User:PiotrGrochowski/Extra-Diatonic Intervals

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