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