Re: [tuning] : The C-Fb-G major triad: Pythag-Just tuning.2

> From: Paul Erlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Monday, December 03, 2001 2:49 PM
> Subject: [tuning] Re: Re : The C-Fb-G major triad: Pythag-Just tuning.2
>
>
> Pythagorean tuning is clearly based on a chain of fifths. The Arabic
> theorists simply lengthened this chain, perhaps to justify scales
> already in use that didn't fit the Greek mold. There is
> circumstantial evidence that they stumbled upon the schisma (much as
> the West did later, around 1420), since they used it to construct,
> essentially, 5-limit just scales.

(Those who have trouble following the Greek terminology here
my get some help from my "Tutorial on ancient Greek Tetrachord-theory"
<http://www.ixpres.com/interval/monzo/aristoxenus/tutorial.htm>).

In my unpublished paper "An Examination of a Possible 5-Limit System
of Boethius" (1997), the conclusion of which appears in my book
_JustMusic: A New Harmony_, I examine the Greek-letter notation
used by Boethius to describe the modes.

There, I assume that the ratios described by Didymus are used for
the diatonic genus, and construct a lattice incorporating all of
the pitches which occur in the complete set of all the modes.

I note how pitches which are separated by a syntonic comma always
have different symbols, but those separated by a skhisma always
have the same symbols.

Boethius's actual theoretical tuning of the diatonic genus supposedly
would have been entirely Pythagorean (3-limit), but as Nicomachus
wrote, c. 100 AD:

>> [Barker 1989, p. 265]
>>
>> ... [the _synemmenon_ tetrachord] begins with its own _trite_
>> a semitone away from _mese_, then, after a tone, has a _paranete_
>> peculiar to itself, then, after another tone, has the _nete
>> synemmene_, which is in all respects of the same tension and sound
>> as _paranete diezeugmenon_.

And this distinction is indicated in Boethius's Greek-letter notation.

Much later (c. 900 AD), Hucbald directly contradicted this:

>> [Babb 1978, p 33]
>>
>> ... the _paranete synemmenon_ is the same in sound as _trite
diezeugmenon_.

which shows that the basic diatonic tuning had changed by his time.

The only tuning system which would have been a good candidate for
Boethius's time which gives pitches matching his descriptions and
notations, is that of Didymus (c. 50 BC - 100 AD), where the
protypical tetrachord is [descending]:

mese A n^0 1:1
> -9:8
lichanos meson G 3^-2 16:9
> -10:9
parhypate meson F 5^-1 8:5
> -16:15
hypate meson E 3^1 3:2

Boethius's description and diagrams of the modal system are
in his Book 4, chapters 16 and 17 [Bower 1989, p 154-160].
A link to the Latin text is given below.

Constructing lattices for each mode notated by Boethius results
in a general lattice for the whole system results in pairs
of pitches which are separated by a skhisma (~2 cents) but notated
with the same letter, one of them occuring with the pair 3^3
(= 27:16 = ~906 cents) and 3^-5 * 5^-1 (= ~904 cents), and another
with the pair 3^2 (= 9:8 = ~204 cents) and 3^-6 * 5^-1 (= ~202 cents).

There are several pairs of pitches separated by a syntonic comma
(= ~21.5 cents), and in every case the two notes have different
symbols representing them.

Taken together, I believe the evidence of the Greek-letter notation
indicates that musicians of the late classical and early post-classical
period (c. 100 BC - 500 AD):

1) utilized a system of pitches related as 5-limit ratios, or
at least tuned closely enough to them to imply them,

2) recognized the "enharmonic equivalence" (in the modern sense)
of pitches separated by a skhisma,

3) recognized a distinction between pitches separated by a
syntonic comma.

If I'm correct that ancient Greek musicians *in practice* (i.e.,
in their actual musical notation) recognized the skhisma, then
it's no surprise at all that later Arab music-theorists picked
it up, as they did so many other ancient Greek concepts.

REFERENCES
----------

Barker, Andrew (ed.). 1989.
_Greek Musical Writings.
Vol. 2: Harmonic and Acoustic Theory._
Cambridge Readings in the Literature of Music,
Cambridge Univ. Press, Cambridge.

Bower, Calvin M. (ed., trans.). 1989.
_Boethius: "Fundamentals of Music"_.
Music Theory Translation Series, ed. Claude Palisca.
Yale University press, new Haven & London.
ISBN: 0-300-03943-3
L.o.C.#: MT5.5.B613

Boethius, Anicius Manlius Severinus. c. 505.
in Friedlein, Godofredus (ed.). 1867.
_Boethii De institutione musica libri quinque_.
B. G. Teubner, Leipzig.
http://www.music.indiana.edu/tml/6th-8th/BOEMUS4_TEXT.html

Hucbald of St. Amand. c. 880.
_De harmonica institutione_.
E. de Coussemaker (ed.), Durand, Paris, 1866-1876.
http://www.music.indiana.edu/tml/9th-11th/HUCHAR_TEXT.html

Warren Babb (trans.). 1978.
_Hucbald, Guido, and John on music : three medieval treatises_.
Edited, with introductions, by Claude V. Palisca;
index of chants by Alejandro Enrique Planchart.
New Haven : Yale University Press.
(Music Theory Translation Series, 3)
ISBN: 0300020406
L.o.C.#: ML170 .H82

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