Re: [tuning] Digest Number 873

On 10 Oct 2000 Paul Erlich wrote:

> > And here's something neat: it looks like this could be a good example
> > to bolster Margo and Dave K's thesis that Golden-derived intervals
> > tend to maximise dissonance.
>
> I'm afraid you've confused golden fractions of an octave with golden
> frequency ratios -- see my post to Joseph of a few minutes ago.

No, I haven't confused them actually - I was under the impression that
Margo and Dave were suggesting that phi-based *additive* generators, like
in Keenan Pepper's noble tuning, led to some similar intervallic
properties as golden *ratio*-based methods, like Dave K's Golden Mediants.

Given the nature of phi, where adding and multiplying can lead to similar
things (1 over phi equals 1 minus phi, for example), I think it's worth
looking into.

Jon wrote,

>No, I haven't confused them actually - I was under the impression that
>Margo and Dave were suggesting that phi-based *additive* generators, like
>in Keenan Pepper's noble tuning, led to some similar intervallic
>properties as golden *ratio*-based methods, like Dave K's Golden Mediants.

Where did you get that impression? I'm pretty confident that Dave, at least,
never suggested such a thing. In fact, many of the "additive" or
"logarithmic" noble generators, like Kornerup's Golden Meantone, result in a
high proportion of _consonant_ intervals.

Jon Wild wrote,

>Given the nature of phi, where adding and multiplying can lead to similar
>things (1 over phi equals 1 minus phi, for example), I think it's worth
>looking into.

Well, the frequency ratios of the noble "additive" or "logarithmic"
generators are 2^phi or 2^noble, which don't share the properties of the
noble ratios themselves. While the noble ratios have continued fraction
representations that end in all 1's, 2^phi, for example, has C.F.:

1 + 1/(1 + 1/(1 + 1/(6 + 1/(1 + 1/(2 + 1/(4 + 1/(1 + 1/(52 + 1/(2 + 1/(5 +
1/(4 + 1/(1 + 1/(106...)))))))))))))

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:

> Jon wrote,
>
> >I was under the impression
> >that Margo and Dave were suggesting that phi-based *additive*
> >generators, like in Keenan Pepper's noble tuning, led to some
> >similar intervallic properties as golden *ratio*-based methods,
> >like Dave K's Golden Mediants.
>
> Where did you get that impression? I'm pretty confident that Dave,
> at least, never suggested such a thing. In fact, many of
> the "additive" or "logarithmic" noble generators, like Kornerup's
> Golden Meantone, result in a high proportion of _consonant_
> intervals.

Alright, I just found where I was remembering this from. In Margo and
Dave's paper they say this:

====================================
|Even while we were engaged in an absorbing dialogue on these plateau
|regions and their mathematical and musical nature, Keenan Pepper, in
|a delightful synchronicity, posted an article to the Tuning List[6]
|on the application of the Golden Ratio or Phi to another area of
|music: the generation of scales with particular relationships
|between scale steps.
|
|One of us recognized that Pepper's Phi-related function could also be
|applied to the problem of finding the region of maximum complexity
|between two simpler ratios, providing a shortcut to the longer
|process of successive approximation by iterating mediants.
|
|In what follows, we show how this function, here termed the "Golden
|Mediant", can be used to locate regions of maximum complexity.
====================================

Apologies to Margo and Dave if I read more into that than they meant
to suggest.

Anyway, I still think it was a cool scale :)