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== Intervals ==
== Intervals ==
16edo can be notated with conventional notation, including the staff, note names, relative notation, etc. in two ways. The first preserves the <u>melodic</u> meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works - this shouldn't be surprising as conventional interval arithmetic is designed for meantone/(super)pythagorean systems and 16edo is neither - 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".)
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—this shouldn't be surprising as conventional interval arithmetic is designed for meantone/(super)pythagorean systems and 16edo is neither—e.g. {{nowrap|M2 + M2}} isn't M3, and {{nowrap|D + M2}} isn't E. Chord names are different because {{dash|C, E, G|med}} is not {{dash|P1, M3, P5|med}}. (But see below in "Chord Names".)


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. This approach may seem bizarre at first.  However, it carries over the way interval arithmetic and chord names work from diatonic notation. 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. This approach may seem bizarre at first.  However, it carries over the way interval arithmetic and chord names work from diatonic notation. Furthermore, conventional 12edo music can be directly translated to 16edo "on the fly".
Line 49: Line 49:
| 28/27, 27/26
| 28/27, 27/26
| aug 1, dim 2nd
| aug 1, dim 2nd
| D#, Eb
| D♯, E♭
| dim 1, aug 2nd
| dim 1, aug 2nd
| Db, E#
| D♭, E♯
| subminor 2nd
| subminor 2nd
| min 2nd
| min 2nd
Line 69: Line 69:
| 8/7
| 8/7
| major 2nd
| major 2nd
| E#
| E♯
| minor 2nd
| minor 2nd
| Eb
| E♭
| supermajor 2nd,<br>septimal whole-tone
| supermajor 2nd,<br>septimal whole-tone
| perf 2nd
| perf 2nd
Line 79: Line 79:
| 19/16, 32/27, 6/5
| 19/16, 32/27, 6/5
| minor 3rd
| minor 3rd
| Fb
| F♭
| major 3rd
| major 3rd
| F#
| F♯
| minor 3rd
| minor 3rd
| min 3rd
| min 3rd
Line 99: Line 99:
| 13/10, 35/27
| 13/10, 35/27
| aug 3rd,<br>dim 4th
| aug 3rd,<br>dim 4th
| F#, Gb
| F♯, G♭
| dim 3rd,<br>aug 4th
| dim 3rd,<br>aug 4th
| Fb, G#
| F♭, G♯
| sub-4th,<br>supermajor 3rd
| sub-4th,<br>supermajor 3rd
| min 4th
| min 4th
Line 119: Line 119:
| 7/5, 10/7
| 7/5, 10/7
| aug 4th,<br>dim 5th
| aug 4th,<br>dim 5th
| G#, Ab
| G♯, A♭
| dim 4th,<br>aug 5th
| dim 4th,<br>aug 5th
| Gb, A#
| G♭, A♯
| tritone
| tritone
| aug 4th,<br>dim 5th
| aug 4th,<br>dim 5th
Line 139: Line 139:
| 20/13, 54/35
| 20/13, 54/35
| aug 5th,<br>dim 6th
| aug 5th,<br>dim 6th
| A#, Bb
| A♯, B♭
| dim 5th,<br>aug 6th
| dim 5th,<br>aug 6th
| Ab, B#
| A♭, B♯
| super-5th,<br>subminor 6th
| super-5th,<br>subminor 6th
| maj 5th
| maj 5th
Line 159: Line 159:
| 5/3, 27/16, 32/19
| 5/3, 27/16, 32/19
| major 6th
| major 6th
| B#
| B♯
| minor 6th
| minor 6th
| Bb
| B♭
| major 6th
| major 6th
| maj 6th
| maj 6th
Line 169: Line 169:
| 7/4
| 7/4
| minor 7th
| minor 7th
| Cb
| C♭
| major 7th
| major 7th
| C#
| C♯
| subminor 7th,<br>septimal minor 7th
| subminor 7th,<br>septimal minor 7th
| perf 7th
| perf 7th
Line 189: Line 189:
| 27/14, 52/27
| 27/14, 52/27
| aug 7th,<br>dim 8ve
| aug 7th,<br>dim 8ve
| C#, Db
| C♯, D♭
| dim 7th,<br>aug 8ve
| dim 7th,<br>aug 8ve
| Cb, D#
| C♭, D♯
| supermajor 7th
| supermajor 7th
| maj 7th
| maj 7th
Line 205: Line 205:
| octave
| octave
|}
|}
<nowiki>*</nowiki> based on treating 16edo as a 2.5.7.13.19.27.45 subgroup temperament; other approaches are possible.
<nowiki />* Based on treating 16edo as a 2.5.7.13.19.27.45 subgroup temperament; other approaches are possible.


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


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 the octave ([[2/1]]) as the "[[decave]]".
Mos scales like mavila[7] (or "inverse/anti-diatonic" which reverses step sizes of diatonic from LLsLLLs to ssLsssL in the heptatonic variation) can work as an alternative to the traditional diatonic scale, while maintaining conventional A-G ♯/♭ notation as described above. Alternatively, one can utilize the Mavila[9] MOS, for a sort of "hyper-diatonic" scale of 7 large steps and 2 small steps. [[Armodue_theory|Armodue notation]] of 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 the octave ([[2/1]]) as the "[[decave]]".


