3-step-size golden scale

Hi Dan, I sent this scale to the tuning list a while ago and I'm not sure
if you remember it - seems related to what you and Gene have been talking
about here. It's from Clampitt's dissertation on "pairwise well-formed
scales", which necessarily have 3 step-sizes. (He identifies a fair number
of world scales that have 3 step-sizes - I'll see if I can dig it up if
you want). Anyway, here's the scale as described in my post to the list a
little over a year ago.

best wishes --Jon

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Date: Tue, 10 Oct 2000 04:28:13 -0400 (EDT)
From: jon wild <wild@fas.harvard.edu>
To: tuning@egroups.com

I don't think I've seen a scale quite like this mentioned here before. I
found it alluded to in passing, in a dissertation on "Pairwise Well-Formed
Scales" by David Clampitt. It's the limiting case of a series of other
scales which he looks at briefly. The scale uses an *additive* generator
of phi+1, taken mod 3phi+2. Clampitt never shows the scale explicitly but
I'll give it here, first in the "generated" order:

0, phi+1, 2phi+2, 1, phi+2, 2phi+3, 2

and here reordered within the 3phi+2 octave:

0, 1, 2, phi+1, phi+2, 2phi+2, 2phi+3, (3phi+2)

the seconds are {phi-1, 1, phi},
the thirds are {phi, 2, phi+1},
the fourths are {phi+1, 2phi, phi+2}
the fifths are {phi+2, 2phi, 2phi+1}
the sixths are {2phi+1, 3phi, 2phi+2}
the sevenths are {2phi+2, 3phi+1, 2phi+3}
the 8ve is 3phi+2

In cents we get:

0 175 350 458 633 917 1092 1200
m m s m L m s

As a whole, it doesn't sound too much like anything I'm familiar with (it
does have the quarter-tone neutral third though). But there's so much
self-similarity here it makes my head hurt to think about it... if we call
the 3 shades of each generic interval s m and L, then we have:

m2:s2 = L2:m2 = L3:L2 = L5:L3 = 8ve:L5 = s6:s4 = s4:s3 = phi !!

also,

8ve:L3 = m5:m2 = 2phi

and

s6:s3 = L3:L2 = phi+1

and you can find many more such relations...

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