Aristoxenos and temperament

I seem to be a member of a dwindling band who still think Aristoxenos
did not invent ET or intend it for Greek music. I will grant grant that
his statements that the Tone is the difference between Fifth and the
Fourth and that the Fourth is equal to 2 and 1/2 such tone formally
describe 12-fold divisions of the octave according to our contemporary
understanding. However, I don't think that he ever intended to divide
the octave into 12, 24, 36, 48, 72 or 144 equal steps, the latter
division being necessary to include the hemiolic chromatic genus (3/8 +
3/8 + 7/4 tones) and a couple of genera rejected as unmelodic in one
temperament.

In the first place, the octave was less important to Aristoxenos than
the the tetrachord, its interior lesser intervals, and the species of
the fifth. Secondly, he spoke only of parts of the Tone and held that
numbers were unnecessary to define tunings as this basic interval and
its subdivisions into fourths, thirds, 3/8's, and halves could be
determined by the trained ear alone. He also said that the tunings of
the intervals within the tetrachordal framework were potentially
infinite and that he was describing only the best-known varieties. In
this I believe is reason to credit Aristoxenos with being the first
psychoacoustician rather than the inventor of temperament.

The description of Aristoxenos's tunings in terms of a fourth divided
into 30 segments is due to a later writer, Cleonides. Granted, this
device does suggest a equal temperament of 72 tones/octave or, to
Kathleen Schlesinger, a tuning based on the divisions of the 9/8 as a
means of approximating the just ratios of her "Harmoniai." I believe,
however, both interpretations are funamentally wrong.

Greek mathematicians always used a decimal-coded sexigesimal notation
(base-60) originally derived from the Sumerians via the Babylonians. In
this system, the 2/1 is conveniently written as 120/60 and the harmonic
framework 2/1 3/2 4/3 1/1 thus becomes the series of string lengths 120
90 80 and 60 units. The ratio 120/90 is 4/3 and 120-90 = 30. I believe
this is the source of Cleonides's 30 parts to the fourth and the main
reason we interpret Aristoxenos's tunings as equal temperament.

This mode of writing the string lengths corresponding to the notes of
the various scales was used by Eratosthenes, the director of the Library
at Alexandria and the first to accurately measure the earth's
circumference, to describe his own tunings, which are numerical
approximations of Aristoxenos's genera. Eratosthenes, however, made the
fundamental mistake of subtracting Cleonides's "parts" from those of a
real string of 120 units to define his genera. While this linear
approximation works reasonably well for the enharmonic (120 117 114 90
or 3 + 3 + 24 parts), the soft and hemiolic chromatic genera are
distorted rather badly and higher primes are
introduced into what was probably meant to be a 3, or at most 7, limit
tuning. Ptolemy perpetuated this misunderstanding in his catalog of
historical tunings in the Harmonics and as result Aristoxenos's tunings
are often misinterpreted by novice xenharmonists.

The final reason I have for not believing that Aristoxenos invented ET
to describe his tunings is that none of the Greek theorists, many of
whom are quite strict Aristoxenians or who combined his and Ptolemy's
ideas, ever seem to have computed the string lengths for even 12-tone
ET. Such computations were well within the capabilities of Greek
mathematics; Archytas himself, who introduced 7 limit tunings into Greek
theory, was famed for a 3-dimensional construction for extracting the
cube root of 2, a solution to the Delian problem of the duplication of
the cube. Tables of square and cube roots are known from the Babylonians
and Greek mathematicians were certainly capable of taking the the
required number of square roots of the cube root of two if they thought
doing so was musically significant.

As a parting shot, let me say that I think that the precision of these
notated tunings is quite illusory given the vocal basis of Greek music
and the mechanical simplicity of Greek instruments. Equal temperament
would have been close enough for contemporary performances in
Aristoxenos's tunings and probably for the JI-based ones of Archytas,
Didymos, and Ptolemy as well.

--- In tuning@y..., John Chalmers <JHCHALMERS@U...> wrote:

Secondly, he spoke only of parts of the Tone and held that
> numbers were unnecessary to define tunings as this basic interval and
> its subdivisions into fourths, thirds, 3/8's, and halves could be
> determined by the trained ear alone.

By "numbers" he presumably meant positive integers and their ratios.
The Greeks were perfectly capable of understanding the idea of dividing an octave (or tetrachord) into equal parts, but would not call that using numbers.

> Greek mathematicians always used a decimal-coded sexigesimal notation
> (base-60) originally derived from the Sumerians via the Babylonians.

Hardly always.

> The final reason I have for not believing that Aristoxenos invented ET
> to describe his tunings is that none of the Greek theorists, many of
> whom are quite strict Aristoxenians or who combined his and Ptolemy's
> ideas, ever seem to have computed the string lengths for even 12-tone
> ET.

They probably wouldn't, and even if they did, they would mostly likely use a geometrical approach to the problem and not Babylonian style tables. This whole thing is supposed to be a theoretical framework for understanding what in practice you do by ear, something which is very Greek; why expect them to treat the problem the way we would?

> As a parting shot, let me say that I think that the precision of these
> notated tunings is quite illusory given the vocal basis of Greek music
> and the mechanical simplicity of Greek instruments. Equal temperament
> would have been close enough for contemporary performances in
> Aristoxenos's tunings and probably for the JI-based ones of Archytas,
> Didymos, and Ptolemy as well.

That I seriously doubt; it is quite clear the different microtonal flavorings were of great importance to them.

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., John Chalmers <JHCHALMERS@U...> wrote:
>
> > Equal temperament
> > would have been close enough for contemporary performances in
> > Aristoxenos's tunings and probably for the JI-based ones of
Archytas,
> > Didymos, and Ptolemy as well.
>
> That I seriously doubt; it is quite clear the different microtonal
>flavorings were of great importance to them.

i think john meant 72-tone or 144-tone equal tempermant, not 12. john?