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5- TO 9-TONE, OCTAVE-REPEATING SCALES

FROM WILSON'S GOLDEN HORAGRAMS OF THE SCALE TREE

David J. Finnamore

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Sorted by Convergent Pair & Generator Size · Sorted by Step Number & Order
Matrix Menu of Scales by Step Order & L:s Value

There are sixty-four Golden generators shown numerically on Wilson's Scale Tree. He drew horagrams for the first thirty-two, numbering them from left to right across the 5th level of the tree. Or maybe better said, he numbers them by generator size at level five. The problem with that method is that it locks you in to that level. If you add another level to explore more horagrams, your numbering changes. I've decided to obviate that by numbering them by closeness of the convergent pairs to the root of the tree. That way, as new levels are explored, the numbers append rather than sifting and shifting.

Theoretically, the number of golden horagrams is infinite - it's just a question of how far you want to keep branching the tree. But the further you go out, the smaller the differences get between scales from horagrams generated by adjacent pairs. At level 5, the difference can average as little 6 cents for 5-tone scales. The average distance between the generators of adjacent horagrams is about 12 cents. It may not be worth going beyond that. On the other hand, the left side of the tree has more air between the branches than the right. And over on the right lies a region of generators in the much ballyhooed neighborhood of the fourth. There are only three horagrams on level 5 that fall between full-comma and zero-comma. 

There are sixty-three 5- to 9-tone scales provided by the first 32 golden horagrams. The gold scales are in gold color and bold font style. Interestingly, there is exactly one gold scale for each type, and, using my numbering scheme, it's always in the lowest numbered horagram which provides a given type. Menu of Scales Sorted by Step Number & Order Horagram: 1 · 2 · 3 · 4 · 5 · 6 · 7 · 8 · 9 · 10 · 11 · 12 · 13 · 14 · 15 · 16 17 · 18 · 19 · 20 · 21 · 22 · 23 · 24 · 25 · 26 · 27 · 28 · 29 · 30 · 31 · 32 Steps Order Horagram #s Wilson's #s 5 LLLLs 3, 13, 21, 26 6, 7, 8, 5 5 ssssL 5, 9, 17, 25 3, 2, 1, 4 5 sLsLL 1, 8, 12, 16, 20, 24, 31, 32 22, 19, 18, 23, 17, 24, 21, 20 5 LsLss 4, 6, 10, 15, 18, 23, 27, 28 27, 30, 31, 26, 32, 25, 29, 28 6 LLLLLs 5, 25 3, 4 6 sssssL 9, 17 2, 1 7 LLLLLLs 9 2 7 ssssssL 17 1 7 sLLsLLL 4, 15, 23, 28 27, 26, 25, 28 7 LssLsss 6, 10, 18, 27 30, 31, 32, 29 7 sLsLsLL 2, 14, 22, 30 11, 10, 9, 12 7 LsLsLss 7, 11, 19, 29 14, 15, 16, 13 8 LLLLLLLs 17 1 8 LLsLLsLs 1, 16, 24, 32 22, 23, 24, 21 8 ssLssLsL 8, 12, 20, 31 19, 18, 17, 20 9 sLLLsLLLL 6, 27 30, 29 9 LsssLssss 10, 18 31, 32 9 sLsLsLsLL 3, 26 6, 5 9 LsLsLsLss 13, 21 7, 8 5-tone scales: 24 6-tone scales: 4 7-tone scales: 18 8-tone scales: 9 9-tone scales: 8

Matrix Menu of Scales
by Step Order & L:s Value

L:s		5				6		7						8			9
		L L L L s	s s s s L	s L s L L	L s L s s	L L L L L s	s s s s s L	L L L L L L s	s s s s s s L	s L L s L L L	L s s L s s s	s L s L s L L	L s L s L s s	L L L L L L L s	L L s L L s L s	s s L s s L s L	s L L L s L L L L	L s s s L s s s s	s L s L s L s L L	L s L s L s L s s
1.276				24	23
1.382		26	25	16	15					28	27	30	29		32	31
1.618		3	5	1	4	5	9	9	17	4	6	2	7	17	1	8	6	10	3	13
1.724				32	28
2.382				31	27
2.618		13	9	8	6	25	17			15	10	14	11		16	12	27	18	26	21
3.618		21	17	12	10					23	18	22	19		24	20
4.618				20	18

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