
- making 16-edo is a truly xenharmonic system.&lt;br /&gt;
16edo chords can be named using ups and downs. Using diatonic interval names, chord names bear little relationship to the sound: a minor chord (spelled {{dash|A, C, E|med}}) sounds like [[4:5:6]], the classical major triad, and a major chord (spelled {{dash|C, E, G|med}}) sounds like [[10:12:15]], a classical minor triad! Instead, using antidiatonic names, the chord names will match the sound&mdash;but finding the name from the spelling follows the rules of antidiatonic rather than diatonic interval arithmetic.
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;
 
&lt;br /&gt;
{| class="wikitable center-all"
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;
|-
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;
! rowspan="2" | Chord
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;
! rowspan="2" | JI ratios
Lemba family (3 2 3 3 2 3, 2 1 2 1 2 2 1 2 1 2)&lt;br /&gt;
! colspan="6" | Name
&lt;br /&gt;
|-
About Mavila Paul Erlich writes, &amp;quot;Like the conventional 12-tet diatonic and pentatonic&lt;br /&gt;
! colspan="3" | Diatonic
(meantone) scales, these arise from tempering out a unison vector from Fokker periodicity&lt;br /&gt;
! colspan="3" | Antidiatonic
blocks. Only in 16-EDO, that unison vector is 135:128, instead of 81:80.&amp;quot;&lt;br /&gt;
|-
&lt;br /&gt;
| {{dash|0, 5, 9|med}}
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;
| 4:5:6
&lt;br /&gt;
| D F A
&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;
| Dm
&lt;br /&gt;
| D minor
0. 1/1 C or 1&lt;br /&gt;
| D F A
1. 75.00 cents C# Dbb or 1*&lt;br /&gt;
| D
2. 150.00 cents Cx Db or 2&lt;br /&gt;
| D major
3. 225.00 cents D or 2*&lt;br /&gt;
|-
4. 300.00 cents D# Ebb or 3&lt;br /&gt;
| {{dash|0, 4, 9|med}}
5. 375.00 cents Dx Eb or 3*&lt;br /&gt;
| 10:12:15
6. 450.00 cents E Fb or 4&lt;br /&gt;
| D F♯ A
7. 525.00 cents F or 5&lt;br /&gt;
| D
8. 600.00 cents F# Gbb or 5*&lt;br /&gt;
| D major
9. 675.00 cents Fx Gb or 6&lt;br /&gt;
| D F♭ A
10. 750.00 cents G Abb or 6*&lt;br /&gt;
| Dm
11. 825.00 cents G# Ab or 7&lt;br /&gt;
| D minor
12. 900.00 cents A or 7*&lt;br /&gt;
|-
13. 975.00 cents A# Bbb or 8&lt;br /&gt;
| {{dash|0, 4, 8|med}}
14. 1050.00 cents Ax Bb or 8*&lt;br /&gt;
| 5:6:7
15. 1125.00 cents B Cb or 9&lt;br /&gt;
| D F♯ A♯
16. 2/1 C or 1&lt;br /&gt;
| Daug
&lt;br /&gt;
| D augmented
&lt;br /&gt;
| D F♭ A♭
1 octave into 8 equal parts = 2 2 2 2 2 2 2 2 = 3/4 tone Neutral Second Progression&lt;br /&gt;
| Ddim
2 octaves into 8 equal parts = 4 4 4 4 4 4 4 4 = Classic Minor Third Progression&lt;br /&gt;
| D diminished
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;
| {{dash|0, 5, 10|med}}
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;
| D F A♭
7 octaves into 8 equal parts = 14 14 14 14 14 14 14 14 = 21/4 tone or Neutral Seventh Progression&lt;br /&gt;
| Ddim
8 octaves into 8 equal parts = 16 16 16 16 16 16 16 16 = Octave Progression&lt;br /&gt;
| D diminished
9 octaves into 8 equal parts = 18 18 18 18 18 18 18 18 = Ninth Progression&lt;br /&gt;
| D F A♯
&lt;br /&gt;
| Daug
&lt;br /&gt;
| D augmented
&lt;br /&gt;
|-
