source file: m1419.txt
Date: Sun, 17 May 1998 14:17:11 -0700 (PDT)

Subject: Tetrachordal scales

From: John Chalmers <non12@deltanet.com>

Mark: The ancients Greeks in about the end of 5th century BCE analysed the
scales of their own music in terms of two tetrachords and a disjunctive
tone.
If one tunes the diatonic tetrachord (E F G A) to 16/15 x 9/8 x 10/9 or
1/1 16/15 6/5 4/3 as Ptolemy proposed about 160 CE and add another
identical
tetrachord transposed up a 3/2 (B C D e), one obtains the Dorian mode,
the principal mode used by the Greeks. The C or Lydian mode of this scale
is our major mode in JI and it does seemingly have dissimilar tetrachords.
However, if you look at the actual sequence of intervals, you will observe
that their are two 16/15 x 9/8 x 10/9 tetrachords and a 9/8 tone in a
cyclically permuted order.

(To get the natural minor mode, use the tetrachord 10/9 x9/8 x 16/15. In
JI, the A (Hypodorian) mode of the 16/15 x 9/8 x 10/9 tetrachord is not the
same as the natural minor scale as it is in Pythagorean or 12-tet.)

In addition to many diatonic tunings, the Greeks  had two other principal
genera in the 4th century BCE. The chromatic genus consisted of roughly
two semitones and a minor third  ( E F Gb A), e.g. 16/15 x 25/24 x 6/5 (a
later tuning of Eratosthenes), and the enharmonic genus contained two
microtones and a major third, the most harmonious of these tunings being
that of Archytas who defined the genus as  28/27 x 36/35 x 5/4 (E F- Gbb-
A).

Tetrachordally-analyzable scales are found in Islamic, European, and Indian
musics today.

--John