5-limit harmony, tritriadic scales

Re 5-limit harmony: Five and seven limit scales apparently go
back to Archytas (about 390 BCE) in Greece, though I'm not sure
how "harmonic" their treatment was. Aristoxenos (circa 330) complained
that the incomposite ditone in the enharmonic genus was being narrowed,
a process called "sweetening" presumably from the 81/64 to the 5/4.
This sounds very much like Archytas's 28/27 x 36/35 x 5/4
enharmonic tetrachord. However, in the diatonic genus, Archytas's tuning
was septimal,( 28/27 x 8/7 x9/8) skipping 5 altogether, and Aristoxenos
admits that this is a well known tuning (as 1/3 + 1 1/6 + 1 tones).

Ptolemy, nearly 500 years later, implies that this is the most
common diatonic tetrachord, though the Pythagorean 3-limit one is
also in use. His own Syntonic Diatonic, 16/15 x 9/8 x 10/9 may be
contrasted with the earlier Didymos's 16/15 x 10/9 x 9/8 tuning. This
latter scale is rather non-harmonic in our sense. It, Pythagorean
(Ditone Diatonic) and Archytas's diatonic (Ptolemy's Middle Soft
Diatonic, Tonic Diatonic, etc.) all have many 32/27 and 81/64 thirds
(the major scale has only 1). This suggests to me a possible avoidance
of 5 limit thirds until Ptolemy's time. This is also true of his mixed
modes which have predominantly septimal and pythagorean intervals.

Traditionally, the development of 5 limit harmony is ascribed to
late medieval English singers (fa-bourdon,etc.). I don't know of any data
supporting 5 or 7 limit harmony in Greece though some late writers
describe 5 and 7 limit intervals as "paraphonies" to distinguish them
from consonances (symphonies) and dissonances (diaphonies). Paraphonies
include, however, the tritone as well as thirds and sixths.

Tritriadics: I would tend to agree that the derivation of the major and
natural minor modes in JI from three triads to be somewhat artificial,
despite Rameau and Ellis (i.e. his 56 tonal modes). However, it is still
useful conceptually, even if comma tweaks be desired in the ii and
VII (in minor) chords or the whole scale mapped into some convenient ET.
The derivation is more plausible in Meantone, as these chords would
be acceptable.

Other "fixes" are possible: one might do as Blackwood recommended for
some ET's, just avoid progressions with ii and VII. One might also make
use of the 117 tones of Ellis's "Duodenarium," the 56 skhismatically
related tones of his "unequally just" tuning or 53-tet.

--John


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