{| class="wikitable center-all"
{| class="wikitable center-all"
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| 75
| 75
| aug unison, minor 2nd
| aug unison, minor 2nd
| 1#, 2b
| 1♯, 2♭
|-
|-
| 2
| 2
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| 225
| 225
| aug 2nd, minor 3rd
| aug 2nd, minor 3rd
| 2#, 3b
| 2♯, 3♭
|-
|-
| 4
| 4
| 300
| 300
| major 3rd, dim 4th
| major 3rd, dim 4th
| 3, 4bb
| 3, 4𝄫
|-
|-
| 5
| 5
| 375
| 375
| minor 4th
| minor 4th
| 4b
| 4♭
|-
|-
| 6
| 6
| 450
| 450
| major 4th,<br>dim 5th
| major 4th,<br>dim 5th
| 4, 5b
| 4, 5♭
|-
|-
| 7
| 7
| 525
| 525
| aug 4th, minor 5th
| aug 4th, minor 5th
| 4#, 5
| 4♯, 5
|-
|-
| 8
| 8
| 600
| 600
| aug 5th, dim 6th
| aug 5th, dim 6th
| 5#, 6b
| 5♯, 6♭
|-
|-
| 9
| 9
| 675
| 675
| perfect 6th, dim 7th
| perfect 6th, dim 7th
| 6, 7bb
| 6, 7𝄫
|-
|-
| 10
| 10
| 750
| 750
| aug 6th, minor 7th
| aug 6th, minor 7th
| 6#, 7b
| 6♯, 7♭
|-
|-
| 11
| 11
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| 900
| 900
| aug 7th, minor 8th
| aug 7th, minor 8th
| 7#, 8b
| 7♯, 8♭
|-
|-
| 13
| 13
| 975
| 975
| major 8th, dim 9th
| major 8th, dim 9th
| 8, 9bb
| 8, 9𝄫
|-
|-
| 14
| 14
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| 1125
| 1125
| major 9th, dim 10ve
| major 9th, dim 10ve
| 9#, 1b
| 9♯, 1♭
|-
|-
| 16
| 16
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|}
|}


===Sagittal notation===
=== Sagittal notation ===
This notation uses the same sagittal sequence as [[21edo#Sagittal notation|21-EDO]].
This notation uses the same sagittal sequence as [[21edo#Sagittal notation|21-EDO]].


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== Chord names ==
== Chord names ==
16edo chords can be named using ups and downs. Using harmonic (circle-of-fifths) interval names, the names are easy to find, but they bear little relationship to the sound: a minor chord (spelled A-C-E) sounds like [[4:5:6]], the classical major triad, and a major chord (spelled C-E-G) sounds like [[10:12:15]], a classical minor triad! Instead, using melodic names, the chord names will match the sound&mdash;but finding the name from the spelling is more complicated (see below).
16edo chords can be named using ups and downs. Using harmonic (circle-of-fifths) interval names, the names are easy to find, but they 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 melodic names, the chord names will match the sound&mdash;but finding the name from the spelling is more complicated (see below).


{| class="wikitable center-all"
{| class="wikitable center-all"
|-
|-
! | chord
! rowspan="2" | Chord
! | JI ratios
! rowspan="2" | JI ratios
! colspan="3" | harmonic name
! colspan="6" | Name
! colspan="3" | melodic name
|-
|-
| 0-5-9
! colspan="3" | Harmonic
! colspan="3" | Melodic
|-
| {{dash|0, 5, 9|med}}
| 4:5:6
| 4:5:6
| D F A
| D F A
Line 340: Line 342:
| D major
| D major
|-
|-
| 0-4-9
| {{dash|0, 4, 9|med}}
| 10:12:15
| 10:12:15
| D F# A
| D F♯ A
| D
| D
| D major
| D major
| D Fb A
| D F♭ A
| Dm
| Dm
| D minor
| D minor
|-
|-
| 0-4-8
| {{dash|0, 4, 8|med}}
| 5:6:7
| 5:6:7
| D F# A#
| D F♯ A♯
| Daug
| Daug
| D augmented
| D augmented
| D Fb Ab
| D F♭ A♭
| Ddim
| Ddim
| D diminished
| D diminished
|-
|-
| 0-5-10
| {{dash|0, 5, 10|med}}
|  
|  
| D F Ab
| D F A♭
| Ddim
| Ddim
| D diminished
| D diminished
| D F A#
| D F A♯
| Daug
| Daug
| D augmented
| D augmented
|-
|-
| 0-5-9-13
| {{dash|0, 5, 9, 13|med}}
| 4:5:6:7
| 4:5:6:7
| D F A C#
| D F A C♯
| Dm(M7)
| Dm(M7)
| D minor-major
| D minor-major
| D F A Cb
| D F A C♭
| D7
| D7
| D seven
| D seven
|-
|-
| 0-5-9-12
| {{dash|0, 5, 9, 12|med}}
|  
|  
| D F A Bb
| D F A Bb
| Dm(b6)
| Dm(♭6)
| D minor flat-six
| D minor flat-six
| D F A B#
| D F A B♯
| D6
| D6
| D six
| D six
|-
|-
| 0-5-9-14
| {{dash|0, 5, 9, 14|med}}
|  
|  
| D F A C
| D F A C
Line 394: Line 396:
| D major seven
| D major seven
|-
|-
| 0-4-9-13
| {{dash|0, 4, 9, 13|med}}
|  
|  
| D F# A C#
| D F♯ A C♯
| DM7
| DM7
| D major seven
| D major seven
| D Fb A Cb
| D F♭ A C♭
| DM7
| DM7
| D minor seven
| D minor seven
|}
|}
Alterations are always enclosed in parentheses, additions never are. An up or down immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). See [[Ups and Downs Notation #Chords and Chord Progressions]] for more examples.
 
