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{{Infobox ET
{{Infobox ET
| Prime factorization = 2<sup>4</sup>
| Prime factorization = 2<sup>4</sup>
| Step size = 75¢
| Step size = 75.000¢
| Fifth = 9\16 (675¢)
| Fifth = 9\16 (675¢)
| Major 2nd = 2\16 (150¢)
| Major 2nd = 2\16 (150¢)
| Semitones = -1:3 (-75¢:225¢)
| Semitones = -1:3 (-75¢ : 225¢)
| Consistency = 7
| Consistency = 7
| Monotonicity = 7
}}
}}
{{todo|add introduction}}
'''16edo''' is the [[equal division of the octave]] into sixteen narrow chromatic semitones each of 75 [[cent]]s exactly.


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


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


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".
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 [[16edo#Hexadecaphonic Notation|below]]
Alternatively, one can use Armodue nine-nominal notation; see [[#Hexadecaphonic Notation]]


{| class="wikitable"
{| class="wikitable center-all"
|-
|-
! Degree
! Degree
Line 40: Line 39:
! Interval <br> Names <br> Simplified
! Interval <br> Names <br> Simplified
|-
|-
| style="text-align:center;" | 0
| 0
| style="text-align:center;" | 0
| 0
| style="text-align:center;" | 1/1
| 1/1
| style="text-align:center;" | unison
| unison
| style="text-align:center;" | D
| D
| style="text-align:center;" | unison
| unison
| style="text-align:center;" | D
| D
| style="text-align:center;" | unison
| unison
| style="text-align:center;" | unison
| unison
|-
|-
| style="text-align:center;" | 1
| 1
| style="text-align:center;" | 75
| 75
| style="text-align:center;" | 28/27, 27/26
| 28/27, 27/26
| style="text-align:center;" | aug 1, dim 2nd
| aug 1, dim 2nd
| style="text-align:center;" | D#, Eb
| D#, Eb
| style="text-align:center;" | dim 1, aug 2nd
| dim 1, aug 2nd
| style="text-align:center;" | Db, E#
| Db, E#
| style="text-align:center;" | subminor 2nd
| subminor 2nd
| style="text-align:center;" | min 2nd
| min 2nd
|-
|-
| style="text-align:center;" | 2
| 2
| style="text-align:center;" | 150
| 150
| style="text-align:center;" | 35/32
| 35/32
| style="text-align:center;" | minor 2nd
| minor 2nd
| style="text-align:center;" | E
| E
| style="text-align:center;" | major 2nd
| major 2nd
| style="text-align:center;" | E
| E
| style="text-align:center;" | neutral 2nd
| neutral 2nd
| style="text-align:center;" | maj 2nd
| maj 2nd
|-
|-
| style="text-align:center;" | 3
| 3
| style="text-align:center;" | 225
| 225
| style="text-align:center;" | 8/7
| 8/7
| style="text-align:center;" | major 2nd
| major 2nd
| style="text-align:center;" | E#
| E#
| style="text-align:center;" | minor 2nd
| minor 2nd
| style="text-align:center;" | Eb
| Eb
| style="text-align:center;" | supermajor 2nd,<br>septimal whole-tone
| supermajor 2nd,<br>septimal whole-tone
| style="text-align:center;" | perf 2nd
| perf 2nd
|-
|-
| style="text-align:center;" | 4
| 4
| style="text-align:center;" | 300
| 300
| style="text-align:center;" | 19/16, 32/27
| 19/16, 32/27
| style="text-align:center;" | minor 3rd
