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__FORCETOC__
{{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}}
| Prime factorization = 2<sup>4</sup>
{{ED intro}}
| Subgroup = 2.5.7.13.19.27
 
| Step size = 75¢
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.
| Fifth type = [[Mavila]] 9\16 = 675¢
| Major 2nd = 2\16 = 150¢
| Minor 2nd = 3\16 = 225¢
| Augmented 1sn = -1\16 = -75¢
| Common uses = mavila, metallic harmony
| Important MOS = [[mavila]] anti-diatonic 2L5s 2223223 (9\16, 1\1)<br/>[[mavila]] superdiatonic 7L2s 222212221 (9\16, 1\1)<br/>[[gorgo]] 5L1s 555551 (3\16, 1\1)<br/>[[lemba]] 4L2s 332332 (3\16, 1\2)
}}


== Theory ==
== Theory ==
{| class="wikitable" style="text-align:center;"
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.
!
!prime 2
!prime 3
!prime 5
!prime 7
!prime 11
!prime 13
!prime 17
!prime 19
|-
!error (¢)
|0¢
|  -26.96¢
|  -11.3¢
|  +6.2¢
|  -26.3¢
|  -15.6¢
| -30.0¢
|2.5¢
|-
![[relative error]] (%)
|0%
| -36
| -15
|8
| -35
| -21
| -40
|3
|-
![[nearest edomapping]]
|16
|9
|5
|13
|7
|11
|1
|4
|-
![[fifthspan]]
|0
|  +1
|  -3
|  +5
|  -1
|  +3
| -7
|4
|}


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]]).


'''16-EDO''' is the [[equal division of the octave]] into sixteen narrow chromatic semitones each of 75 [[cent]]s exactly. It is not especially good at representing most low-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 12-EDO, giving it four diminished seventh chords exactly like those of [[12edo|12-EDO]], and a diminished triad on each scale step.
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.  


==Intervals==
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.


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 isn't P1 - M3 - P5. (But see below in "Chord Names".)
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]].  


The second approach is to preserve the <u>harmonic</u> 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".
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.  


Alternatively, one can use Armodue nine-nominal notation; see [[16edo#Hexadecaphonic Notation|below]]
16edo works as a tuning for [[extraclassical tonality]], due to its ultramajor third of 450 cents.


{| class="wikitable"
=== Odd harmonics ===
|-
{{Harmonics in equal|16}}
! | Degree
! | Cents
!7mus
! | Approximate


Ratios*
=== Octave stretch ===
! colspan="2" | Melodic names,
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.


major wider than minor
=== Subsets and supersets ===
! colspan="2" | Harmonic names,
Since 16 factors into primes as 2<sup>4</sup>, 16edo has subset edos {{EDOs| 2, 4, and 8 }}.


major narrower
=== Composition theory ===
* [[User:VectorGraphics/16edo theory|Vector's approach]]
* [[Armodue harmony]]


than minor
{{Todo|inline=1| expand }}
! | Interval Names


Just
== Intervals ==
! | Interval
{{Mavila}}


Names
Alternatively, one can use Armodue nine-nominal notation.


Simplified
{| 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
|-
|-
| colspan="3" style="text-align:center;" | 0
| 0
| style="text-align:center;" | 1/1
| 0
| style="text-align:center;" | unison
| 1/1
| style="text-align:center;" | D
| unison
| style="text-align:center;" | unison
| D
| style="text-align:center;" | D
| unison
| style="text-align:center;" | Unison
| D
| style="text-align:center;" | Unison
| unison
| unison
|-
|-
| style="text-align:center;" | 1
| 1
| style="text-align:center;" | 75
| 75
|96 (60<sub>16</sub>)
| 28/27, 27/26
| style="text-align:center;" | 28/27, 27/26
| aug 1, dim 2nd
| style="text-align:center;" | aug 1, dim 2nd
| D♯, E♭
| style="text-align:center;" | D#, Eb
| dim 1, aug 2nd
| style="text-align:center;" | dim 1, aug 2nd
| D♭, E♯
| style="text-align:center;" | Db, E#
| subminor 2nd
| style="text-align:center;" | Subminor 2nd
| min 2nd
| style="text-align:center;" | Min 2nd
|-
|-
| style="text-align:center;" | 2
| 2
| style="text-align:center;" | 150
| 150
|192 (C0<sub>16</sub>)
| 35/32
| style="text-align:center;" | 35/32
| minor 2nd
| style="text-align:center;" | minor 2nd
| E
| style="text-align:center;" | E
| major 2nd
| style="text-align:center;" | major 2nd
| E
| style="text-align:center;" | E
| neutral 2nd
| style="text-align:center;" | Neutral 2nd
| maj 2nd
| style="text-align:center;" | Maj 2nd
|-
|-
| style="text-align:center;" | 3
| 3
| style="text-align:center;" | 225
| 225
|288 (120<sub>16</sub>)
| 8/7
| style="text-align:center;" | 8/7
| major 2nd
| style="text-align:center;" | major 2nd
| E♯
| style="text-align:center;" | E#
| minor 2nd
| style="text-align:center;" | minor 2nd
| E♭
| style="text-align:center;" | Eb
| supermajor 2nd,<br>septimal whole-tone
| style="text-align:center;" | Supermajor 2nd,
| perf 2nd
 
Septimal Whole-Tone
| style="text-align:center;" | Perf 2nd
|-
|-
| style="text-align:center;" | 4
| 4
| style="text-align:center;" | 300
| 300
|384 (180<sub>16</sub>)
| 19/16, 32/27
| style="text-align:center;" | 19/16, 32/27
| minor 3rd
| style="text-align:center;" | minor 3rd
| F♭
| style="text-align:center;" | Fb
| major 3rd
| style="text-align:center;" | major 3rd
| F♯
| style="text-align:center;" | F#
| minor 3rd
| style="text-align:center;" | Minor 3rd
| min 3rd
| style="text-align:center;" | Min 3rd
|-
|-
| style="text-align:center;" | 5
| 5
| style="text-align:center;" | 375
| 375
|480 (1E0<sub>16</sub>)
| 5/4, 16/13, 26/21
| style="text-align:center;" | 5/4, 16/13, 26/21
| major 3rd
| style="text-align:center;" | major 3rd
| F
| style="text-align:center;" | F
| minor 3rd
| style="text-align:center;" | minor 3rd
| F
| style="text-align:center;" | F
| major 3rd
| style="text-align:center;" | Major 3rd
| maj 3rd
| style="text-align:center;" | Maj 3rd
|-
|-
| style="text-align:center;" | 6
| 6
| style="text-align:center;" | 450
| 450
|576 (240<sub>16</sub>)
| 13/10, 35/27
| style="text-align:center;" | 13/10, 35/27
| aug 3rd,<br>dim 4th
| style="text-align:center;" | aug 3rd
| F♯, G♭
 
| dim 3rd,<br>aug 4th
dim 4th
| F♭, G♯
| style="text-align:center;" | F#, Gb
| sub-4th,<br>supermajor 3rd
| style="text-align:center;" | dim 3rd,
| min 4th
 
aug 4th
| style="text-align:center;" | Fb, G#
| style="text-align:center;" | Sub-4th,
 
