Re: High Primes in Greek Theory

Joe and Allison:

I feel that Partch made too much of the occasional use of cetain ratios
off 11 and 13 in Ptolemy's catalog of tunings and in Schlesinger's harmoniai.
There is no convincing evidence that Schlesinger's aulos scales were ever
part of Greek music and little evidence that the Greeks called any intervals
played simultaneously consonant other than the octave, fifth, fourth and their
octave compounds. There is a late author, the reference to whom I cannot now
find who, who called thirds and tritones "paraphonic" because they had
some, but
not all, the aural character of the true consonances. Nevertheless, I
think it
exaggeration to claim that the Greeks used ratios of 11, 13, 17, 19,
etc. as
consonances in the modern sense or had assimilated them to their musical system.

Basically, I think that the appearance of high primes in Eratosthenes's
tunings is a fortunate artifact of his use of sexigesimal notation to
express Aristoxenos's tetrachords as his enharmonic and chromatic
tunings are identical to those of Aristoxenos expressed in this manner.
Ptolemy, who was primarily an astronomer/astrology, also used this convention.

Eratosthenes, I believe, erroneously tried to express Cleonides's
versions of
Aristoxenos's genera as fractions of a string of 120 parts. The 120
rather than
100 or some other convenient length was chosen, IMO, because of the
Greek tradition
of doing serious mathematics in the decimal-coded sexigesimal (base-60)
notation
deriving from Babylonian astronomy rather than in pure decimals. Happily
the
tetrachordal framework can be expressed this way in whole numbers as
120 90 80 60, giving a lower fourth (120/90) divided by integers into 30
parts
in accordance with Cleonides's descriptions. When one subtracts
Cleonides's
parts from 120, one gets string lengths whose successive ratios involve
the higher primes automatically as seen below from Ptolemy's catalog.

Enharmonic: 3 + 3 + 24 parts to the Fourth,.
120 117 114 90 or 40/39 x 39/ 38 x 19/15,
50 + 50 + 400 cents vs 44 + 45 + 409,
virtually the same as the quasi-Pythagorean tuning
of Boethius: 512/499 x 499/486 x 81/64 or
45 + 46+ 408.

Soft Chromatic: 4 + 4 + 22 parts
120 116 112 90 or 30/29 x 29/28 x 56/45
59 + 61 + 379 cents, closer to an enharmonic
genus than the canonical 67 + 67 + 367 cents

Hemiolic Chromatic: 4.5 + 4.5 + 21 parts
120 115.5 111 90 or 80/77 x 77/74 x 37/30
66 + 69 + 363, a very good Soft Chromatic tuning!
A closer fit would be the non-extant 5 + 5 + 20 parts
120 115 110 90 or 24/23 x 23/22 x 11/9,
74 + 77 + 347 cents for the canonical 75+ 75+ 350, but this
tuning appears only in Winnington-Ingram's discussion, AFAIK.
Aristoxenos, IMO, wanted to stress that the pyknotic
intervals are audibly intermediate between the enharmonic
and intense Chromatic ones and hence described them as
being 4.5 twelfths of tones wide rather than 5/12.

Tonal Chromatic 6 +6 + 18 parts, identical with Eratosthenes's tuning
120 114 108 90 or 20/19 x 19/18 x 6/5
89 + 94 + 316 for 100 + 100 + 300
Compare to the Pythagorean 256/243 x 2187/2048 x 32/27.

Soft Diatonic : 6 + 9 + 15 parts
120 114 105 90 or 20/19 x 38/35 x 7/6
89 + 142 + 267 for the canonical 10 + 150 + 250 cents.

Intense Diatonic: 6 + 12 +12 parts
120 114 102 90 or 20/19 x 19/17 x 17/15
89+ 193 + 217 for the Pythagorean 90 + 204 + 204.
Eratosthenes proposes the Pythagorean tuning instead of the
pseudo-aristoxenian version above.

There is evidence independent of these scales for the widespread use of
3/4 tone intervals and these may be approximated by the intervals 12/11
and 11/10. Although Robert Erickson derives the 3/4 tone pyknon
of the hemiolic chromatic by 7 limit intervals, I think it was soon
discovered
that 12/11 is good approximation to this interval. Hence it would be
quite
natural for Ptolemy to use it to notate a neutral diatonic tetrachord as
120 110 100 90 when he heard it played. That it was a Egyptian or Near
Eastern
folk tuning is suggested by his epithets, "rather foreign and rustic."
Similar
tetrachords are still heard today from Greece and Nubia to Iran (G.
Bilalis thinks
that Greek Orthodox liturgical music uses the 9 + 9 + 12 part
tetrachord in 72-tet).

