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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. 


Menu of Horagrams
Sorted by Convergent Pair
(Finnamore)

HoragramWilson'sConvergent
Pairs
Generator
Interval
#1#220/11/2458.36 c
#2#110/11/3331.67 c
#3*#60/11/4259.85 c
#4#271/22/5503.79 c
#5*#30/11/5213.60 c
#6#301/23/7527.15 c
#7#141/32/7354.82 c
#8#191/33/8440.59 c
#9*#20/11/6181.32 c
#10#311/24/9541.38 c
#11#151/33/10366.26 c
#12#181/34/11431.12 c
#13*#71/42/9273.85 c
#14#101/43/11322.27 c
#15#262/55/12495.90 c
#16#232/55/13465.08 c
#17*#10/11/7157.52 c
#18#321/25/11550.96 c
#19#161/34/13373.07 c
#20#171/35/14425.23 c
#21*#81/43/13280.61 c
#22#91/44/15317.17 c
#23#252/57/17491.95 c
#24#242/57/18468.62 c
#25*#41/52/11222.97 c
#26*#51/53/14254.04 c
#27#293/77/16522.72 c
#28#283/78/19506.94 c
#29#132/75/17350.90 c
#30#122/75/18335.18 c
#31#203/87/19443.74 c
#32#213/88/21455.78 c
   

Menu of Horagrams
Sorted by Generator Size
(Wilson)

HoragramWilson'sConvergent
Pairs
Generator
Interval
#17*#10/11/7157.52 c
#9*#20/11/6181.32 c
#5*#30/11/5213.60 c
#25*#41/52/11222.97 c
#26*#51/53/14254.04 c
#3*#60/11/4259.85 c
#13*#71/42/9273.85 c
#21*#81/43/13280.61 c
#22#91/44/15317.17 c
#14#101/43/11322.27 c
#2#110/11/3331.67 c
#30#122/75/18335.18 c
#29#132/75/17350.90 c
#7#141/32/7354.82 c
#11#151/33/10366.26 c
#19#161/34/13373.07 c
#20#171/35/14425.23 c
#12#181/34/11431.12 c
#8#191/33/8440.59 c
#31#203/87/19443.74 c
#32#213/88/21455.78 c
#1#220/11/2458.36 c
#16#232/55/13465.08 c
#24#242/57/18468.62 c
#23#252/57/17491.95 c
#15#262/55/12495.90 c
#4#271/22/5503.79 c
#28#283/78/19506.94 c
#27#293/77/16522.72 c
#6#301/23/7527.15 c
#10#311/24/9541.38 c
#18#321/25/11550.96 c

* MIDI and/or mp3 files available



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

StepsOrderHoragram #sWilson's #s
5LLLLs3, 13, 21, 266, 7, 8, 5
5ssssL5, 9, 17, 253, 2, 1, 4
5sLsLL1, 8, 12, 16, 20, 24, 31, 3222, 19, 18, 23, 17, 24, 21, 20
5LsLss4, 6, 10, 15, 18, 23, 27, 2827, 30, 31, 26, 32, 25, 29, 28
6LLLLLs5, 253, 4
6sssssL9, 172, 1
7LLLLLLs92
7ssssssL171
7sLLsLLL4, 15, 23, 2827, 26, 25, 28
7LssLsss6, 10, 18, 2730, 31, 32, 29
7sLsLsLL2, 14, 22, 3011, 10, 9, 12
7LsLsLss7, 11, 19, 2914, 15, 16, 13
8LLLLLLLs171
8LLsLLsLs1, 16, 24, 3222, 23, 24, 21
8ssLssLsL8, 12, 20, 3119, 18, 17, 20
9sLLLsLLLL6, 2730, 29
9LsssLssss10, 1831, 32
9sLsLsLsLL3, 266, 5
9LsLsLsLss13, 217, 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
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1.276   2423                 
1.38226251615      28273029  3231    
1.6183514599 17462717186 10313
1.724  3228                 
2.382  3127                 
2.618139862517   15101411 1612 27182621
3.61821171210     23182219 2420     
4.618  2018                 

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