[time 505] Music scales and 2-adic numbers

Matti Pitkanen (matpitka@pcu.helsinki.fi)
Fri, 30 Jul 1999 21:32:19 +0300 (EET DST)

Hi Stephen and all,

I was pondering about the model for music experience, reading Pinker's
'How the Mind Works' and drinking cheap wine, when I made curious
discovery: our music experience reflects directly the mapping from reals
to 2-adics!

Just as TGD inspired theory of cs predicts: realities are mapped
to personal p-adicities! Now p is however 2. Music listener inside
us is at the lowest level of the hierarchy of intelligences (but
I love music still(;-)!).

The basic questions were:

a) Why 5-tone scale (pentatonic), 7-tone scale (the western)
and 12-tone ('atonal', chromatic) scales are so fundamental?

b) Why frequencies which are 2^k multiples of fundamental, octaves,
are heard as identical?

What I learned from Pinker was that basic scales corresponds
to multiples of the basic frequency divided by a suitable power
of two such that result is between 1 and 2, that is in single octave.
Even integer multiples given redundant results so that
one can use only odd integers. This implies that the range n=1,..5,
gives C,E,G: the notes in C chord. The range n=1,...,9 gives
pentatonic scale: C,D,E,G,A.. The range n=1,..13 gives 7-tone scale and
n=1,...,23 gives 12-tone or chromatic scale.

Then I realized that the real frequences
can be obtained as canonical images of odd integers regarded
as 2-adic numbers and mapped to real numbers by canonical identification

SUM (k) x(k) 2^k --> SUM(k) x_k2^(-k).

If p-adic number represents odd integer the image is always between
1 and 2 and hence in the basic octave.

For instance:
a) 13= 1+ 2^2+ 2^3--> 1+ 2^(-2)+2^(-3)= 13/8<2.
b) the basic chord CEG corresponds to 1,5,3.

Thus we can make the following conclusions:

a) Chromatic scale, the basic scale of Western music including also
'atonal' music, corresponds to odd integers 1,..,23 regarded as
p-adic integers.

b) The mapping of reals to 2-adics by canonical
identification maps the notes of this scale
to 2-adic integers.

c) p-Adic images of frequencies differing by power of 2 are
equivalent in the sense that they differ by some number
of octaves. Therefore 2-adic integers characterize totally
our musical experience!

d) That frequencies are 2-adic integers provides
a possible explanation for why harmony in the most general Western
sense is based on chromatic scale: these frequencies are integers
multiples of basic frequency, which appear always when system
has time periodicity. They appear always when one approximates
system as collection of harmonic oscillators. Integer
valued spectrum of Hamiltonian is also necessary ingredient
for the existence of p-adic thermodynamics since
Boltzman weights exp(-E/T) are replaced by p^(E/T), which exists
only provided E/T is integer so that E itself must be integer
valued in suitable units.

e) What is perhaps also remarkable is that n=1,,,,,23 is involved in
Western chromatic scale. D=24 is the mystery dimension of string
(much to do with music!) model. D=24 emerges also in quantum TGD
in some contexts! To get the scale of Eastern music
one should allow n to run in the range n=1,....,48.

f) Then comes the bad news. This observation, together with the
fact that our the engineering feats resemble remarkably 2-adic
fractals, suggests that part of our conscious experience corresponds to
2-adic numbers. 2 is smallest possible prime, the representative for the
lowest level of intelligence in the hierarchy of intelligence.



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