Does harmonic complexity reduce to 3-adicity?
I have been updating the chapter Self and Binding of TGD inspired theory of consciousness. The goal is to base the theory from the beginning on ideas like zero energy ontology, hierarchy of Planck constants aand its connection with dark matter, p-adic physics as physics of cognition and intentionality -and in in particular, life as something residing in the interection real and p-adic worlds. The notion of number theoretic entropy makes it possible to assign to rational or even algebraic entanglement probabilities a positive negentropy and this has meant a breakthrough in the understanding of TGD inspired theory of consciousness and life. These updatings are extremely fruitful -not only because they force to throw away the ancient and obsolete stuff- but because the interaction of new ideas with old ones produces something completely unexpected new ideas.
The understanding of music experience is especially fascinating challenge for a consciousness theorists. In TGD framework quite nice vision about the basic aspects of music experience emerges. Octave phenomenon can be reduced to p-adic length scale hypothesis implied if the causal diamonds defined as intersections of future and past directed lightcones and playing key role in quantum TGD have discrete size spectrum coming as octaves of CP2 length. Also the preferred role of certain rational frequencies can be understood in the p-adic context from the fact that sine waves in p-adic context can be defined only by introducing an algebraic extension of p-adic numbers allowing the needed roots of unity. Plane waves are defined p-adically only in discrete subset of points and for harmonics of the fundamental frequency. This implies automatically the preferred role of a limited set of rational valued multiples of the fundamental.
The following little excerpt from the updated chapter provides a inspired vision about one particular aspect of music experience, namely the harmony and allows to interpret Pythagorean vision about music scale in terms of 3-adicity, to deduce a measure for a harmonic measure of the chord, and to replace the Pythagorean tuning of 12-tone system based on perfect fifths with a unique tuning based on 3-adicity.
An interesting question relates to the conditions guaranteing that a chord is experienced as harmonious in the Pythagorean sense. Pythagorean tuning is based on the notion of perfect fifths identified as a scaling by 3/2 producing the sequence C,G,D,A,E,.. In this tuning major-C scale corresponds to ratios C= 1/1, D=9/8, E=81/64, F=4/3, G= 3/2, A=27/16, B= 243/128, C=2/1. Eb and F# correspond to ratios 25/33 and 36/29. All notes are expressible as powers of two and three. Since the multiplication of any note by a power of two does not affect the harmony it should be to drop the powers of two from the integers characterizing the notes in the ratio of three notes. For instance, C-E-G reduces 3:34:1, C-Eb-G to 34:1:33, and tritonus C-Eb-F# to 39:1:33.
The problem of Pythagorean tuning is that one cannot represent 2 as an exact integer power of 3/2 and the scalings give infinite number of tones. If the construction starts from Gb then F# and Gb correspond to frequencies, which are not quite identical in Pythagorean tuning. One could make compromize by introducing the geometric mean of F# and Gb but this would bring in 31/2 and would force to leave the world of pure rationals. For string instruments and electronic instruments the Pythagorean tuning is practical but for instruments like piano the transposition of the scale is impossible.
One should be able to characterize a given chord harmonically by a function F(a,b,c), which is symmetric under the permutations of the reduced pitches a, b and c obtained by dropping powers of two and is invariant under over all scaling of the reduce frequencies. The elementary symmetric functions F(a,b,c)=[a2(b+c)+b2(a+c)+c2(a+b)]/abc and G(a,b,c)=[a3+b3+c3]/abc are the simplest functions of this kind. Either of these functions or their product or ratio could be considered as a measure for the harmonic complexity. The value of the denominator abc equals to 3n, n=3,7,12 in the cases considered. The numerator has in all cases 3-adic norm equal to one for both F and G. This suggests that the 3-based logarithm of the 3-adic norm 1/|abc|3=|F|3=|G|3 having the values 3,7, and 12 for C-major, C-minor, and tritonus could serve as the measure for the complexity. It is indeed smallest for major and largest for tritonus. 3-adic norm for the product 1/a1a2...an of n tones of the chord defines a measure of complexity in more general case. A good guess is that the 3-adic norms of the elementary symmetric functions give rise to the same measure.
For the chords C-E-G, F-A-C, and G-H-D appearing as basic chords in C- major scale the values of the harmonic measure are 3, 2, and 8. This means that the basic chords are not harmonically equivalent in Pythagorean system whereas in equally tempered system they would be. One might think that this explains why the tonic is remembered. The anomalously low value for F-A-C relates to the fact that it is only tone for which the power of 3 is negative. Situation changes of F is identified as a minimal power of 3 giving F equivalent with Pythagorean F within the resolution of ear to pitch which is about |Δ f/f|= 4.3 per cent . F=35/28 gives |Δ f/f|= 4.8 per cent. This F would give for F-A-C the harmonic measure 8 which equals to that for G. This looks more reasonable than the purely Pythagorean value. This definition would also allow to find a unique choice of powers of three for 12-chord system. For instance, F# is favored over Gb since it corresponds to a positive power of 3.
To sum up, music seems to provide a possible manifestations of 3-adicity, and the proposed measure of harmonic complexity might provide a manner to construct also a theory of aesthetically pleasing harmonic progressions.