| minor 3rd
| style="text-align:center;" | Fb
| Fb
| style="text-align:center;" | major 3rd
| major 3rd
| style="text-align:center;" | F#
| F#
| style="text-align:center;" | minor 3rd
| minor 3rd
| style="text-align:center;" | min 3rd
| min 3rd
|-
|-
| style="text-align:center;" | 5
| 5
| style="text-align:center;" | 375
| 375
| style="text-align:center;" | 5/4, 16/13, 26/21
| 5/4, 16/13, 26/21
| style="text-align:center;" | major 3rd
| major 3rd
| style="text-align:center;" | F
| F
| style="text-align:center;" | minor 3rd
| minor 3rd
| style="text-align:center;" | F
| F
| style="text-align:center;" | major 3rd
| major 3rd
| style="text-align:center;" | maj 3rd
| maj 3rd
|-
|-
| style="text-align:center;" | 6
| 6
| style="text-align:center;" | 450
| 450
| style="text-align:center;" | 13/10, 35/27
| 13/10, 35/27
| style="text-align:center;" | aug 3rd,<br>dim 4th
| aug 3rd,<br>dim 4th
| style="text-align:center;" | F#, Gb
| F#, Gb
| style="text-align:center;" | dim 3rd,<br>aug 4th
| dim 3rd,<br>aug 4th
| style="text-align:center;" | Fb, G#
| Fb, G#
| style="text-align:center;" | sub-4th,<br>supermajor 3rd
| sub-4th,<br>supermajor 3rd
| style="text-align:center;" | min 4th
| min 4th
|-
|-
| style="text-align:center;" | 7
| 7
| style="text-align:center;" | 525
| 525
| style="text-align:center;" | 19/14, 27/20, 52/35, 256/189
| 19/14, 27/20, 52/35, 256/189
| style="text-align:center;" | perfect 4th
| perfect 4th
| style="text-align:center;" | G
| G
| style="text-align:center;" | perfect 4th
| perfect 4th
| style="text-align:center;" | G
| G
| style="text-align:center;" | wide 4th
| wide 4th
| style="text-align:center;" | maj 4th
| maj 4th
|-
|-
| style="text-align:center;" | 8
| 8
| style="text-align:center;" | 600
| 600
| style="text-align:center;" | 7/5, 10/7
| 7/5, 10/7
| style="text-align:center;" | aug 4th,<br>dim 5th
| aug 4th,<br>dim 5th
| style="text-align:center;" | G#, Ab
| G#, Ab
| style="text-align:center;" | dim 4th,<br>aug 5th
| dim 4th,<br>aug 5th
| style="text-align:center;" | Gb, A#
| Gb, A#
| style="text-align:center;" | tritone
| tritone
| style="text-align:center;" | aug 4th,<br>dim 5th
| aug 4th,<br>dim 5th
|-
|-
| style="text-align:center;" | 9
| 9
| style="text-align:center;" | 675
| 675
| style="text-align:center;" | 28/19, 40/27, 35/26, 189/128
| 28/19, 40/27, 35/26, 189/128
| style="text-align:center;" | perfect 5th
| perfect 5th
| style="text-align:center;" | A
| A
| style="text-align:center;" | perfect 5th
| perfect 5th
| style="text-align:center;" | A
| A
| style="text-align:center;" | narrow 5th
| narrow 5th
| style="text-align:center;" | min 5th
| min 5th
|-
|-
| style="text-align:center;" | 10
| 10
| style="text-align:center;" | 750
| 750
| style="text-align:center;" | 20/13, 54/35
| 20/13, 54/35
| style="text-align:center;" | aug 5th,<br>dim 6th
| aug 5th,<br>dim 6th
| style="text-align:center;" | A#, Bb
| A#, Bb
| style="text-align:center;" | dim 5th,<br>aug 6th
| dim 5th,<br>aug 6th
| style="text-align:center;" | Ab, B#
| Ab, B#
