History of the Diatonic Scale - Aristoxenus and Ptolemy

Manuel Op de Coul wrote:

> 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.

I thought this old chestnut concerning Aristoxenus had been laid to rest;
clearly I was wrong. The adoption, in the 16th century of Aristoxenus as the
father of equal temperament tells us much about the ideas of 16th century
musici looking for antique sources to justify their intuitions about lute
and viol tunings, but it says little or nothing of value regarding the ideas
of Aristoxenus himself. In constructing his tetrachords, Aristoxenus accepts
both the 4/3 as the boundary of the tetrachord, and the 9/8 as the tone of
disjunction; if you doubt this, I suggest you read Aristoxenus, together
with Barker's footnotes. Now this leads to a contradiction in the
description of the various tetrachords: the diatonic, for instance consists
of two tones and a semitone, but since he has already accepted the 9/8 tone,
(9/8)^2 * sqrt(9/8) must equal 4/3, which they obviously do not. Don't take
my word for this; if you're short of time, might I suggest one passage in
particular: Elementa Harmonica, Book II, 56-7. Here Aristoxenus attempts to
give a practical proof that two-and-a-half tones equals a fourth, with a
proof that depends on tuning a fifth correctly; the fifth, however, must be
flat by a Pythagorean comma in order to make the proof work. But Aristoxenus
never indicates any such drastic flattening, nor does he even once, in the
whole treatise, ever suggest tempering intervals. Trying to pick your way
through this and other arithmetical contradictions can lead, I am convinced,
to only one honest conclusion: that Aristoxenus should be understood as
providing us with essentially vague, musicianly descriptions, which are
distorted if we begin to squint too hard at the details, and try to endow
them with more mathematical rigour than he ever countenanced -- this is the
problem with later apologists from Cleonides to the 16th century.
Aristoxenus himself implicitly encourages this attitude, when he states
that, in the manner of his teacher Aristotle, music theory should be
regarded as an autonomous discipline, relying on its own concepts rather
than being subordinate to, say, acoustics or to the arithmetic of the
rationals. I don't personally endorse this approach, but I don't think it's
intellectually dishonourable; but twisting Aristoxenus in directions he
never intended is a distinctly more dubious pastime.

If, for the sake of argument, we follow Aristotle and Aristoxenus in taking
music theory as an autonomous discipline, then of course we will think in
terms of adding or subtracting intervals, rather than multiplying or
dividing ratios. Aristoxenus's diatonic tetrachord, with its two equal-sized
tones, would appear to be nothing other than the standard Pythagorean
diatonic (the "ditonic diatonic" in Ptolemy's list) -- the only difference
being that Aristoxenus eschews the Pythagorean ratio description. In this
case, we can argue our way from Pythagoras to Aristoxenus along the
following path:

9/8 * 9/8 * 256/243 4/3

log(9/8) + log(9/8) + log(256/243) 4/3

tone + tone + limma diatessaron (fourth)

and if we are to take Aristoxenus as a faithful reporter of contemporary
musicians' thinking, the limma was taken to be half of a tone, so:

tone + tone + semitone fourth.

But we must note that this was the informal thinking of people who are
concerned with their art rather than with theoretical niceties; it was
certainly not a matter of sophisticated mathematical reasoning far in
advance of its time, involoving irrational quantities in the specification
of the semitone, whether (9/8)^(1/2) or 2^(1/12); nor does Aristoxenus
provide any such reasoning.

> 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.

What is a 2-limit approximation of a diatonic scale. The "2-limit" gives us
nothing but octaves. Or did you mean 2^1/x, in which case the "limit"
terminology is out of place.

> 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?

Reading the first two books of Ptolemy's Harmonics certainly doesn't suggest
a mystical approach to the description of Greek tuning practices; Ptolemy
remains faithful to the approach he gives at the beginning: the testing of
all reasoning by empirical means. The superparticulars 2/1 (which is also
multiple) 3/2, 4/3 and 9/8 were not challenged by any Greek theorist;
Ptolemy reasoned that all melodics (i.e. conjunct intervals in tetrachords)
should similarly consist of superparticulars, on the assumption that an
observed pattern should not be thrown aside at the first hint of
uncertainty, but should be explored further to see if it will yield further
results that stand up to empirical testing. We can carry out this project
with greater rigour than Ptolemy, who could only rely upon his ears. But we
see in various places that he certainly didn't treat the superparticular
principle as a sacred truth: witness his readiness to grant that the intense
diatonic tetrachord used in singing was approximated by the still more
easily tunable ditonic diatonic, even though the latter involved a
non-superparticular ration (namely the 256/243). (Another of Ptolemy's
departures from any doctrinal presentation is the listing of the six tunings
customarily used in his time by practising musicians: four of the six
combine two different tetrachordal tunings.)

> 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!

Leopold Mozart, being representative of a 17th/18th-century string playing
tradition, specified 1/6th-comma meantone as the tuning which expert players
should employ. His son, it seems, endorsed this (see the Chesnut article
"Mozart's Teaching of Intonation" in JAMS, c.1980 -- sorry about the vague
date), and with it the corollory of tuning flats higher than sharps that
mean-tone tunings share with just intonation. Since string players are
limited by the tuning of their open strings, they cannot incorporate any
irreversable comma shifts (as opposed to some local fluctuation), but the
use of any meantone tuning removes this problem, by means of their different
characteristic compromises. As for a Mozart keyboard piece, equal
temperament was in the process of supplanting well-tempered systems for
keyboards in the German-speaking lands during Mozart's lifetime -- what of
it. Just intonation of the type requiring comma shifts (i.e. prohibiting
intervals such as 40/27) has two obvious home grounds: a capella singing, in
which it can be employed intuitively, and in electro-acoustic music, where
it can be applied by calculation. Any other application requires special
efforts; Johnston string quartets, yes, but Mozart, no.

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