Re: Boethius and "nine comma" division?

Hello, there, Monz and Paul and everyone, and this is a quick caution
on interpreting a mention of "nine commas to a whole-tone" as
_necessarily_ implying a 55-tET kind of system.

It could also imply an approximation for conventional Pythagorean
tuning, or possibly a 53-tET outlook of one kind or another.

If such a concept occurred in Boethius, I would tend to follow the
reading of an approximation of usual Pythagorean tuning, unless there
were some reason to favor another explanation.

In classic Pythagorean theory, as in the mathematics of the tuning,
the comma is, of course, not equal to any _aliquot_ or neat fractional
portion of the tone such as 1/8 or 1/9, in fact being somewhere
between these values. We can confirm this by taking the Pythagorean
comma of 531441:524288 as ~23.46 cents, and dividing this by the 9:8
whole-tone at ~203.91 cents.

However, in some early 15th-century theory, at least, the "ninefold"
division is given as a convenient approximation.

We can reach this approximation by disregarding a slight discrepancy
between two units of measure in late medieval theory: the Pythagorean
comma, and the _diaschisma_ in the sense of an interval equal to
precisely half of a usual diatonic semitone or limma at 256:243. Since
the limma is ~90.22 cents, that gives us a diaschisma of ~45.11 cents.

In one late medieval tradition, based on the familiar division of the
tone into a limma or diatonic semitone plus an apotome or chromatic
semitone (2187:2048, ~113.69 cents), the apotome is further subdivided
into limma and comma (the two semitones differing by the comma):

~23.46 ~90.22
531441:524288 256:243
comma limma
|--------|-----------------------------|
|-----------------------------|--------------------------------------|
limma apotome
256:243 2187:2048
~90.22 ~113.69

In this medieval approach, the next step is to divide each limma into
two equal diaschismas, giving us the result that a tone is equal to
"four diaschismas plus a comma":

~45.11 ~45.11 ~23.46 ~45.11 ~45.11
|--------------|--------------|--------|-------------|--------------|
diaschisma diaschisma comma diaschisma diaschisma
|-----------------------------|--------|----------------------------|
limma comma limma

Now for the approximation: once we have a tone defined as four
diaschismas plus a comma, we might note -- as theorists did by the
earlier 15th century, at least -- that the diaschisma is quite close
in size to two commas.

Of course, as orthodox Pythagorean mathematics recognizes, they are
not _precisely equal_. A diaschisma is ~45.11 cents, while two
Pythagorean commas of ~23.46 cents each add up to ~46.92 cents.

However, these two measures are rather close, close enough at any rate
for a rule a thumb that the limma or diatonic semitone (the minor
semitone) is equal to about 4/9-tone, and the apotome or chromatic
semitone (the major semitone) to about 5/9-tone.

I would not take such an approximation, in itself, as evidence for
53-tET or the like, although we might indeed define 53-tET as the
tuning where a whole-tone is in fact equal to precisely nine commas,
with a limma (five fifths down) of 4/9-tone and an apotome (seven
fifths up) of 5/9 tone. Here the medieval-like "diaschisma" of
2/9-tone is equal to precisely two commas as well as half of a limma.

By the 17th century, 53-tET is being described, with Mercator and Kircher
often mentioned as early advocates.

The "nine commas" question raises a more general point: other
divisions of the tone into an even number of commas or dieses can also
have more than one reading. Paul, you have discussed this very
important point.

Thus the division of the tone into five equal dieses can describe
either 29-tET (one reading of Marchettus of Padua, 1318), or some
precise or approximate 31-tET (Vicentino, 1555, who may have
implemented his division with a 31-note cycle of 1/4-comma meantone).

The division of the tone into three thirdtones can likewise describe
either 17-tET or 19-tET.

The division into seven parts might describe either 41-tET or 43-tET.

Sometimes this kind of situation can be further complicated by a
theorist's not-so-conventional use of terminology.

Marchettus, for example, in presenting his equal (or unequal?)
fivefold division of the tone, says that an "enharmonic" semitone is
equal to two parts, and a "diatonic" semitone to three parts.

This language, read in the familiar way, can quite naturally suggest a
tuning system leaning toward ratios of 5, in the manner of Renaissance
meantone or 5-limit JI systems, with large diatonic semitones and
small chromatic semitones as a characteristic feature.

However, we find that Marchettus actually defines his "enharmonic
semitone" as the usual limma (e.g. B-C), and the "diatonic semitone"
as the usual apotome (e.g. Bb-B).

In more conventional language, he is describing a "diatonic semitone"
(i.e. limma) of two parts, and a larger "chromatic semitone"
(i.e. apotome) of three parts.

This definition of the limma as _smaller_ than the apotome, to use his
own more familiar terms, suggests some approximation and possibly some
accentuation of the usual Pythagoraen tuning, moving if anything in
the opposite direction from that of Renaissance 5-limit systems (just
or meantone).

Returning to the "nine comma" question, just how far back the
approximation of the Pythagorean limma and apotome as 4/9-tone and
5/9-tone goes is a very interesting question.

Another question: might 55-tET have had appeal for theorists around
1700, or a bit later, in part because it neatly reverses an earlier
tradition, as you have pointed out, Paul, by keeping the ninefold
division but defining the _diatonic_ semitone as 5/9-tone and the
chromatic semitone as 4/9-tone.

One article, in a book on music theory as I recall, traces how the
"fivefold division" of Marchettus was used by some Renaissance theorists
who did the same thing: the diatonic semitone (in the usual meaning) was
now 3/5-tone, and the chromatic semitone 2/5-tone. Here we are talking
about sources around the late 15th century or thereabouts, if I am
correct, some time before Vicentino reaches a similar result in his
treatise of 1555.