Supermajor 3rd
| style="text-align:center;" | Min 4th
|-
|-
| style="text-align:center;" | 7
| 7
| style="text-align:center;" | 525
| 525
|672 (2A0<sub>16</sub>)
| 19/14, 27/20, 35/26, 256/189
| style="text-align:center;" | 19/14, 27/20, 52/35, 256/189
| perfect 4th
| style="text-align:center;" | perfect 4th
| G
| style="text-align:center;" | G
| perfect 4th
| style="text-align:center;" | perfect 4th
| G
| style="text-align:center;" | G
| wide 4th
| style="text-align:center;" | Wide 4th
| maj 4th
| style="text-align:center;" | Maj 4th
|-
|-
| style="text-align:center;" | 8
| 8
| style="text-align:center;" | 600
| 600
|768 (300<sub>16</sub>)
| 7/5, 10/7
| style="text-align:center;" | 7/5, 10/7
| aug 4th,<br>dim 5th
| style="text-align:center;" | aug 4th
| G♯, A♭
 
| dim 4th,<br>aug 5th
dim 5th
| G♭, A♯
| style="text-align:center;" | G#, Ab
| tritone
| style="text-align:center;" | dim 4th,
| aug 4th,<br>dim 5th
 
aug 5th
| style="text-align:center;" | Gb, A#
| style="text-align:center;" | Tritone
| style="text-align:center;" | Aug 4th,
 
Dim 5th
|-
|-
| style="text-align:center;" | 9
| 9
| style="text-align:center;" | 675
| 675
|864 (360<sub>16</sub>)
| 28/19, 40/27, 52/35, 189/128
| style="text-align:center;" | 28/19, 40/27, 35/26, 189/128
| perfect 5th
| style="text-align:center;" | perfect 5th
| A
| style="text-align:center;" | A
| perfect 5th
| style="text-align:center;" | perfect 5th
| A
| style="text-align:center;" | A
| narrow 5th
| style="text-align:center;" | Narrow 5th
| min 5th
| style="text-align:center;" | Min 5th
|-
|-
| style="text-align:center;" | 10
| 10
| style="text-align:center;" | 750
| 750
|960 (3C0<sub>16</sub>)
| 20/13, 54/35
| style="text-align:center;" | 20/13, 54/35
| aug 5th,<br>dim 6th
| style="text-align:center;" | aug 5th
| A♯, B♭
 
| dim 5th,<br>aug 6th
dim 6th
| A♭, B♯
| style="text-align:center;" | A#, Bb
| super-5th,<br>subminor 6th
| style="text-align:center;" | dim 5th, aug 6th
| maj 5th
| style="text-align:center;" | Ab, B#
| style="text-align:center;" | Super-5th,
 
Subminor 6th
| style="text-align:center;" | Maj 5th
|-
|-
| style="text-align:center;" | 11
| 11
| style="text-align:center;" | 825
| 825
|1056 (420<sub>16</sub>)
| 8/5, 13/8, 21/13
| style="text-align:center;" | 8/5, 13/8, 21/13
| minor 6th
| style="text-align:center;" | minor 6th
| B
| style="text-align:center;" | B
| major 6th
| style="text-align:center;" | major 6th
| B
| style="text-align:center;" | B
| minor 6th
| style="text-align:center;" | Minor 6th
| min 6th
| style="text-align:center;" | Min 6th
|-
|-
| style="text-align:center;" | 12
| 12
| style="text-align:center;" | 900
| 900
|1152 (480<sub>16</sub>)
| 27/16, 32/19
| style="text-align:center;" | 27/16, 32/19
| major 6th
| style="text-align:center;" | major 6th
| B♯
| style="text-align:center;" | B#
| minor 6th
| style="text-align:center;" | minor 6th
| B♭
| style="text-align:center;" | Bb
| major 6th
| style="text-align:center;" | Major 6th
| maj 6th
| style="text-align:center;" | Maj 6th
|-
|-
| style="text-align:center;" | 13
| 13
| style="text-align:center;" | 975
| 975
|1248 (4E0<sub>16</sub>)
| 7/4
| style="text-align:center;" | 7/4
| minor 7th
| style="text-align:center;" | minor 7th
| C♭
| style="text-align:center;" | Cb
| major 7th
| style="text-align:center;" | major 7th
| C♯
| style="text-align:center;" | C#
| subminor 7th,<br>septimal minor 7th
| style="text-align:center;" | Subminor 7th,
| perf 7th
 
Septimal Minor 7th
| style="text-align:center;" | Perf 7th
|-
|-
| style="text-align:center;" | 14
| 14
| style="text-align:center;" | 1050
| 1050
|1344 (540<sub>16</sub>)
| 64/35
| style="text-align:center;" | 64/35
| major 7th
| style="text-align:center;" | major 7th
| C
| style="text-align:center;" | C
| minor 7th
| style="text-align:center;" | minor 7th
| C
| style="text-align:center;" | C
| neutral 7th
| style="text-align:center;" | Neutral 7th
| min 7th
| style="text-align:center;" | Min 7th
|-
|-
| style="text-align:center;" | 15
| 15
| style="text-align:center;" | 1125
| 1125
|1440 (5A0<sub>16</sub>)
| 27/14, 52/27
| style="text-align:center;" | 27/14, 52/27
| aug 7th,<br>dim 8ve
| style="text-align:center;" | aug 7th
| C♯, D♭
 
| dim 7th,<br>aug 8ve
dim 8ve
| C♭, D♯
| style="text-align:center;" | C#, Db
| supermajor 7th
| style="text-align:center;" | dim 7th,
| maj 7th
 
aug 8ve
| style="text-align:center;" | Cb, D#
| style="text-align:center;" | Supermajor 7th
| style="text-align:center;" | Maj 7th
|-
|-
| style="text-align:center;" | 16
| 16
| style="text-align:center;" | 1200
| 1200
|1536 (600<sub>16</sub>)
| 2/1
| style="text-align:center;" | 2/1
| 8ve
| style="text-align:center;" | 8ve
| D
| style="text-align:center;" | D
| 8ve
| style="text-align:center;" | 8ve
| D
| style="text-align:center;" | D
| octave
| style="text-align:center;" | Octave
| octave
| style="text-align:center;" | Octave
|}
|}
*based on treating 16-EDO 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.


==Chord Names==
== Notation ==
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.


