Third-tones, 19-tet, Greek Music

I beg to differ but I think M. Joel Mandelbaum's dissertation is
very original, especially his combined error function for evaluating
temperaments in terms of the 5-limit consonances and particularly his
discovery and compositional use of QEIS scales (moments of symmetry
(well-formed scales) where the intervals sizes are 1 and 2 units of
the temperament).

The cited literature is also very complete, especially for its day
(1961). From it, I learned of Novaro's, Brun's, and other researchers'
works. Other writers on 19 such as Ariel are still virtually
unknown to the xenharmonic community and would have completely so were
it not for Mandelbaum's pioneering work.

It is a shame that someone has not reprinted this book, though I
imagine it is still available through Xerox University Microfilms
in Ann Arbor, MI. I'd love to hear a performance of his pieces on
a synth rather than the two old pianos he was forced to use.

As for the ancient Greek use of 1/3 tones, they are found in
Aristoxenos's Soft Chromatic genus (chroma malakon). This tetrachord
has the ascending intervals 1/3 1/3 11/6 tones (66.67 66.67
366.67 cents). It fits quite well into 19-tet as 1+1+6 degrees.
The other Aristoxenian tetrachord found in 19-tet is the Intense
Diatonic as 2+3+3 (permuted as 3+3+2, 3+2+3 also).

There is also a non-canonical chromatic tetrachord 2+2+4, which
resembles the Byzantine and Islamic genera 16/15 x 15/14 x 7/6 and
14/13 x 13/12 x 8/7. This is the closest approximation possible in
19-tet to the Intense Chromatic 100 100 300 cents.

A just interval approximating 1/3 tone is found in Archytas's and
Plolemy's tetrachords as 28/27. Ptolemy gives 28/27 x 15/14 x 6/5
(Soft Chromatic) and both he and Archytas offer 28/27 x 8/7 x 9/8
(as Ptolemy's Soft Diatonic, A's Diatonic). Archytas's chromatic
28/27 x 243/224 x 32/27 is similar to Ptolemy's. In 19-tet
these tunings would be 1+2+5 and 1+4+3, the first being quite
good, the second recognizable.

However, there is no evidence that the Greeks used any kind of
equal temperament; the ascription of 12, 24, 36, 72...-tet to
Aristoxenos is a modern interpretation. If the Greeks had wanted
0 mod 12 ET's, they were more than capable of calculating them.
Extracting 19th roots would have been much more difficult, but they
could have probably managed by using the mesolabium or some numerical
procedure.


BTW, a transcription of the Costeley Chanson has been published
in 19-tet. Have you ever played and recorded it?


--John


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