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{{interwiki
{{interwiki
| de = 16edo
| de = 16-EDO
| en = 16edo  
| en = 16edo
| es =  
| es = 16 EDO
| ja = 16平均律
| ja = 16平均律
}}
}}
{{Infobox ET}}
{{Infobox ET}}
'''16edo''' is the [[equal division of the octave]] into sixteen narrow chromatic semitones each of 75 [[cent]]s exactly.  
{{ED intro}}
 
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.


== Theory ==
== Theory ==
16edo is not especially good at representing most low-odd-limit musical intervals, but it has a [[7/4]] which is only six cents sharp, and a [[5/4]] which is only eleven cents flat. Four steps of it gives the 300 cent minor third interval, the same of that 12edo, giving it four diminished seventh chords exactly like those of [[12edo]], and a diminished triad on each scale step.
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.
 
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]]).
 
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 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.
 
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]].
 
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.
 
16edo works as a tuning for [[extraclassical tonality]], due to its ultramajor third of 450 cents.  


=== Odd harmonics ===
=== Odd harmonics ===
{{Harmonics in equal|16}}
{{Harmonics in equal|16}}
=== Octave stretch ===
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.
=== Subsets and supersets ===
Since 16 factors into primes as 2<sup>4</sup>, 16edo has subset edos {{EDOs| 2, 4, and 8 }}.
=== Composition theory ===
* [[User:VectorGraphics/16edo theory|Vector's approach]]
* [[Armodue harmony]]
{{Todo|inline=1| expand }}


== Intervals ==
== Intervals ==
16edo can be notated with conventional notation, including the staff, note names, relative notation, etc. in two ways. The first preserves the <u>melodic</u> meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G is not P1 - M3 - P5. (But see below in "Chord Names".)
{{Mavila}}
 
The second approach is to preserve the *harmonic* meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 16edo "on the fly".


Alternatively, one can use Armodue nine-nominal notation; see [[#Hexadecaphonic Notation]]
Alternatively, one can use Armodue nine-nominal notation.


