Archytas and 28:27 (was: Re: [crazy_music] Some thirdtones in JI)

Hi Margo,

It's very interesting to me that right at this moment you
sent this detailed post about the 28:27 ratio.

I began a post on the tuning list a couple of days ago
in answer to a question posed by Alison Monteith about
ancient Greek "higher-limit" ratios. I haven't finished
it yet because it has turned into quite an exploration
of what I know about Archytas, Eratosthenes, Didymus,
and Ptolemy (in chronological order).

So I've just been looking thru my notes on Archytas. He
used 28:27 as the bottom interval in all three of his
genera: diatonic, chromatic, and enharmonic. His tetrachords
are the only ones extant to have been recorded by an
ancient Greek which have this trait. Audibly, it has
the effect of making his genera sound totally different
from all other ancient Greek systems.

All other theorists used either one size of lowest interval
for the diatonic and another for both chromatic and enharmonic,
or used three different sized intervals, one for each genus.
Aristoxenus and his rationalizer Ptolemy recorded several
different varieties ("shades") of both diatonic and chromatic.

So 28:27 was tremendously important in Archytas's system,
as the only tetrachordal interval besides the 9:8 "tone of
disjunction" which was common to all three genera. Audibly,
those two intervals would be the ones heard over and over
again during the course of a melody.

In fact, if the style were a sophisticated one using much
modulation (the same way as in the later medieval European
method of "mutation"), the 9:8 would frequently "disappear",
but the 28:27 would *always* be present no matter which
combination of tetrachord, mode, and genus.

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

----- Original Message -----
From: mschulter <MSCHULTER@...>
To: <crazy_music@yahoogroups.com>
Sent: Monday, July 09, 2001 5:31 PM
Subject: [crazy_music] Some thirdtones in JI

> Hello, there, everyone, and here's a JI example featuring thirdtones
> at 28:27 (~62.96 cents), the "7-flavor semitone" as it's called in
> neo-Gothic music.
>
> While I posted a similar example of "circumambulation" by a chain of
> whole-tones last month, this one features an "expansive/intensive"
> quality, with major thirds expanding to fifths, major sixths to
> octaves, and major seconds to fourths, by way of ascending 28:27
> semitones. One melodic step in each voice gets altered by a
> Pythagorean comma in order to make the octave an even 2:1 -- not to
> exclude other possibilities!
>
> MIDI example: <http://value.net/~mschulter/sesci002.mid>
>
> In the music of Gothic Europe around 1200-1400, cadences of this
> general kind occur based on a standard Pythagorean tuning, but here
> the "7-flavor" JI (ratios of 2-3-7, e.g. thirds at 9:7 and 7:6) and
> the extra-compact 28:27 semitone maybe give the music a different
> nuance.
>
> As I hear them, 13th-14th century cadences are as compelling and
> beautiful as anything in later European styles, and they seem open to
> many intonational variations, from pure Pythagorean or 2-3-7 JI to
> some wild Sethareanized tunings.
>
> However, one the most appealing ways of approaching these progressions
> is through just intonation, and the 28:27 semitone or "thirdtone"
> feels to me very congenial to the character of this music.
>
> Peace and love,
>
> Margo

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Joe!
I have always found the 28/27 one of my favorite "semitones" even
after 25 years. ( I know 62 cents is stretching, oops shrinking, the
idea of semitone quite a bit). I assumed others also share this love
like Archytas.
In the tetrachordal combinations with the 9/8 and 8/7 I have spent
countless hours that few other tetrachords have rivaled. One other come
close (including inversions) Just for those who like to wallow in such
objects of pure beauty.

7/6 12/11 22/21

BTW both of these can be found in the 22 tone Eikosany 1-3-7-9-11-15.
The further advantage of such a condensed array of pitch organization is
that you always have such peripheral added features.

monz wrote:

> Hi Margo,
>
>
> It's very interesting to me that right at this moment you
> sent this detailed post about the 28:27 ratio.
>
> I began a post on the tuning list a couple of days ago
> in answer to a question posed by Alison Monteith about
> ancient Greek "higher-limit" ratios. I haven't finished
> it yet because it has turned into quite an exploration
> of what I know about Archytas, Eratosthenes, Didymus,
> and Ptolemy (in chronological order).
>
> So I've just been looking thru my notes on Archytas. He
> used 28:27 as the bottom interval in all three of his
> genera: diatonic, chromatic, and enharmonic. His tetrachords
> are the only ones extant to have been recorded by an
> ancient Greek which have this trait. Audibly, it has
> the effect of making his genera sound totally different
> from all other ancient Greek systems.
>
> All other theorists used either one size of lowest interval
> for the diatonic and another for both chromatic and enharmonic,
> or used three different sized intervals, one for each genus.
> Aristoxenus and his rationalizer Ptolemy recorded several
> different varieties ("shades") of both diatonic and chromatic.
>
> So 28:27 was tremendously important in Archytas's system,
> as the only tetrachordal interval besides the 9:8 "tone of
> disjunction" which was common to all three genera. Audibly,
> those two intervals would be the ones heard over and over
> again during the course of a melody.
>
> In fact, if the style were a sophisticated one using much
> modulation (the same way as in the later medieval European
> method of "mutation"), the 9:8 would frequently "disappear",
> but the 28:27 would *always* be present no matter which
> combination of tetrachord, mode, and genus.
>
>
>
> -monz
> http://www.monz.org
> "All roads lead to n^0"
>
>
>
>
> ----- Original Message -----
> From: mschulter <MSCHULTER@...>
> To: <crazy_music@yahoogroups.com>
> Sent: Monday, July 09, 2001 5:31 PM
> Subject: [crazy_music] Some thirdtones in JI
>
>
> > Hello, there, everyone, and here's a JI example featuring thirdtones
>
> > at 28:27 (~62.96 cents), the "7-flavor semitone" as it's called in
> > neo-Gothic music.
> >
> > While I posted a similar example of "circumambulation" by a chain of
>
> > whole-tones last month, this one features an "expansive/intensive"
> > quality, with major thirds expanding to fifths, major sixths to
> > octaves, and major seconds to fourths, by way of ascending 28:27
> > semitones. One melodic step in each voice gets altered by a
> > Pythagorean comma in order to make the octave an even 2:1 -- not to
> > exclude other possibilities!
> >
> > MIDI example: <http://value.net/~mschulter/sesci002.mid>
> >
> > In the music of Gothic Europe around 1200-1400, cadences of this
> > general kind occur based on a standard Pythagorean tuning, but here
> > the "7-flavor" JI (ratios of 2-3-7, e.g. thirds at 9:7 and 7:6) and
> > the extra-compact 28:27 semitone maybe give the music a different
> > nuance.
> >
> > As I hear them, 13th-14th century cadences are as compelling and
> > beautiful as anything in later European styles, and they seem open
> to
> > many intonational variations, from pure Pythagorean or 2-3-7 JI to
> > some wild Sethareanized tunings.
> >
> > However, one the most appealing ways of approaching these
> progressions
> > is through just intonation, and the 28:27 semitone or "thirdtone"
> > feels to me very congenial to the character of this music.
> >
> > Peace and love,
> >
> > Margo
>
>
>
>
>
>
>
>
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-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com

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