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="External theory links"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;External theory links&lt;/h1&gt;
| {{dash|0, 5, 9, 13|med}}
&lt;br /&gt;
| 4:5:6:7
&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 - translation, anyone?&lt;br /&gt;
| D F A C♯
&lt;br /&gt;
| Dm(M7)
&lt;!-- ws:start:WikiTextRemoteImageRule:12:&amp;lt;img src=&amp;quot;http://ronsword.com/images/ESG_sm.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 161px; width: 120px;&amp;quot; /&amp;gt; --&gt;&lt;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;" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:12 --&gt;&lt;br /&gt;
| D minor-major
Sword, Ronald. &amp;quot;Hexadecaphonic Scales for Guitar.&amp;quot; IAAA Press, UK-USA. First Ed: Feb, 2010. (superfourth tuning).&lt;br /&gt;
| D F A C♭
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;
| D7
&lt;br /&gt;
| D seven
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Compositions"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Compositions&lt;/h1&gt;
|-
&lt;br /&gt;
| {{dash|0, 5, 9, 12|med}}
&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; by Herman Miller&lt;br /&gt;
|
&lt;a class="wiki_link_ext" href="http://www.jeanpierrepoulin.com/mp3/Armodue78.mp3" rel="nofollow"&gt;Armodue78&lt;/a&gt; by &lt;a class="wiki_link_ext" href="http://www.jeanpierrepoulin.com/" rel="nofollow" target="_blank"&gt;Jean-Pierre Poulin&lt;/a&gt;&lt;br /&gt;
| D F A Bb
&lt;br /&gt;
| Dm(♭6)
&lt;a class="wiki_link_ext" href="http://ronsword.com/sounds/16chordscale_improv.mp3" rel="nofollow" target="_blank"&gt;Chord-scale Improvisation in 16-tet&lt;/a&gt; by Ron Sword&lt;br /&gt;
| D minor flat-six
&lt;a class="wiki_link_ext" href="http://www.ronsword.com/sounds/ron_sword_16_improv.mp3" rel="nofollow" target="_blank"&gt;Chromatic 16-tet Improvisation&lt;/a&gt; by Ron Sword&lt;br /&gt;
| D F A B♯
&lt;a class="wiki_link_ext" href="http://www.ronsword.com/sounds/Ron%20Sword%20-%2016-tone%20acoustic%20improvisation.mp3" rel="nofollow" target="_blank"&gt;16-tet Acoustic Improvisation&lt;/a&gt; by Ron Sword&lt;br /&gt;
| D6
&lt;a class="wiki_link_ext" href="http://www.ronsword.com/sounds/ronsword_miracle528_part3.mp3" rel="nofollow" target="_blank"&gt;16-tet Magic Drone&lt;/a&gt; by Ron Sword&lt;/body&gt;&lt;/html&gt;</pre></div>
| D six
|-
| {{dash|0, 5, 9, 14|med}}
|
| D F A C
| Dm7
| D minor seven
| D F A C
| DM7
| D major seven
|-
| {{dash|0, 4, 9, 13|med}}
|
| D F♯ A C♯
| DM7
| D major seven
| D F♭ A C♭
| DM7
| D minor seven
|}
 
Alterations are always enclosed in parentheses, additions never are. An up or down immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord {{dash|6, 1, 3, 5, 7, 9, 11, 13}}). See [[Ups and downs notation #Chords and chord progressions]] for more examples.
 
Using antidiatonic names, if you're used to diatonic interval arithmetic, you can do antidiatonic interval arithmetic by following the simple guideline that qualities are '''reversed''' from standard diatonic. As in, just as adding two major seconds gives you a major third in 12edo, adding two minor seconds gives a minor third in 16edo.
 