{| class="wikitable center-all"
{| class="wikitable center-all"
|-
|-
! Degree
! rowspan="2" | Degree
! [[Cent]]s
! rowspan="2" | [[Cent]]s
! Approximate <br> Ratios*
! rowspan="2" | Approximate<br>ratios*
! colspan="2" | Melodic names, <br> major wider than minor
! colspan="6" | Names
! colspan="2" | Harmonic names, <br> major narrower <br> than minor
|-
! Interval Names <br> Just
! colspan="2" | Antidiatonic
! Interval <br> Names <br> Simplified
! colspan="2" | Diatonic
! Just
! Simplified
|-
|-
| 0
| 0
Line 45: Line 71:
| 28/27, 27/26
| 28/27, 27/26
| aug 1, dim 2nd
| aug 1, dim 2nd
| D#, Eb
| D♯, E♭
| dim 1, aug 2nd
| dim 1, aug 2nd
| Db, E#
| D♭, E♯
| subminor 2nd
| subminor 2nd
| min 2nd
| min 2nd
Line 65: Line 91:
| 8/7
| 8/7
| major 2nd
| major 2nd
| E#
| E♯
| minor 2nd
| minor 2nd
| Eb
| E♭
| supermajor 2nd,<br>septimal whole-tone
| supermajor 2nd,<br>septimal whole-tone
| perf 2nd
| perf 2nd
Line 75: Line 101:
| 19/16, 32/27
| 19/16, 32/27
| minor 3rd
| minor 3rd
| Fb
| F♭
| major 3rd
| major 3rd
| F#
| F♯
| minor 3rd
| minor 3rd
| min 3rd
| min 3rd
Line 95: Line 121:
| 13/10, 35/27
| 13/10, 35/27
| aug 3rd,<br>dim 4th
| aug 3rd,<br>dim 4th
| F#, Gb
| F♯, G♭
| dim 3rd,<br>aug 4th
| dim 3rd,<br>aug 4th
| Fb, G#
| F♭, G♯
| sub-4th,<br>supermajor 3rd
| sub-4th,<br>supermajor 3rd
| min 4th
| min 4th
Line 103: Line 129:
| 7
| 7
| 525
| 525
| 19/14, 27/20, 52/35, 256/189
| 19/14, 27/20, 35/26, 256/189
| perfect 4th
| perfect 4th
| G
| G
Line 115: Line 141:
| 7/5, 10/7
| 7/5, 10/7
| aug 4th,<br>dim 5th
| aug 4th,<br>dim 5th
| G#, Ab
| G♯, A♭
| dim 4th,<br>aug 5th
| dim 4th,<br>aug 5th
| Gb, A#
| G♭, A♯
| tritone
| tritone
| aug 4th,<br>dim 5th
| aug 4th,<br>dim 5th
Line 123: Line 149:
| 9
| 9
| 675
| 675
| 28/19, 40/27, 35/26, 189/128
| 28/19, 40/27, 52/35, 189/128
| perfect 5th
| perfect 5th
| A
| A
Line 135: Line 161:
| 20/13, 54/35
| 20/13, 54/35
| aug 5th,<br>dim 6th
| aug 5th,<br>dim 6th
| A#, Bb
| A♯, B♭
| dim 5th,<br>aug 6th
| dim 5th,<br>aug 6th
| Ab, B#
| A♭, B♯
| super-5th,<br>subminor 6th
| super-5th,<br>subminor 6th
| maj 5th
| maj 5th
Line 155: Line 181:
| 27/16, 32/19
| 27/16, 32/19
| major 6th
| major 6th
| B#
| B♯
| minor 6th
| minor 6th
| Bb
| B♭
| major 6th
| major 6th
| maj 6th
| maj 6th
Line 165: Line 191:
| 7/4
| 7/4
| minor 7th
| minor 7th
| Cb
| C♭
| major 7th
| major 7th
| C#
| C♯
| subminor 7th,<br>septimal minor 7th
| subminor 7th,<br>septimal minor 7th
| perf 7th
| perf 7th
Line 185: Line 211:
| 27/14, 52/27
| 27/14, 52/27
| aug 7th,<br>dim 8ve
| aug 7th,<br>dim 8ve
| C#, Db
| C♯, D♭
| dim 7th,<br>aug 8ve
| dim 7th,<br>aug 8ve
| Cb, D#
| C♭, D♯
| supermajor 7th
| supermajor 7th
| maj 7th
| maj 7th
Line 201: Line 227:
| octave
| octave
|}
|}
<nowiki>*</nowiki> based on treating 16edo as a 2.5.7.13.19.27 subgroup temperament; other approaches are possible.
<nowiki />* Based on treating 16edo as a 2.5.7.13.19.27 subgroup temperament; other approaches are possible.
 
== JI approximation ==
=== Selected just intervals by error ===
The following table shows how [[15-odd-limit intervals]] are represented in 16edo (ordered by absolute error).
 
{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! | Interval, complement
! | Error (abs., in [[cent]]s)
|-
| [[12/11]], [[11/6]]
| 0.637
|-
| [[13/10]], [[20/13]]
| 4.214
|-
| '''[[8/7]], [[7/4]]'''
| '''6.174'''
|-
| [[13/11]], [[22/13]]
| 10.790
|-
| '''[[5/4]], [[8/5]]'''
| '''11.314'''
|-
| [[13/12]], [[24/13]]
| 11.427
|-
| [[15/11]], [[22/15]]
| 11.951
|-
| [[9/7]], [[14/9]]
| 14.916
|-
| [[11/10]], [[20/11]]
| 15.004
|-
| '''[[16/13]], [[13/8]]'''
| '''15.528'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[9/8]], [[16/9]]
| 21.090
|-
| [[14/13]], [[13/7]]
| 21.702
|-
| [[15/13]], [[26/15]]
| 22.741
|-
| '''[[11/8]], [[16/11]]'''
| '''26.318'''
|-
| '''[[4/3]], [[3/2]]'''
| '''26.955'''
|-
| [[11/9]], [[18/11]]
| 27.592
|-
| [[15/14]], [[28/15]]
| 30.557
|-
| [[10/9]], [[9/5]]
| 32.404
|-
| [[14/11]], [[11/7]]
| 32.492
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[16/15]], [[15/8]]
| 36.731
|}
{{15-odd-limit|16}}
 
It's worth noting that the 525-cent 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]]


== Notation ==
== Notation ==
16edo 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. The Armodue model uses a 4-line staff for 16edo.
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.