| style="text-align:center;" | super-5th,<br>subminor 6th
| super-5th,<br>subminor 6th
| style="text-align:center;" | maj 5th
| maj 5th
|-
|-
| style="text-align:center;" | 11
| 11
| style="text-align:center;" | 825
| 825
| style="text-align:center;" | 8/5, 13/8, 21/13
| 8/5, 13/8, 21/13
| style="text-align:center;" | minor 6th
| minor 6th
| style="text-align:center;" | B
| B
| style="text-align:center;" | major 6th
| major 6th
| style="text-align:center;" | B
| B
| style="text-align:center;" | minor 6th
| minor 6th
| style="text-align:center;" | min 6th
| min 6th
|-
|-
| style="text-align:center;" | 12
| 12
| style="text-align:center;" | 900
| 900
| style="text-align:center;" | 27/16, 32/19
| 27/16, 32/19
| style="text-align:center;" | major 6th
| major 6th
| style="text-align:center;" | B#
| B#
| style="text-align:center;" | minor 6th
| minor 6th
| style="text-align:center;" | Bb
| Bb
| style="text-align:center;" | major 6th
| major 6th
| style="text-align:center;" | maj 6th
| maj 6th
|-
|-
| style="text-align:center;" | 13
| 13
| style="text-align:center;" | 975
| 975
| style="text-align:center;" | 7/4
| 7/4
| style="text-align:center;" | minor 7th
| minor 7th
| style="text-align:center;" | Cb
| Cb
| style="text-align:center;" | major 7th
| major 7th
| style="text-align:center;" | C#
| C#
| style="text-align:center;" | subminor 7th,<br>septimal minor 7th
| subminor 7th,<br>septimal minor 7th
| style="text-align:center;" | perf 7th
| perf 7th
|-
|-
| style="text-align:center;" | 14
| 14
| style="text-align:center;" | 1050
| 1050
| style="text-align:center;" | 64/35
| 64/35
| style="text-align:center;" | major 7th
| major 7th
| style="text-align:center;" | C
| C
| style="text-align:center;" | minor 7th
| minor 7th
| style="text-align:center;" | C
| C
| style="text-align:center;" | neutral 7th
| neutral 7th
| style="text-align:center;" | min 7th
| min 7th
|-
|-
| style="text-align:center;" | 15
| 15
| style="text-align:center;" | 1125
| 1125
| style="text-align:center;" | 27/14, 52/27
| 27/14, 52/27
| style="text-align:center;" | aug 7th,<br>dim 8ve
| aug 7th,<br>dim 8ve
| style="text-align:center;" | C#, Db
| C#, Db
| style="text-align:center;" | dim 7th,<br>aug 8ve
| dim 7th,<br>aug 8ve
| style="text-align:center;" | Cb, D#
| Cb, D#
| style="text-align:center;" | supermajor 7th
| supermajor 7th
| style="text-align:center;" | maj 7th
| maj 7th
|-
|-
| style="text-align:center;" | 16
| 16
| style="text-align:center;" | 1200
| 1200
| style="text-align:center;" | 2/1
| 2/1
| style="text-align:center;" | 8ve
| 8ve
| style="text-align:center;" | D
| D
| style="text-align:center;" | 8ve
| 8ve
| style="text-align:center;" | D
| D
| style="text-align:center;" | octave
| octave
| style="text-align:center;" | octave
| octave
|}
|}
<nowiki>*</nowiki> based on treating 16-EDO as a 2.5.7.13.19.27 subgroup temperament; other approaches are possible.
<nowiki>*</nowiki> based on treating 16edo as a 2.5.7.13.19.27 subgroup temperament; other approaches are possible.
 
