RE: 3d. tone and golden ratios

Chowning's _Stria_ uses a tuning that incorporates the golden ratio as the
frequency ratio between notes. It's a strange-sounding interval since it's
not close to any simple-integer ratios.

As for the Kornerup, the golden ratio appears in the logarithmic (cents)
measurements, rather than as a frequency ratio -- this may be of interest
(from two weeks ago):

Date: Thu, 26 Aug 1999 15:01:05 -0400
From: "Paul H. Erlich" <PErlich@xxxxxxxxxxxxx.xxxx
Subject: RE: Fractal Scales

I wrote,

>> One (loosely speaking) fractal scale that sounds very nice (it is
virtually
>> the optimal tuning for diatonic triadic music) is Kornerup's golden
meantone
>> tuning.

Rick McGowan wrote,

>Interesting. Have you a pointer to a web page or a ratio list, where I
>could get formula for construction of such a tuning?

Constructing the tuning is easy. Consider the pentatonic scale. The minor
third and major second are, as we know, in the golden ratio, and two minor
thirds plus three major second complete the octave. So we have the
equations:

t=s*g (where g is the golden ratio, (sqrt(5)+1)/2)
2*t+3*s=1200

solving the second equation for t

t=600-1.5*s

and plugging in to the first

600-1.5*s=s*g (now use g=1.618034)
600=s*3.118034
s=192.4289

So the perfect fifth is (1200+192.4289)/2 = 696.2145 cents.

The whole tuning is constructed by chaining this fifth over and over again.
For example, the 19 tones centered around D would be

D 0
D# 73.5013
Eb 118.9276
E 192.4289
E# 265.9303
F 311.3566
F# 384.8579
Gb 430.2842
G 503.7855
G# 577.2868
Ab 622.7132
A 696.2145
A# 769.7158
Bb 815.1421
B 888.6434
Cb 934.0697
C 1007.5711
C# 1081.0724
Db 1126.4987
(D 1200)