16edo: Difference between revisions
remove table "Differences between distributionally-even scales and smaller edos" |
clean up table, add new harmonic template, remove 7mus |
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== Theory == | == Theory == | ||
{{ | {{primes 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. | '''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. | ||
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! Degree | ! Degree | ||
! [[Cent]]s | ! [[Cent]]s | ||
! Approximate <br> Ratios* | ! Approximate <br> Ratios* | ||
! colspan="2" | Melodic names, <br> major wider than minor | ! colspan="2" | Melodic names, <br> major wider than minor | ||
| Line 41: | Line 40: | ||
! Interval <br> Names <br> Simplified | ! Interval <br> Names <br> Simplified | ||
|- | |- | ||
| style="text-align:center;" | 0 | | style="text-align:center;" | 0 | ||
| style="text-align:center;" | 0 | | style="text-align:center;" | 0 | ||
| Line 49: | Line 47: | ||
| style="text-align:center;" | unison | | style="text-align:center;" | unison | ||
| style="text-align:center;" | D | | style="text-align:center;" | D | ||
| style="text-align:center;" | | | style="text-align:center;" | unison | ||
| style="text-align:center;" | | | style="text-align:center;" | unison | ||
|- | |- | ||
| style="text-align:center;" | 1 | | style="text-align:center;" | 1 | ||
| style="text-align:center;" | 75 | | style="text-align:center;" | 75 | ||
| style="text-align:center;" | 28/27, 27/26 | | style="text-align:center;" | 28/27, 27/26 | ||
| style="text-align:center;" | aug 1, dim 2nd | | style="text-align:center;" | aug 1, dim 2nd | ||
| Line 60: | Line 57: | ||
| style="text-align:center;" | dim 1, aug 2nd | | style="text-align:center;" | dim 1, aug 2nd | ||
| style="text-align:center;" | Db, E# | | style="text-align:center;" | Db, E# | ||
| style="text-align:center;" | | | style="text-align:center;" | subminor 2nd | ||
| style="text-align:center;" | | | style="text-align:center;" | min 2nd | ||
|- | |- | ||
| style="text-align:center;" | 2 | | style="text-align:center;" | 2 | ||
| style="text-align:center;" | 150 | | style="text-align:center;" | 150 | ||
| style="text-align:center;" | 35/32 | | style="text-align:center;" | 35/32 | ||
| style="text-align:center;" | minor 2nd | | style="text-align:center;" | minor 2nd | ||
| Line 71: | Line 67: | ||
| style="text-align:center;" | major 2nd | | style="text-align:center;" | major 2nd | ||
| style="text-align:center;" | E | | style="text-align:center;" | E | ||
| style="text-align:center;" | | | style="text-align:center;" | neutral 2nd | ||
| style="text-align:center;" | | | style="text-align:center;" | maj 2nd | ||
|- | |- | ||
| style="text-align:center;" | 3 | | style="text-align:center;" | 3 | ||
| style="text-align:center;" | 225 | | style="text-align:center;" | 225 | ||
| style="text-align:center;" | 8/7 | | style="text-align:center;" | 8/7 | ||
| style="text-align:center;" | major 2nd | | style="text-align:center;" | major 2nd | ||
| Line 82: | Line 77: | ||
| style="text-align:center;" | minor 2nd | | style="text-align:center;" | minor 2nd | ||
| style="text-align:center;" | Eb | | style="text-align:center;" | Eb | ||
| style="text-align:center;" | | | style="text-align:center;" | supermajor 2nd,<br>septimal whole-tone | ||
| style="text-align:center;" | perf 2nd | |||
| style="text-align:center;" | | |||
|- | |- | ||
| style="text-align:center;" | 4 | | style="text-align:center;" | 4 | ||
| style="text-align:center;" | 300 | | style="text-align:center;" | 300 | ||
| style="text-align:center;" | 19/16, 32/27 | | style="text-align:center;" | 19/16, 32/27 | ||
| style="text-align:center;" | minor 3rd | | style="text-align:center;" | minor 3rd | ||
| Line 95: | Line 87: | ||
| style="text-align:center;" | major 3rd | | style="text-align:center;" | major 3rd | ||
| style="text-align:center;" | F# | | style="text-align:center;" | F# | ||
| style="text-align:center;" | | | style="text-align:center;" | minor 3rd | ||
| style="text-align:center;" | | | style="text-align:center;" | min 3rd | ||
|- | |- | ||
| style="text-align:center;" | 5 | | style="text-align:center;" | 5 | ||
| style="text-align:center;" | 375 | | style="text-align:center;" | 375 | ||
| style="text-align:center;" | 5/4, 16/13, 26/21 | | style="text-align:center;" | 5/4, 16/13, 26/21 | ||