==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 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).


{| class="wikitable"
{| class="wikitable center-all"
|-
|-
! | chord
! | chord
Line 223: Line 221:
! colspan="3" | melodic name
! colspan="3" | melodic name
|-
|-
| style="text-align:center;" | 0-5-9
| 0-5-9
| style="text-align:center;" | 4:5:6
| 4:5:6
| style="text-align:center;" | D F A
| D F A
| style="text-align:center;" | Dm
| Dm
| style="text-align:center;" | D minor
| D minor
| style="text-align:center;" | D F A
| D F A
| style="text-align:center;" | D
| D
| style="text-align:center;" | D major
| D major
|-
|-
| style="text-align:center;" | 0-4-9
| 0-4-9
| style="text-align:center;" | 10:12:15
| 10:12:15
| style="text-align:center;" | D F# A
| D F# A
| style="text-align:center;" | D
| D
| style="text-align:center;" | D major
| D major
| style="text-align:center;" | D Fb A
| D Fb A
| style="text-align:center;" | Dm
| Dm
| style="text-align:center;" | D minor
| D minor
|-
|-
| style="text-align:center;" | 0-4-8
| 0-4-8
| style="text-align:center;" | 5:6:7
| 5:6:7
| style="text-align:center;" | D F# A#
| D F# A#
| style="text-align:center;" | Daug
| Daug
| style="text-align:center;" | D augmented
| D augmented
| style="text-align:center;" | D Fb Ab
| D Fb Ab
| style="text-align:center;" | Ddim
| Ddim
| style="text-align:center;" | D diminished
| D diminished
|-
|-
| style="text-align:center;" | 0-5-10
| 0-5-10
| style="text-align:center;" |  
|  
| style="text-align:center;" | D F Ab
| D F Ab
| style="text-align:center;" | Ddim
| Ddim
| style="text-align:center;" | D diminished
| D diminished
| style="text-align:center;" | D F A#
| D F A#
| style="text-align:center;" | Daug
| Daug
| style="text-align:center;" | D augmented
| D augmented
|-
|-
| style="text-align:center;" | 0-5-9-13
| 0-5-9-13
| style="text-align:center;" | 4:5:6:7
| 4:5:6:7
| style="text-align:center;" | D F A C#
| D F A C#
| style="text-align:center;" | Dm(M7)
| Dm(M7)
| style="text-align:center;" | D minor-major
| D minor-major
| style="text-align:center;" | D F A Cb
| D F A Cb
| style="text-align:center;" | D7
| D7
| style="text-align:center;" | D seven
| D seven
|-
|-
| style="text-align:center;" | 0-5-9-12
| 0-5-9-12
| style="text-align:center;" |  
|  
| style="text-align:center;" | D F A Bb
| D F A Bb
| style="text-align:center;" | Dm(b6)
| Dm(b6)
| style="text-align:center;" | D minor flat-six
| D minor flat-six
| style="text-align:center;" | D F A B#
| D F A B#
| style="text-align:center;" | D6
| D6
| style="text-align:center;" | D six
| D six
|-
|-
| style="text-align:center;" | 0-5-9-14
| 0-5-9-14
| style="text-align:center;" |  
|  
| style="text-align:center;" | D F A C
| D F A C
| style="text-align:center;" | Dm7
| Dm7
| style="text-align:center;" | D minor seven
| D minor seven
| style="text-align:center;" | D F A C
| D F A C
| style="text-align:center;" | DM7
| DM7
| style="text-align:center;" | D major seven
| D major seven
|-
|-
| style="text-align:center;" | 0-4-9-13
| 0-4-9-13
| style="text-align:center;" |  
|  
| style="text-align:center;" | D F# A C#
| D F# A C#
| style="text-align:center;" | DM7
| DM7
| style="text-align:center;" | D major seven
| D major seven
| style="text-align:center;" | D Fb A Cb
| D Fb A Cb
| style="text-align:center;" | DM7
| DM7
| style="text-align:center;" | 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|Ups and Downs Notation - Chords and Chord Progressions]] for more 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 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:
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:
Line 388: Line 386:
|}
|}


==Selected just intervals by error==
== JI approximation ==
The following table shows how [[Just-24|some prominent just intervals]] are represented in 16-EDO (ordered by absolute error).
=== Selected just intervals by error ===
 
The following table shows how [[15-odd-limit intervals]] are represented in 16edo (ordered by absolute error).
===Best direct mapping, even if inconsistent===
 
{| class="wikitable"
|-
! | Interval, complement
! | Error (abs., in [[cent|cents]])
|-
| 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;" | [[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]]
| 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;" | [[15/14|15/14]], [[28/15|28/15]]
| style="text-align:center;" | 30.557
|-
| style="text-align:center;" | [[10/9|10/9]], [[9/5|9/5]]
| style="text-align:center;" | 32.404
|-
| 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;" | [[18/13|18/13]], [[13/9|13/9]]
| style="text-align:center;" | 36.618
|-
| style="text-align:center;" | [[16/15|16/15]], [[15/8|15/8]]
| style="text-align:center;" | 36.731
|}
 