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).
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"
{| class="wikitable center-all"
|-
|-
! | chord
! Degree
! | JI ratios
! Cents
! colspan="3" | harmonic name
! colspan="2" | Mavila[9] notation
! colspan="3" | melodic name
|-
|-
| style="text-align:center;" | 0-5-9
| 0
| style="text-align:center;" | 4:5:6
| 0
| style="text-align:center;" | D F A
| unison
| style="text-align:center;" | Dm
| 1
| style="text-align:center;" | D minor
| style="text-align:center;" | D F A
| style="text-align:center;" | D
| style="text-align:center;" | D major
|-
|-
| style="text-align:center;" | 0-4-9
| 1
| style="text-align:center;" | 10:12:15
| 75
| style="text-align:center;" | D F# A
| aug unison, minor 2nd
| style="text-align:center;" | D
| 1♯, 2♭
| style="text-align:center;" | D major
| style="text-align:center;" | D Fb A
| style="text-align:center;" | Dm
| style="text-align:center;" | D minor
|-
|-
| style="text-align:center;" | 0-4-8
| 2
| style="text-align:center;" | 5:6:7
| 150
| style="text-align:center;" | D F# A#
| major 2nd
| style="text-align:center;" | Daug
| 2
| style="text-align:center;" | D augmented
| style="text-align:center;" | D Fb Ab
| style="text-align:center;" | Ddim
| style="text-align:center;" | D diminished
|-
|-
| style="text-align:center;" | 0-5-10
| 3
| style="text-align:center;" |
| 225
| style="text-align:center;" | D F Ab
| aug 2nd, minor 3rd
| style="text-align:center;" | Ddim
| 2♯, 3♭
| style="text-align:center;" | D diminished
| style="text-align:center;" | D F A#
| style="text-align:center;" | Daug
| style="text-align:center;" | D augmented
|-
|-
| style="text-align:center;" | 0-5-9-13
| 4
| style="text-align:center;" | 4:5:6:7
| 300
| style="text-align:center;" | D F A C#
| major 3rd, dim 4th
| style="text-align:center;" | Dm(M7)
| 3, 4𝄫
| style="text-align:center;" | D minor-major
| style="text-align:center;" | D F A Cb
| style="text-align:center;" | D7
| style="text-align:center;" | D seven
|-
|-
| style="text-align:center;" | 0-5-9-12
| 5
| style="text-align:center;" |
| 375
| style="text-align:center;" | D F A Bb
| minor 4th
| style="text-align:center;" | Dm(b6)
| 4♭
| style="text-align:center;" | D minor flat-six
| style="text-align:center;" | D F A B#
| style="text-align:center;" | D6
| style="text-align:center;" | D six
|-
|-
| style="text-align:center;" | 0-5-9-14
| 6
| style="text-align:center;" |
| 450
| style="text-align:center;" | D F A C
| major 4th,<br>dim 5th
| style="text-align:center;" | Dm7
| 4, 5♭
| style="text-align:center;" | D minor seven
| style="text-align:center;" | D F A C
| style="text-align:center;" | DM7
| style="text-align:center;" | D major seven
|-
|-
| style="text-align:center;" | 0-4-9-13
| 7
| style="text-align:center;" |
| 525
| style="text-align:center;" | D F# A C#
| aug 4th, minor 5th
| style="text-align:center;" | DM7
| 4♯, 5
| style="text-align:center;" | D major seven
| style="text-align:center;" | D Fb A Cb
| style="text-align:center;" | DM7
| style="text-align:center;" | 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|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:
 
{| class="wikitable" style="text-align:center;"
!initial question
!reverse everything
!do the math
!reverse again
|-
|-
|M2 + M2
| 8
|m2 + m2
| 600
|dim3
| aug 5th, dim 6th
|aug3
| 5♯, 6♭
|-
|-
|D to F#
| 9
|D to Fb
| 675
|dim3
| perfect 6th, dim 7th
|aug3
| 6, 7𝄫
|-
|-
|D to F
| 10
|D to F
| 750
|m3
| aug 6th, minor 7th
|M3
| 6♯, 7♭
|-
|-
|Eb + m3
| 11
|E# + M3
| 825
|G##
| major 7th
|Gbb
| 7
|-
|-
|Eb + P5
| 12
|E# + P5
| 900
|B#
| aug 7th, minor 8th
|Bb
| 7♯, 8♭
|-
|-
|A minor chord
| 13
|A major chord
| 975
|A C# E
| major 8th, dim 9th
|A Cb E
| 8, 9𝄫
|-
|-
|Eb major chord
| 14
|E# minor chord
| 1050
|E# G# B#
| minor 9th
|Eb Gb Db
| 9
|-
|-
|Gm7 = G + m3 + P5 + m7
| 15
|G + M3 + P5 + M7
| 1125
|G B D F#
| major 9th, dim 10ve
|G B D Fb
| 9♯, 1♭
|-
|-
|Ab7aug = Ab + M3 + A5 + m7
| 16
|A# + m3 + d5 + M7
| 1200
|A# C# E G##
| 10ve (Decave)
|Ab Cb E Gbb
| 1
|-
|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
|}
|}


==Selected just intervals by error==
=== Sagittal notation ===
The following table shows how [[Just-24|some prominent just intervals]] are represented in 16-EDO (ordered by absolute error).
This notation uses the same sagittal sequence as [[21edo #Sagittal notation|21edo]].


===Best direct mapping, even if inconsistent===
<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>


{| class="wikitable"
=== 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.
! | Interval, complement
 
! | Error (abs., in [[cent|cents]])
For resources on the Armodue theory, see the [[Armodue]] on this wiki
|-
 
| style="text-align:center;" | [[12/11|12/11]], [[11/6|11/6]]
== Chord names ==
| style="text-align:center;" | 0.637
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.
|-
 
