RE: [tuning] Re: The Golden Mediant: Complex ratios and metastabl e intervals

David Keenan wrote,

>Long time no post.

Too long. Good to have you back. How do you like my new, harmonic-entropy
evaluation of equal temperaments?

>It seems likely to me that this formula for the limit of these iterated
>mediants (these other "noble numbers") has been published elsewhere. Has
>anyone seen it?

I believe this formula:

> (i + Phi * m)
> GoldenMediant(i/j, m/n) = -------------
> (j + Phi * n)

is quite similar to something presented in the appendix to one or both of
Manfred Schroeder's Springer "pop-math" books, and Wilson applied it to the
scale tree, as you'll see at http://www.anaphoria.com/hrgm01.html

I wrote,

>>I believe this formula:
>>
>>> (i + Phi * m)
>>> GoldenMediant(i/j, m/n) = -------------
>>> (j + Phi * n)
>>
>>is quite similar to something presented in the appendix to one or both of
>>Manfred Schroeder's Springer "pop-math" books,

Dave Keenan wrote,

>I'm guessing it's in his 'Number Theory in Science and Communication'. The
>whole book sounds great. The review says it's written to be understandable
>by people with only (advanced?) high school math (which is of course what
>Paul means by "pop-math").

I looked in the other book, _Fractals and Chaos_, and there it is, it's the
appendix. What a great book.

>Thanks. I hadn't seen this before. And even if I had I probably wouldn't
>have deciphered it!

>Does anyone want to suggest another name for it rather than "golden
>mediant"? Would anyone prefer "noble mediant"?

I would -- "golden mediant" might make one think simply of "golden section"
or dividng the segment into the proporion 1:Phi.

>I think it's important to make it very clear that the golden mediant is
>being applied to two completely different musical purposes.

>1. Melodic or logarithmic (fractions of an octave) (e.g. Wilson, Kornerup,
>Lamothe, Pepper, Stearns)

>2. Harmonic or linear (interval ratios) (e.g. Schulter, Keenan)

Yup!

>Could someone please explain to me the significance of golden mediants (or
>noble numbers) in the first application? At present, it just looks like
>numerology to me. Is it somehow related to Rothenberg propriety?

Yes, it is. More directly, it's related to MOS scales. Look in the archives
from a few months ago, and spend more time staring at Wilson's papers.

>This is a point I failed to make in Margo's and my paper. Sorry.

Where is this paper?

Dave Keenan wrote,

>I already did that, and failed to understand anything of the following
kind:
>e.g. "The noble mediant between two fractions of an octave 2^(i/j) and
>2^(m/n) gives scales which have special property X(i,j,m,n)."

It is the generator which will produce MOS scales whenever the number of
notes belongs to the Fibonacci-like series j, n, j+n, . . . and these MOS
scales will all be proper (with a possible exception for the first few),
since the ratio of the small to large steps in these MOS scales always
approaches Phi.

>I tried that too. It just gives me a headache. How come there's no text to
>provide the keys to all these beautiful but obscure diagrams?

That's Wilson for you!

Oh yeah, the Schroeder book is actually called _Fractals, Chaos, and Power
Laws_. You'll love it, I guarantee.