[celestial-tuning] common musical intervals in Babylonian math texts

This post was originally for celestial-tuning,
but it got so math-heavy that I decided to send a
copy to tuning-math.

> From: monz <joemonz@yahoo.com>
> To: <celestial-tuning@yahoogroups.com>
> Sent: Friday, June 22, 2001 5:08 PM
> Subject: Re: [celestial-tuning] Sumerian tuning
>
> I actually did find a paper on the web somewhere which was
> devoted to a discussion of the syntonic comma in ancient
> Babylonian mathematics. It was more in relation to other
> uses than music (can't remember what right now, mainly
> architecture and building). Try a search on "comma" and
> "Babylonian", see what you find.

> ----- Original Message -----
> From: <JGill99@imajis.com>
> To: <celestial-tuning@yahoogroups.com>
> Sent: Saturday, June 23, 2001 2:49 PM
> Subject: [celestial-tuning] Sumerian Links
>

> I found a couple of links searching. One relates to
> the Sumerians and Babylonians in relation to their
> mathematics, with some materials involving their
> music (cross-referenced abstracts of existing articles
> and books):
>
> http://math.truman.edu/~thammond/history/Babylonia.html#ff19

Thanks for this link, Jay! Looks like a lot of good reading
on ancient math.

I found the article I was referring to:
http://www.metrum.org/measures/stereometric.htm

It's part of _A History of Measures_ by Livio C. Stecchini.

The author's native language appears to be Italian, because
this article (in English) is *full* of typos, so its already
difficult discussion is even harder to follow than it need be.

This article, as I correctly remembered a few days ago,
is about the volumes, weights, and specific gravities
of various amounts of bricks. As the author says in
section 7:

> Considering the importance of bricks in the life of
> Mesopotamia it is not surprising that the units of
> volume and weight were so well adjusted the problem
> of measuring and transporting bricks.

Here are some extracts with the discussion of the Greek
terms which appear in music-theory.

Near the end of section 2:

> (a cube that contains 300 double qa with a six-finger
> edge, less a discrepancy komma or 81/80),

(A "qa" is a measurement of area.)

Stecchini explicitly equates the "komma" with 81:80.

From section 4:

> A massiqtu of 60 qa has a base of 24 x 24 fingers and
> a height of 22� (that is, 24 minus a diesis).

(A "finger" is a measurement of length.)

Thus, this "diesis" is 24 : 22&1/2 = 16:15.

In the next paragraph, however:

> The tablet indicates that when one came to the sheqel,
> the unit used to weigh the media of exchange, one
> reckoned by referring to the basic sheqel of 9 grams:
> here the unit is a double sheqel of 18 grams reduced
> of a diesis; it is equal to two sheqels of 8.4 grams.

Thus, this is a different "diesis": 9 : 8.4 = 15:14.

Then in section 6 we find:

> ... we find a relation 1:3 3/5 between the two amounts
> of seed, with a discrepancy diesis (3 3/8 * 16/15 = 3 3/5).

So here again the "diesis" is 16:15.

At the end of section 6:

> The text describes a near-cube with a basis of 4 x 4 fingers
> and a height of 3 3/4. If it were a perfect cube the qa
> would be 64 cubic fingers with an excess of a diesis;

The volume of the given shape is 60 cubic fingers, so this
diesis is again 64:60 = 16:15.

> 7. Neugebauer and Sachs have shown that a brick
> measuring 15 x 10 x 5 fingers was considered a typical
> brick, but they have not explained why such a brick
> should have been considered typical. The explanation
> is that the brick has the volume of a royal qa:
> 750 cubic fingers by the barley cubit equal
> 216 cubic fingers by the great cubit. However,
> the brick is calculated with an excess of a leimma
> over the volume of the qa, for the purpose of obtaining
> a brick measured by be convenient figures 15 x 10 x 5
> barley fingers. Deducting a leimma the volume of a
> brick becomes 720 cubic fingers. Since 720 = 216 x 3 1/3,
> the relation between royal qa and normal qa is calculated
> as 1:3 1/3, a relation frequently used instead of the
> less convenient relation 1:3 3/8, with a resulting
> discrepancy komma. The dimensions of the brick are
> such that 6 occupy the area of a square cubit and
> 36 have the volume of a cubic cubit. Below it will
> appear that these bricks were counted by the dozen,
> and that a dozen of bricks is as much as a man can
> carry. Calculating by the great cubit each brick is
> a royal qa with an excess of a leimma, so that 120 bricks,
> or 10 dozens, make exactly a cube of great cubit.

Note that what Stecchini is calling a "leimma" is *not* the
same as the Pythagorean musical interval of that name (it's
also the Pythagorean "chromatic semitone"), which is the
ratio 256:243, which translates into ~90.2249957 cents.

This "leimma" is 750:720 = 25:24, the familiar 5-limit
"chromatic semitone", which translates into ~70.6724269 cents.

This "discrepancy komma" is again the same ratio
as the musical syntonic comma: 3&3/8 : 3&1/3 = 81:80.

Note that Stecchini also mentions a "diesis" here. I'm a
little confused about exactly how big this is. It seems to
describe the difference between the "mina stereometric brutta"
and the "regular mina", and I gather from the discussion that
the number of sheqels contained in these minai are respectively
63&2/3 and 60, which gives the ratio 191:180, ~102.690878 cents.

But I could be wrong here. The ratio between the two
specific gravities is 2.4 : 2.25 = 16:15. The earlier
mentions of diesis usually show it to be 16:15.

Actually, my other calculations may also be wrong.
I think a very close study of this article (as well as the
others in the series by Stecchini) is necessary to be sure
about what these terms really mean.

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

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Wow... look what I found in the appendix to Stecchini's
series of articles on ancient measurement:

http://www.metrum.org/measures/appendix.htm

> I have given to the discrepancies names derived from
> the accidentals of musical scales, because there is
> a close correlation between units of measures and
> ancient musical scales. This is made clear by Chinese
> musical treatises. Actually I have found that the
> reading of the Greek musical treatises or of the
> similar Chinese ones, which must have been derived
> from a common source, is the best preparation for the
> understanding of the arithmetic of ancient metrology.
>
> The arithmetic of discrepancies is essential to the
> understanding of the development of problems in
> cuneiform mathematical texts. It is disputed among
> musicologists whether musical scales have a physiological
> or conventional origin. The evidence I have gathered
> indicates that musical scales were derived from the
> arrangement of the units of volume. The ancients used
> to arrange the units of public reference standards in
> a series, in ascending or descending order. The relation
> among the contents of the basic units of volume appears
> to have been adopted as determining the basic tetrachord.

Stecchini then gives examples, and quite a bit of further
discussion on music and tuning.

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

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