More on 2/7-comma temperament extraordinaire

Hello, everyone, and I'd like to write about my excitement with the
temperament extraordinaire based on Zarlino's 2/7-comma meantone,
which looks like one of my favorite 12-note circulating systems.

In this tuning, eight fifths (F-C#) are tempered in Zarlino's
meantone, with fifths of around 695.81 cents, or 6.14 cents narrow;
the other four fifths are tempered equally wide at around 708.38
cents, or 6.42 cents larger than pure.

Possibly the concept of a "squirrel-like" or "squirrely" tuning circle
might help to explain some of the things I like about this system,
although some of the advantages are shared with a regular 2/7-comma
tuning in 12 or more notes.

First of all, notes and sonorities within the F-C# range are identical
to those in a regular Zarlino 2/7-comma meantone, with an effect which
Mark Lindley finds especially pleasing in modes where the arithmetic
division of the fifth (string ratio 6:5:4, frequency ratio 10:12:15)
is common -- Dorian (D-D), Phrygian (E-E), and Aeolian (A-A). Within
this range, major and minor thirds are equally impure by 1/7 comma
(~3.07 cents). The effect is for me very stately.

Secondly, the major thirds Bb-D and E-G# are within 10 cents of pure:
as in any "8-4" temperament extraordinaire (eight regular meantone
fifths, others equally wide), these thirds are precisely 300 cents
smaller than the regular meantone fifth, or here about 395.81 cents
(9.50 cents wide). They are almost as good, with respect to 4:5
(~386.31 cents), as in 1/7-comma meantone (~395.53 cents, ~9.22 cents
wide). This can be especially helpful in E Phrygian, where sonorities
with a major third (E-G#-B) are often used at closes.

Thirdly, all five of the remaining major thirds are _larger_ than
Pythagorean, the "squirrel" factor so pleasing for neo-medieval styles
in this part of the circle, and also for some 16th-century music
calling for diminished fourths. For example, I find the diminished
fourth G#-C (~420.95 cents) very pleasing for a Spanish piece
audaciously using the sonority E-G#-C in approaching the final cadence
on E.

Also, the 7-limit approximations at the farthest side of the circle
are almost as good as in 22-EDO: for example, Eb-Bb-Db-F (or, in a
conventional meantone spelling, Eb-Bb-C#-F) gives ~0-708-983-1417
cents as an approximation of 4:6:7:9 (~0-709-982-1418 in 22-EDO).

Of course, a temperament extraordinaire with even more heavily
tempered meantone fifths (as with Wilson's metameantone) would yield
even better results for Bb-D and E-G# with respect to 4:5, and even
better 7-limit approximations (including a virtually pure 7:9, as has
been observed). However, one could argue that the extra tempering
would compromise the quality of the usual meantone sonorities; Lindley
suggests that around 2/7-comma is the upper end of the optimal range
(although one might choose 1/3-comma or 19-EDO for a regular 19-note
circulating system, for example).

As long as conventional Renaissance music stays in the range of Bb-G#,
apart from sonorities such as diminished fourths (e.g. C#-F or G#-C),
the system is pretty much interchangeable with a regular 2/7-comma
meantone, with all usually spelled major thirds staying within 10
cents of 4:5.

The obvious weak point is Eb-G, with a third slightly larger than
Pythagorean (~408.38 cents) hardly interchangeable with the intended
meantone size in a Renaissance setting. However, many 16th-century
pieces stay within a Bb-G# range, or can be transposed with such a
range. For a neo-medieval style, of course, Eb-G is a fine resource.

Anyway, thanks to Gene Ward Smith and Bob Wendell for the dialogue
which led me to experiment with the temperament extraordinaire
approach; and also to Paul Erlich for his 22-EDO and paultone
discussions, some of which seem curiously relevant to a system of this
type.

Most appreciatively,

Margo Schulter
mschulter@value.net