| style="text-align:center;" | [[13/10|13/10]], [[20/13|20/13]]
{| class="wikitable center-all"
| style="text-align:center;" | 4.214
|-
| style="text-align:center;" | '''[[8/7|8/7]], [[7/4|7/4]]'''
| style="text-align:center;" | '''6.174'''
|-
| style="text-align:center;" | [[13/11|13/11]], [[22/13|22/13]]
| style="text-align:center;" | 10.790
|-
| style="text-align:center;" | '''[[5/4|5/4]], [[8/5|8/5]]'''
| style="text-align:center;" | '''11.314'''
|-
| style="text-align:center;" | [[13/12|13/12]], [[24/13|24/13]]
| style="text-align:center;" | 11.427
|-
| style="text-align:center;" | [[15/11|15/11]], [[22/15|22/15]]
| style="text-align:center;" | 11.951
|-
| style="text-align:center;" | [[9/7|9/7]], [[14/9|14/9]]
| style="text-align:center;" | 14.916
|-
| style="text-align:center;" | [[11/10|11/10]], [[20/11|20/11]]
| style="text-align:center;" | 15.004
|-
| style="text-align:center;" | '''[[16/13|16/13]], [[13/8|13/8]]'''
| style="text-align:center;" | '''15.528'''
|-
| style="text-align:center;" | [[6/5|6/5]], [[5/3|5/3]]
| style="text-align:center;" | 15.641
|-
| style="text-align:center;" | [[7/5|7/5]], [[10/7|10/7]]
| style="text-align:center;" | 17.488
|-
| style="text-align:center;" | [[9/8|9/8]], [[16/9|16/9]]
| style="text-align:center;" | 21.090
|-
| style="text-align:center;" | [[14/13|14/13]], [[13/7|13/7]]
| style="text-align:center;" | 21.702
|-
|-
| style="text-align:center;" | [[15/13|15/13]], [[26/15|26/15]]
! rowspan="2" | Chord
| style="text-align:center;" | 22.741
! rowspan="2" | JI ratios
! colspan="6" | Name
|-
|-
| style="text-align:center;" | '''[[11/8|11/8]], [[16/11|16/11]]'''
! colspan="3" | Diatonic
| style="text-align:center;" | '''26.318'''
! colspan="3" | Antidiatonic
|-
|-
| style="text-align:center;" | '''[[4/3|4/3]], [[3/2|3/2]]'''
| {{dash|0, 5, 9|med}}
| style="text-align:center;" | '''26.955'''
| 4:5:6
| D F A
| Dm
| D minor
| D F A
| D
| D major
|-
|-
| style="text-align:center;" | [[11/9|11/9]], [[18/11|18/11]]
| {{dash|0, 4, 9|med}}
| style="text-align:center;" | 27.592
| 10:12:15
| D F♯ A
| D
| D major
| D F♭ A
| Dm
| D minor
|-
|-
| style="text-align:center;" | [[15/14|15/14]], [[28/15|28/15]]
| {{dash|0, 4, 8|med}}
| style="text-align:center;" | 30.557
| 5:6:7
| D F♯ A♯
| Daug
| D augmented
| D F♭ A♭
| Ddim
| D diminished
|-
|-
| style="text-align:center;" | [[10/9|10/9]], [[9/5|9/5]]
| {{dash|0, 5, 10|med}}
| style="text-align:center;" | 32.404
|
| D F A♭
| Ddim
| D diminished
| D F A♯
| Daug
| D augmented
|-
|-
| style="text-align:center;" | [[14/11|14/11]], [[11/7|11/7]]
| {{dash|0, 5, 9, 13|med}}
| style="text-align:center;" | 32.492
| 4:5:6:7
| D F A C♯
| Dm(M7)
| D minor-major
| D F A C♭
| D7
| D seven
|-
|-
| style="text-align:center;" | [[7/6|7/6]], [[12/7|12/7]]
| {{dash|0, 5, 9, 12|med}}
| style="text-align:center;" | 33.129
|
| D F A Bb
| Dm(♭6)
| D minor flat-six
| D F A B♯
| D6
| D six
|-
|-
| style="text-align:center;" | [[18/13|18/13]], [[13/9|13/9]]
| {{dash|0, 5, 9, 14|med}}
| style="text-align:center;" | 36.618
|
| D F A C
| Dm7
| D minor seven
| D F A C
| DM7
| D major seven
|-
|-
| style="text-align:center;" | [[16/15|16/15]], [[15/8|15/8]]
| {{dash|0, 4, 9, 13|med}}
| style="text-align:center;" | 36.731
|
| D F♯ A C♯
| DM7
| D major seven
| D F♭ A C♭
| DM7
| D minor seven
|}
|}


===Patent val mapping===
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"
Examples can be found at the bottom of the page.
|-
 
! | Interval, complement
== Approximation to JI ==
! | Error (abs., in [[cent|cents]])
=== Selected just intervals by error ===
|-
{{Q-odd-limit intervals|16}}
| style="text-align:center;" | [[12/11|12/11]], [[11/6|11/6]]
| style="text-align:center;" | 0.637
|-
| style="text-align:center;" | [[13/10|13/10]], [[20/13|20/13]]
| style="text-align:center;" | 4.214
|-
| style="text-align:center;" | '''[[8/7|8/7]], [[7/4|7/4]]'''
| style="text-align:center;" | '''6.174'''
|-
| style="text-align:center;" | [[13/11|13/11]], [[22/13|22/13]]
| style="text-align:center;" | 10.790
|-
| style="text-align:center;" | '''[[5/4|5/4]], [[8/5|8/5]]'''
| style="text-align:center;" | '''11.314'''
|-
| style="text-align:center;" | [[13/12|13/12]], [[24/13|24/13]]
| style="text-align:center;" | 11.427
|-
| style="text-align:center;" | [[15/11|15/11]], [[22/15|22/15]]
| style="text-align:center;" | 11.951
|-
| style="text-align:center;" | [[11/10|11/10]], [[20/11|20/11]]
| style="text-align:center;" | 15.004
|-
| style="text-align:center;" | '''[[16/13|16/13]], [[13/8|13/8]]'''
| style="text-align:center;" | '''15.528'''
|-
| style="text-align:center;" | [[6/5|6/5]], [[5/3|5/3]]
| style="text-align:center;" | 15.641
|-
| style="text-align:center;" | [[7/5|7/5]], [[10/7|10/7]]
| style="text-align:center;" | 17.488
|-
| style="text-align:center;" | [[14/13|14/13]], [[13/7|13/7]]
| style="text-align:center;" | 21.702
|-
| style="text-align:center;" | [[15/13|15/13]], [[26/15|26/15]]
| style="text-align:center;" | 22.741
|-
| style="text-align:center;" | '''[[11/8|11/8]], [[16/11|16/11]]'''
| style="text-align:center;" | '''26.318'''
|-
| style="text-align:center;" | '''[[4/3|4/3]], [[3/2|3/2]]'''
| style="text-align:center;" | '''26.955'''
|-
| style="text-align:center;" | [[11/9|11/9]], [[18/11|18/11]]
| style="text-align:center;" | 27.592
|-
| style="text-align:center;" | [[14/11|14/11]], [[11/7|11/7]]
| style="text-align:center;" | 32.492
|-
| style="text-align:center;" | [[7/6|7/6]], [[12/7|12/7]]
| style="text-align:center;" | 33.129
|-
| style="text-align:center;" | [[16/15|16/15]], [[15/8|15/8]]
| style="text-align:center;" | 38.269
|-
| style="text-align:center;" | [[18/13|18/13]], [[13/9|13/9]]
| style="text-align:center;" | 38.382
|-
| style="text-align:center;" | [[10/9|10/9]], [[9/5|9/5]]
| style="text-align:center;" | 42.596
|-
| style="text-align:center;" | [[15/14|15/14]], [[28/15|28/15]]
| style="text-align:center;" | 44.443
|-
| style="text-align:center;" | [[9/8|9/8]], [[16/9|16/9]]
| style="text-align:center;" | 53.910
|-
| style="text-align:center;" | [[9/7|9/7]], [[14/9|14/9]]
| style="text-align:center;" | 60.084
|}


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.
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.]]
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[[:File:16ed2-001.svg|16ed2-001.svg]]
[[:File:16ed2-001.svg|16ed2-001.svg]]


==Hexadecaphonic 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 supports Mavila temperament, where the mapping of major and minor is reversed. The temperament could be popular for its 150-cent "3/4-tone" equal division of the traditional 300-cent minor third.
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).


