Salutations to Anaphoria (for Kraig Grady)

Hello, there, Kraig, and many greetings to Anaphoria!

In response to your recent mention of my interest in the territory of
tuning schemes with two chains of fifths at some interval apart such
as a Pythagorean comma or septimal comma, etc., maybe I should offer a
few comments about the qualities of some of these schemes.

Of course, I would be delighted if this information might give the
diplomatic channels of Anaphoria a quick summary on this neo-Gothic
type of approach.

Generally these schemes, as I apply them to practical music, seem
especially to feature intervals of the following kinds:

(1) Thirds with ratios of around 9:7 (~435.08 cents) and
7:6 (~266.87 cents), and corresponding sixths;

(2) Often, thirds with more complex ratios such as the
regular Pythagorean 81:64 (~407.82 cents) and 32:27
(~294.13 cents); or around 14:11 (~417.51 cents)
and 13:11 (~289.21 cents);

(3) Often, "supraminor/submajor" or "semineutral" thirds
with ratios of around 17:14 (~336.13 cents) and 21:17
(~365.83 cents).

In a just intonation (JI) approach, two 12-note Pythagorean chains
(Eb-G#) at the distance of a Pythagorean or septimal comma yield pure
fifths and fourths, plus intervals of types (1) and (2).

An interesting JI variation on this theme is the Pythagorean
"tricomma" tuning with the two chains at the distance of three
Pythagorean commas, or a "tricomma" (~70.38 cents), yielding all three
types of thirds or sixths, albeit with the 7-based ratios not so
precise, varying from just by about 7.42 cents.

In the area of temperaments with fifths gently wider than pure, the
"e-based" tuning with fifths at around 704.61 cents also yields all
three types of intervals in a 24-note regular version. Here the two
chains are at the distance of the diesis, about 55.28 cents.

Since this discussion has mentioned 12-tET chains, I should add that
24-out-of-36-tET, with two such chains at 1/6-tone (33-1/3 cents)
apart, yields excellent approximations of both the 7-based and
submajor/supraminor thirds.

What I would like to emphasize especially is that practice has played
a central role in the development and use of these tunings.

For example, when I first tuned the "e-based" temperament, I did it on
a general kind of intuition, and in actually playing it, came upon the
intervals which I was to describe as "supraminor/submajor" thirds,
evidently choosing these names with a bit of help from Scala's list
of terms, intnam.par.

If one hears something new and likes it, of course, then that
experience can become part of one's "theory," available for use in
devising or appreciating new scales.

Again, Kraig, thank you for your contributions here, which often have
an invigorating effect on me.

Most appreciatively,

Margo Schulter
mschulter@value.net