That is, reversing means exchanging major for minor, aug for dim, and sharp for flat. Perfect and natural are unaffected.
 
Examples can be found at the bottom of the page.
 
== Approximation to JI ==
=== Selected just intervals by error ===
{{Q-odd-limit intervals|16}}
 
It's worth noting that the 525{{c}} interval is almost exactly halfway in between 4/3 and 11/8, making it very discordant, although playing this in the context of a larger chord, and with specialized timbres, can make this less noticeable.
 
[[File:16ed2-001.svg|alt=alt : Your browser has no SVG support.]]
 
[[:File:16ed2-001.svg|16ed2-001.svg]]
 
== Octave theory ==
The scale supports the diminished temperament with its 1/4 octave period, though its generator size, equal to its step size of 75{{c}}, is smaller than ideal. Its very flat 3/2 of 675{{c}} [[support]]s Mavila temperament, where the mapping of major and minor is reversed. The temperament could be popular for its 150{{c}} "3/4-tone" equal division of the traditional 300{{c}} minor third.
 
16edo is also a tuning for the [[Jubilismic clan|no-threes 7-limit temperament tempering out 50/49]]. This has a period of a half-octave (600{{c}}), and a generator of a flat septimal major 2nd, for which 16edo uses 3\16. For this, there are mos scales of sizes 4, 6, and 10; extending this temperament to the full 7-limit can produce either Lemba or Astrology (16edo supports both, but is not a very accurate tuning of either).
 
16edo is also a tuning for the no-threes 7-limit temperament tempering out [http://x31eq.com/cgi-bin/uv.cgi?uvs=%5B-19%2C7%2C1%3E&limit=2_5_7 546875:524288], which has a flat major third as generator, for which 16-EDO provides 5\16 octaves. For this, there are MOS of sizes 7, 10, and 13; these are shown below under "'''Magic family of scales'''".
 
[[Easley Blackwood Jr]] writes of 16edo:
 
"''16 notes: This tuning is best thought of as a combination of four intertwined diminished seventh chords. Since 12-note tuning can be regarded as a combination of three diminished seventh chords, it is plain that the two tunings have elements in common. The most obvious difference in the way the two tunings sound and work is that triads in 16-note tuning, although recognizable, are too discordant to serve as the final harmony in cadences. Keys can still be established by successions of altered subdominant and dominant harmonies, however, and the Etude is based mainly upon this property. The fundamental consonant harmony employed is a minor triad with an added minor seventh.''"
 
From a harmonic series perspective, if we take 13\16 as a 7/4 ratio approximation, sharp by 6.174{{c}}, and take the 300{{c}} minor third as an approximation of the harmonic 19th ([[19/16]], approximately 297.5{{c}}), that can combine with the approximation of the harmonic seventh to form a 16:19:28 triad .
 
The interval between the 28th &amp; 19th harmonics, 28:19, measures approximately 671.3{{c}}, which is 3.7{{c}} away from 16edo's "narrow fifth". Another voicing for this chord is 14:16:19, which features 19:14 as the outer interval (528.7{{c}} just, 525.0{{c}} in 16edo). A perhaps more consonant open voicing is 7:16:19
 
== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|edo=16}}
 
=== Commas ===
16et [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes [[val]] {{val| 16 25 37 45 55 59 }}.)
 
{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime<br>limit]]
! [[Ratio]]<ref group=note>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 5
| [[135/128]]
| {{monzo| -7 3 1 }}
| 92.18
| Layobi
| Mavila comma, major chroma
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, major diesis
|-
| 5
| [[3125/3072]]
| {{monzo| -10 -1 5 }}
| 29.61
| Laquinyo
| Magic comma
|-
| 5
| [[6115295232/6103515625|(20 digits)]]
| {{monzo| 23 6 -14 }}
| 3.34
| Sasepbiru
| [[Vishnuzma]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quartertone
|-
| 7
| [[525/512]]
| {{monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64827/64000]]
| {{monzo| -9 3 -3 4 }}
| 22.23
| Laquadzo-atrigu
| Squalentine comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[1029/1024]]
| {{monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[6144/6125]]
| {{monzo| 11 1 -3 -2 }}
| 5.36
| Sarurutrigu
| Porwell comma
|-
| 11
| [[121/120]]
| {{monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[385/384]]
| {{monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[3025/3024]]
| {{monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|}
 