| style="text-align:center;" | major 3rd | | style="text-align:center;" | major 3rd | ||
| Line 106: | Line 97: | ||
| style="text-align:center;" | minor 3rd | | style="text-align:center;" | minor 3rd | ||
| style="text-align:center;" | F | | style="text-align:center;" | F | ||
| style="text-align:center;" | | | style="text-align:center;" | major 3rd | ||
| style="text-align:center;" | | | style="text-align:center;" | maj 3rd | ||
|- | |- | ||
| style="text-align:center;" | 6 | | style="text-align:center;" | 6 | ||
| style="text-align:center;" | 450 | | style="text-align:center;" | 450 | ||
| style="text-align:center;" | 13/10, 35/27 | | style="text-align:center;" | 13/10, 35/27 | ||
| style="text-align:center;" | aug 3rd | | style="text-align:center;" | aug 3rd,<br>dim 4th | ||
dim 4th | |||
| style="text-align:center;" | F#, Gb | | style="text-align:center;" | F#, Gb | ||
| style="text-align:center;" | dim 3rd, | | style="text-align:center;" | dim 3rd,<br>aug 4th | ||
aug 4th | |||
| style="text-align:center;" | Fb, G# | | style="text-align:center;" | Fb, G# | ||
| style="text-align:center;" | | | style="text-align:center;" | sub-4th,<br>supermajor 3rd | ||
| style="text-align:center;" | min 4th | |||
| style="text-align:center;" | | |||
|- | |- | ||
| style="text-align:center;" | 7 | | style="text-align:center;" | 7 | ||
| style="text-align:center;" | 525 | | style="text-align:center;" | 525 | ||
| style="text-align:center;" | 19/14, 27/20, 52/35, 256/189 | | style="text-align:center;" | 19/14, 27/20, 52/35, 256/189 | ||
| style="text-align:center;" | perfect 4th | | style="text-align:center;" | perfect 4th | ||
| Line 134: | Line 117: | ||
| style="text-align:center;" | perfect 4th | | style="text-align:center;" | perfect 4th | ||
| style="text-align:center;" | G | | style="text-align:center;" | G | ||
| style="text-align:center;" | | | style="text-align:center;" | wide 4th | ||
| style="text-align:center;" | | | style="text-align:center;" | maj 4th | ||
|- | |- | ||
| style="text-align:center;" | 8 | | style="text-align:center;" | 8 | ||
| style="text-align:center;" | 600 | | style="text-align:center;" | 600 | ||
| style="text-align:center;" | 7/5, 10/7 | | style="text-align:center;" | 7/5, 10/7 | ||
| style="text-align:center;" | aug 4th | | style="text-align:center;" | aug 4th,<br>dim 5th | ||
dim 5th | |||
| style="text-align:center;" | G#, Ab | | style="text-align:center;" | G#, Ab | ||
| style="text-align:center;" | dim 4th, | | style="text-align:center;" | dim 4th,<br>aug 5th | ||
aug 5th | |||
| style="text-align:center;" | Gb, A# | | style="text-align:center;" | Gb, A# | ||
| style="text-align:center;" | | | style="text-align:center;" | tritone | ||
| style="text-align:center;" | | | style="text-align:center;" | aug 4th,<br>dim 5th | ||
|- | |- | ||
| style="text-align:center;" | 9 | | style="text-align:center;" | 9 | ||
| style="text-align:center;" | 675 | | style="text-align:center;" | 675 | ||
| style="text-align:center;" | 28/19, 40/27, 35/26, 189/128 | | style="text-align:center;" | 28/19, 40/27, 35/26, 189/128 | ||
| style="text-align:center;" | perfect 5th | | style="text-align:center;" | perfect 5th | ||
| Line 162: | Line 137: | ||
| style="text-align:center;" | perfect 5th | | style="text-align:center;" | perfect 5th | ||
| style="text-align:center;" | A | | style="text-align:center;" | A | ||
| style="text-align:center;" | | | style="text-align:center;" | narrow 5th | ||
| style="text-align:center;" | | | style="text-align:center;" | min 5th | ||
|- | |- | ||
| style="text-align:center;" | 10 | | style="text-align:center;" | 10 | ||
| style="text-align:center;" | 750 | | style="text-align:center;" | 750 | ||
| style="text-align:center;" | 20/13, 54/35 | | style="text-align:center;" | 20/13, 54/35 | ||
| style="text-align:center;" | aug 5th | | style="text-align:center;" | aug 5th,<br>dim 6th | ||
dim 6th | |||
| style="text-align:center;" | A#, Bb | | style="text-align:center;" | A#, Bb | ||
| style="text-align:center;" | dim 5th, aug 6th | | style="text-align:center;" | dim 5th,<br>aug 6th | ||
| style="text-align:center;" | Ab, B# | | style="text-align:center;" | Ab, B# | ||
| style="text-align:center;" | | | style="text-align:center;" | super-5th,<br>subminor 6th | ||
| style="text-align:center;" | maj 5th | |||
| style="text-align:center;" | | |||
|- | |- | ||
| style="text-align:center;" | 11 | | style="text-align:center;" | 11 | ||