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 #/b 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 16-EDO "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 octaves as 2/1, "[[decave]]".
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.


{| class="wikitable center-all"
{| class="wikitable center-all"
Line 302: Line 238:
! Degree
! Degree
! Cents
! Cents
! colspan="2" | Mavila[9] Notation
! colspan="2" | Mavila[9] notation
|-
|-
| 0
| 0
Line 312: Line 248:
| 75
| 75
| aug unison, minor 2nd
| aug unison, minor 2nd
| 1#, 2b
| 1♯, 2♭
|-
|-
| 2
| 2
Line 322: Line 258:
| 225
| 225
| aug 2nd, minor 3rd
| aug 2nd, minor 3rd
| 2#, 3b
| 2♯, 3♭
|-
|-
| 4
| 4
| 300
| 300
| major 3rd, dim 4th
| major 3rd, dim 4th
| 3, 4bb
| 3, 4𝄫
|-
|-
| 5
| 5
| 375
| 375
| minor 4th
| minor 4th
| 4b
| 4♭
|-
|-
| 6
| 6
| 450
| 450
| major 4th,<br>dim 5th
| major 4th,<br>dim 5th
| 4, 5b
| 4, 5♭
|-
|-
| 7
| 7
| 525
| 525
| aug 4th, minor 5th
| aug 4th, minor 5th
| 4#, 5
| 4♯, 5
|-
|-
| 8
| 8
| 600
| 600
| aug 5th, dim 6th
| aug 5th, dim 6th
| 5#, 6b
| 5♯, 6♭
|-
|-
| 9
| 9
| 675
| 675
| perfect 6th, dim 7th
| perfect 6th, dim 7th
| 6, 7bb
| 6, 7𝄫
|-
|-
| 10
| 10
| 750
| 750
| aug 6th, minor 7th
| aug 6th, minor 7th
| 6#, 7b
| 6♯, 7♭
|-
|-
| 11
| 11
Line 367: Line 303:
| 900
| 900
| aug 7th, minor 8th
| aug 7th, minor 8th
| 7#, 8b
| 7♯, 8♭
|-
|-
| 13
| 13
| 975
| 975
| major 8th, dim 9th
| major 8th, dim 9th
| 8, 9bb
| 8, 9𝄫
|-
|-
| 14
| 14
Line 382: Line 318:
| 1125
| 1125
| major 9th, dim 10ve
| major 9th, dim 10ve
| 9#, 1b
| 9♯, 1♭
|-
|-
| 16
| 16
Line 390: Line 326:
|}
|}


=== Armodue theory (4-line staff) ===
=== Sagittal notation ===
[http://www.armodue.com/ricerche.htm Armodue]: Pierpaolo Beretta's website for his "Armodue" theory for 16edo (esadekaphonic), including compositions.
This notation uses the same sagittal sequence as [[21edo #Sagittal notation|21edo]].
 
<imagemap>
File:16-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 471 0 631 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 471 106 [[Fractional_3-limit_notation#Bad-fifths_limma-fraction_notation | limma-fraction notation]]
default [[File:16-EDO_Sagittal.svg]]
</imagemap>
 
=== Armodue notation (4-line staff) ===
[http://www.armodue.com/ricerche.htm Armodue]: Pierpaolo Beretta's website for his Armodue theory for 16edo (esadekaphonic), including compositions.