=== Rank-2 temperaments ===
* [[List of 16et rank two temperaments by badness]]
 
{| class="wikitable center-1 center-2"
|+ Table of temperaments by generator
|-
! Periods<br>per 8ve
! Generator
! Temperaments
|-
| 1
| 1\16
| [[Valentine]], [[slurpee]]
|-
| 1
| 3\16
| [[Gorgo]]
|-
| 1
| 5\16
| [[Magic]]/[[muggles]]
|-
| 1
| 7\16
| [[Mavila]]/[[armodue]]
|-
| 2
| 1\16
| [[Bipelog]]
|-
| 2
| 3\16
| [[Lemba]], [[astrology]]
|-
| 4
| 1\16
| [[Diminished (temperament)|Diminished]]/[[demolished]]
|-
| 8
| 1\16
| [[Semidim]]
|}
 
== Scales ==
* {{Main|List of MOS scales in {{PAGENAME}}}}
Important mosses include:
* [[magic]] anti-diatonic 3L4s 1414141 (5\16, 1\1)
* [[magic]] superdiatonic 3L7s 1311311311 (5\16, 1\1)
* [[magic]] chromatic 11121121112 3L10s (5\16, 1\1)
* [[mavila]] anti-diatonic 2L5s 2223223 (9\16, 1\1)
* [[mavila]] superdiatonic 7L2s 222212221 (9\16, 1\1)
* [[gorgo]] 5L1s 333331 (3\16, 1\1)
* [[lemba]] 4L2s 332332 (3\16, 1\2)
 
 
'''Mavila'''
 
{| class="wikitable"
|-
| [5]:
| 5 2 5 2 2
|
|-
| [7]:
| 3 2 2 3 2 2 2
|[[File:MavilaAntidiatonic16edo.mp3]]
|-
| [9]:
| 1 2 2 2 1 2 2 2 2
|[[File:MavilaSuperdiatonic16edo.mp3]]
|}
See also [[Mavila Temperament Modal Harmony]].
 
'''Diminished'''
 
{| class="wikitable"
|-
| [8]:
| 1 3 1 3 1 3 1 3
|[[File:htgt16edo.mp3]]
|-
| [12]:
| 1 1 2 1 1 2 1 1 2 1 1 2
|
|}
 
'''Magic'''
 
[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/Astrology'''
 
[4]: 3 5 3 5
 
[6]: 3 2 3 3 2 3
 
[10]: 2 1 2 1 2 2 1 2 1 2
 
== Metallic harmony ==
In 16edo, triadic harmony can be based on on heptatonic sevenths (or seconds) rather than thirds. For instance, 16edo approximates 7/4 well enough to use
 
it in place of the usual 3/2, and in Mavila[7] this 7/4 approximation shares an interval class with a well-approximated 11/6 (at 1050{{c}}). Stacking these two intervals reaches 2025{{c}}, or a minor 6th plus an octave. Thus the out-of-tune 675{{c}} interval is bypassed, and all the dyads in the triad are consonant.
 
Depending on whether the Mavila[7] major 7th or minor 7th is used, one of two triads is produced: a small one, {{nowrap|{{dash|0, 975, 2025{{c}}}}}}, and a large one, {{nowrap|{{dash|0, 1050, 2025{{c}}}}}}. William Lynch, a major proponent of this style of harmony, calls these two triads "hard" and "soft", respectively. In addition, two other "symmetrical" triads are also obvious possible chords: a narrow symmetrical triad at {{nowrap|{{dash|0, 975, 1950{{c}}}}}}, and a wide symmetrical triad at {{nowrap|{{dash|0, 1050, 2100{{c}}}}}}. These are sort of analogous to "diminished" and "augmented" triads. The characteristic buzzy/metallic sound of these seventh-based triads inspired William Lynch to call them "Metallic triads".
 
