Phi and/or fractal scales

I was able to unpack some files and locate more references to work on
scales derived from phi or exhibiting some sort of self-similar or
fractal order. Most of what I will say is condensed and paraphrased
from Xenharmonikon 15, Autumn 1993 (available from the Just Intonation
Network and Frog Peak Music). I recommend this issue and XH13 to anyone
seriously interested in scales of these types. One might play them with
timbres generated as described in William Sethares's book "Tuning,
Timbre, Spectrum, Scale."

Phi Tonality:

Walter O'Connell (with help from Brian McLaren) updated an article
originally written for Die Reihe in the 1960's and privately distributed
after Die Reihe ceased publication. The article is "The Tonality of the
Golden Section," and in it O'Connell proposed a timbre based on powers
of phi and an ultrachromatic gamut which is the equal division of phi
into twenty five parts (25 degrees of 36-tet is a very good
approximation to phi/1). Out of this set of pitches, he selected modes
with 7,9, 11 and 14 tones.

Another early inventor of phi-based scales is David E. Schroer, who used
the 2nd and 3rd order difference tones between two pitches differing by
phi to generate the intervals. Schroer called his system
"Aureotonality," but I have been unable to find out much more about it.

In my introductory remarks (Notes and Comments) , I mentioned a scale of
5 tones within Phi invented by Lorne Temes, who in 1970 was a student in
Toronto. Temes's scale was approximately 1.0 1.169 1.236 1.309 1.382
and 1.618 (phi). These intervals correspond to 1/1, phi^3/(phi^2+1),
2/phi, phi^2/2, (phi^2+1)/phi^2, and phi. The tone phi^3/(phi^2+1) is
proposed as the dominant. If anyone knows Temes or his/her present
whereabouts, I'd appreciate the information.

Self-Similar Scales:

Brian McLaren, in his article "The Uses and Characteristics of Non-Just
Non-Equal-Tempered Scales," discusses a number of different kinds of
schemes to generate new types of scales, a topic he began in XH13
(1991) with "General Methods for Generating Musical Scales." The type of
scale he terms "integrated self-similar" is generated from a single
relation. For example, the "E Scale" is constructed by computing the
cents span of the interval e/1 (2.71828....), or 1731.234 cents. This
interval and each successive quotient is then divided by e to obtain the
scale 1731.234, 636.885, 234.297, 86.193, 31.709, 11.66 and 4.29 cents.
Similarly, McLaren starts with pi, sqr(3), sqr(5), sqr(7) and mentions
using the square roots of 11, 13, 17, 19, 23, and so on.

These scales differ from the examples I have posted or sent by private
email in that McLaren uses a constant divisor and I a constant
operation. In my case, I started with an interval such as pi, phi, or
3/1 and repeatedly took the square, cube or higher root. McLaren
selected an interval, converted it to cents, and iteratively divided it
by the same interval. I stopped the process when the intervals come too
small and added a final 0 tone. McLaren generally does not.

Later in the article, McLaren defines "Integrated non-self-similar"
scales. The example he gives is obtained by first computing the cents
value of the sqr(3), 950.9775. This is divided by the sqr(5), the result
of this by the sqr(7), and so on. The remainder of his paper is
devoted to scales generated by infinite continued fractions, metal bar
spectra and other techniques.

I might add that Brink McGoogy has also proposed using certain
irrational intervals to construct scales in an article in XH15. Brink
composes and performs "Xenphionic" music and
writes neo-Dadaistic prose, some of which he has collected as "xyzonia,
a book of devices."

--John

Thanks, John, for all the additional information on "Golden Ratio" scales.
Seem's that with my 7-tone selection inside phi I have re-invented O'Connels
wheel. Well, perhaps that is helpful in having a fresh look at it.
Thanks again,
Heinz