ancient numbering systems (was: "wolf" pack, rat pack [Isacoff])

> From: genewardsmith <genewardsmith@juno.com>
> To: <tuning@yahoogroups.com>
> Sent: Sunday, March 10, 2002 9:20 PM
> Subject: [tuning] Re: "wolf" pack, rat pack [Isacoff]
>
>
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
>
> > this is also quite important. why did the greeks and romans
> > obsess over rational scales to the degree that they did? *this* is
> > why. others will hang me for saying it, but it's plain as day.
>
> Since the time of Eudoxus, the Greeks had a rigorous definition
> of, in effect, the positive real numbers from his theory of
> ratios. However, in practical terms they had a hard time
> dealing with rational numbers, much less irrational ones.

very good point, Gene. it should be mentioned that a big
part of the reason why ancient people (except for the
Sumerians and Babylonians) had such difficulty in dealing
with math computations involving anything other than integers,
is because of their cumbersome numerical notation.

everyone here is familiar with Roman numerals. the ancient
Greek and Hebrew numeral systems worked in a similar way,
something that we (in hindsight today) can see as a sort
of mixture of the way Roman and Arabic numerals work.

just for completeness, here's a quick recap of Roman numerals:

1 I
5 V
10 X
50 L
100 C
500 D
1000 M

symbols are simply added together, and individual symbols
are repeated as necessary until the next symbol is reached,
thus:

2 II
3 III
4 IIII

6 VI
7 VII
8 VIII
9 VIIII

etc.

a much later development, from medieval times, was
to eliminate the four-fold repetition of digits by
using subtraction from the next higher symbol for
any number which would contain them, thus:

old new

4 = IIII IV
9 = VIIII IX
14 = XIIII XIV
19 = XVIIII XIX
99 = LXXXXVIIII XCIX

so that whenever larger digits follow smaller ones,
subtraction of those two instead of addition is indicated.

the Greeks had two numerical systems: acrophonic and
alphabetic. the acrophonic system was older, and very
similar to the Roman system ... even a bit simpler,
using distinct symbols to represent only

1 I iota
5 P pi
10 D delta
100 H eta
1,000 X xi
10,000 M mu

(of course, "pi" and "delta" should look different...
these are the Roman ASCII equivalents. also note that
"gamma" came to be used for 5 instead of "pi", probably
because of similiarity of appearance.)

the top diagram here gives a good explanation:
http://www.clas.ufl.edu/users/rhatch/HIS-SCI-STUDY-GUIDE/0025_ancientGreekNu
merals.html

the Greek alphabetic system used the 24 letters of the
Phoenician alphabet -- which became the regular Greek
alphabet -- plus 3 auxilliary letters (digamma, koppi,
sampi) to represent numbers.

in Greek, the first 9 letters of the alphabet, usually
(but not always) with a stroke mark after the letter,
represent the digits 0...9, then the next 9
letters[-plus-stroke] represent the tens from 10...90,
then then next 9 letters[-plus-stroke] represent the
hundreds 100...900 :

1 alpha
2 beta
3 gamma
4 delta
5 epsilon
6 digamma
7 zeta
8 eta
9 theta

10 iota
20 kappa
30 lambda
40 mu
50 nu
60 xi
70 omicron
80 pi
90 koppa

100 rho
200 sigma
300 tau
400 upsilon
500 phi
600 chi
700 psi
800 omega
900 sampi

then by putting a stroke mark *before* rather than after
the letter, each of these would be multiplied by 1,000.
for these larger numbers the stroke-mark is obviously
mandatory. see:
http://mkatz.web.wesleyan.edu/grk202/vocabulary/greek_numerals.html

(and note that there's a typo there: 21 should have a
kappa followed by an alpha followed by a stroke ... the
alpha is missing.)

this Greek usage is clearly derived from that of Hebrew,
probably by way of Phoenician adoption of the Hebrew system,
which is how the Greeks got their alphabet in the first
place.

the Hebrew alphabet has 22 letters, alef...tet representing
digits 1...9, yod...tzade representing tens 10...90, and
qof...tav representing the hundreds 100...400.

see the explanation about halfway down this page:
http://www.jewfaq.org/alephbet.htm

1 alef
2 bet
3 gimel
4 dalet
5 he
6 vav
7 zayin
8 chet
9 tet

10 yod
20 kaf [or khaf]
30 lamed
40 mem [2 forms]
50 nun [2 forms]
60 samech
70 ayin
80 pe [or fe]
90 tzade [2 forms]

100 qof
200 resh
300 shin
400 tav

perhaps i should also note that the ancient Egyptians
knew how to use fractions, but it was also a very
cumbersome system, based on successive division into
unit fractions.

in any case, it should be easy for any of us, who are
used to the simple 9-digit positional "Arabic" system,
that doing *any* type of calculations with these systems
is much more difficult than with our system. it's not
at all surprising to me that so many music-theorists tried
very hard to find a way to represent tuning systems
mathematically by using nothing but integers.

this really gets at the root of what i said in my last
post about this: the formulation of the mathematical
concepts of roots-and-powers, and logarithms, around 1600,
enabled theorists to once again use simple integer
calculations, but now to describe *irrational* tuning systems.

> They unfortunately never learned from the Babylonians how
> to write numbers to a base.

hmmm ... well, Ptolemy certainly did! his _Almagest_ gives
all the astronomical data with "minutes" and "seconds" in
base_60 for the decimal part. i'm not sure now about the
ratios in his musical treatise ... i'll have to look again.

but in general, yes, you're correct that most Greek
math eschewed the use of the sophisticated base_60 numbering
of the Babylonians, and that fractional parts would not
be calculated with such ease and precision again until
the invention, again around 1600, of the regular
"decimal point" system that we use today.

> > > Also, the contrasting of that approach with that of
> > > Rousseau was very interesting.
> >
> > can you summarize rousseau's approach? maybe i should just
> > buy the book if i want to find out about this? gene?
>
> Rousseau protruded himself into music theory, and one of
> the funny bits in Isacoff is the description of Rameau
> at the premire of Rousseau's opera, ghosted by Philidor.
>
> I never read Rousseau on music, but it sounds just like
> you'd think from Isacoff's description--melody is an
> expression of untamed erotic fervor, and is better than
> harmony, which is too darned civilized and calculating.
> Italian opera is better than French because it is more
> emotional, partly because the French language is better
> at logic than emotion, and partly because the Italians
> care first and formost about melody, not harmony.

sweeping generalizations! which is probably at least partly
why modern music-theory is so much more indebted to Rameau
than Rousseau.

-monz

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