For translations of parts of the Armodue pages see the [[Armodue]] on this wiki
For resources on the Armodue theory, see the [[Armodue]] on this wiki


=== Chord names ===
== Chord names ==
16edo chords can be named using ups and downs. Using harmonic interval names, the names are easy to find, but they bear little relationship to the sound. 4:5:6 is a minor chord and 10:12:15 is a major chord! Using melodic names, the chord names will match the sound, but finding the name is much more complicated (see below).
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.


{| class="wikitable center-all"
{| class="wikitable center-all"
|-
|-
! | chord
! rowspan="2" | Chord
! | JI ratios
! rowspan="2" | JI ratios
! colspan="3" | harmonic name
! colspan="6" | Name
! colspan="3" | melodic name
|-
! colspan="3" | Diatonic
! colspan="3" | Antidiatonic
|-
|-
| 0-5-9
| {{dash|0, 5, 9|med}}
| 4:5:6
| 4:5:6
| D F A
| D F A
Line 414: Line 364:
| D major
| D major
|-
|-
| 0-4-9
| {{dash|0, 4, 9|med}}
| 10:12:15
| 10:12:15
| D F# A
| D F♯ A
| D
| D
| D major
| D major
| D Fb A
| D F♭ A
| Dm
| Dm
| D minor
| D minor
|-
|-
| 0-4-8
| {{dash|0, 4, 8|med}}
| 5:6:7
| 5:6:7
| D F# A#
| D F♯ A♯
| Daug
| Daug
| D augmented
| D augmented
| D Fb Ab
| D F♭ A♭
| Ddim
| Ddim
| D diminished
| D diminished
|-
|-
| 0-5-10
| {{dash|0, 5, 10|med}}
|  
|  
| D F Ab
| D F A♭
| Ddim
| Ddim
| D diminished
| D diminished
| D F A#
| D F A♯
| Daug
| Daug
| D augmented
| D augmented
|-
|-
| 0-5-9-13
| {{dash|0, 5, 9, 13|med}}
| 4:5:6:7
| 4:5:6:7
| D F A C#
| D F A C♯
| Dm(M7)
| Dm(M7)
| D minor-major
| D minor-major
| D F A Cb
| D F A C♭
| D7
| D7
| D seven
| D seven
|-
|-
| 0-5-9-12
| {{dash|0, 5, 9, 12|med}}
|  
|  
| D F A Bb
| D F A Bb
| Dm(b6)
| Dm(♭6)
| D minor flat-six
| D minor flat-six
| D F A B#
| D F A B♯
| D6
| D6
| D six
| D six
|-
|-
| 0-5-9-14
| {{dash|0, 5, 9, 14|med}}
|  
|  
| D F A C
| D F A C
Line 468: Line 418:
| D major seven
| D major seven
|-
|-
| 0-4-9-13
| {{dash|0, 4, 9, 13|med}}
|  
|  
| D F# A C#
| D F♯ A C♯
| DM7
| DM7
| D major seven
| D major seven
| D Fb A Cb
| D F♭ A C♭
| DM7
| DM7
| D minor seven
| 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 6-1-3-5-7-9-11-13). See [[Ups and Downs Notation #Chords and Chord Progressions]] for more examples.


Using melodic names, interval arithmetic is done using a simple trick: first reverse everything, then perform normal arithmetic, then reverse everything again. Reversing means exchanging major for minor, aug for dim, and sharp for flat. Perfect and natural are unaffected. Examples:
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.


{| class="wikitable" style="text-align:center;"
Examples can be found at the bottom of the page.
!initial question
 
