Self Similar Scales: Further Fractal Flavors

Self Similar Scales: Further Fractal Flavors
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Based on the overwhelmingly positive responses to the Self Similar E,
Phi and Pi scales,

/tuning/topicId_18578.html#18578

I give to my friendly list members another possibility not based on
constants, but a simple division scheme.

This time let's create a Self Similar (fractal) scale based on
iterative division of the interval 3/1 @ 1901.955 cents, into 4
parts, with a 6/1 boundary @ 3101.955.

1. If we take the cents value for 3/1 @ 1901.955, and divide it by
4, we get the following sequence:

475.489, 950.978 (475.489*2), 1426.466 (475.489*3), 1901.955
(475.489*4)

2. Then we divide 475.489 by 4 giving the below sequence:

118.872, 237.744 (118.872*2), 356.617 (118.872*3)

3. Next, we take 118.872 cents divided by 4 and obtain the
following:

29.718, 59.436 (29.718*2), 89.154 (29.718*3)

4. If we carry this out two more times we get the remaining
members of the descending series (stopping here with consideration
for 768 tuning unit synths as well):

7.430, 14.859, 22.289

1.857, 3.715, 5.572

5. Arranged in descending order we have the first half of our
scale:

1901.955
1426.466
950.978
475.489
356.617
237.744
118.872
89.154
59.436
29.718
22.289
14.859
7.430
5.572
3.715
1.857
0.000

6. Now let's impose our 6/1 border @ 3101.955 cents, and invert
the above series by subtracting each member's value from 3101.955,
obtaining the below sequence, and the upper half of our scale:

1200.000
1675.489
2150.978
2626.466
2745.338
2864.211
2983.083
3012.801
3042.519
3072.237
3079.666
3087.096
3094.525
3096.383
3098.240
3100.098
3101.955

7. Lastly, let's combine the two sequences together and sort them,
obtaining our final scale:

33 Tone Self Similar Scale Based on Iterative Division of 3/1 into 4
Parts
Cents Consecutive
0
1.857 1.857
3.715 1.857
5.572 1.857
7.430 1.857
14.859 7.430
22.289 7.430
29.718 7.430
59.436 29.718
89.154 29.718
118.872 29.718
237.744 118.872
356.617 118.872
475.489 118.872
950.978 475.489
1200.000 249.022
1426.466 226.466
1675.489 249.022
1901.955 226.466
2150.978 249.022
2626.466 475.489
2745.338 118.872
2864.211 118.872
2983.083 118.872
3012.801 29.718
3042.519 29.718
3072.237 29.718
3079.666 7.430
3087.096 7.430
3094.525 7.430
3096.383 1.857
3098.240 1.857
3100.098 1.857
3101.955 1.857

These kinds of scales are brimming with fractal-like Self Similar
properties. They are not only interesting from a mathematical point
of view, but are also incredibly fun to play. These really do fit
into Robert Walker's aptly named "recreational mathematics" category,
which is indeed the special reason why I like to explore such scales.
They allow me to learn about new sounds and tuning possibilities, and
have boundless fun doing so!

Again, if you chose to create any music with these, let me hear what
you come up with, as I would be delighted to check it out. And if any
portion of the above is unclear, let me know and I'll be glad to help
clarify.

Thanks,

Jacky Ligon