Re: 27, 28, 29, and 34 -- 29-tET advocacy

Hello, there, everyone, and as an advocate of 29-tET in a neo-Gothic
kind of setting inspired by Western European styles of the 13th-14th
centuries, I'd like to suggest that this tuning has a very different
character from the more "Renaissance-like" 31-tET, and that both
tunings are excellent when used for music making the most of their
distinct qualities.

Whether Marchettus of Padua (1318) may have been suggesting for
singers something like an _equal_ "fivefold division" of the tone
remains a topic of much debate.

Whatever the possible answers, depending in part on whether one
interprets certain passages in Marchettus as monochord divisions of
the tone into five _unequal_ parts, 29-tET is a beautiful tuning for a
neo-Gothic style, and one that seems to fit the interval aesthetics he
apparently espouses, with very wide major thirds and fifths expanding
respectively to fifths and octaves.

The regular major and minor thirds at 10/29 octave and 7/29 octave are
around 413.79 cents and 289.66 cents in a kind of "accentuated
Pythagorean" feeling. Minor thirds give excellent approximations of a
just 13:11 (~289.21 cents), while major thirds aren't too far from
14:11 (~417.51 cents).

Cadential resolutions featuring major thirds expanding to fifths, and
minor thirds contracting to unisons, have concise diatonic semitones
of 2/29 octave, or ~82.76 cents, considerably narrower and more
efficient than in 12-tET.

We also get diminished fourths and augmented seconds at 9/29 octave
and 8/29 octave, around 372.41 cents and 331.03 cents. These approach
the region of 21:17 (~365.83 cents) and 17:14 (~336.13 cents), while
also on the outskirts of the region of 5:4 (~386.31 cents) and 6:5
(~315.64 cents).

Typical progressions involving these intervals resolve by a melodic
motion of a chromatic semitone (3/29 octave, ~124.14 cents) in one or
more of the voices, as with the outer voices of this example (here C4
shows middle C):

Bb3 B3
F#3 E3
Eb3 E3

Both families of 29-tET thirds found in a regular 12-note tuning are
rather complex and active, lending themselves to neo-Gothic cadential
progressions.

Renaissance music, or other styles based on ratios of 5, would suggest
to me a different type of tuning, with 31-tET or 1/4-comma meantone an
obvious choice. These tunings divide the whole-tone into five equal or
nearly equal parts, but in a very different way.

With a larger set of 29-tET, we get another family of intervals which
is one of the special attractions: intervals very close to 15:13
(~247.74 cents), 13:10 (~454.21 cents), and 26:15 (952.26 cents) --
somewhere between the familiar categories of major second and minor
third; major third and fourth; and major sixth and minor seventh.

These intervals have 29-tET approximations at 6/29 octave (~248.28
cents), 11/29 octave (~455.17 cents), and 23/29 octave (~951.72
cents).

Here there's scope for much fun and creativity: is that near-15:13 a
large major second that might expand to a fourth, or a small minor
third about to contract to a unison? Similarly, will that near-26:15
act like a large and expansive major sixth seeking the octave, or a
small minor seventh seeking contraction to a fifth?

Using an asterisk (*) to show a note raised by a diesis of 1/29 octave
or 1/5-tone, I'll give an example of these alternate resolutions:

D3 C4 D3 D*3
B*3 C4 B*3 A*3
E*3 F3 E*3 D*3
or

In the first resolution, the E*3-B*3-D3 sonority at a rounded
0-703-952 cents acts like an outer minor seventh contracting to a
fifth and an upper minor third contracting to a unison.

In the second, the same sonority suggests an outer major sixth
expanding to an octave and an upper major second expanding to a
fourth.

One thing in common between 29-tET and 24-tET is that both tunings
offer this kind of family of intervals -- precisely 250, 450, and 950
cents in 24-tET.

For whatever reason, I've noticed that lots of discussions of 29-tET
tend to approach it from other perspectives; but I'd say that the
general "accentuated Pythagorean" structure, the rather gentle
"submajor/supraminor" thirds, and the intervals about midway between
usual Pythagorean or 12-tET categories that I've just described, all
contribute to a "mood" for the tuning.

Take all of these features together with the near-just fifths,
tempered by only about 1.49 cents in the wide direction, and we have a
very impressive and beautiful tuning, one deserving lots of
enthusiastic advocacy.

Most appreciatively,

Margo Schulter
mschulter@value.net