!reverse everything
== Approximation to JI ==
!do the math
=== Selected just intervals by error ===
!reverse again
{{Q-odd-limit intervals|16}}
|-
 
|M2 + M2
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.
|m2 + m2
 
|dim3
[[File:16ed2-001.svg|alt=alt : Your browser has no SVG support.]]
|aug3
 
|-
[[:File:16ed2-001.svg|16ed2-001.svg]]
|D to F#
|D to Fb
|dim3
|aug3
|-
|D to F
|D to F
|m3
|M3
|-
|Eb + m3
|E# + M3
|G##
|Gbb
|-
|Eb + P5
|E# + P5
|B#
|Bb
|-
|A minor chord
|A major chord
|A C# E
|A Cb E
|-
|Eb major chord
|E# minor chord
|E# G# B#
|Eb Gb Db
|-
|Gm7 = G + m3 + P5 + m7
|G + M3 + P5 + M7
|G B D F#
|G B D Fb
|-
|Ab7aug = Ab + M3 + A5 + m7
|A# + m3 + d5 + M7
|A# C# E G##
|Ab Cb E Gbb
|-
|what chord is D F A#?
|D F Ab
|D + m3 + d5
|D + M3 + A5 = Daug
|-
|what chord is C E Gb Bb?
|C E G# B#
|C + M3 + A5 + A7
|C + m3 + d5 + d7 = Cdim7
|-
|C major scale = C + M2 + M3
+ P4 + P5 + M6 + M7 + P8
|C + m2 + m3 + P4
+ P5 + m6 + m7 + P8
|C Db Eb F
G Ab Bb C
|C D# E# F
G A# B# C
|-
|C minor scale = C + M2 + m3
+ P4 + P5 + m6 + m7 + P8
|C + m2 + M3 + P4
+ P5 + M6 + M7 + P8
|C Db E F
G A B C
|C D# E F
G A B C
|-
|what scale is A B# Cb D
E F Gb A?
|A Bb C# D
E F G# A
|A + m2 + M3 + P4
+ P5 + m6 + M7
|A + M2 + m3 + P4
+ P5 + M6 + m7 = A dorian
|}


== Octave theory ==
== 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.


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 [[support]]s 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.
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 [[Jubilismic clan|no-threes 7-limit temperament tempering out 50/49]]. This has a period of a half-octave (600¢), 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'''".
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'''".
Line 582: Line 457:
"''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.''"
"''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 cents, and take the 300-cent minor third as an approximation of the harmonic 19th ([[19/16]], approximately 297.5 cents), that can combine with the approximation of the harmonic seventh to form a 16:19:28 triad .
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 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
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