16-EDO 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 16-EDO 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 (16-EDO 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'''".


16-EDO is also a tuning for the no-threes 7-limit temperament tempering out 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:


[[Easley Blackwood Jr]] writes of 16-EDO:
"''16 notes: This tuning is best thought of as a combination of four intertwined diminished seventh chords. Since 12-note tuning can be regarded as a combination of three diminished seventh chords, it is plain that the two tunings have elements in common. The most obvious difference in the way the two tunings sound and work is that triads in 16-note tuning, although recognizable, are too discordant to serve as the final harmony in cadences. Keys can still be established by successions of altered subdominant and dominant harmonies, however, and the Etude is based mainly upon this property. The fundamental consonant harmony employed is a minor triad with an added minor seventh.''"


"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 .


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 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 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


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


==Hexadecaphonic Notation==
=== Commas ===
16et [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes [[val]] {{val| 16 25 37 45 55 59 }}.)


16-EDO notation can be easy utilizing Goldsmith's Circle of keys, nominals, and respective notation. The nominals for a 6 line staff can be switched for Wilson's Beta and Epsilon additions to A-G. The Armodue model uses a 4-line staff for 16-EDO.
{| class="commatable wikitable center-all left-3 right-4 left-6"
 
Moment of Symmetry Scales like Mavila [7] (or "Inverse/Anti-Diatonic" which reverses step sizes of diatonic from LLsLLLs to ssLsssL in the heptatonic variation) can work as an alternative to the traditional diatonic scale, 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 16-EDO keyboard. If the 9-note "Enneatonic" MOS is adopted as a notational basis for 16-EDO, 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|Decave]]".
 
{| class="wikitable"
|-
|-
! | Degree
! [[Harmonic limit|Prime<br>limit]]
! | Cents
! [[Ratio]]<ref group=note>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! colspan="2" | Mavila[9] Notation
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
|-
| style="text-align:center;" | 0
| 5
| style="text-align:center;" | 0
| [[135/128]]
| style="text-align:center;" | unison
| {{monzo| -7 3 1 }}
| style="text-align:center;" | 1
| 92.18
| Layobi
| Mavila comma, major chroma
|-
|-
| style="text-align:center;" | 1
| 5
| style="text-align:center;" | 75
| [[648/625]]
| style="text-align:center;" | aug unison, minor 2nd
| {{monzo| 3 4 -4 }}
| style="text-align:center;" | 1#, 2b
| 62.57
| Quadgu
| Diminished comma, major diesis
|-
|-
| style="text-align:center;" | 2
| 5
| style="text-align:center;" | 150
| [[3125/3072]]
| style="text-align:center;" | major 2nd
| {{monzo| -10 -1 5 }}
| style="text-align:center;" | 2
| 29.61
| Laquinyo
| Magic comma
|-
|-
| style="text-align:center;" | 3
| 5
| style="text-align:center;" | 225
| [[6115295232/6103515625|(20 digits)]]
| style="text-align:center;" | aug 2nd, minor 3rd
| {{monzo| 23 6 -14 }}
| style="text-align:center;" | 2#, 3b
| 3.34
| Sasepbiru
| [[Vishnuzma]]
|-
|-
| style="text-align:center;" | 4
| 7
| style="text-align:center;" | 300
| [[36/35]]
| style="text-align:center;" | major 3rd, dim 4th
| {{monzo| 2 2 -1 -1 }}
| style="text-align:center;" | 3, 4bb
| 48.77
| Rugu
| Mint comma, septimal quartertone
|-
|-
| style="text-align:center;" | 5
| 7
| style="text-align:center;" | 375
| [[525/512]]
| style="text-align:center;" | minor 4th
| {{monzo| -9 1 2 1 }}
| style="text-align:center;" | 4b
| 43.41
| Lazoyoyo
| Avicennma
|-
|-
| style="text-align:center;" | 6
| 7
| style="text-align:center;" | 450
| [[50/49]]
| style="text-align:center;" | major 4th,
| {{monzo| 1 0 2 -2 }}
 
| 34.98
dim 5th
| Biruyo
| style="text-align:center;" | 4, 5b
| Jubilisma
|-
|-
| style="text-align:center;" | 7
| 7
| style="text-align:center;" | 525
| [[64827/64000]]
| style="text-align:center;" | aug 4th, minor 5th
| {{monzo| -9 3 -3 4 }}
| style="text-align:center;" | 4#, 5
| 22.23
| Laquadzo-atrigu
| Squalentine comma
|-
|-
| style="text-align:center;" | 8
| 7
| style="text-align:center;" | 600
| [[3125/3087]]
| style="text-align:center;" | aug 5th, dim 6th
| {{monzo| 0 -2 5 -3 }}
| style="text-align:center;" | 5#, 6b
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
|-
| style="text-align:center;" | 9
| 7
| style="text-align:center;" | 675
| [[126/125]]
| style="text-align:center;" | perfect 6th, dim 7th
| {{monzo| 1 2 -3 1 }}
| style="text-align:center;" | 6, 7bb
| 13.79
| Zotrigu
| Starling comma
|-
|-
| style="text-align:center;" | 10
| 7
| style="text-align:center;" | 750
| [[1029/1024]]
| style="text-align:center;" | aug 6th, minor 7th
| {{monzo| -10 1 0 3 }}
| style="text-align:center;" | 6#, 7b
| 8.43
| Latrizo
| Gamelisma
|-
|-
| style="text-align:center;" | 11
| 7
| style="text-align:center;" | 825
| [[6144/6125]]
| style="text-align:center;" | major 7th
| {{monzo| 11 1 -3 -2 }}
| style="text-align:center;" | 7
| 5.36
| Sarurutrigu
| Porwell comma
|-
|-
| style="text-align:center;" | 12
| 11
| style="text-align:center;" | 900
| [[121/120]]
| style="text-align:center;" | aug 7th, minor 8th
| {{monzo| -3 -1 -1 0 2 }}
| style="text-align:center;" | 7#, 8b
| 14.37
| Lologu
| Biyatisma
|-
|-
| style="text-align:center;" | 13
| 11
| style="text-align:center;" | 975
| [[176/175]]
| style="text-align:center;" | major 8th, dim 9th
| {{monzo| 4 0 -2 -1 1 }}
| style="text-align:center;" | 8, 9bb
| 9.86
| Lorugugu
| Valinorsma
|-
|-
| style="text-align:center;" | 14
| 11
| style="text-align:center;" | 1050
| [[385/384]]
| style="text-align:center;" | minor 9th
| {{monzo| -7 -1 1 1 1 }}
| style="text-align:center;" | 9
| 4.50
| Lozoyo
| Keenanisma
|-
|-
| style="text-align:center;" | 15
| 11
| style="text-align:center;" | 1125
| [[441/440]]
| style="text-align:center;" | major 9th, dim 10ve
| {{monzo| -3 2 -1 2 -1 }}
| style="text-align:center;" | 9#, 1b
| 3.93
| Luzozogu
| Werckisma
|-
|-
| style="text-align:center;" | 16
| 11
| style="text-align:center;" | 1200
| [[3025/3024]]
| style="text-align:center;" | 10ve (Decave)
| {{monzo| -4 -3 2 -1 2 }}
| style="text-align:center;" | 1
| 0.57
| Loloruyoyo
| Lehmerisma
|}
|}


