Self-similar scales

There's a discussion between Paul Erlich and Dan Stearns about what
qualifies as a generalized "silver" scale. Well, even if they aren't
properly "silver", those scales where ratios of intervals are equal are
interesting. In particular, the two scales Paul Hahn mentions here:

<http://www.egroups.com/message/tuning/12670>

Which is a copy of these two messages:

<http://www.egroups.com/message/tuning/4497>
<http://www.egroups.com/message/tuning/4498>

The second, "silver fifth" scale comes from the series

1/2, 2/5, 5/12, 12/29, 29/70 . .

according to Paul Erlich. That seems to be fourth/octave, then.

We know that fourth/octave = tone/fourth, and octave=2*fourth+tone. Call
tone t and fourth f, and we have

t/f = f/(t+2f) = x

x = 1/(x+2)

which can be rewritten

x = 1 + 0x
------
2 + 1x

That starts to look like these golden "horagrams" or whatever, where x
would be phi. 0/1 is the first step in the series. Another expression
would be

x = 2 + 1x
------
5 + 2x

which gives the same value. So this seems to work.

The first scale comes from a fifth/octave equal to a semitone/tone. Set
tone=t, semitone=s, that gives

s/t = (3t+s)/(5t+2s) = y

y = 3 + y
------
5 + 2y

suggesting the series starts 1/2, 3/5

And that's as far as I've got.

I can show this y in terms of the x(above) and phi

y = 2phi - 1
-------- - 1
x + 1

but I don't think that's at all significant.

I do, however, note that there is a meantone scale with self similarity.
I didn't appreciate that before, obviously not reading Paul Hahn's
messages properly. So, you could tune 12 notes to a tone, and have 74
notes to the octave on a Halberstadt keyboard. Well, not immediately
inviting, but nice to know it's there.

Graham

Graham Breed wrote,

> There's a discussion between Paul Erlich and Dan Stearns about what
qualifies as a generalized "silver" scale.

I noticed that when (S+1)/(L/s) = S, you get a nice compromise of the
Phi convergence that sort of splits the difference between 1/4 and 1/5
comma meantone.

Here's the P/((5+S*7))*(3+S*4) generator (where "P" = 1200 and "S" =
sqrt(2)):

0 194 389 503 697 892 1086 1200
0 194 308 503 697 892 1006 1200
0 114 308 503 697 811 1006 1200
0 194 389 583 697 892 1086 1200
0 194 389 503 697 892 1006 1200
0 194 308 503 697 811 1006 1200
0 114 308 503 617 811 1006 1200

Here's Paul Erlich's standard pentachordal where (S+1)/(L/s) = S:

0 105 210 390 495 600 705 885 990 1095 1200
0 105 285 390 495 600 779 885 990 1095 1200
0 179 285 390 495 674 779 885 990 1095 1200
0 105 210 315 495 600 705 810 915 1021 1200
0 105 210 390 495 600 705 810 915 1095 1200
0 105 285 390 495 600 705 810 990 1095 1200
0 179 285 390 495 600 705 885 990 1095 1200
0 105 210 315 421 526 705 810 915 1021 1200
0 105 210 315 421 600 705 810 915 1095 1200
0 105 210 315 495 600 705 810 990 1095 1200

--Dan Stearns