Matti Pitkanen (email@example.com)
Sun, 1 Aug 1999 08:19:29 +0300 (EET DST)
On Fri, 30 Jul 1999, Matti Pitkanen wrote:
> 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.
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
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
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, ...?
> 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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