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"

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