Re: M3 choices in 53-tet: reply to Paul Erlich

Hello, there, and recently Paul Erlich wrote:

> 22-tET, 34-tET, 41-tET, and 53-tET all widen the fifth compared to
> 12-tone, and compared to just (except 53TET, which is ALMOST exactly
> right!). All support the syntonic comma (i.e., show net microtonal
> motion) in the progression I-vi-ii-V-I. All have two major second
> intervals: four fifths do NOT equal a major third.

In reference to 53-tet, I'm tempted to state the matter from a
slightly different angle, doubtless a bit more "medievalist" and
Pythagorean. Both Pythagorean and 53-tet do have regular or "ditonic"
major thirds equal precisely to four fifths, but _additionally_
support two other M3-like intervals, close to 5:4 and 9:7.

Please let me emphasize that this statement doesn't really contradict
Paul's, because for someone using 53-tet to play music where an
analysis like "I-vi-ii-V-I" would be likely, I'd guess that the small
M3 close to 5:4 would be taken as the norm. This version of M3 at
17/53 octave is a scale step smaller than what I would call the
"regular" M3 at 18/53 octave (four 31/53 octave fifths minus two
octaves), which in turn is a step smaller than the "large" M3 at 19/53
octave (~9:7).

Interestingly, a Pythagorean tuning if extended to 17 notes (16
fifths) yields intervals very close to these:

Pythagorean 53-tet

Small M3 octave - 8 fifths 17/53 octave
(tetratonic) 8192:6561 (~384.36 cents) (~384.91 cents)
(compare with 5:4, ~386.31 cents)

Regular M3 4 5ths - 2 octaves 18/53 octave
(ditonic) 81:64 (~407.82 cents) (~407.55 cents)

Large M3 16 5ths - 9 octaves 19/53 octave
(octotonic) 43046721:33554432
(~431.28 cents) (~430.19 cents)
(compare with 9:7, ~435.08 cents)

To describe these three M3 alternatives in either Pythagorean or
53-tet, I've borrowed some medieval terms. The regular M3 is
"ditonic," equal to precisely two 9:8 whole-tones in Pythagorean, and
two 9/53-octave whole-tones in 53-tet. The small M3 is "tetratonic,"
equal to an octave minus _four_ whole tones -- in 53-tet, that is, to
53 - 36 or 17 scale steps. The large M3 is "octotonic," being equal to
eight 9:8 whole-tones minus one octave (or 16 fifths minus nine
octaves) in Pythagorean; and likewise 8 9/53-octave whole-tones minus
one octave (72 - 53) or 19 scale steps in 53-tet.

In either of these very similar tuning systems, the small major third
is also equal to a regular tone (9:8 or 9/53 octave) plus a _tonus
minor_ or "small whole-tone" very close to 10:9 (65536:59049 or
~180.45 cents in Pythagorean, 8/53 octave or ~181.13 cents in
53-tet). The large major third is also equal to a regular tone plus a
_tonus major_ or "large whole-tone" very close to 8:7 (4782969:4194304
or ~227.37 cents in Pythagorean, 10/53 octave or ~226.42 cents in
53-tet).

> (I have long considered 53-tET to be a beautiful thing, since its
> base microtonal interval falls nicely between a Pythagorean comma
> and a syntonic comma, but I've never written music in it...)

Yes, at about 22.64 cents, 1/53-octave is not far from the mean of the
Pythagorean comma (531441:524288, ~23.46 cents) and the syntonic comma
(81:80, ~21.51 cents). I find it interesting that while a pure
extended Pythagorean tuning gives a slightly better approximation of
9:7 for the large M3, 53-tet is a bit closer to 5:4 for the small M3.

Most respectfully,

Margo Schulter
mschulter@value.net