==16 Tone Piano Layout Based on the Mavila[7]/"Anti-diatonic" Scale==
=== Rank-2 temperaments ===
[[File:16-EDO-PIano-Diagram.png|alt=16-EDO-PIano-Diagram.png|748x293px|16-EDO-PIano-Diagram.png]]
* [[List of 16et rank two temperaments by badness]]
 
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".
 
==Rank two temperaments==
[[List_of_16et_rank_two_temperaments_by_badness|List of 16et rank two temperaments by badness]]


Temperaments listed by generator size:
{| class="wikitable center-1 center-2"
 
|+ Table of temperaments by generator
{| class="wikitable"
|-
|-
! | Periods
! Periods<br>per 8ve
 
! Generator
per octave
! Temperaments
! | Generator
! | Temperaments
|-
|-
| | 1
| 1
| | 1\16
| 1\16
| style="text-align:center;" | [[Valentine|Valentine]], [[Slurpee|slurpee]]
| [[Valentine]], [[slurpee]]
|-
|-
| | 1
| 1
| | 3\16
| 3\16
| style="text-align:center;" | [[Gorgo|Gorgo]]
| [[Gorgo]]
|-
|-
| | 1
| 1
| | 5\16
| 5\16
| style="text-align:center;" | Messed-up [[Magic|magic]]/muggles
| [[Magic]]/[[muggles]]
|-
|-
| | 1
| 1
| | 7\16
| 7\16
| style="text-align:center;" | [[Mavila|Mavila]]/armodue
| [[Mavila]]/[[armodue]]
|-
|-
| | 2
| 2
| | 1\16
| 1\16
| style="text-align:center;" | [[Bipelog|Bipelog]]
| [[Bipelog]]
|-
|-
| | 2
| 2
| | 3\16
| 3\16
| style="text-align:center;" | [[Lemba|Lemba]], [[Astrology|astrology]]
| [[Lemba]], [[astrology]]
|-
|-
| | 4
| 4
| | 1\16
| 1\16
| style="text-align:center;" | [[Diminished|Diminished]]/demolished
| [[Diminished (temperament)|Diminished]]/[[demolished]]
|-
|-
| | 8
| 8
| | 1\16
| 1\16
| style="text-align:center;" |
| [[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 834: Line 656:
{| class="wikitable"
{| class="wikitable"
|-
|-
| | [5]:
| [5]:
| | 5 2 5 2 2
| 5 2 5 2 2
| |  
|  
|-
|-
| | [7]:
| [7]:
| | 3 2 2 3 2 2 2
| 3 2 2 3 2 2 2
| | [[File:MavilaAntidiatonic16edo.mp3]]
|[[File:MavilaAntidiatonic16edo.mp3]]
|-
|-
| | [9]:
| [9]:
| | 1 2 2 2 1 2 2 2 2
| 1 2 2 2 1 2 2 2 2
| | [[File:MavilaSuperdiatonic16edo.mp3]]
|[[File:MavilaSuperdiatonic16edo.mp3]]
|}
|}
See also [[Mavila_Temperament_Modal_Harmony|Mavila Temperament Modal Harmony]].
See also [[Mavila Temperament Modal Harmony]].


'''Diminished'''
'''Diminished'''
Line 852: Line 674:
{| class="wikitable"
{| class="wikitable"
|-
|-
| | [8]:
| [8]:
| | 1 3 1 3 1 3 1 3
| 1 3 1 3 1 3 1 3
| | [[File:htgt16edo.mp3]]
|[[File:htgt16edo.mp3]]
|-
|-
| | [12]:
| [12]:
| | 1 1 2 1 1 2 1 1 2 1 1 2
| 1 1 2 1 1 2 1 1 2 1 1 2
| |  
|  
|}
|}


Line 885: Line 707:
[10]: 2 1 2 1 2 2 1 2 1 2
[10]: 2 1 2 1 2 2 1 2 1 2


==Metallic Harmony in 16 EDO==
== 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]].''


Because 16edo doesn't approximate 3/2 well at all, triadic harmony based on heptatonic thirds isn't a great option for typical harmonic timbres.
== Diagrams ==
'''16-tone piano layout based on the mavila[7]/antidiatonic scale'''


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
This Layout places mavila[7] on the black keys and mavila[9] on the white keys, according to antidiatonic notation.


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


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".
'''Interleaved edos'''


===MOS scales supporting Metallic Harmony in 16edo===
A visualization of 16edo being two interleaved copies of [[8edo]] and four interleaved copies of [[4edo]].
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.


Another possible MOS scales for this approach would be Lemba[6], which gives two each of the soft, hard, and narrow symmetric triads.
[[File:16edo_wheel_01.png|alt=16edo wheel 01.png|325x325px|16edo wheel 01.png]]


See [[Metallic_Harmony|Metallic Harmony]].
=== Lumatone mapping ===


==Commas==
See: [[Lumatone mapping for 16edo]]
16 EDO [[tempering_out|tempers out]] the following [[comma]]s. (Note: This assumes [[val|val]] &lt; 16 25 37 45 55 59 |.)


