From: Christian.Radoux@skynet.be
Newsgroups: sci.math
Subject: Re: number theoretic (or statistical?) basis of music theory and harmony
Date: Sun, 03 May 1998 17:46:50 0600
In article <1998050319314300.PAA16049@ladder03.news.aol.com>#1/1,
kjinnovatn@aol.com (Kjinnovatn) wrote:
>
> Here's a little puzzler that I've been wondering about: If you investigate
the
> numbertheoretic basis of music theory, it all hinges on the fact that
certain
> simple fractional powers of 2 "accidentally" happen to be very close to
simple
> fractions. For example,
> 2^(7/12) ~= 3/2 (within 0.1%)
> from which it follows that the first and fifth notes on the chromatic scale
> have a harmonic frequency ratio of 3/2. This also implies that the first and
> fourth have a simple harmonic ratio of 4/3:
> 2^(5/12) = 2^(17/12) ~= 2/(3/2) = 4/3
> The harmony between the first and third is based on the approximate
> relationship
> 2^(1/3) ~= 5/4 (within 0.8%)
> and the firstsixth harmony follows from
> 2^(3/4) ~= 5/3 (within 0.9%)
> These relationships form the basis of the chromatic scale and all of chord
> theory.
>
> The question is, are these happy little "accidents" just pure coincidence,
or
> do they actually have some basis in number theory? For example, could the
above
> relationships be the leading term in some sort of rapidly converging
expansion
> for the 12th roots of 2  maybe something like a continuing fraction
> expansion? (Considering that the first term seems to typically be within 1%
of
> the solution, such an expansion might have practical computational
> applications.)
>
> Of course, the explanation might have more to do with statistics than number
> theory; i.e. if you just pick any handful of random numbers some might
happen
> to be close to harmonic ratios.
>
> Here are a couple of similar such "coincidences" to ponder:
> 3^(1/6) ~= 6/5 (within 0.1%)
> 3^(5/6) ~= 5/2 (within 0.1%)
> Anybody have any ideas?
>
>
As long as you consider the classical temperated scale used since J.S. Bach,
you gave yourself a very good answer.
Let us add a little remark : "Why" 12 notes in this scale ?
Saying that
3^y is a close neighbour of 2^x
resp.
5^a " " " " " 2^b
lead to compute
ln 3 / ln 2 = 1 + 1

1 + 1

2 + 1

2 + 1

3 + 1

...
resp.
ln 5 / ln 2 = 2 + 1

3 + 1

9 + 1

...
If you truncate this continued fraction expansion of
ln 3 / ln 2 before the "3"
resp.
ln 5 / ln 2 before the "9",
you get
19/12
resp.
7/3
i.e. the relations well known by every beginner in harmony.
JeanPhilippe Rameau and the young Leonhard Euler (Tentamen novae theoriae
musicae, ex certissimis harmoniae principiis dilucide expositae, 1731) have
written early beautiful pages on that topic.
It is possible to adapt this kind of computaion to other scales.
With best regards from an (amateur) pianist !
email : Christian.Radoux@skynet.be
URL : http://users.skynet.be/radoux
P.S. From this URL, you can download (for free, of course), amongst many
other programs, a little software for DOS : gammes.exe (sorry, it's in
French...). It gives and plays major and minor scales (bohemian, oriental,
and so on), the main chords,.... Sometimes, it doesn't work under the dual
boot of Windows 95 and you may have to adapt the speed of lecture (the
program is written in TurboPascal 7, and I use only the internal speaker)
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From: Einar Andreas Rdland
Newsgroups: sci.math
Subject: Re: number theoretic (or statistical?) basis of music theory and harmony
Date: Thu, 07 May 1998 00:10:53 +0200
Kjinnovatn wrote:
>
> Here's a little puzzler that I've been wondering about: If you investigate the
> numbertheoretic basis of music theory, it all hinges on the fact that certain
> simple fractional powers of 2 "accidentally" happen to be very close to simple
> fractions. For example,
> 2^(7/12) ~= 3/2 (within 0.1%)
[snip]
> 2^(5/12) = 2^(17/12) ~= 2/(3/2) = 4/3
(Because this is 2/2^(7/12), the error is as for 2^(7/12): 0.1%.)
> 2^(1/3) ~= 5/4 (within 0.8%)
> 2^(3/4) ~= 5/3 (within 0.9%)
So, the basic quotients that should be approximated are 3:2 (with 4:3
being the same with just the 2 shifted one octave), 5:3, and 5:4.
Now, if we want the scale to go as powers of 2^(1/n) for some n, ie.
notes are 2^(m/n) for various integer (or near integer) values of m, we
may look at the logarithms:
If p/q is the quotient (eg. 3/2), we want p/q=2^(m/n). The logarithm of
this gives log_2(p/q)=ln(p/q)/ln(2)=m/n. So, for the p/q of interest
(ie. the above three), we want to find an n such that n*log_2(p/q) in
near an integer. So, how well do these match for n=12?
p/q=3/2 gives 12*log_2(3/2)=7.02 (a very good match!)
p/q=5/3 gives 12*log_2(5/3)=8.84 (not very close)
p/q=5/4 gives 12*log_2(5/4)=3.86 (not very close)
So, it appares that the 12tone scale is very good for 3:2 (and 4:3),
but not at the other relations.
Now, about the 0.8% and 0.9% differences you note, remember that
2^(1/12)=1.06, so by picking a suitable power you can always (ie. in the
worst case) get within 3%.
Now, to find "good" scales so that 2^(m/n) for various integer values of
m give good approximations to the three rationals, I took a look at
n*log_2(p/q) and the distance from the closest integer and summed this
over the three rational numbers (p/q=3/2,5/3,5/4). This gives a sequence
S_n. The lowest values of this (with S_n The question is, are these happy little "accidents" just pure coincidence, or
> do they actually have some basis in number theory? For example, could the above
As shown above, the relation is not very good except for 3:2. How good
is this? To evaluate this, you may look at s_n=Dist(n*log_2(3/2),Z)*n
(ie. n times the distance from n*... to the nearest integer). This will
bounce up and down, but if we look at the smallest values of s_n (ie. so
that s_ninfty}n*s_n is limited. If we list s_n with s_n,
Einar Andreas Rdland writes:
[...]
> Now, to find "good" scales so that 2^(m/n) for various integer values of
> m give good approximations to the three rationals, I took a look at
> n*log_2(p/q) and the distance from the closest integer and summed this
> over the three rational numbers (p/q=3/2,5/3,5/4). This gives a sequence
> S_n. The lowest values of this (with S_n n<100):
>
> n=3: S=0.490 n*S=1.47
> n=12: S=0.313 n*S=3.75
> n=19: S=0.233 n*S=4.43
> n=34: S=0.223 n*S=7.57
> n=53: S=0.124 n*S=6.59
>
> So, how about that 53tone scale :)
See also:
Viggo Brun: Euclidean algorithms and musical theory.
Enseignement Math. 10#2 (1964), 125137.
(Based on a 10minute lecture at the 1962 ICM in Stockholm.
He also noted that 53tone scale, along with 31, 72 and 74,
and gives some background about systems that were actually
used in music, or had been suggested earlier in music theory.)
Enjoy, Gerhard

* Gerhard Niklasch * spam totally unwelcome
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