Re: Aristoxenos again

Gene: Perhaps all is an unconscious exaggeration, though base-60 is used
in all the documents I've seen. Neugebauer says the
alphabetic-decimal-coded sexigesimal notation was used extensively in
mathematical, astronomical, economic and literary texts. Of course, for
expressing numbers less than 60 and simple fractions greater than 1/60,
it might be hard to tell which system was in use <g>.

I do think, however, the facile equivalence of 120/60 to 2/1 may have
influenced Cleonides and Eratosthenes and explains both Eratosthenes's
genera and Ptolemy's misunderstanding of Aristoxenos's tetrachords.

Conceiving of musical space geometrically isn't quite the same as
computing the string lengths numerically for 72 or 144 tets to
demonstrate them on the monochord or constructing a diagram with
correctly calculated tempered distances.

I just don't think the idea ever occurred to the theorists or if it did,
they didn't think it was worth the trouble to actually do it. The most
comprehensive lists of tunings are in Ptolemy's catalog and he uses
base-60 to express the string lengths and interprets Aristoxenos's
tunings as parts a real string.

By equal temperament, I meant the appropriate temperament from the set
24, 36, 48, 72, 144, not just 12-tet. These could express the microtonal
inflections quite well. In fact, some of Aristoxenos's less well-known
genera - 1/3 + 7/6 + 1 tone and 1/3 + 2/3 + 3/2 tone - are quite close
to Archytas's diatonic (28/27 x 8/7 x 9/8) and chromatic (28/27 x
243/224 x 32/27). One could express Archytas's enharmonic as 4 + 3 + 23
parts as degrees of 72-tet with little distortion.

I really doubt that in performance on a lyra made from a tortoise shell
and strung with natural gut strings or even on a professional's kithara
whose gut strings were attached not with precision-made, worm-gear metal
tuning machines, but with greasy nape-skin, cord, and wooden levers,
that one could tell the difference between strict JI and the nearest ET.
The tuning of auloi would be even more problematic as both ancient
writers and modern re-creators and players have stated. Ptolemy also
said that singers sang different intervals from the instrumental tuning
in some modes. While we don't know much about Greek practice, in
classical or Roman times, I think it unlikely that singers were were
completely accurate and never expressively altered their pitches.

That's not to say that we with our electronic technology shouldn't tune
accurately and at least listen to the theoretical scales.

--John

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

> I do think, however, the facile equivalence of 120/60 to 2/1 may
have
> influenced Cleonides and Eratosthenes and explains both
Eratosthenes's
> genera and Ptolemy's misunderstanding of Aristoxenos's tetrachords.
>

hi john, probably in the old days, i might have given you a hard
time over that. ;)

>
>
> I really doubt that in performance on a lyra made from a tortoise
shell
> and strung with natural gut strings or even on a professional's
kithara
> whose gut strings were attached not with precision-made, worm-gear
metal
> tuning machines, but with greasy nape-skin, cord, and wooden levers,
> that one could tell the difference between strict JI and the
nearest ET.

i don't know why certain people get pissy with me when i tell them to
try getting a rough measurement ( a 20$ korg cromatic tuner, even )
of this or that. i see 20 cent variations poping up on guitars
reguarly. or add a little phase shift or reverb to your synthesizer
and well the deviations adds up quickly.

.
>
best, buzzy^

hi John,

the talk i gave at the El Paso Microhoot (which Erv Wilson
attended and of which Brian McLaren has a video) was all
about my speculations on how the Sumerians might possibly
have been able to calculate a very accurate approximation
to 12edo in base-60 math.

i have much of it beginning about halfway down on this page:
http://www.ixpres.com/interval/monzo/sumerian/sumeriantuning.htm

also, for those who haven't seen it, here's much paper
speculating on the rational possibilities of Aristoxenos's
tuning (this page is much in need of editing, but still interesting):
http://www.ixpres.com/interval/monzo/aristoxenus/318tet.htm

-monz

----- Original Message -----
From: "John Chalmers" <JHCHALMERS@UCSD.EDU>
To: <tuning@yahoogroups.com>
Sent: Sunday, June 16, 2002 7:50 PM
Subject: [tuning] Re: Aristoxenos again

> Gene: Perhaps all is an unconscious exaggeration, though base-60 is used
> in all the documents I've seen. Neugebauer says the
> alphabetic-decimal-coded sexigesimal notation was used extensively in
> mathematical, astronomical, economic and literary texts. Of course, for
> expressing numbers less than 60 and simple fractions greater than 1/60,
> it might be hard to tell which system was in use <g>.
>
> I do think, however, the facile equivalence of 120/60 to 2/1 may have
> influenced Cleonides and Eratosthenes and explains both Eratosthenes's
> genera and Ptolemy's misunderstanding of Aristoxenos's tetrachords.
>
> Conceiving of musical space geometrically isn't quite the same as
> computing the string lengths numerically for 72 or 144 tets to
> demonstrate them on the monochord or constructing a diagram with
> correctly calculated tempered distances.
>
> I just don't think the idea ever occurred to the theorists or if it did,
> they didn't think it was worth the trouble to actually do it. The most
> comprehensive lists of tunings are in Ptolemy's catalog and he uses
> base-60 to express the string lengths and interprets Aristoxenos's
> tunings as parts a real string.
>
> By equal temperament, I meant the appropriate temperament from the set
> 24, 36, 48, 72, 144, not just 12-tet. These could express the microtonal
> inflections quite well. In fact, some of Aristoxenos's less well-known
> genera - 1/3 + 7/6 + 1 tone and 1/3 + 2/3 + 3/2 tone - are quite close
> to Archytas's diatonic (28/27 x 8/7 x 9/8) and chromatic (28/27 x
> 243/224 x 32/27). One could express Archytas's enharmonic as 4 + 3 + 23
> parts as degrees of 72-tet with little distortion.
>
> I really doubt that in performance on a lyra made from a tortoise shell
> and strung with natural gut strings or even on a professional's kithara
> whose gut strings were attached not with precision-made, worm-gear metal
> tuning machines, but with greasy nape-skin, cord, and wooden levers,
> that one could tell the difference between strict JI and the nearest ET.
> The tuning of auloi would be even more problematic as both ancient
> writers and modern re-creators and players have stated. Ptolemy also
> said that singers sang different intervals from the instrumental tuning
> in some modes. While we don't know much about Greek practice, in
> classical or Roman times, I think it unlikely that singers were were
> completely accurate and never expressively altered their pitches.
>
> That's not to say that we with our electronic technology shouldn't tune
> accurately and at least listen to the theoretical scales.
>
> --John