===Patent val mapping===


{| class="wikitable"
{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
|-
! | Interval, complement
! | Interval, complement
! | Error (abs., in [[cent|cents]])
! | Error (abs., in [[cent]]s)
|-
|-
| style="text-align:center;" | [[12/11|12/11]], [[11/6|11/6]]
| [[12/11]], [[11/6]]
| style="text-align:center;" | 0.637
| 0.637
|-
|-
| style="text-align:center;" | [[13/10|13/10]], [[20/13|20/13]]
| [[13/10]], [[20/13]]
| style="text-align:center;" | 4.214
| 4.214
|-
|-
| style="text-align:center;" | '''[[8/7|8/7]], [[7/4|7/4]]'''
| '''[[8/7]], [[7/4]]'''
| style="text-align:center;" | '''6.174'''
| '''6.174'''
|-
|-
| style="text-align:center;" | [[13/11|13/11]], [[22/13|22/13]]
| [[13/11]], [[22/13]]
| style="text-align:center;" | 10.790
| 10.790
|-
|-
| style="text-align:center;" | '''[[5/4|5/4]], [[8/5|8/5]]'''
| '''[[5/4]], [[8/5]]'''
| style="text-align:center;" | '''11.314'''
| '''11.314'''
|-
|-
| style="text-align:center;" | [[13/12|13/12]], [[24/13|24/13]]
| [[13/12]], [[24/13]]
| style="text-align:center;" | 11.427
| 11.427
|-
|-
| style="text-align:center;" | [[15/11|15/11]], [[22/15|22/15]]
| [[15/11]], [[22/15]]
| style="text-align:center;" | 11.951
| 11.951
|-
|-
| style="text-align:center;" | [[11/10|11/10]], [[20/11|20/11]]
| [[9/7]], [[14/9]]
| style="text-align:center;" | 15.004
| 14.916
|-
|-
| style="text-align:center;" | '''[[16/13|16/13]], [[13/8|13/8]]'''
| [[11/10]], [[20/11]]
| style="text-align:center;" | '''15.528'''
| 15.004
|-
|-
| style="text-align:center;" | [[6/5|6/5]], [[5/3|5/3]]
| '''[[16/13]], [[13/8]]'''
| style="text-align:center;" | 15.641
| '''15.528'''
|-
|-
| style="text-align:center;" | [[7/5|7/5]], [[10/7|10/7]]
| [[6/5]], [[5/3]]
| style="text-align:center;" | 17.488
| 15.641
|-
|-
| style="text-align:center;" | [[14/13|14/13]], [[13/7|13/7]]
| [[7/5]], [[10/7]]
| style="text-align:center;" | 21.702
| 17.488
|-
|-
| style="text-align:center;" | [[15/13|15/13]], [[26/15|26/15]]
| [[9/8]], [[16/9]]
| style="text-align:center;" | 22.741
| 21.090
|-
|-
| style="text-align:center;" | '''[[11/8|11/8]], [[16/11|16/11]]'''
| [[14/13]], [[13/7]]
| style="text-align:center;" | '''26.318'''
| 21.702
|-
|-
| style="text-align:center;" | '''[[4/3|4/3]], [[3/2|3/2]]'''
| [[15/13]], [[26/15]]
| style="text-align:center;" | '''26.955'''
| 22.741
|-
|-
| style="text-align:center;" | [[11/9|11/9]], [[18/11|18/11]]
| '''[[11/8]], [[16/11]]'''
| style="text-align:center;" | 27.592
| '''26.318'''
|-
|-
| style="text-align:center;" | [[14/11|14/11]], [[11/7|11/7]]
| '''[[4/3]], [[3/2]]'''
| style="text-align:center;" | 32.492
| '''26.955'''
|-
|-
| style="text-align:center;" | [[7/6|7/6]], [[12/7|12/7]]
| [[11/9]], [[18/11]]
| style="text-align:center;" | 33.129
| 27.592
|-
|-
| style="text-align:center;" | [[16/15|16/15]], [[15/8|15/8]]
| [[15/14]], [[28/15]]
| style="text-align:center;" | 38.269
| 30.557
|-
|-
| style="text-align:center;" | [[18/13|18/13]], [[13/9|13/9]]
| [[10/9]], [[9/5]]
| style="text-align:center;" | 38.382
| 32.404
|-
|-
| style="text-align:center;" | [[10/9|10/9]], [[9/5|9/5]]
| [[14/11]], [[11/7]]
| style="text-align:center;" | 42.596
| 32.492
|-
|-
| style="text-align:center;" | [[15/14|15/14]], [[28/15|28/15]]
| [[7/6]], [[12/7]]
| style="text-align:center;" | 44.443
| 33.129
|-
|-
| style="text-align:center;" | [[9/8|9/8]], [[16/9|16/9]]
| [[18/13]], [[13/9]]
| style="text-align:center;" | 53.910
| 36.618
|-
|-
| style="text-align:center;" | [[9/7|9/7]], [[14/9|14/9]]
| [[16/15]], [[15/8]]
| style="text-align:center;" | 60.084
| 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.
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.
Line 557: Line 476:
[[:File:16ed2-001.svg|16ed2-001.svg]]
[[:File:16ed2-001.svg|16ed2-001.svg]]