== Commas ==
== Regular temperament properties ==
16et [[tempers out]] the following [[comma]]s. (Note: This assumes [[val]] {{val| 16 25 37 45 55 59 }}.)
=== 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"
{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
|-
![[Harmonic limit|Prime<br>Limit]]
! [[Harmonic limit|Prime<br>limit]]
![[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Ratio]]<ref group=note>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
![[Monzo]]
! [[Monzo]]
![[Cent]]s
! [[Cent]]s
![[Color name]]
! [[Color name]]
! Name
! Name
|-
|-
| 5
| 5
|[[135/128]]
| [[135/128]]
|{{monzo| -7 3 1 }}
| {{monzo| -7 3 1 }}
| 92.18
| 92.18
| Layobi
| Layobi
| Major chroma
| Mavila comma, major chroma
|-
|-
| 5
| 5
|[[648/625]]
| [[648/625]]
|{{monzo| 3 4 -4 }}
| {{monzo| 3 4 -4 }}
| 62.57
| 62.57
| Quadgu
| Quadgu
| Major diesis
| Diminished comma, major diesis
|-
|-
| 5
| 5
|[[3125/3072]]
| [[3125/3072]]
|{{monzo| -10 -1 5 }}
| {{monzo| -10 -1 5 }}
| 29.61
| 29.61
| Laquinyo
| Laquinyo
Line 620: Line 499:
|-
|-
| 5
| 5
|[[6115295232/6103515625|(20 digits)]]
| [[6115295232/6103515625|(20 digits)]]
|{{monzo| 23 6 -14 }}
| {{monzo| 23 6 -14 }}
| 3.34
| 3.34
| Sasepbiru
| Sasepbiru
|[[Vishnuzma]]
| [[Vishnuzma]]
|-
|-
| 7
| 7
|[[36/35]]
| [[36/35]]
|{{monzo| 2 2 -1 -1 }}
| {{monzo| 2 2 -1 -1 }}
| 48.77
| 48.77
| Rugu
| Rugu
| Septimal quartertone
| Mint comma, septimal quartertone
|-
|-
| 7
| 7
|[[525/512]]
| [[525/512]]
|{{monzo| -9 1 2 1 }}
| {{monzo| -9 1 2 1 }}
| 43.41
| 43.41
| Lazoyoyo
| Lazoyoyo
Line 641: Line 520:
|-
|-
| 7
| 7
|[[50/49]]
| [[50/49]]
|{{monzo| 1 0 2 -2 }}
| {{monzo| 1 0 2 -2 }}
| 34.98
| 34.98
| Biruyo
| Biruyo
Line 648: Line 527:
|-
|-
| 7
| 7
|[[64827/64000]]
| [[64827/64000]]
|{{monzo| -9 3 -3 4 }}
| {{monzo| -9 3 -3 4 }}
| 22.23
| 22.23
| Laquadzo-atrigu
| Laquadzo-atrigu
| Squalentine
| Squalentine comma
|-
|-
| 7
| 7
|[[3125/3087]]
| [[3125/3087]]
|{{monzo| 0 -2 5 -3 }}
| {{monzo| 0 -2 5 -3 }}
| 21.18
| 21.18
| Triru-aquinyo
| Triru-aquinyo
| Gariboh
| Gariboh comma
|-
|-
| 7
| 7
|[[126/125]]
| [[126/125]]
|{{monzo| 1 2 -3 1 }}
| {{monzo| 1 2 -3 1 }}
| 13.79
| 13.79
| Zotrigu
| Zotrigu
Line 669: Line 548:
|-
|-
| 7
| 7
|[[81029/1024]]
| [[1029/1024]]
|{{monzo| -10 1 0 3 }}
| {{monzo| -10 1 0 3 }}
| 8.43
| 8.43
| Latrizo
| Latrizo
Line 676: Line 555:
|-
|-
| 7
| 7
|[[6144/6125]]
| [[6144/6125]]
|{{monzo| 11 1 -3 -2 }}
| {{monzo| 11 1 -3 -2 }}
| 5.36
| 5.36
| Sarurutrigu
| Sarurutrigu
Line 683: Line 562:
|-
|-
| 11
| 11
|[[121/120]]
| [[121/120]]
|{{monzo| -3 -1 -1 0 2 }}
| {{monzo| -3 -1 -1 0 2 }}
| 14.37
| 14.37
| Lologu
| Lologu
Line 690: Line 569:
|-
|-
| 11
| 11
|[[176/175]]
| [[176/175]]
|{{monzo| 4 0 -2 -1 1 }}
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| 9.86
| Lorugugu
| Lorugugu
Line 697: Line 576:
|-
|-
| 11
| 11
|[[385/384]]
| [[385/384]]
|{{monzo| -7 -1 1 1 1 }}
| {{monzo| -7 -1 1 1 1 }}
| 4.50
| 4.50
| Lozoyo
| Lozoyo
Line 704: Line 583:
|-
|-
| 11
| 11
|[[441/440]]
| [[441/440]]
|{{monzo| -3 2 -1 2 -1 }}
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| 3.93
| Luzozogu
| Luzozogu
Line 711: Line 590:
|-
|-
| 11
| 11
|[[3025/3024]]
| [[3025/3024]]
|{{monzo| -4 -3 2 -1 2 }}
| {{monzo| -4 -3 2 -1 2 }}
| 0.57
| 0.57
| Loloruyoyo
| Loloruyoyo
| Lehmerisma
| Lehmerisma
|}
|}
<references />
== Rank-2 temperaments ==
*[[List of 16et rank two temperaments by badness]]
Important mosses include:
*[[magic]] anti-diatonic 3L4s 1414141 (5\16, 1\1)
*[[magic]] superdiatonic 3L7s 1311311311 (5\16, 1\1)
* Pathological [[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)
* Pathological [[1L 12s]] 4 1 1 1 1 1 1 1 1 1 1 1 (1\16, 1\1)
* Pathological [[1L 13s]] 3 1 1 1 1 1 1 1 1 1 1 1 1 1 (1\16, 1\1)
* Pathological [[2L 12s]] 2 1 1 1 1 1 1 2 1 1 1 1 1 1 (1\16, 1\2)
* Pathological [[1L 14s]] 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 (1\16, 1\1)


