Re: A proposal to simplify the notation of EDOs with bad fifths
Posted: Thu Jun 11, 2020 4:33 am
Here's all the fractions up to 72-edo:Dave Keenan wrote: ↑Tue Dec 03, 2019 12:01 pm I call it "group number" because many EDOs (but not all) have a whole-number value for this, like the element groups on the chemistry periodic table.
Group number EDOs Colour ------------------------------------------------- 0 1 Black I 3, 6 Gold II 8 Gold 2.5 13 Gold 2.8 18 Gold III 23b Gold - fifth = 14\23 - 13\23 is group 17.5 IV 5, 10, 15, 20, 25, 30 Gold 4.86 42 Gold - group = 4 + 6/7 V 37 Gold 5.2 32 Gold 5.33 59 Green - group = 5 + 1/3 - formerly Gold 5.5 27, 54 Green 5.71 49 Green - group = 5 + 5/7 5.8 71 Green VI 22, 44, 66 Green 6.25 61 Green 6.4 39 Blue 6.57 56 Blue - group = 6 + 4/7 VII 17, 34, 51, 68 Blue 7.43 63 Blue - group = 7 + 3/7 7.6 46 Blue VIII 29, 58 Magenta 8.29 70 Magenta - group = 8 + 2/7 8.5 41 Grey 8.8 53 Grey - actual 2:3 fifth is group X where X is the number of apotomes in a 9:16 minor seventh IX 65 Grey X 12, 24, 36, 48, 60, 72 Orange XI 67 Yellow 11.2 55 Yellow 11.5 43 Yellow XII 31, 62 Cyan 12.4 50 Cyan 12.57 69 Cyan - group = 12 + 4/7 XIII 19, 38, 57 Cyan 13.43 64 Purple - group = 13 + 3/7 13.6 45 Purple XIV 26, 52 Purple 14.5 33 Rose 14.8 40 Rose XV 47 Rose XVI 7, 14, 21, 28, 35 Rose XVII 37b Rose - fifth = 21\37 - 22\37 is group V 17.5 23 Rose XVIII 16 Rose XIX 9 Rose XX 11 Rose XXI 17b White - fifth = 9\17 - 10\17 is group VII XXII 2, 4 White XXIII 7bb White - fifth = 3\7 - 4\7 is group XVI - 5\7 (7b) is group 4/7 = 0.57 XXIV 1b White - 0¢ fifthMost of the non-integer groups are in fifths or sevenths, which was kind of to be expected. (Halves, thirds, and quarters also show up occasionally.) The integer groups also often happen to fall on important EDOs—groups I through XX are all the fifths of these EDOs in order: 6, 8, 23b, 5, 37, 22, 17, 29, 65, 12, 67, 31, 19, 26, 47, 7, 37b, 16, 9, and 11.
The function's derivative is actually continuous at 700¢ (it's 0.72 groups per cent on both sides).
(also why is the fact that the exact color boundaries were never specified—even Trojan's was, after some searching—bothering me so much)
59 and 61 also overlap.