==Hexadecaphonic Octave Theory==
== Hexadecaphonic Octave Theory ==


The scale supports the diminished temperament with its 1/4 octave period, though its generator size, equal to its step size of 75 cents, is smaller than ideal. Its very flat 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.
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.


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 [[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).


16-EDO 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'''".


[[Easley Blackwood Jr]] writes of 16-EDO:
[[Easley Blackwood Jr]] writes of 16edo:


"16 notes: This tuning is best thought of as a combination of four intertwined diminished seventh chords. Since 12-note tuning can be regarded as a combination of three diminished seventh chords, it is plain that the two tunings have elements in common. The most obvious difference in the way the two tunings sound and work is that triads in 16-note tuning, although recognizable, are too discordant to serve as the final harmony in cadences. Keys can still be established by successions of altered subdominant and dominant harmonies, however, and the Etude is based mainly upon this property. The fundamental consonant harmony employed is a minor triad with an added minor seventh."
"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."
Line 573: Line 492:
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 cents, which is 3.7 cents away from 16edo's "narrow fifth". Another voicing for this chord is 14:16:19, which features 19:14 as the outer interval (528.7 cents just, 525.0 cents in 16edo). A perhaps more consonant open voicing is 7:16:19.


==Hexadecaphonic Notation==
== Hexadecaphonic Notation ==


16-EDO notation can be easy utilizing Goldsmith's Circle of keys, nominals, and respective notation. The nominals for a 6 line staff can be switched for Wilson's Beta and Epsilon additions to A-G. The Armodue model uses a 4-line staff for 16-EDO.
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.


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]]".
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]]".


{| class="wikitable"
{| class="wikitable center-all"
|-
|-
! | Degree
! Degree
! | Cents
! Cents
! colspan="2" | Mavila[9] Notation
! colspan="2" | Mavila[9] Notation
|-
|-
| style="text-align:center;" | 0
| 0
| style="text-align:center;" | 0
| 0
| style="text-align:center;" | unison
| unison
| style="text-align:center;" | 1
| 1
|-
|-
| style="text-align:center;" | 1
| 1
| style="text-align:center;" | 75
| 75
| style="text-align:center;" | aug unison, minor 2nd
| aug unison, minor 2nd
| style="text-align:center;" | 1#, 2b
| 1#, 2b
|-
|-
| style="text-align:center;" | 2
| 2
| style="text-align:center;" | 150
| 150
| style="text-align:center;" | major 2nd
| major 2nd
| style="text-align:center;" | 2
| 2
|-
|-
| style="text-align:center;" | 3
| 3
| style="text-align:center;" | 225
| 225
| style="text-align:center;" | aug 2nd, minor 3rd
| aug 2nd, minor 3rd
| style="text-align:center;" | 2#, 3b
| 2#, 3b
|-
|-
| style="text-align:center;" | 4
| 4
| style="text-align:center;" | 300
| 300
| style="text-align:center;" | major 3rd, dim 4th
| major 3rd, dim 4th
| style="text-align:center;" | 3, 4bb
| 3, 4bb
|-
|-
| style="text-align:center;" | 5
| 5
| style="text-align:center;" | 375
| 375
| style="text-align:center;" | minor 4th
| minor 4th
| style="text-align:center;" | 4b
| 4b
|-
|-
| style="text-align:center;" | 6
| 6
| style="text-align:center;" | 450
| 450
| style="text-align:center;" | major 4th,
| major 4th,<br>dim 5th
 
