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

*Sun, 1 Aug 1999 08:19:29 +0300 (EET DST)*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Matti Pitkanen: "[time 509] Re: [time 504] Observation models"**Previous message:**Stephen P. King: "[time 507] Re: [time 504] Observation models"

On Fri, 30 Jul 1999, Matti Pitkanen wrote:

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*> Hi Stephen and all,
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*>
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*> I was pondering about the model for music experience, reading Pinker's
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*> 'How the Mind Works' and drinking cheap wine, when I made curious
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*> discovery: our music experience reflects directly the mapping from reals
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*> to 2-adics!
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*>
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*> Just as TGD inspired theory of cs predicts: realities are mapped
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*> to personal p-adicities! Now p is however 2. Music listener inside
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*> us is at the lowest level of the hierarchy of intelligences (but
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*> I love music still(;-)!).
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*>
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*>
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*>
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*> The basic questions were:
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*>
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*> a) Why 5-tone scale (pentatonic), 7-tone scale (the western)
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*> and 12-tone ('atonal', chromatic) scales are so fundamental?
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*>
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*> b) Why frequencies which are 2^k multiples of fundamental, octaves,
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*> are heard as identical?
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*>
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*> What I learned from Pinker was that basic scales corresponds
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*> to multiples of the basic frequency divided by a suitable power
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*> of two such that result is between 1 and 2, that is in single octave.
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*> Even integer multiples given redundant results so that
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*> one can use only odd integers. This implies that the range n=1,..5,
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*> gives C,E,G: the notes in C chord. The range n=1,...,9 gives
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*> pentatonic scale: C,D,E,G,A.. The range n=1,..13 gives 7-tone scale and
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*> n=1,...,23 gives 12-tone or chromatic scale.
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I checked these statements using even chromatic scale

(omega_m/omega0)= x^m, x= 2^(1/12) and finding best rational

approximation as a small rational of form n/2^k.

Unfortunately the statements do not quite hold true (should I accuse

Pinker or cheap wine?).

a) Basic chord CEG and pentatonic scale are OK.

b) 7-tone scale Cmajor corresponds to small odd integers

C=1,G=3,E=5,D=9,A=11,F=13,B=15 .

Everything is fine except that Bb= 7 is lacking and

corresponds to 'blue' note ('Norwegian woods' by Beatles applies

this scale). In blues Bb appears often: as does also Eb=25.

The notes Gb=29 and Ab=19 appearing in standard Aminor scale

when one goes 'upwards' corresspond to rather large integers:

somehow notes which bring in 'nontrivial' emotion (Pyre Cmajor

is killingly boring) seem to correspond to large integers.

c) Chromatic scale corresponds to

C=1,G=3,E=5, Bb=7,D=9,A=11,

F=13,H=15,Db= 17,Ab=19, Eb=25,Gb= 29 give

12-tone or chromatic scale. Note that the integers n=21,23,27

are not present in the set.

A slightly better fit is given by the integers

1,3,5,7,9, 13,15,17,19 and 25,27, 29: now

the integers 11,21,23 are lacking from the set.

d) All keys are obtained as subscales using chromatic scale although

the ratios of the frequencies are not exactly the same since

canonical identification does not commute with scalings:

in practice instruments use this scale and this

cause pains for the people having absolute ear.

e) One could of course criticize this representation: by using

sufficiently rough resolution one can arrange that

p-adic integers corresponding to notes of the basic scale are

reasonably small. One could also claim that good explanation

for the lacking integers is needed.

f) The strongest support for the 2-adicity of music experience

is that octaves are head as similar. All real frequencies omeag>=omega0

correspond to finite p-adic integers in good approximation (binary

cutoff for map to 2-adics?) so that this part of argument is

not strong. Note however that harmony is p-adically indeed

same as what one means with harmony in real cases: frequencies

with are integer multiplies of basic frequency coexist peacefully.

g) The overall frequency scale (basic tone)

and hence basic p-adic time scale can be chosen freely.

This can be understood if the p-adic time scale is related to

genuine p-adic length scale by

equation $T= L_p/v$, such that $v$ is

some (very low) characteristic velocity which adapts to different

values depending on key. Vecolity related to some wave propation

phenomenon must be in question. Sound waves, em waves, ...?

Best,

MP

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*> Then I realized that the real frequences
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*> can be obtained as canonical images of odd integers regarded
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*> as 2-adic numbers and mapped to real numbers by canonical identification
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*>
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*> SUM (k) x(k) 2^k --> SUM(k) x_k2^(-k).
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*>
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*> If p-adic number represents odd integer the image is always between
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*> 1 and 2 and hence in the basic octave.
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*>
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*> For instance:
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*> a) 13= 1+ 2^2+ 2^3--> 1+ 2^(-2)+2^(-3)= 13/8<2.
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*> b) the basic chord CEG corresponds to 1,5,3.
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*>
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*>
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*> Thus we can make the following conclusions:
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*>
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*>
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*> a) Chromatic scale, the basic scale of Western music including also
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*> 'atonal' music, corresponds to odd integers 1,..,23 regarded as
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*> p-adic integers.
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*>
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*> b) The mapping of reals to 2-adics by canonical
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*> identification maps the notes of this scale
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*> to 2-adic integers.
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*>
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*> c) p-Adic images of frequencies differing by power of 2 are
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*> equivalent in the sense that they differ by some number
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*> of octaves. Therefore 2-adic integers characterize totally
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*> our musical experience!
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*>
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*> d) That frequencies are 2-adic integers provides
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*> a possible explanation for why harmony in the most general Western
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*> sense is based on chromatic scale: these frequencies are integers
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*> multiples of basic frequency, which appear always when system
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*> has time periodicity. They appear always when one approximates
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*> system as collection of harmonic oscillators. Integer
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*> valued spectrum of Hamiltonian is also necessary ingredient
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*> for the existence of p-adic thermodynamics since
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*> Boltzman weights exp(-E/T) are replaced by p^(E/T), which exists
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*> only provided E/T is integer so that E itself must be integer
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*> valued in suitable units.
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*>
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*> e) What is perhaps also remarkable is that n=1,,,,,23 is involved in
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*> Western chromatic scale. D=24 is the mystery dimension of string
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*> (much to do with music!) model. D=24 emerges also in quantum TGD
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*> in some contexts! To get the scale of Eastern music
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*> one should allow n to run in the range n=1,....,48.
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*>
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*>
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*> f) Then comes the bad news. This observation, together with the
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*> fact that our the engineering feats resemble remarkably 2-adic
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*> fractals, suggests that part of our conscious experience corresponds to
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*> 2-adic numbers. 2 is smallest possible prime, the representative for the
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*> lowest level of intelligence in the hierarchy of intelligence.
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*>
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*>
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*> Best,
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*> MP
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