More on 22-tone equal

I wish to clarify my statements regarding the tone of 22-tone equal.

This temperament does not merge 10:9 (182 cents) and 9:8 (204 cents) in
the sense that it does not merge the two at a value intermediate between
the two( as _do_ the harmonically usable temperaments) but it does merge
them at a value quite divergent from both, namely at 218 cents. So in
one sense it does not merge (or compromise between) the two species of
tone, but in another sense it does merge them, at a very unfortunate
excentric value.

In other terms, 22-tone equal eliminates the comma only to break the
back of the cycle of fifths, harmonically considered. It also reduces
the diatonic semitone from 112 cents to 55, and the chromatic semitone
it swells enormously from 71 cents to 164, presuming the fifths are held
consonant at 709 cents. I suppose someone will announce they adore music
without any consonant fifths, and if possible, without any other
consonances either, but no matter. It would be hard to find a more
grotesque temperament for any purpose of melody, whether diatonic,
chromatic or enharmonic.

On the other hand... this system is _so_ outrageous that perhaps one
should give it the benefit of the doubt! Perhaps it will find wide use
as an example of what exquisitely mistuned intervals sound like.

All harmonically usable temperaments: 12- 19- 31- 43- 50- 55-tone equal,
etc merge the two species of tone at a value intermediate between 10:9 &
9:8. This permits the cycles of the fifth, major third and minor third
to be integrated within a single coherent system. These temperaments
divide the tone into 2,3,5,7,8 & 9 equal parts respectively.

It is interesting that no usable temperament divides the tone into four
equal parts. Both 22- and 24-tone equal do so, and both are dogs, though
for very different reasons. There is also a gap in the sequence at 6.

It occurs to me that I might make my points concerning 22-tone equal
easier to understand by giving at least the first few fifths of the
system. I start on C, purely out of reverence for tradition:

etc F C G D
A E B F# etc
491 0 709 218
927 427 1145 654

Note the very narrow diatonic semitone between E & F, and the very wide
chromatic semitone between F & F#. It occurs to me that Pythagoreans
bored with their chosen system, and who want to _really_ get hold of
some mistuned intervals, might love all this.


SMTPOriginator: tuning@eartha.mills.edu
From: "Paul H. Erlich"
Subject: Gibson's modes
PostedDate: 12-12-97 14:41:37
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