More on Golden Meantone scales

Here's a challenge. What I was doing was deriving tunings, taken to
19-tone, using the golden ratio for various triads, including the major
chord and the root-fifth-octave. Then I realized....

Let's try some tunings, taken to 9-tone (31 if you're really daring), using
three adjacent harmonics going as far as the tenth. I'll use C as a
fundamental. Once again, the ratio of step size (NOT frequency) is
1.618:1.000.

1) C - C' - G'
2) C - G - C'
3) G - C - E
4) C - E - G
5) E - G - A# (Euler's tritone = 2.618)
6) G - A# - C
7) A# - C - D
8) C - D - E (there will be greater and lesser major seconds as in just
intonation)

I'll get to work on this tonight.

~DaW~

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--- In tuning@y..., "Danny Wier" <dawier@y...> wrote:
> Here's a challenge. What I was doing was deriving tunings, taken to
> 19-tone, using the golden ratio for various triads, including the
major
> chord and the root-fifth-octave. Then I realized....
>
> Let's try some tunings, taken to 9-tone (31 if you're really
daring), using
> three adjacent harmonics going as far as the tenth. I'll use C as a
> fundamental. Once again, the ratio of step size (NOT frequency) is
> 1.618:1.000.
>
> 1) C - C' - G'
> 2) C - G - C'
> 3) G - C - E
> 4) C - E - G
> 5) E - G - A# (Euler's tritone = 2.618)
> 6) G - A# - C
> 7) A# - C - D
> 8) C - D - E (there will be greater and lesser major seconds as in
just
> intonation)
>
> I'll get to work on this tonight.

These will all be Golden Horagrams of the Scale Tree:

http://www.anaphoria.com/sctree.PDF
http://www.anaphoria.com/line.PDF
http://www.anaphoria.com/hrgm.PDF

Should anyone be interested, here's a generalized Golden scale
algorithm that I came up with independent of Erv Wilson's Horagrams:

x = p/((A+Phi*B))*(a+Phi*b)

where

p = any given periodicity

Phi = the Fibonacci constant (L/s = Phi)

a/A, b/B = the two adjacent fractions of any given A, B, ... Fibonacci
series where A and B are relatively prime and A is always the number
of small steps and B the number of large steps

x = a generalized Golden generator

Though I was inspired in a small way by Dave Keenan and Margo
Schulter's weighted mediant paper, it was Kornerup via McLaren in the
latter's A Brief History of Microtonality in the Twentieth Century
that got me thinking in these terms--specifically I was interested in
finding a way to look at scales that could be generalized and was not
tied to either ET or JI thinking.

Note that this generalization accounts for all Messiaen like instances
of fractional periodicity, all instances of Bohlen-Pierce like
non-octave scales, and the internal ordering of scale steps.

--Dan Stearns