| style="text-align:center;" | 825 | | style="text-align:center;" | 825 | ||
| style="text-align:center;" | 8/5, 13/8, 21/13 | | style="text-align:center;" | 8/5, 13/8, 21/13 | ||
| style="text-align:center;" | minor 6th | | style="text-align:center;" | minor 6th | ||
| Line 188: | Line 157: | ||
| style="text-align:center;" | major 6th | | style="text-align:center;" | major 6th | ||
| style="text-align:center;" | B | | style="text-align:center;" | B | ||
| style="text-align:center;" | | | style="text-align:center;" | minor 6th | ||
| style="text-align:center;" | | | style="text-align:center;" | min 6th | ||
|- | |- | ||
| style="text-align:center;" | 12 | | style="text-align:center;" | 12 | ||
| style="text-align:center;" | 900 | | style="text-align:center;" | 900 | ||
| style="text-align:center;" | 27/16, 32/19 | | style="text-align:center;" | 27/16, 32/19 | ||
| style="text-align:center;" | major 6th | | style="text-align:center;" | major 6th | ||
| Line 199: | Line 167: | ||
| style="text-align:center;" | minor 6th | | style="text-align:center;" | minor 6th | ||
| style="text-align:center;" | Bb | | style="text-align:center;" | Bb | ||
| style="text-align:center;" | | | style="text-align:center;" | major 6th | ||
| style="text-align:center;" | | | style="text-align:center;" | maj 6th | ||
|- | |- | ||
| style="text-align:center;" | 13 | | style="text-align:center;" | 13 | ||
| style="text-align:center;" | 975 | | style="text-align:center;" | 975 | ||
| style="text-align:center;" | 7/4 | | style="text-align:center;" | 7/4 | ||
| style="text-align:center;" | minor 7th | | style="text-align:center;" | minor 7th | ||
| Line 210: | Line 177: | ||
| style="text-align:center;" | major 7th | | style="text-align:center;" | major 7th | ||
| style="text-align:center;" | C# | | style="text-align:center;" | C# | ||
| style="text-align:center;" | | | style="text-align:center;" | subminor 7th,<br>septimal minor 7th | ||
| style="text-align:center;" | perf 7th | |||
| style="text-align:center;" | | |||
|- | |- | ||
| style="text-align:center;" | 14 | | style="text-align:center;" | 14 | ||
| style="text-align:center;" | 1050 | | style="text-align:center;" | 1050 | ||
| style="text-align:center;" | 64/35 | | style="text-align:center;" | 64/35 | ||
| style="text-align:center;" | major 7th | | style="text-align:center;" | major 7th | ||
| Line 223: | Line 187: | ||
| style="text-align:center;" | minor 7th | | style="text-align:center;" | minor 7th | ||
| style="text-align:center;" | C | | style="text-align:center;" | C | ||
| style="text-align:center;" | | | style="text-align:center;" | neutral 7th | ||
| style="text-align:center;" | | | style="text-align:center;" | min 7th | ||
|- | |- | ||
| style="text-align:center;" | 15 | | style="text-align:center;" | 15 | ||
| style="text-align:center;" | 1125 | | style="text-align:center;" | 1125 | ||
| style="text-align:center;" | 27/14, 52/27 | | style="text-align:center;" | 27/14, 52/27 | ||
| style="text-align:center;" | aug 7th | | style="text-align:center;" | aug 7th,<br>dim 8ve | ||
dim 8ve | |||
| style="text-align:center;" | C#, Db | | style="text-align:center;" | C#, Db | ||
| style="text-align:center;" | dim 7th, | | style="text-align:center;" | dim 7th,<br>aug 8ve | ||
aug 8ve | |||
| style="text-align:center;" | Cb, D# | | style="text-align:center;" | Cb, D# | ||
| style="text-align:center;" | | | style="text-align:center;" | supermajor 7th | ||
| style="text-align:center;" | | | style="text-align:center;" | maj 7th | ||
|- | |- | ||
| style="text-align:center;" | 16 | | style="text-align:center;" | 16 | ||
| style="text-align:center;" | 1200 | | style="text-align:center;" | 1200 | ||
| style="text-align:center;" | 2/1 | | style="text-align:center;" | 2/1 | ||
| style="text-align:center;" | 8ve | | style="text-align:center;" | 8ve | ||
| Line 249: | Line 207: | ||
| style="text-align:center;" | 8ve | | style="text-align:center;" | 8ve | ||
| style="text-align:center;" | D | | style="text-align:center;" | D | ||
| style="text-align:center;" | | | style="text-align:center;" | octave | ||
| style="text-align:center;" | | | style="text-align:center;" | octave | ||
|} | |} | ||
*based on treating 16-EDO as a 2.5.7.13.19.27 subgroup temperament; other approaches are possible. | *based on treating 16-EDO as a 2.5.7.13.19.27 subgroup temperament; other approaches are possible. | ||