History of the Diatonic Scale

From: PAULE

I do agree that in most cases the diatonic scale existed without a
standardized tuning. I believe that the creation of similar tetrachords was
the only guiding principle in most cases, since I do not believe that
5-limit or higher ratios are relevant to purely melodic music. The 3-limit
(and even 2-limit) approximations have a wide range of tolerance when
harmony is not an issue, or when inharmonic instruments make the value of
just intonation questionable. Many of the pentatonic scales of Southeast
Asia and Africa exhibit two identical or nearly identical "trichords" each
spanning a perfect fourth. A pentatonic scale with only two step sizes, such
as in China or Thailand, obeys the principle of two identical trichords in
every octave species.

What is "mystical" about Ptolemy is that he insisted on superparticular
ratios and a geocentric universe with purely circular motions as if these

were some sort of revealed truths. They ended up requiring exceedingly
complicated models to approach the fairly simple realities. It is just as
fair to criticize his views as it is to criticize Aristotelian physics.
Where would the world be if no one had ever criticized Aristotelian physics?
If a schoolteacher were to teach Aristotelian physics today, would it not be
fair to criticize them? If the Greeks had not their absolutist
Pythagorean/Platonic insistence on ratios in explaining everything, to the
extent that the discovery of irrational numbers was a forbidden secret,
would they not have progressed farther in mathematics and science? In
particular, would some mathematician not have interpreted Aristoxenus' ideas
in terms of the twelfth root of two? I'm sorry if it's not politically
correct to identify the intellectual stumbling blocks of past civilizations.

Matt, you could be given a Mozart string piece which would have to descend
by, say, seven commas from beginning to end, if it is to have all chords in
just intonation and no comma shifts in sustained tones (this is actually a
typical scenario). Even though the beginning and ending keys may be notated
exactly, you would say that the piece ends in a distincly different key than
it began. Would your analysis reflect the musical reality better than the
traditional analysis? I think not! What if the piece was a keyboard piece?
Forget it!

Perhaps your platonic ideal is just intonation, and mine is a fixed diatonic
scale. Perhaps we both need to loosen up a bit, and admit that both are
sometimes compromised in favor of the other. The point of compromise can
depend on such factors as the tempo of the piece. Slow tempi demand just
intonation, while fast tempi demand symmetrical, digestible melodic
patterns. The best musicians intuitively understand both ideals and how to
compromise between them according to the musical situation. An analysis that
purports to describe pitch in more accurate terms than what is actually
notated would have to take these issues into account, or else it is nothing
more than an exercise in Pythagorean mysticism.

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John Chalmers wrote:
>
> 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.

Ptolemy rejects the 81/64 ditone as a possible melodic (i.e. an interval
between two adjacent degrees) in his "rational systems", since it is epimeric
(i.e. not superparticular). The only superparticular melodic approaching this
magnitude is the 5/4. However, as I've said before, Ptolemy distinguishes
between the generic pattern arising in singing, and that which players of
kithara and lyra prefer to follow: namely, the intense diatonic (10/9, 9/8,
16/15 downwards) in the former case, and the ditonic diatonic (9/8, 9/8,
256/243) in the latter. However, the 5/4 is only a composite interval made up
of two melodics in the intense diatonic; it is only available as a melodic in
the context of the enharmonic tetrachord, and of the few sources we have,
there appears to be a consensus in favour of 5/4 as the leading ratio, which
is endorsed by Archytas, Didymus and Ptolemy, although they differ in their
division of the remaining pyknon. One of the grounds on which Ptolemy
objected to Archytas's divisions was the retention of 28/27 as the following
(i.e. lowest) interval in all three genera, a feature which was contrary to
practice (and thus unacceptable, in the context of Ptolemy, because it is
evidence of reasoning which has not been submitted to empirical testing).
Ptolemy also provides us with important evidence for a trend in favour of the
more tense tetrachords ("more tense" meaning that the mutable inner degrees
of the tetrachord were tuned higher in relation to the fixed outer degrees),
since he tells us that his enharmonic and soft chromatic, though constructed
in accordance with reason, were unfamiliar to experience (by his time), and
so the intense chromatic is the softest of all the tetrachords featured in
Ptolemy's careful description of the tunings that were actually used in his
time (it appears in the tropoi kithara tuning, mixed, of course, with the
even diatonic).

--
Jonathan Walker
Queen's University Belfast
mailto:kollos@cavehill.dnet.co.uk
http://www.music.qub.ac.uk/~walker/



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