Temperaments listed by generator size:
=== Rank-2 temperaments ===
* [[List of 16et rank two temperaments by badness]]


{| class="wikitable center-1 center-2"
{| class="wikitable center-1 center-2"
|+ Table of temperaments by generator
|-
|-
! Periods<br>per octave
! Periods<br>per 8ve
! Generator
! Generator
! Temperaments
! Temperaments
Line 745: Line 609:
| 1
| 1
| 1\16
| 1\16
|[[Valentine]], [[slurpee]]
| [[Valentine]], [[slurpee]]
|-
|-
| 1
| 1
| 3\16
| 3\16
|[[Gorgo]]
| [[Gorgo]]
|-
|-
| 1
| 1
| 5\16
| 5\16
|[[magic]]/muggles
| [[Magic]]/[[muggles]]
|-
|-
| 1
| 1
| 7\16
| 7\16
|[[Mavila]]/armodue
| [[Mavila]]/[[armodue]]
|-
|-
| 2
| 2
| 1\16
| 1\16
|[[Bipelog]]
| [[Bipelog]]
|-
|-
| 2
| 2
| 3\16
| 3\16
|[[Lemba]], [[astrology]]
| [[Lemba]], [[astrology]]
|-
|-
| 4
| 4
| 1\16
| 1\16
|[[Diminished]]/demolished
| [[Diminished (temperament)|Diminished]]/[[demolished]]
|-
|-
| 8
| 8
| 1\16
| 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'''
'''Mavila'''
Line 832: Line 708:


== Metallic harmony ==
== 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


Because 16edo does not approximate 3/2 well at all, triadic harmony based on heptatonic thirds is not a great option for typical harmonic timbres.
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.
 
However, 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 cents). Stacking these two intervals reaches 2025¢, or a minor 6th plus an octave. Thus the out-of-tune 675¢ 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, 0-975-2025¢, and a large one, 0-1050-2025¢. 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 0-975-1950¢, and a wide symmetrical triad at 0-1050-2100¢. 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".
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 ===
=== 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¢. In Mavila[9], hard and soft triads cease to share a triad class, as 975¢ is a major 8th, while 1050¢ is a minor 9th; the triads may still be used, but parallel harmonic motion will function differently.
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.
Another possible MOS scales for this approach would be Lemba[6], which gives two each of the soft, hard, and narrow symmetric triads.
Line 851: Line 724:
'''16-tone piano layout based on the mavila[7]/antidiatonic scale'''
'''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. As you can see, flats are higher than naturals and sharps are lower, as per the "harmonic notation" above. Simply swap sharps with flats for "melodic notation".[[File:16-EDO-PIano-Diagram.png|alt=16-EDO-PIano-Diagram.png|748x293px|16-EDO-PIano-Diagram.png]]
This Layout places mavila[7] on the black keys and mavila[9] on the white keys, according to antidiatonic notation.


'''Un-annotated diagram'''
[[File:16-EDO-PIano-Diagram.png|alt=16-EDO-PIano-Diagram.png|748x293px|16-EDO-PIano-Diagram.png]]


Please explain this image.
'''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]]
[[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 ==
== Music ==
{{Catrel|16edo tracks}}
{{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]]
; [[Beheld]]
* [https://www.youtube.com/watch?v=kzPeVB2mncc Nebulous vibe]
* [https://www.youtube.com/watch?v=kzPeVB2mncc ''Nebulous vibe'']


; [[City of the Asleep]]
; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/huckleberry-regional-preserve Huckleberry Regional Preserve]
* [https://cityoftheasleep.bandcamp.com/track/huckleberry-regional-preserve ''Huckleberry Regional Preserve'']
* [https://cityoftheasleep.bandcamp.com/track/illegible-red-ink Illegible Red Ink]
* [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]]
; [[E8 Heterotic]]
Line 873: Line 842:


