Examples of Self-Similar Scales

There are many ways of constructing self-similar or quasi-fractal
scales. One way is to approximate a Cantor Dust ( see Mandelbroit) by
iteratively dividing a given interval into 3 parts and skipping the
middle one. For example, one can choose the octave (1200 cents) and
divide it into 3 intervals as 0 400 800 1200. Next, one divides the two
outside 400 cent intervals into 3 parts giving 0 133 266.7 400 cents and
800 933 1067 1200. The next iteration yields 0 44 89 133 267 311 355 400
and 800 844 889 933 1067 1111 1156 and 1200 cents.

In more detail:
cents decimal Approx. Ratio
0 1.0 1/1
44.44 1.0260 40/39
88.88 1.05269 20/19
133.33 1.0801 14/13
266.67 1.1665 7/6
311.11 1.1969 6/5
355.55 1.2280 27/2
400.0 1.2599 5/4
800.0 1.5874 8/5
844.44 1.6287 13/8
888.88 1.6710 5/3
933.33 1.7145 12/7
1066.67 1.8517 13/7
1111.11 1.8999 19/10
1155.55 1.9493 39/20
1200.0 2.0000 2/1

Another method is to choose an interval and iteratively apply some
function whose value is
always less than that of its argument (roots, logs, etc.). For example,
one can select the 3/1
and take successive square roots. This action results in a scale with
decreasing intervals which quickly become too small to hear. In the
following example, I have stopped the process when the intervals became
less than 2 cents and have added the "tonic."

# Decimals Cents
1 1 0
2 1.002148 3.7147559
3 1.0043007 7.4295117
4 1.0086198 14.859023
5 1.017314 29.718047
6 1.0349278 59.436094
7 1.0710755 118.87219
8 1.1472027 237.74438
9 1.316074 475.48875
10 1.7320508 950.9775
11 3.0000 1901.955

Brian McLaren has composed with similar set of tones starting with the
square root of 3 and cycling at that interval (as a pseudo-octave).

Many other starting intervals are possible -- pi, e, phi, sqr(5), etc.
The scale need not repeat at any interval of equivalence.

One may also fill the larger gaps in self-similar scales by the same
process as was used to generate the scale. In the case of the first
scale, one might treat 400 cents as the interval of equivalence and fill
in the 400 cent gap in the middle of the scale. Alternatively, one
might continue the process one step further to get 0 14.81 29.63 44.44
88.89 103.7 118.5 133.33 and use this last interval (1/9 octave) as the
I. of E., optionally ignoring the 2/1

I presented a several examples of scales of these types in my tutorial
at the 1996 ICMC
meeting and, IIRC, sent them to Manuel Op de Coul. (are they in Scala?).

--John

John Chalmers wrote:
> I presented a several examples of scales of these types in my
> tutorial at the 1996 ICMC meeting and, IIRC, sent them to
> Manuel Op de Coul. (are they in Scala?).

No, at that time I first skipped them because I didn't find
them very interesting, and apparently forgot to include them later.
There's only one in the archive: iter_fifth.scl.

Manuel Op de Coul coul@ezh.nl