{| class="wikitable"
== Interval arithmetic examples ==
|-
These examples show the correspondence between interval arithmetic using diatonic and antidiatonic notation.
! | [[Ratio]]
{| class="wikitable" style="text-align: center;"
! | [[Monzo]]
! colspan="2" |Diatonic (i.e. 12edo)
! | [[Cent|Cents]]
! colspan="2" |Antidiatonic (i.e. 16edo)
![[Color notation/Temperament Names|Color Name]]
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | 135/128
| |<nowiki> | -7 3 1 </nowiki>&gt;
| style="text-align:right;" | 92.18
| style="text-align:center;" |Layobi
| style="text-align:center;" | Major Chroma
| style="text-align:center;" | Major Limma
| style="text-align:center;" | Pelogic Comma
|-
| style="text-align:center;" | 648/625
| |<nowiki> | 3 4 -4 </nowiki>&gt;
| style="text-align:right;" | 62.57
| style="text-align:center;" |Quadgu
| style="text-align:center;" | Major Diesis
| style="text-align:center;" | Diminished Comma
| style="text-align:center;" |
|-
|-
| style="text-align:center;" | 3125/3072
! Question
| |<nowiki> | -10 -1 5 </nowiki>&gt;
! Result
| style="text-align:right;" | 29.61
! Question
| style="text-align:center;" |Laquinyo
! Result
| style="text-align:center;" | Small Diesis
| style="text-align:center;" | Magic Comma
| style="text-align:center;" |
|-
|-
| style="text-align:center;" | 36/35
| M2 + M2
| |<nowiki> | 2 2 -1 -1 </nowiki>&gt;
| aug3
| style="text-align:right;" | 48.77
| m2 + m2
| style="text-align:center;" |Rugu
| dim3
| style="text-align:center;" | Septimal Quarter Tone
| style="text-align:center;" |
| style="text-align:center;" |
|-
|-
| style="text-align:center;" | 525/512
| D to F♯
| |<nowiki> | -9 1 2 1 </nowiki>&gt;
| aug3
| style="text-align:right;" | 43.41
| D to F♭
| style="text-align:center;" |Lazoyoyo
| dim3
| style="text-align:center;" | Avicennma
| style="text-align:center;" | Avicenna's Enharmonic Diesis
| style="text-align:center;" |
|-
|-
| style="text-align:center;" | 50/49
| D to F
| |<nowiki> | 1 0 2 -2 </nowiki>&gt;
| M3
| style="text-align:right;" | 34.98
| D to F
| style="text-align:center;" |Biruyo
| m3
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| style="text-align:center;" |
|-
|-
| style="text-align:center;" | 64827/64000
| E♭ + m3
| |<nowiki> | -9 3 -3 4 </nowiki>&gt;
| Gbb
| style="text-align:right;" | 22.23
| E♯ + M3
| style="text-align:center;" |Laquadzo-atrigu
| G♯♯
| style="text-align:center;" | Squalentine
| style="text-align:center;" |
| style="text-align:center;" |
|-
|-
| style="text-align:center;" | 3125/3087
| E♭ + P5
| |<nowiki> | 0 -2 5 -3 </nowiki>&gt;
| B♭
| style="text-align:right;" | 21.18
| E♯ + P5
| style="text-align:center;" |Triru-aquinyo
| B♯
| style="text-align:center;" | Gariboh
| style="text-align:center;" |
| style="text-align:center;" |
|-
|-
| style="text-align:center;" | 126/125
| A minor chord
| |<nowiki> | 1 2 -3 1 </nowiki>&gt;
| A C♭ E
| style="text-align:right;" | 13.79
| A major chord
| style="text-align:center;" |Zotrigu
| A C♯ E
| style="text-align:center;" | Septimal Semicomma
| style="text-align:center;" | Starling Comma
| style="text-align:center;" |
|-
|-
| style="text-align:center;" | 1029/1024
| E♭ major chord
| |<nowiki> | -10 1 0 3 </nowiki>&gt;
| E♭ G♭ D♭
| style="text-align:right;" | 8.43
| E♯ minor chord
| style="text-align:center;" |Latrizo
| E♯ G♯ B♯
| style="text-align:center;" | Gamelisma
| style="text-align:center;" |
| style="text-align:center;" |
|-
|-
| style="text-align:center;" | 6144/6125
| Gm7 = G + m3 + P5 + m7
| |<nowiki> | 11 1 -3 -2 </nowiki>&gt;
| G B D F♭
| style="text-align:right;" | 5.36
| G + M3 + P5 + M7
| style="text-align:center;" |Sarurutrigu
| G B D F♯
| style="text-align:center;" | Porwell
| style="text-align:center;" |
| style="text-align:center;" |
|-
|-
| style="text-align:center;" |
| A♭7aug = A♭ + M3 + A5 + m7
| |<nowiki> | 23 6 -14 </nowiki>&gt;
| A♭ C♭ E Gbb
| style="text-align:right;" | 3.34
| A♯ + m3 + d5 + M7
| style="text-align:center;" |Sasepbiru
| A♯ C♯ E G♯♯
| style="text-align:center;" | Vishnuzma
| style="text-align:center;" | Semisuper
| style="text-align:center;" |
|-
|-
| style="text-align:center;" | 121/120
| what chord is D F A♯?
| |<nowiki> | -3 -1 -1 0 2 </nowiki>&gt;
| D + M3 + A5 = Daug
| style="text-align:right;" | 14.37
| D F A♭
| style="text-align:center;" |Lologu
| D + m3 + d5
| style="text-align:center;" | Biyatisma
| style="text-align:center;" |
| style="text-align:center;" |
|-
|-
| style="text-align:center;" | 176/175
| what chord is C E G♭ B♭?
| |<nowiki> | 4 0 -2 -1 1 </nowiki>&gt;
| C + m3 + d5 + d7 = Cdim7
| style="text-align:right;" | 9.86
| C E G♯ B♯
| style="text-align:center;" |Lorugugu
| C + M3 + A5 + A7
| style="text-align:center;" | Valinorsma
| style="text-align:center;" |
| style="text-align:center;" |
|-
|-
| style="text-align:center;" | 385/384
| C major scale = C + M2 + M3<br>+ P4 + P5 + M6 + M7 + P8
| |<nowiki> | -7 -1 1 1 1 </nowiki>&gt;
| C D♯ E♯ F<br>G A♯ B♯ C
| style="text-align:right;" | 4.50
| C + m2 + m3 + P4<br>+ P5 + m6 + m7 + P8
| style="text-align:center;" |Lozoyo
| C D♭ E♭ F<br>G A♭ B♭ C
| style="text-align:center;" | Keenanisma
| style="text-align:center;" |
| style="text-align:center;" |
|-
|-
| style="text-align:center;" | 441/440
| C minor scale = C + M2 + m3<br>+ P4 + P5 + m6 + m7 + P8
| |<nowiki> | -3 2 -1 2 -1 </nowiki>&gt;
| C D♯ E F<br>G A B C
| style="text-align:right;" | 3.93
| C + m2 + M3 + P4<br>+ P5 + M6 + M7 + P8
| style="text-align:center;" |Luzozogu
| C D♭ E F<br>G A B C
| style="text-align:center;" | Werckisma
| style="text-align:center;" |
| style="text-align:center;" |
|-
|-
| style="text-align:center;" | 3025/3024
| what scale is A B♯ C♭ D<br>E F G♭ A?
| |<nowiki> | -4 -3 2 -1 2 </nowiki>&gt;
| A + M2 + m3 + P4<br>+ P5 + M6 + m7 = A dorian
| style="text-align:right;" | 0.57
| A B♭ C♯ D<br>E F G♯ A
| style="text-align:center;" |Loloruyoyo
| A + m2 + M3 + P4<br>+ P5 + m6 + M7
| style="text-align:center;" | Lehmerisma
| style="text-align:center;" |
| style="text-align:center;" |
|}
|}