=== MOS scales supporting metallic harmony in 16edo ===
The ssLsssL mode of Mavila[7] contains two hard triads on degrees 1 and 4 and two soft triads on degrees 2 and 6. The other three chords are wide symmetrical triads 0-1050-2025{{c}}. In Mavila[9], hard and soft triads cease to share a triad class, as 975{{c}} is a major 8th, while 1050{{c}} is a minor 9th; the triads may still be used, but parallel harmonic motion will function differently.
 
Another possible MOS scales for this approach would be Lemba[6], which gives two each of the soft, hard, and narrow symmetric triads.
 
''See: [[Metallic Harmony]].''
 
== Diagrams ==
'''16-tone piano layout based on the mavila[7]/antidiatonic scale'''
 
This Layout places mavila[7] on the black keys and mavila[9] on the white keys, according to antidiatonic notation.
 
[[File:16-EDO-PIano-Diagram.png|alt=16-EDO-PIano-Diagram.png|748x293px|16-EDO-PIano-Diagram.png]]
 
'''Interleaved edos'''
 
A visualization of 16edo being two interleaved copies of [[8edo]] and four interleaved copies of [[4edo]].
 
[[File:16edo_wheel_01.png|alt=16edo wheel 01.png|325x325px|16edo wheel 01.png]]
 
=== Lumatone mapping ===
 
See: [[Lumatone mapping for 16edo]]
 
== Interval arithmetic examples ==
These examples show the correspondence between interval arithmetic using diatonic and antidiatonic notation.
{| class="wikitable" style="text-align: center;"
! colspan="2" |Diatonic (i.e. 12edo)
! colspan="2" |Antidiatonic (i.e. 16edo)
|-
! Question
! Result
! Question
! Result
|-
| M2 + M2
| aug3
| m2 + m2
| dim3
|-
| D to F♯
| aug3
| D to F♭
| dim3
|-
| D to F
| M3
| D to F
| m3
|-
| E♭ + m3
| Gbb
| E♯ + M3
| G♯♯
|-
| E♭ + P5
| B♭
| E♯ + P5
| B♯
|-
| A minor chord
| A C♭ E
| A major chord
| A C♯ E
|-
| E♭ major chord
| E♭ G♭ D♭
| E♯ minor chord
| E♯ G♯ B♯
|-
| Gm7 = G + m3 + P5 + m7
| G B D F♭
| G + M3 + P5 + M7
| G B D F♯
|-
| A♭7aug = A♭ + M3 + A5 + m7
| A♭ C♭ E Gbb
| A♯ + m3 + d5 + M7
| A♯ C♯ E G♯♯
|-
| what chord is D F A♯?
| D + M3 + A5 = Daug
| D F A♭
| D + m3 + d5
|-
| what chord is C E G♭ B♭?
| C + m3 + d5 + d7 = Cdim7
| C E G♯ B♯
| C + M3 + A5 + A7
|-
| C major scale = C + M2 + M3<br>+ P4 + P5 + M6 + M7 + P8
| C D♯ E♯ F<br>G A♯ B♯ C
| C + m2 + m3 + P4<br>+ P5 + m6 + m7 + P8
| C D♭ E♭ F<br>G A♭ B♭ C
|-
| C minor scale = C + M2 + m3<br>+ P4 + P5 + m6 + m7 + P8
| C D♯ E F<br>G A B C
| C + m2 + M3 + P4<br>+ P5 + M6 + M7 + P8
| C D♭ E F<br>G A B C
|-
| what scale is A B♯ C♭ D<br>E F G♭ A?
| A + M2 + m3 + P4<br>+ P5 + M6 + m7 = A dorian
| A B♭ C♯ D<br>E F G♯ A
| A + m2 + M3 + P4<br>+ P5 + m6 + M7
|}
 