Winnington-Ingram thinks that the Spondeion, the old Greek libation mode,
later the Spondeiakos tropos, was a neutral-third pentatonic that gradually
became hexatonic and finally heptatonic by splitting the 3/4 tone interval
in each tetrachord. His tuning for the Spondeion would thus be 12/11 x
11/9 x 9/8
x 12/11 x 11/9. After the 12/11's were divided the heptatonic form
became identical
to Aristoxenos's Hemiolic Chromatic. He further suggests that 11/9 and 11/8
may have occurred in the accompaniment, perhaps as paraphonies.

As for the other uses of high-limit ratios in Ptolemy's tunings, I think
he was simply
trying to express the Intense Chromatic and enharmonic with
superparticular ratios.
His tuning for the Intense Chromatic is 22/21 x 12/11 x 7/6, nearly the
same as his (or Eratosthenes's) version of Aristoxenos's Soft Diatonic.
His enharmonic, 46/45 x 24/23
x 5/4, represents a tripartite division of the pyknon and the
recombination of two of the resulting intervals. The same process
accounts for the 11 in the Intense Chromatic:
8 /7 becomes 24/23 x 23/22 x 22/21. Recombining the 24/23 and 23/22 generates
the 12/11 as the 2nd interval of the tetrachord. The tripartite
division may represent
a stylistic peculiarity of Hellenistic or Alexandrian music which
Ptolemy applied to the
extinct enharmonic genus as well as the still played Intense Chromatic.

Didymos's 32/31 x 31/30 division of the 16/15 is simply the bipartite
melodic division
of the lowest interval in the chromatic and diatonic genera. As this
prime is
used nowhere else in Didymos's tunings, I don't think that it represents
the conscious appreciation of 31-ness, but rather an arithmetic
convenience similar in spirit to
Aristides Quintilianus's division of the whole tone as 18/17 x 17/16.

--John

--- In tuning@y..., John Chalmers <JHCHALMERS@U...> wrote:
> Joe and Allison:
>
> I feel that Partch made too much of the occasional use of cetain
ratios
> off 11 and 13 in Ptolemy's catalog of tunings and in Schlesinger's
harmoniai.
> There is no convincing evidence that Schlesinger's aulos scales
were ever
> part of Greek music

While I tend to agree, Kraig Grady has a dissenting viewpoint on this
issue. Kraig?

> There is evidence independent of these scales for the widespread
use of
> 3/4 tone intervals and these may be approximated by the intervals
12/11
> and 11/10. Although Robert Erickson derives the 3/4 tone pyknon
> of the hemiolic chromatic by 7 limit intervals, I think it was soon
> discovered
> that 12/11 is good approximation to this interval. Hence it would
be
> quite
> natural for Ptolemy to use it to notate a neutral diatonic
tetrachord as
> 120 110 100 90 when he heard it played. That it was a Egyptian or
Near
> Eastern
> folk tuning is suggested by his epithets, "rather foreign and
rustic."
> Similar
> tetrachords are still heard today from Greece and Nubia to Iran (G.
> Bilalis thinks
> that Greek Orthodox liturgical music uses the 9 + 9 + 12 part
> tetrachord in 72-tet).
>
> Winnington-Ingram thinks that the Spondeion, the old Greek libation
mode,
> later the Spondeiakos tropos, was a neutral-third pentatonic that
gradually
> became hexatonic and finally heptatonic by splitting the 3/4 tone
interval
> in each tetrachord. His tuning for the Spondeion would thus be
12/11 x
> 11/9 x 9/8
> x 12/11 x 11/9. After the 12/11's were divided the heptatonic form
> became identical
> to Aristoxenos's Hemiolic Chromatic. He further suggests that 11/9
and 11/8
> may have occurred in the accompaniment, perhaps as paraphonies.
>
> As for the other uses of high-limit ratios in Ptolemy's tunings, I
think
> he was simply
> trying to express the Intense Chromatic and enharmonic with
> superparticular ratios.
> His tuning for the Intense Chromatic is 22/21 x 12/11 x 7/6, nearly
the
> same as his (or Eratosthenes's) version of Aristoxenos's Soft
Diatonic.
> His enharmonic, 46/45 x 24/23
> x 5/4, represents a tripartite division of the pyknon and the
> recombination of two of the resulting intervals. The same process
> accounts for the 11 in the Intense Chromatic:
> 8 /7 becomes 24/23 x 23/22 x 22/21. Recombining the 24/23 and
23/22 generates
> the 12/11 as the 2nd interval of the tetrachord. The tripartite
> division may represent
> a stylistic peculiarity of Hellenistic or Alexandrian music which
> Ptolemy applied to the
> extinct enharmonic genus as well as the still played Intense
Chromatic.
>
> Didymos's 32/31 x 31/30 division of the 16/15 is simply the
bipartite
> melodic division
> of the lowest interval in the chromatic and diatonic genera. As
this
> prime is
> used nowhere else in Didymos's tunings, I don't think that it
represents
> the conscious appreciation of 31-ness, but rather an arithmetic
> convenience similar in spirit to
> Aristides Quintilianus's division of the whole tone as 18/17 x
17/16.