Most respectfully,

Margo Schulter
mschulter@value.net

--- In tuning@y..., mschulter <MSCHULTER@V...> wrote:

> Another question: might 55-tET have had appeal for theorists around
> 1700, or a bit later, in part because it neatly reverses an earlier
> tradition, as you have pointed out, Paul, by keeping the ninefold
> division but defining the _diatonic_ semitone as 5/9-tone and the
> chromatic semitone as 4/9-tone.

Yes, I believe the traditional "nine commas to a tone" reckoning had
some influence on this; though of course in an 18th century context,
it implies a 55-part, rather than 53-part, division of the octave.
>
> One article, in a book on music theory as I recall, traces how the
> "fivefold division" of Marchettus was used by some Renaissance
theorists
> who did the same thing: the diatonic semitone (in the usual
meaning) was
> now 3/5-tone, and the chromatic semitone 2/5-tone. Here we are
talking
> about sources around the late 15th century or thereabouts, if I am
> correct, some time before Vicentino reaches a similar result in his
> treatise of 1555.

Fascinating.

Hello Margo,

> From: mschulter <MSCHULTER@VALUE.NET>
> To: <tuning@yahoogroups.com>
> Sent: Wednesday, October 10, 2001 12:52 PM
> Subject: [tuning] Re: Boethius and "nine comma" division?
>
>
> Hello, there, Monz and Paul and everyone, and this is a quick caution
> on interpreting a mention of "nine commas to a whole-tone" as
> _necessarily_ implying a 55-tET kind of system.
>
> It could also imply an approximation for conventional Pythagorean
> tuning, or possibly a 53-tET outlook of one kind or another.

Agreed.

> If such a concept occurred in Boethius, I would tend to follow the
> reading of an approximation of usual Pythagorean tuning, unless there
> were some reason to favor another explanation.
>
> In classic Pythagorean theory, as in the mathematics of the tuning,
> the comma is, of course, not equal to any _aliquot_ or neat fractional
> portion of the tone such as 1/8 or 1/9, in fact being somewhere
> between these values. We can confirm this by taking the Pythagorean
> comma of 531441:524288 as ~23.46 cents, and dividing this by the 9:8
> whole-tone at ~203.91 cents.

In fact, in searching for confirmation of Ernest McClain's claim
(in _The Pythagorean Plato_, p 161) that Boethius said the tone was
divided into 9 commas, I've found _contra_ McClain that Boethius
actually said that the whole tone is "larger than 8 commas but
smaller than 9".

Boethius was definitely talking strictly about the Pythagorean comma,
ratio 531441:524288.

*However*, he uses incorrect mathematics at this point (the only place
in his entire book where he does so), even tho it doesn't affect the
validity of his statement.

I can see, by doing some calculations myself, that the reason he
employed this erroneous approach was because he would have had to
deal with (531441/524288)^8 and (531441/524288)^9, which would
result in enormous numbers for his ratios, which would be very
difficult to calculate accurately... altho it *could* have been done.

I've gotten so deeply into this investigation that I'm going to put
all the information into a webpage. Stay "tuned". :)

> However, in some early 15th-century theory, at least, the "ninefold"
> division is given as a convenient approximation.
>
> We can reach this approximation by disregarding a slight discrepancy
> between two units of measure in late medieval theory: the Pythagorean
> comma, and the _diaschisma_ in the sense of an interval equal to
> precisely half of a usual diatonic semitone or limma at 256:243. Since
> the limma is ~90.22 cents, that gives us a diaschisma of ~45.11 cents.
>
> In one late medieval tradition, based on the familiar division of the
> tone into a limma or diatonic semitone plus an apotome or chromatic
> semitone (2187:2048, ~113.69 cents), the apotome is further subdivided
> into limma and comma (the two semitones differing by the comma):
>
> ~23.46 ~90.22
> 531441:524288 256:243
> comma limma
> |--------|-----------------------------|
> |-----------------------------|--------------------------------------|
> limma apotome
> 256:243 2187:2048
> ~90.22 ~113.69
>
> In this medieval approach, the next step is to divide each limma into
> two equal diaschismas, giving us the result that a tone is equal to
> "four diaschismas plus a comma":
>
> ~45.11 ~45.11 ~23.46 ~45.11 ~45.11
> |--------------|--------------|--------|-------------|--------------|
> diaschisma diaschisma comma diaschisma diaschisma
> |-----------------------------|--------|----------------------------|
> limma comma limma
>
> Now for the approximation: once we have a tone defined as four
> diaschismas plus a comma, we might note -- as theorists did by the
> earlier 15th century, at least -- that the diaschisma is quite close
> in size to two commas.
>
> Of course, as orthodox Pythagorean mathematics recognizes, they are
> not _precisely equal_. A diaschisma is ~45.11 cents, while two
> Pythagorean commas of ~23.46 cents each add up to ~46.92 cents.
>
> However, these two measures are rather close, close enough at any rate
> for a rule a thumb that the limma or diatonic semitone (the minor
> semitone) is equal to about 4/9-tone, and the apotome or chromatic
> semitone (the major semitone) to about 5/9-tone.

In fact, Margo, the 5-fold division of a whole-tone seems to go all
the way back to Philolaus (Archytas's teacher, c. 350 BC), at least
according to Boethius.

And don't forget that Marchettus, whom you bring up in connection
with this 5-fold division, measured his 5 dieses by means of a
prior *9-fold* division of the whole-tone!

Much more to come from me on this.

love / peace / harmony ...

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

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com