==Armodue Theory (4-line staff)==
== Music ==
[http://www.armodue.com/ricerche.htm Armodue]: Pierpaolo Beretta's website for his "Armodue" theory for 16edo (esadekaphonic), including compositions.
{{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'']


Translations of parts of the Armodue pages can be found [[Armodue|here]] on this wiki.
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/IfVvjoRqqNk ''16edo jam''] (2025)
* [https://www.youtube.com/watch?v=cUgbkkIvy0g ''Waltz in 16edo''] (2025)


==Images==
; [[E8 Heterotic]]
[[File:16edo_wheel_01.png|alt=16edo wheel 01.png|325x325px|16edo wheel 01.png]]
* [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)


==Books/Literature==
; [[William Lynch]]
Sword, Ronald. "Thesaurus of Melodic Patterns and Intervals for 16-Tones" IAAA Press, USA. First Ed: August, 2011
* [[:File:Mavila_Jazz_Rhodes_1.mp3|''Mavila Jazz Groove'']]
* [[:File:mavila4.mp3|''Cold, Dark Night for a Dance'']]


Sword, Ronald. "Hexadecaphonic Scales for Guitar." IAAA Press, UK-USA. First Ed: Feb, 2010. (superfourth tuning)
; [[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)


Sword, Ronald. "Esadekaphonic Scales for Guitar." IAAA Press, UK-USA. First Ed: April, 2009. (semi-diminished fourth tuning)
; [[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}})


==Compositions==
; [[Nae Ayy]]
[https://cityoftheasleep.bandcamp.com/track/huckleberry-regional-preserve Huckleberry Regional Preserve] by City of the Asleep
* [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'']


[https://cityoftheasleep.bandcamp.com/track/illegible-red-ink Illegible Red Ink] by City of the Asleep
; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=LXsZIbT6wpM ''Edolian - Seventhic''] (2020)
* [https://www.youtube.com/watch?v=UrQPr7V9feA ''Finality''] (2021)


[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=660895&songID=12370649&showPlayer=true Prenestyna Highway] by [http://fiale.tk Fabrizio Fulvio Fausto Fiale]
; [[Jean-Pierre Poulin]]
* [http://www.jeanpierrepoulin.com/mp3/Armodue78.mp3 ''Armodue78'']


[http://lastsacrament.bandcamp.com/album/enantiodromia Enantiodromia (album)] by Last Sacrament
; [[Ron Sword]]
* [https://soundcloud.com/ron-sword/mavila-fog ''The Foggy Road from Pasadena'']{{dead link}}


[http://www.cdbaby.com/cd/aeternamusic Tribute to Armodue] by Aeterna
; [[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]


[http://www.io.com/%7Ehmiller/midi/16tet.mid Etude in 16-tone equal tuning] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/16tet.mp3 play] ([http://soonlabel.com/xenharmonic/archives/2604 organ version]) by [[Herman_Miller|Herman Miller]]
; [[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'']


[https://soundcloud.com/ron-sword/mavila-fog The Foggy Road from Pasadena] by Ron Sword
; [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/05%20-%205.%2016%20octave.mp3 Comets Over Flatland 5]{{dead link}}


[http://www.jeanpierrepoulin.com/mp3/Armodue78.mp3 Armodue78] by [[Jean-Pierre_Poulin|Jean-Pierre Poulin]]
; [[Woyten]]
* [https://www.youtube.com/watch?v=LLgClI8pyNw ''Don't Take Five''] (2021)


[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Fiale/ffffiale+palestrinamortafantasiaquasiunasonata.mp3 Palestrina Morta, fantasia quasi una sonata] by [http://fiale.tk Fabrizio Fulvio Fausto Fiale]
; [[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]


[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/05%20-%205.%2016%20octave.mp3 Comets Over Flatland 5] by [[Randy_Winchester|Randy Winchester]]
; [[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'']


[http://micro.soonlabel.com/16-ET/20120527-16-malathion.mp3 Malathion] by [http://chrisvaisvil.com/?p=2358#comments Chris Vaisvil]
; [[Zewen Senpai]]
* [https://www.youtube.com/watch?v=QOzBGd64Pi4 ''Simple Ambient Study No. 1'']


[http://micro.soonlabel.com/16-ET/20130216_16edo_vesta.mp3 Being of Vesta] by [http://chrisvaisvil.com/?p=3061 Chris Vaisvil]
== Notes ==
<references group=note/>


[http://micro.soonlabel.com/simultaneous-tunings/20130607_thin_ice_christiane.mp3 Thin Ice] by [[Chris_Vaisvil|Chris Vaisvil]] ; [http://chrisvaisvil.com/thin-ice-for-alto-female-choir-harp-and-percussion-in-adaptive-ji-16-edo-and-8-edo/ information on the composition]
== See also ==
* [[57ed12]] - octave stretched version of 16edo; 57ed12 improves 3.5.11.13.17 but damages 2.7


[[:File:Mavila_Jazz_Rhodes_1.mp3|Mavila Jazz Groove]] by [https://soundcloud.com/pianofreak96 William Lynch]
=== Approaches ===
* [[User:VectorGraphics/16edo theory|Vector's approach]]
* [[Armodue theory]]
** [[Armodue armonia]]


[[:File:mavila4.mp3|Cold, Dark Night for a Dance ]]by [https://soundcloud.com/pianofreak96 William Lynch]
== References ==
<references />


[https://soundcloud.com/fff-fiale/in-sospensione-neutra In Sospensione Neutra by Fabrizio Fulvio Fiale]
== 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}}


<span style="display: block; height: 1px; left: 0px; overflow: hidden; position: absolute; top: 3102px; width: 1px;"><span style="font-family: Palatino; font-size: medium;">[http://x31eq.com/cgi-bin/uv.cgi?uvs=%5B-19%2C7%2C1%3E&limit=2_5_7 546875:524288]</span>
[[Category:Teentuning]]
[[Category:Listen]]
[[Category:Mavila]]
[[Category:Guitar]]
[[Category:Pages with internal sound examples]]


[[Category:16-tone]]
{{Todo|cleanup}}
[[Category:16edo]]
[[Category:Equal divisions of the octave]]
[[Category:guitar]]
[[Category:image]]
[[Category:listen]]
[[Category:mavila]]
[[Category:teentuning]]
[[Category:theory]]
[[Category:todo:unify_precision]]
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