I am in complete agreement with John Chalmers here.

John and Paul! ( I won't say it!)

Paul Erlich wrote:

> --- In tuning@y..., John Chalmers <JHCHALMERS@U...> wrote:
>
> > There is no convincing evidence that Schlesinger's aulos scales
> were ever
> > part of Greek music

well there is the existence of the instruments themselves that
Schlesinger had in her possession as well as the instrument Jim French
had rebuilt based on measurements. Considering that we are talking about
such a long period in a Polytheistic Culture. It seem more than probable
that there were many many approaches to tuning, many i assume were not
documented.

>
> > There is evidence independent of these scales for the widespread
> use of
> > 3/4 tone intervals and these may be approximated by the intervals
> 12/11
> > and 11/10.

Unless their is major omissions in Greek Musical Writings:II Barker.
Cambridge Univ. Press. It seem intervals were thought more in terms of
ratios than the divisions of Aristoxenus, who appears to be the
exception more than the rule.

> Although Robert Erickson derives the 3/4 tone pyknon
> > of the hemiolic chromatic by 7 limit intervals, I think it was soon
> > discovered
> > that 12/11 is good approximation to this interval. Hence it would
> be
> > quite
> > natural for Ptolemy to use it to notate a neutral diatonic
> tetrachord as
> > 120 110 100 90 when he heard it played. That it was a Egyptian or
> Near
> > Eastern
> > folk tuning is suggested by his epithets, "rather foreign and
> rustic."
> > Similar
> > tetrachords are still heard today from Greece and Nubia to Iran (G.
> > Bilalis thinks
> > that Greek Orthodox liturgical music uses the 9 + 9 + 12 part
> > tetrachord in 72-tet).

I think it is common to think of Greece as part of Europe when it is
just as much a part of the Mid East and North Africa. IN fact , I can
not tell you how much African vocal music I find sounding Greek to me.
The Kora resembles the Lyre in more ways than one. At the time of the
greeks most of Europe was still occupying its time with human sacrifice.

> > Winnington-Ingram thinks that the Spondeion, the old Greek libation
> mode,
> > later the Spondeiakos tropos, was a neutral-third pentatonic that
> gradually
> > became hexatonic and finally heptatonic by splitting the 3/4 tone
> interval
> > in each tetrachord. His tuning for the Spondeion would thus be
> 12/11 x
> > 11/9 x 9/8
> > x 12/11 x 11/9. After the 12/11's were divided the heptatonic form
> > became identical
> > to Aristoxenos's Hemiolic Chromatic. He further suggests that 11/9
> and 11/8
> > may have occurred in the accompaniment, perhaps as paraphonies.
> >
> > As for the other uses of high-limit ratios in Ptolemy's tunings, I
> think
> > he was simply
> > trying to express the Intense Chromatic and enharmonic with
> > superparticular ratios.
> > His tuning for the Intense Chromatic is 22/21 x 12/11 x 7/6, nearly
> the
> > same as his (or Eratosthenes's) version of Aristoxenos's Soft
> Diatonic.
> > His enharmonic, 46/45 x 24/23
> > x 5/4, represents a tripartite division of the pyknon and the
> > recombination of two of the resulting intervals. The same process
> > accounts for the 11 in the Intense Chromatic:
> > 8 /7 becomes 24/23 x 23/22 x 22/21. Recombining the 24/23 and
> 23/22 generates
> > the 12/11 as the 2nd interval of the tetrachord. The tripartite
> > division may represent
> > a stylistic peculiarity of Hellenistic or Alexandrian music which
> > Ptolemy applied to the
> > extinct enharmonic genus as well as the still played Intense
> Chromatic.
> >
> > Didymos's 32/31 x 31/30 division of the 16/15 is simply the
> bipartite
> > melodic division
> > of the lowest interval in the chromatic and diatonic genera. As
> this
> > prime is
> > used nowhere else in Didymos's tunings, I don't think that it
> represents
> > the conscious appreciation of 31-ness, but rather an arithmetic
> > convenience similar in spirit to
> > Aristides Quintilianus's division of the whole tone as 18/17 x
> 17/16.
>
> I am in complete agreement with John Chalmers here.