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 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:


{| class="wikitable" style="text-align:center;"
{| class="wikitable" style="text-align: center;"
!initial question
|-
!reverse everything
! Initial question
!do the math
! Reverse everything
!reverse again
! Do the math
! Reverse again
|-
|-
|M2 + M2
| M2 + M2
|m2 + m2
| m2 + m2
|dim3
| dim3
|aug3
| aug3
|-
|-
|D to F#
| D to F♯
|D to Fb
| D to F♭
|dim3
| dim3
|aug3
| aug3
|-
|-
|D to F
| D to F
|D to F
| D to F
|m3
| m3
|M3
| M3
|-
|-
|Eb + m3
| E♭ + m3
|E# + M3
| E♯ + M3
|G##
| G♯♯
|Gbb
| Gbb
|-
|-
|Eb + P5
| E♭ + P5
|E# + P5
| E♯ + P5
|B#
| B♯
|Bb
| B♭
|-
|-
|A minor chord
| A minor chord
|A major chord
| A major chord
|A C# E
| A C♯ E
|A Cb E
| A C♭ E
|-
|-
|Eb major chord
| E♭ major chord
|E# minor chord
| E♯ minor chord
|E# G# B#
| E♯ G♯ B♯
|Eb Gb Db
| E♭ G♭ D♭
|-
|-
|Gm7 = G + m3 + P5 + m7
| Gm7 = G + m3 + P5 + m7
|G + M3 + P5 + M7
| G + M3 + P5 + M7
|G B D F#
| G B D F♯
|G B D Fb
| G B D F♭
|-
|-
|Ab7aug = Ab + M3 + A5 + m7
| A♭7aug = A♭ + M3 + A5 + m7
|A# + m3 + d5 + M7
| A♯ + m3 + d5 + M7
|A# C# E G##
| A♯ C♯ E G♯♯
|Ab Cb E Gbb
| A♭ C♭ E Gbb
|-
|-
|what chord is D F A#?
| what chord is D F A♯?
|D F Ab
| D F A♭
|D + m3 + d5
| D + m3 + d5
|D + M3 + A5 = Daug
| D + M3 + A5 = Daug
|-
|-
|what chord is C E Gb Bb?
| what chord is C E G♭ B♭?
|C E G# B#
| C E G♯ B♯
|C + M3 + A5 + A7
| C + M3 + A5 + A7
|C + m3 + d5 + d7 = Cdim7
| C + m3 + d5 + d7 = Cdim7
|-
|-
|C major scale = C + M2 + M3
| C major scale = C + M2 + M3<br>+ P4 + P5 + M6 + M7 + P8
+ P4 + P5 + M6 + M7 + P8
| C + m2 + m3 + P4<br>+ P5 + m6 + m7 + P8
|C + m2 + m3 + P4
| C D♭ E♭ F<br>G A♭ B♭ C
+ P5 + m6 + m7 + P8
| C D♯ E♯ F<br>G A♯ B♯ C
|C Db Eb F
G Ab Bb C
|C D# E# F
G A# B# C
|-
|-
|C minor scale = C + M2 + m3
| C minor scale = C + M2 + m3<br>+ P4 + P5 + m6 + m7 + P8
+ P4 + P5 + m6 + m7 + P8
| C + m2 + M3 + P4<br>+ P5 + M6 + M7 + P8
|C + m2 + M3 + P4
| C D♭ E F<br>G A B C
+ P5 + M6 + M7 + P8
| C D♯ E F<br>G A B C
|C Db E F
G A B C
|C D# E F
G A B C
|-
|-
|what scale is A B# Cb D
| what scale is A B♯ C♭ D<br>E F G♭ A?
E F Gb A?
| A B♭ C♯ D<br>E F G♯ A
|A Bb C# D
| A + m2 + M3 + P4<br>+ P5 + m6 + M7
E F G# A
| A + M2 + m3 + P4<br>+ P5 + M6 + m7 = A dorian
|A + m2 + M3 + P4
+ P5 + m6 + M7
|A + M2 + m3 + P4
+ P5 + M6 + m7 = A dorian
|}
|}


Line 507: Line 499:


== 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 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.


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