Re: Tuning with Phi as interval ratio

Hello, there, this is just a note on a neo-Gothic tuning based on Phi
as an _interval ratio_, in contrast to various tuning systems such as
those of Thorwald Kornerup, Keenan Pepper, and Dan Stearns where a
Phi-based function is applied to logarithmic divisions of the octave,
or to ratios between logarithmic sizes of intervals (as measured, for
example, in cents).

This tuning is defined as the regular tuning having Phi as its
augmented fifth, an interval of ~1.61803398874989484820459:1 or
approximately 833.090 cents, a kind of "superminor sixth" quite close
to 34:21 (~834.175 cents).

Since an augmented fifth is made up of precisely four whole-tones --
and is thus termed a _tetratonus_ by Jacobus of Liege (c. 1325) --
this gives us a regular major second of ~208.273 cents. From here we
can determine the rest of the tuning, with a fifth of ~704.136 cents.

Does this look at all familiar? Curiously, Keenan Pepper's "Noble
Fifth" tuning based on the Phi-based mediant between 4/7-octave and
3/5-octave (weighted toward the latter) yields a fifth of ~704.096
cents.

In discussing the "Noble Fifth" temperament, devised as Keenan Pepper
has emphasized from the viewpoint of a theoretical "noble generator"
rather than specifically as a neo-Gothic temperament, I noted the very
close approximation of Phi for the augmented fifth.

Thus these two tunings, derived from Phi in radically different ways,
curiously are almost identical.

Again, I'd like to emphasize that these distinct methods of using Phi
should _not_ be confused or equated; here, for whatever reason, they
happen to produce these similar results.

Most respectfully,

Margo Schulter
mschulter@value.net