I think i have brought up the point in the past that the use of their
different forms of "means "has much more historical precedence than the
idea of limits. In fact in the writings of the day i see absolutely no
mention of it.
This is something that Andrew Barker agrees with me on and shows how the
idea of means influenced the tuning of even Archytas.
Ptolemy also "hints" at the relationships between ratios and
planetary motion. When one looks for a finer and finer "gear ratio to
describe a movement, one falls into a Farey series which is based on
Means. The division into means was also widely used by the Persian
theorist who inherited the Greek tradition and preserved until the the
European a 1000 years later was at a point of being "civilized" to catch
up with their Eastern neighbors.
As much of what actually happen and practiced in the Greeks
city-states
was more often than not, kept within the confines of the closely guarded
secrets of various schools and i think i might be naive to think that
what was written was the whole story, but believe they were careful to
"hint " at the actual methods and concerns in order that it could be
recovered by "initiates"

>

-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com

The Wandering Medicine Show
Wed. 8-9 KXLU 88.9 fm

> From: Kraig Grady <kraiggrady@anaphoria.com>
> To: <tuning@yahoogroups.com>
> Sent: Monday, August 06, 2001 3:39 PM
> Subject: Re: [tuning] Re: High Primes in Greek Theory
>
>
> ... <snip> ...
>
> I think it is common to think of Greece as part of Europe when it is
> just as much a part of the Mid East and North Africa. IN fact , I can
> not tell you how much African vocal music I find sounding Greek to me.
> The Kora resembles the Lyre in more ways than one. At the time of the
> greeks most of Europe was still occupying its time with human sacrifice.

I found in my research one day a story about a Greek song having
something to do with lions, which one of the Greek writers said
came from Egypt. This statement was doubted for a couple of
millennia, but sure enough, recent Egyptologists have found
the evidence. (wish I could remember more about this... I'll
post more details if I find my notes on it)

The point being: I support what Kraig says, that ancient Greece
had very close ties to Africa, probably much closer to Africa than
to Europe.

> > > [John Chalmers:]
> > > As for the other uses of high-limit ratios in Ptolemy's tunings,
> > > I think he was simplytrying to express the Intense Chromatic and
> > > enharmonic with superparticular ratios.
> > > <snip>
> >
> > [Paul Erlich:]
> > I am in complete agreement with John Chalmers here.

And for the record, so am I.

> I think i have brought up the point in the past that the use of their
> different forms of "means "has much more historical precedence than the
> idea of limits. In fact in the writings of the day i see absolutely no
> mention of it.
> This is something that Andrew Barker agrees with me on and shows how the
> idea of means influenced the tuning of even Archytas.

In fact, Barker shows that Archytas's use of means was an innovative
idea that was retained for centuries of Greek and Roman theory.
Aristoxenos was indeed the unique exception, altho in a sense
he was using the idea of means too, quantifying it numerically
but *not rationally*.

> Ptolemy also "hints" at the relationships between ratios and
> planetary motion. When one looks for a finer and finer "gear ratio to
> describe a movement, one falls into a Farey series which is based on
> Means.

Now it's *REALLY* interesting that you should post this just now!
Yesterday on the </celestial-tuning>
list, we were having a bit of discussion about both the Antikythera
machine *and* Ptolemy, both of which are relevant here!

(Too much synchronicity lately... it's getting spooky...)

love / peace / harmony ...

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

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