; [[Fabrizio Fiale]]
; [[Fabrizio Fiale]]
* [https://www.soundclick.com/music/songInfo.cfm?songID=12370649 Prenestyna Highway]
* [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://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]
* [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]]
; [[Last Sacrament]]
* ''[http://lastsacrament.bandcamp.com/album/enantiodromia Enantiodromia]'' (album) (from 2013)
* [http://lastsacrament.bandcamp.com/album/enantiodromia ''Enantiodromia''] (album) (from 2013)
* ''[https://lastsacrament.bandcamp.com/album/maniacal-meditations-ep Maniacal Meditations]'' (EP) (2013 EP)
* [https://lastsacrament.bandcamp.com/album/maniacal-meditations-ep ''Maniacal Meditations''] (EP) (2013 EP)


; [[William Lynch]]
; [[William Lynch]]
* [[:File:Mavila_Jazz_Rhodes_1.mp3|Mavila Jazz Groove]]
* [[:File:Mavila_Jazz_Rhodes_1.mp3|''Mavila Jazz Groove'']]
* [[:File:mavila4.mp3|Cold, Dark Night for a Dance ]]
* [[: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]]
; [[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}})
* [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]]
; [[Jean-Pierre Poulin]]
* [http://www.jeanpierrepoulin.com/mp3/Armodue78.mp3 Armodue78]
* [http://www.jeanpierrepoulin.com/mp3/Armodue78.mp3 ''Armodue78'']


; [[Ron Sword]]
; [[Ron Sword]]
* [https://soundcloud.com/ron-sword/mavila-fog The Foggy Road from Pasadena]{{dead link}}
* [https://soundcloud.com/ron-sword/mavila-fog ''The Foggy Road from Pasadena'']{{dead link}}


; [[Chris Vaisvil]]
; [[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/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/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]
* [http://micro.soonlabel.com/simultaneous-tunings/20130607_thin_ice_christiane.mp3 ''Thin Ice''] - [http://chrisvaisvil.com/?p=3354 details]


; [[Stephen Weigel]]
; [[Stephen Weigel]]
* [https://www.youtube.com/watch?v=0t7ZmlmrE0Q Shot Fades the Sum Of]
* [https://www.youtube.com/watch?v=0t7ZmlmrE0Q ''Shot Fades the Sum Of'']
* [https://www.youtube.com/watch?v=2y01AlgOPvk When the Saints go Marching]
* [https://www.youtube.com/watch?v=2y01AlgOPvk ''When the Saints go Marching'']


; [[Randy Winchester]]
; [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/05%20-%205.%2016%20octave.mp3 Comets Over Flatland 5]{{dead link}}
* [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]]
; [[Xotla]]
* [https://www.youtube.com/watch?v=dNBDG4wymN8 Cognitive Climate]
* "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]]
; [[User:Nick_Vuci|Nick Vuci]]
* [https://en.xen.wiki/images/4/44/NickVuci-20220206-16edo-Prelude.mp3 Prelude]
* [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/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/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/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]
* [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 ==
== Further reading ==
Line 921: Line 929:
* Sword, Ron. "Thesaurus of Melodic Patterns and Intervals for 16-Tones" IAAA Press, USA. First Ed: August, 2011{{citation needed}}
* Sword, Ron. "Thesaurus of Melodic Patterns and Intervals for 16-Tones" IAAA Press, USA. First Ed: August, 2011{{citation needed}}


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[[Category:Teentuning]]
[[Category:Teentuning]]
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[[Category:Listen]]
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[[Category:Mavila]]
[[Category:Image]]
[[Category:Guitar]]
[[Category:Guitar]]
[[Category:Pages with internal sound examples]]
[[Category:Pages with internal sound examples]]


{{Todo
{{Todo|cleanup}}
| unify precision
| cleanup
}}
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