User:PiotrGrochowski/Extra-Diatonic Intervals

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Extra-Diatonic Intervals tries to base on 72edo and tries to take into account that superpyth minor seconds can reach 70 cents, although in meantone it's actually an augmented unison. However, superpyth is not suited for a diatonic scale, we can exclude it out of my version and base it on some meantone tuning like 43edo, 74edo or 105edo.

13-limit ratios are based on grosstone and are 69–integer–limit. They are valid to the amount of fifths, and need not necessarily convert to the same interval.

The base intervals come from the circle of fifths from -7 to 7 fifths: diminished octave, diminished fifth, minor second, minor sixth, minor third, minor seventh, fourth, unison, fifth, major second, major sixth, major third, major seventh, augmented fourth and augmented unison. This is unlike Igliashon's Extra-Diatonic Intervals, which only uses -5 to +5 fifths. The reasoning for the expanded range is that a chromatic scale is made of 6 augmented fourths and 6 diminished fifths, as well as 7 minor seconds and 5 augmented unisons. In the other hand, the diminished fourth at -8 fifths is considered a wolf major third in 5-limit, and a septimal major third in 7-limit, so this system names it a high major third instead.

To convert an interval to a name, select the closest 43edo interval, round down if halfway. This project can be used for converting intervals of an entire edo to interval names in this system: https://scratch.mit.edu/projects/250811968/

Like Igliashon's Extra-Diatonic Intervals, this is based on size, not the harmonic function. For example 2 steps in 26edo (92.308 cents) is named augmented unison, although it is not one in its meantone system.

interval 31edo steps 43edo steps 74edo steps 43edo cents fifths 13-limit ratios
unison 0 0 0 0 0 1/1
high unison 1 1 2 27.907 -12 36/35, 45/44, 50/49, 56/55, 64/63
low augmented unison 1 2 3 55.814 +19 27/26, 33/32, 40/39, 49/48, 55/54
augmented unison 2 3 5 83.721 +7 21/20, 22/21, 25/24, 28/27
minor second 3 4 7 111.628 -5 15/14, 16/15, 27/25, 35/33
low neutral second 4 5 9 139.535 -17 12/11, 13/12, 54/49
high neutral second 4 6 10 167.442 +14 11/10, 49/45
major second 5 7 12 195.349 +2 9/8, 10/9, 28/25, 49/44, 55/49
high major second 6 8 14 223.256 -10 8/7
second–third 6 9 15 251.163 +21 15/13, 55/48
low minor third 7 10 17 279.070 +9 7/6, 33/28
minor third 8 11 19 306.977 -3 6/5, 25/21, 32/27
low neutral third 9 12 21 334.884 -15 27/22, 39/32, 40/33, 60/49, 65/54
high neutral third 9 13 22 362.791 +16 11/9, 16/13, 49/40
major third 10 14 24 390.698 +4 5/4, 56/45, 63/50
high major third 11 15 26 418.605 -8 9/7, 14/11, 32/25
third–fourth 12 16 28 446.512 -20 13/10, 64/49
low fourth 12 17 29 474.419 +11 21/16, 35/27, 55/42
fourth 13 18 31 502.326 -1 4/3, 27/20, 66/49
high fourth 14 19 33 530.233 -13 15/11, 65/48
low augmented fourth 14 20 34 558.140 +18 11/8, 18/13, 49/36
augmented fourth 15 21 36 586.047 +6 7/5, 25/18, 45/32
diminished fifth 16 22 38 613.953 -6 10/7, 36/25, 63/44, 64/45
high diminished fifth 17 23 40 641.860 -18 13/9, 16/11
low fifth 17 24 41 669.767 +13 22/15
fifth 18 25 43 697.674 +1 3/2, 40/27, 49/33
high fifth 19 26 45 725.581 -11 32/21, 54/35
fifth–sixth 19 27 46 753.488 +20 20/13, 49/32, 55/36
low minor sixth 20 28 48 781.395 +8 11/7, 14/9, 25/16, 63/40
minor sixth 21 29 50 809.302 -4 8/5, 45/28
low neutral sixth 22 30 52 837.209 -16 13/8, 18/11
high neutral sixth 22 31 53 865.116 +15 33/20, 44/27, 49/30, 64/39
major sixth 23 32 55 893.023 +3 5/3, 27/16, 42/25
high major sixth 24 33 57 920.930 -9 12/7, 56/33
sixth–seventh 25 34 59 948.837 -21 26/15
low minor seventh 25 35 60 976.744 +10 7/4
minor seventh 26 36 62 1004.651 -2 9/5, 16/9, 25/14
low neutral seventh 27 37 64 1032.558 -14 20/11, 65/36
high neutral seventh 27 38 65 1060.465 +17 11/6, 24/13, 49/27
major seventh 28 39 67 1088.372 +5 15/8, 28/15, 50/27, 66/35
diminished octave 29 40 69 1116.279 -7 21/11, 27/14, 40/21, 48/25
high diminished octave 30 41 71 1144.186 -19 39/20, 52/27, 64/33
low octave 30 42 72 1172.093 +12 35/18, 49/25, 55/28, 63/32
octave 31 43 74 1200 0 2/1