Greek triads, Classification of temperaments

I agree with Bill Alves's statements re Greek Music. Some ancient
authors admit that there are intervals which somewhat resemble the
consonances of the 4th, 5th and Octave, but are different. These
intervals are the 3rds, 6ths and the tritone. The 7/6 may have
been an important interval in the enharmonic genus, at least at the
time of Archytas. While the Greeks had a form of heterophonic
accompaniment, I think it's fair to say that there is no good
evidence for triadic harmony or even texture in the ancient world.


The recent discussion on the classification of temperaments,
tunings and scales presupposes that 12-tet is the standard,
that the Fifth is the generator of the scale and that we are
interested in diatonic (7-tone) and chromatic (12-tone) scales.
If say 19-tet and the 12-tone MOS generated by 11 degrees were
the norm, the classification would be somewhat different. Similarly,
one might use a 5-tone subset of 7-tet (Colin Brown's) or a 5-tone
cycle (one of Wilson's patented keyboards). See his articles
in XH2 and XH3, and, of course Bosanquet's original work on the
general theory of the octave and the generalized keyboard.

In the Bosanquetian scheme, temperaments are defined as positive or
negative to various "orders" according to the number of degrees
separating B# from C (or in other words, is the Pythagorean comma
positive or negative and by how many degrees of the temperament?).
Bosanquet calls this parameter "r." The neutral system is 12-tet
where the fifth has 700 cents and the PC (DC) has 0 degrees (r=0).

Another parameter, the number of units separating C from C# defines the
system as "singulary," binary, ternary, etc. according to Wilson's
terminology. This parameter is equivalent to selecting the diatonic
scale as the basic scale in the temperament. For some reason, it
appears to have no accepted name as far as I can tell; one might call
it "s" from "Seven-Fifths' semitone." (Seven-tet would be the
"zero-ary" system where C=C#.)

Relative to 19, the first positive series is 12, 31, 50; relative
to 5, it is 3, 8, 13; and relative to 7, 5, 12, 19. If for some
reason one took 13-tet, 8 degrees and the 8-tone MOS as the norm,
8, 21, 34 and 47 would be the first positive series and 5,18 and
31 the first negative. In each case, the role of the PC is taken
by a chain generators equal to the number of the neutral system
(19, 5,7, 13 etc.) and the

These two parameters are sufficient to classify all temperaments in a
2-D array from which one can deduce a usable notation and keyboard
variety. Thus, this theory is not entirely abstract.

--John


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From: John Chalmers
Subject: phonons, speculation
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