**Matti Pitkanen** (*matpitka@pcu.helsinki.fi*)

*Fri, 30 Jul 1999 21:32:19 +0300 (EET DST)*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Previous message:**Matti Pitkanen: "[time 504] Re: [time 503] Re: [time 500] Observation models"

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.

Best,

MP

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