== Music ==
{{Catrel| 16edo tracks }}
 
; [[Abnormality]]
* [https://www.youtube.com/watch?v=zao6E8GdQh0 ''it's not not opposite day''] (2023)
* [https://www.youtube.com/watch?v=1pa3dztk8o0 ''nightfall''] (2024)
 
; [[Beheld]]
* [https://www.youtube.com/watch?v=kzPeVB2mncc ''Nebulous vibe'']
 
; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/huckleberry-regional-preserve ''Huckleberry Regional Preserve'']
* [https://cityoftheasleep.bandcamp.com/track/illegible-red-ink ''Illegible Red Ink'']
 
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/IfVvjoRqqNk ''16edo jam''] (2025)
* [https://www.youtube.com/watch?v=cUgbkkIvy0g ''Waltz in 16edo''] (2025)
 
; [[E8 Heterotic]]
* [https://youtu.be/a8Jgb_XIj7c "Hexed"]
 
; [[Fabrizio Fiale]]
* [https://www.soundclick.com/music/songInfo.cfm?songID=12370649 ''Prenestyna Highway'']
* [https://www.soundclick.com/music/songInfo.cfm?songID=7715803 ''Palestrina Morta, fantasia quasi una sonata'']
* [https://soundcloud.com/fff-fiale/in-sospensione-neutra ''In Sospensione Neutra'']
 
; [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fuga-a3-in-16et ''Fuga a3 in 16ET'']
 
; [[Last Sacrament]]
* [http://lastsacrament.bandcamp.com/album/enantiodromia ''Enantiodromia''] (album) (from 2013)
* [https://lastsacrament.bandcamp.com/album/maniacal-meditations-ep ''Maniacal Meditations''] (EP) (2013 EP)
 
; [[William Lynch]]
* [[:File:Mavila_Jazz_Rhodes_1.mp3|''Mavila Jazz Groove'']]
* [[:File:mavila4.mp3|''Cold, Dark Night for a Dance'']]
 
; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=vIWxP_C0aUM ''Mavila Fugue'']
* [https://www.youtube.com/watch?v=KYkmT46oGhw ''Canon at the Semitone on The Mother's Malison Theme'', for Cor Anglais and Violin] ([https://www.youtube.com/watch?v=I6BUauD8EaE for Organ])
* [https://www.youtube.com/watch?v=P7LUSRd1kMg ''Canon on Twinkle Twinkle Little Star'', for Organ] (2023) ([https://www.youtube.com/watch?v=QHJYyqge_JQ for Baroque Oboe and Viola])
* [https://www.youtube.com/shorts/I4-URAGgQMQ ''Baroque Micropiece in 16edo''] (2024)
 
; [[Herman Miller]]
* [http://www.io.com/%7Ehmiller/midi/16tet.mid ''Etude in 16-tone equal tuning'']{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/16tet.mp3 play]{{dead link}} ([http://soonlabel.com/xenharmonic/archives/2604 organ version]{{dead link}})
 
; [[Nae Ayy]]
* [https://www.youtube.com/watch?v=H74psBvdeT4 ''Rambling'']
* [https://www.youtube.com/watch?v=OAhV8ol2Hbw ''a n g e r y'']
* [https://www.youtube.com/watch?v=-MboZelse90 ''Maundering'']
 
; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=LXsZIbT6wpM ''Edolian - Seventhic''] (2020)
* [https://www.youtube.com/watch?v=UrQPr7V9feA ''Finality''] (2021)
 
; [[Jean-Pierre Poulin]]
* [http://www.jeanpierrepoulin.com/mp3/Armodue78.mp3 ''Armodue78'']
 
; [[Ron Sword]]
* [https://soundcloud.com/ron-sword/mavila-fog ''The Foggy Road from Pasadena'']{{dead link}}
 
; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/16-ET/20120527-16-malathion.mp3 ''Malathion''] - [http://chrisvaisvil.com/?p=2358 details]
* [http://micro.soonlabel.com/16-ET/20130216_16edo_vesta.mp3 ''Being of Vesta''] - [http://chrisvaisvil.com/?p=3061 details]
* [http://micro.soonlabel.com/simultaneous-tunings/20130607_thin_ice_christiane.mp3 ''Thin Ice''] - [http://chrisvaisvil.com/?p=3354 details]
 
; [[Stephen Weigel]]
* [https://www.youtube.com/watch?v=0t7ZmlmrE0Q ''Shot Fades the Sum Of'']
* [https://www.youtube.com/watch?v=2y01AlgOPvk ''When the Saints go Marching'']
 
; [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/05%20-%205.%2016%20octave.mp3 Comets Over Flatland 5]{{dead link}}
 
; [[Woyten]]
* [https://www.youtube.com/watch?v=LLgClI8pyNw ''Don't Take Five''] (2021)
 
; [[Xotla]]
* "Robotic Dialogue" from ''Microtones & Garden Gnomes'' (2017) [https://xotla.bandcamp.com/track/robotic-dialogue-16edo Bandcamp] | [https://youtu.be/sFxny2JNGpo?si=8MKPuIMCR_Xx1DTi YouTube]
* "Cognitive Climate" from Science Fraction (2022) [https://open.spotify.com/track/52v382I0OUotQjHo0pPoXs Spotify] | [https://xotla.bandcamp.com/track/cognitive-climate-16edo Bandcamp] | [https://youtu.be/dNBDG4wymN8?si=XGbpNkRp3qUo0Xgb YouTube]
 
; [[User:Nick_Vuci|Nick Vuci]]
* [https://en.xen.wiki/images/4/44/NickVuci-20220206-16edo-Prelude.mp3 ''Prelude'']
* [https://en.xen.wiki/images/9/9a/NickVuci-20231102-16edo-SofterForJ.mp3 ''Softer for J'']
* [https://en.xen.wiki/images/4/48/NickVuci-20220306-16edo-Invention.mp3 ''2-Part Invention'']
* [https://en.xen.wiki/w/User:Nick_Vuci#Modal_Studies ''Mavila Modal Studies'']
* [https://en.xen.wiki/images/c/c6/NV-20210526-16NEJI128-SerialismDubstepSketch.mp3 ''EDM based on a tone row'']
 
; [[Zewen Senpai]]
* [https://www.youtube.com/watch?v=QOzBGd64Pi4 ''Simple Ambient Study No. 1'']
 
== Notes ==
<references group=note/>
 
== See also ==
* [[57ed12]] - octave stretched version of 16edo; 57ed12 improves 3.5.11.13.17 but damages 2.7
 
=== Approaches ===
* [[User:VectorGraphics/16edo theory|Vector's approach]]
* [[Armodue theory]]
** [[Armodue armonia]]
 
== References ==
<references />
 
== Further reading ==
* [[Sword, Ron]]. ''[https://ronsword.bigcartel.com/product/esadekaphonic-scales-for-guitar Hexadecaphonic Scales for Guitar: A Microtonal Guitar Method Book, for Theory, Scales, and Information on the Sixteen Equal Division Octave System]''. 2009. (semi-diminished fourth tuning)
* Sword, Ron. ''[http://www.metatonalmusic.com/books.html Hexadecaphonic Scales for Guitar: Theory, Scales and Information on the Sixteen Equal Division Octave system]''. 2010? (superfourth tuning)
* Sword, Ron. "Thesaurus of Melodic Patterns and Intervals for 16-Tones" IAAA Press, USA. First Ed: August, 2011{{citation needed}}
 
[[Category:Teentuning]]
[[Category:Listen]]
[[Category:Mavila]]
[[Category:Guitar]]
[[Category:Pages with internal sound examples]]
 
{{Todo|cleanup}}
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