ratios in ancient Greek tetrachords (was: Byzantine and Serb Church Music)

Hi Alison,

My apologies for the long delay in sending this response.
What I have below was all written a day or two after your
post (nearly a month ago), and I've held onto it all this
time because it wasn't finished. At this point I don't know
when it will be, so I'm sending what I have.

> From: Alison Monteith <alison.monteith3@which.net>
> To: <tuning@yahoogroups.com>
> Sent: Saturday, July 07, 2001 12:32 AM
> Subject: Re: [tuning] Re: Byzantine and Serb Church Music
>

>
> Would I be somewhere near the truth in surmising that
> the 11 in the early tetrachords came less from the
> audible effect of 11 than from the mathematical process
> of splitting 12 and 10?

Well... yes and no.

Ptolemy is the earliest writer I know of who used 11 as
a factor in his tuning ratios. As I write in my book:
http://www.ixpres.com/interval/monzo/book/book.htm

> Claudius Ptolemais ... was interested in synchronizing
> the two disparate schools of thought in ancient Greek
> music theory: the Pythagorean, which believed that
> everything harmonious was due to numerical relationships
> (but which was close-minded in that it only accepted
> powers of 3), and the Aristoxenean, which eschewed numbers
> completely,but which also based consonance on the faculty
> of the ear to recognize "smooth" sounds, which we know
> are the result of small-number relationships.

I'm sure that I'll eventually rewrite that last bit, because
we recognize as consonant many intervals which are *not*
small-number ratios, but the small-number ratios still
provide at least a rough guidepost to what we perceive as
the basic consonant intervallic _gestalts_; see Paul Erlich's
"On Harmonic Entropy":
http://www.ixpres.com/interval/td/erlich/entropy.htm

Again from my book:

> Ptolemy's achievement was to recognize relationships
> of the smallest prime numbers which were compatible
> with the required asesthetic effect for any given
> musical system.

(Hmmm... I didn't cite Partch here, but I know that this
is pretty much a quote from _Genesis of a Music_. In any
case, Partch clearly expressed that this was precisely the
reason for his admiration of Ptolemy.)

Ptolemy was indeed a strong supporter of using 5 as a
factor in measuring the "Tense Diatonic" genus on the
monochord (the "Ptolemaic Sequence" of which Partch
speaks so fondly, and which, after Zarlino's and Lippius's
advocacy, became the foundation of harmonic theory in
Europe after c. 1550).

So yes, when Ptolemy needed to find a small-number ratio
which split the 6:5 in half arithmetically, 11 would do
the trick. But note that he felt the need to find that
ratio *because* dozens of different notes, closely related
in pitch-height, which gave a certain audible effect which
we generally associate with 11-limit ratios, *were already
being used in practice*.

The historical situation is not quite as simple as Partch
makes it look in his graph "Diagram 3 - Chronology of the
Recognition of Intervals (_Genesis_, 2nd ed., p 92).
For one thing, Partch only illustrates odd-factors up
to 13, when in fact ancient Greek writers used primes up
to 31, and possibly odd-limits which were even larger.
Secondly, Partch includes Eratosthenes in his graph but
does not give him credit for recognizing 13, when in fact
not only did he utilize 13, but also 19.

Eratosthenes based his system on the prime-factors
2, 3, 5, 13, and 19.

Eratosthenes used certain 19-limit ratios, but audibly
they are basically identical to Pythagorean pitches
and intervals which were already in use in theory and
practice in his time, so his employment of 19 could only
be numerological.

Eratosthenes makes use of a 3==19 xenharmonic bridge
in his scale system, by employing "standard" Pythagorean
3-limit pitches in his diatonic genus, but substituting
19-limit ratios which are very close by in pitch in his
chromatic and enharmonic genera. For example, one trio
of notes (which constitutes one pair of pitches) is:

2 3 5 13 19

| 8 -5 0 0 0 | = 256/243 diatonic trite synemmenenon
- | 2 0 1 0 -1 | = 20/19 chr. trite & enh. paranete syn.
-------------------
| 6 -5 -1 0 1 | = 1216/1215 = ~1.424297941 cents

The bottom row gives the measurement of this particular bridge.
With a pitch-height or interval distance of less than
1 & 1/2 cents, it's very unlikely that anyone ever *heard*
any difference between these notes. (We'd have to use 843-EDO
to have an equal-temperament which gives a degree separating
these two pitches.) So the only reason Eratosthenes could
possibly have for introducing the 19-limit ratios is
numerological considerations.

That this is so, is borne out by an examination of the
string-lengths of Eratosthenes's chromatic and enharmonic
tetrachords (descending in pitch, as was the custom in
ancient Greece, and using the convention of naming "mese"
as "A" and using the _tetrachord meson_ for illustration):

diatonic: 192 : 216 : 243 : 256
A G F E

chromatic: 15 : 18 : 19 : 20
A F# F E

enharmonic: 30 : 38 : 39 : 40
A F Fv E

So what's going on here is that Eratosthenes cleverly found
a way, by making use of the 3==19 bridge, to substitute 6:5
in place of the usual Pythagorean 32:27 as the characteristic
interval in his chromatic genus, and still have his chromatic
and enharmonic tetrachords employ superparticular ratios,
without introducing pitches that deviated too far from the
regular Pythagorean ones in use.

His diatonic "F" is the regular Pythagorean 256/243, but
his chromatic and enharmonic "F"s are both 20/19... and
there is no audible difference.

The chromatic A:F# 15:18 reduces to a 5:6 "minor 3rd",
and the 19-limit "F" is a neat arithmetical mean between
the F#:E 9:10 whole-tone.

Likewise in the enharmonic, the A:F 30:38 reduces to 15:19,
a 19-limit "major 3rd" of ~409.2443014 cents virtually
indistinguishable from the usual Pythagorean 64:81 "major 3rd",
and 39 is an arithmetical mean between the F:E 38:40 = 19:20
semitone. (And, as seen above, this 19:20 is nearly the
same as the usual Pythagorean 243:256.)

See also my post from Wed Dec 22, 1999 2:50 am:
</tuning/topicId_7159.html#7159>,
for more on Eratosthenes's use of 19 in his tetrachord divisions.

Partch mentions that "isolated ratios with even larger
prime numbers" than 13 "are found in the scales compiled
by Ptolemy". As we've seen, Eratosthenes was no slouch
when it came to using both 13 and, even more prominently,
19. But Ptolemy did indeed use a whole slew of higher
numbers in his rationalizations of the scales in use
in his day. His system is based on prime-factors
2, 3, 5, 7, 11, and 23

[long delay in sending post because info on Ptolemy was
never added here. Hopefully I'll eventually post it.]

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

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