| 4, 5b
dim 5th
| style="text-align:center;" | 4, 5b
|-
|-
| style="text-align:center;" | 7
| 7
| style="text-align:center;" | 525
| 525
| style="text-align:center;" | aug 4th, minor 5th
| aug 4th, minor 5th
| style="text-align:center;" | 4#, 5
| 4#, 5
|-
|-
| style="text-align:center;" | 8
| 8
| style="text-align:center;" | 600
| 600
| style="text-align:center;" | aug 5th, dim 6th
| aug 5th, dim 6th
| style="text-align:center;" | 5#, 6b
| 5#, 6b
|-
|-
| style="text-align:center;" | 9
| 9
| style="text-align:center;" | 675
| 675
| style="text-align:center;" | perfect 6th, dim 7th
| perfect 6th, dim 7th
| style="text-align:center;" | 6, 7bb
| 6, 7bb
|-
|-
| style="text-align:center;" | 10
| 10
| style="text-align:center;" | 750
| 750
| style="text-align:center;" | aug 6th, minor 7th
| aug 6th, minor 7th
| style="text-align:center;" | 6#, 7b
| 6#, 7b
|-
|-
| style="text-align:center;" | 11
| 11
| style="text-align:center;" | 825
| 825
| style="text-align:center;" | major 7th
| major 7th
| style="text-align:center;" | 7
| 7
|-
|-
| style="text-align:center;" | 12
| 12
| style="text-align:center;" | 900
| 900
| style="text-align:center;" | aug 7th, minor 8th
| aug 7th, minor 8th
| style="text-align:center;" | 7#, 8b
| 7#, 8b
|-
|-
| style="text-align:center;" | 13
| 13
| style="text-align:center;" | 975
| 975
| style="text-align:center;" | major 8th, dim 9th
| major 8th, dim 9th
| style="text-align:center;" | 8, 9bb
| 8, 9bb
|-
|-
| style="text-align:center;" | 14
| 14
| style="text-align:center;" | 1050
| 1050
| style="text-align:center;" | minor 9th
| minor 9th
| style="text-align:center;" | 9
| 9
|-
|-
| style="text-align:center;" | 15
| 15
| style="text-align:center;" | 1125
| 1125
| style="text-align:center;" | major 9th, dim 10ve
| major 9th, dim 10ve
| style="text-align:center;" | 9#, 1b
| 9#, 1b
|-
|-
| style="text-align:center;" | 16
| 16
| style="text-align:center;" | 1200
| 1200
| style="text-align:center;" | 10ve (Decave)
| 10ve (Decave)
| style="text-align:center;" | 1
| 1
|}
|}


==16 Tone Piano Layout Based on the Mavila[7]/"Anti-diatonic" Scale==
== 16-tone piano layout based on the mavila[7]/antidiatonic scale ==
[[File:16-EDO-PIano-Diagram.png|alt=16-EDO-PIano-Diagram.png|748x293px|16-EDO-PIano-Diagram.png]]
[[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. 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".
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]]


Important MOSes include:
== Rank-2 temperaments ==
* [[List of 16et rank two temperaments by badness]]


Important mosses include:
* [[magic]] anti-diatonic 3L4s 1414141 (5\16, 1\1)
* [[magic]] anti-diatonic 3L4s 1414141 (5\16, 1\1)
* [[magic]] superdiatonic 3L7s 1311311311 (5\16, 1\1)
* [[magic]] superdiatonic 3L7s 1311311311 (5\16, 1\1)
Line 690: Line 606:
* [[gorgo]] 5L1s 333331 (3\16, 1\1)
* [[gorgo]] 5L1s 333331 (3\16, 1\1)
* [[lemba]] 4L2s 332332 (3\16, 1\2)
* [[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 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 [[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 [[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)
* 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:
Temperaments listed by generator size:


