diatonic scale

Re Hermann Pedtke's keyboard: The keyboard described by Pedtke
in his Xenharmonikon 6 article on meantone (Summer 1977)is the
two-manual Scalatron, not the generalized keyboard version. I think
Brian is referring to Hans Luedtke's two patents for keyboards with
hexagonal digitals, #2,061,364 of Nov. 17, 193 and # 2,003,894
of June 4, 1935.

I am unaware of the Paris keyboard.

RE the diatonic scale: I think our emphasis is a bit misplaced.
As I understand European musical history, Pythagorean tuning
and the rather artificial system of modes began to break down
during the Middle Ages as Musica Ficta tended to erase the
modal differences and the intonation softened towards just thirds
and sixths, at least in English music, according to Walter of
Odington.

This tweaking went on for several centuries and eventually in
the 18th century Rameau analysed the diatonic scale in terms of triads
and theory caught up with practice. However Barbour showed that tuners
in the 15th and 16th centuries, when Pythagorean tuning was being
replaced by meantone, 12-TET and various irregular systems, devised
complete chromatic octaves (12 notes) not merely the seven diatonic
tones. Thus the diatonic scale was never viewed wholly in isolation
from its modes and transpositions.

No doubt by this time the diatonic scales were considered to be
constructed of triads as all the chromatic tones were harmonized
by triads, but most probably musicians thought in terms of five or
six chords, not just three primary ones, though a specialist on early
music could advise me here. I think it was Rameau's insight that the scale
could be construed as the union of just three primary triads rather
than 5 (including the conjugate ones on degrees 3 and 6) that led
to the myth that it was deliberately constructed with this
in mind. (I'm ignoring the supertonic triad deliberately.)

It seems to me to be a coincidence that Ptolemy's Intense Diatonic
in the Greek Lydian mode provides 4:5:6 triads on 1/1, 4/3 and 3/2.
There is little evidence that the Greeks were aware of this fact
and there were other diatonic tunings in favor without this property.
The tuning does NOT have a mode with 3 minor (10:12:15) triads on
these roots, however and neither does any other Greek tuning. It seems
therefore, that the major-minor system presupposes some sort of tuning
with acceptable thirds that ignores the syntonic comma. Meantone, ET
and various irregular systems do just this.

(Redfield's just scale does have this property as it consists of three
minor triads on 1/1, 4/3 and 3/2. Every mode of this scale contains a
note differing by a comma from a Ptolemaic mode. The generating tetrachord
is 10/9 x 9/8 x16/15 rather than the Greek 16/15 x 9/8 x 10/9. )

I must say that I think Rameau's insight is very valuable for
contemporary
scale builders. I generated a number of analogous scales using various
kinds of triads and mixtures in the 60's, but did very little with them
save play a few on Partch's chromelodeon until I spoke with Max Mathews
and saw David Lewin's article on Generalize Tonal Functions. As most
scale builders soon find out, it is no easy task to find a harmonically
generated scale with the melodic strength and logic of the diatonic
scale.

--John


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