{| class="wikitable"
{| class="wikitable center-1 center-2"
|-
|-
! | Periods
! Periods<br>per octave
 
! 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|astrology]]
|-
|-
| | 4
| 4
| | 1\16
| 1\16
| style="text-align:center;" | [[Diminished|Diminished]]/demolished
| [[Diminished]]/demolished
|-
|-
| | 8
| 8
| | 1\16
| 1\16
| style="text-align:center;" |  
|  
|}
|}


Line 742: 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 760: 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 793: 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 ==


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.
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.


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
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
Line 803: Line 717:
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, 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".


===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¢. 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.
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|Metallic Harmony]].
See [[Metallic Harmony]].


==Commas==
== Commas ==
16 EDO [[tempers out]] the following [[comma]]s. (Note: This assumes [[val]] {{val| 16 25 37 45 55 59 }}.)
16et [[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"
Line 820: Line 734:
! [[Cent]]s
! [[Cent]]s
! [[Color name]]
! [[Color name]]
! Name(s)
! Name
|-
|-
| 5
| 5
Line 827: Line 741:
| 92.18
| 92.18
| Layobi
| Layobi
| Major Chroma, Major Limma, Pelogic Comma
| Major chroma
|-
|-
| 5
| 5
Line 834: Line 748:
| 62.57
| 62.57
| Quadgu
| Quadgu
| Major Diesis, Diminished Comma
| Major diesis
|-
|-
| 5
| 5
Line 841: Line 755:
| 29.61
| 29.61
| Laquinyo
| Laquinyo
| Small Diesis, Magic Comma
| Magic comma
|-
|-
| 5
| 5
Line 848: Line 762:
| 3.34
| 3.34
| Sasepbiru
| Sasepbiru
| [[Vishnuzma]], Semisuper
| [[Vishnuzma]]
|-
|-
| 7
| 7
Line 855: Line 769:
| 48.77
| 48.77
| Rugu
| Rugu
| Septimal Quarter Tone
| Septimal quartertone
|-
|-
| 7
| 7
Line 862: Line 776:
| 43.41
| 43.41
| Lazoyoyo
| Lazoyoyo
| Avicennma, Avicenna's Enharmonic Diesis
| Avicennma
|-
|-
| 7
| 7
Line 869: Line 783:
| 34.98
| 34.98
| Biruyo
| Biruyo
| Tritonic Diesis, Jubilisma
| Jubilisma
|-
|-
| 7
| 7
Line 890: Line 804:
| 13.79
| 13.79
| Zotrigu
| Zotrigu
| Septimal Semicomma, Starling Comma
| Starling comma
|-
|-
| 7
| 7
Line 904: Line 818:
| 5.36
| 5.36
| Sarurutrigu
| Sarurutrigu
| Porwell
| Porwell comma
|-
|-
| 11
| 11
Line 943: Line 857:
<references/>
<references/>


==Armodue Theory (4-line staff)==
== Armodue theory (4-line staff) ==
[http://www.armodue.com/ricerche.htm Armodue]: Pierpaolo Beretta's website for his "Armodue" theory for 16edo (esadekaphonic), including compositions.
[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 translations of parts of the Armodue pages see the [[Armodue]] on this wiki.


==Images==
== Diagrams ==
[[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]]


==Books/Literature==
== Books/literature ==
 
* Sword, Ronald. "Thesaurus of Melodic Patterns and Intervals for 16-Tones" IAAA Press, USA. First Ed: August, 2011
* Sword, Ronald. "Thesaurus of Melodic Patterns and Intervals for 16-Tones" IAAA Press, USA. First Ed: August, 2011
* Sword, Ronald. "Hexadecaphonic Scales for Guitar." IAAA Press, UK-USA. First Ed: Feb, 2010. (superfourth tuning)
* Sword, Ronald. "Hexadecaphonic Scales for Guitar." IAAA Press, UK-USA. First Ed: Feb, 2010. (superfourth tuning)
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