Dear Brian,
according to music I like to make some minor comments.
Pythagoras found out that the mathematical music intervals of fifth, fourth, major third as well as the octave. If we built a ton scale from the harmonic series, as Pythagoras may wanted to achieve, there is a certain problem that prevents this.
The problem is that one cannot factorize the harmonic series such that 12 triads (chords) with their respective root notes derived from building fifths (around the cycle of fiths) results in a fifth that is what it was at the beginning. The final fifth is 23,46 cent to sharp and every ear can hear it.
So, on a musical instrument tuned with Pythagorean fifths, there are no equally distant half-steps to change musical keys.
Therefore equal temperament was introduced. The main instrument for this kind of tuning is the guitar. If one compares tuning the guitar with the standard method and with the 5/7 flageolett method, one recognizes that with the latter, the high e-string will be 21,5 cent too flat, destroying the possibility to change musical keys without having strong dissonances.
These dissonances result from out-of-phase frequencies within the plugged tones, since every plugged tone contains more or less the 'complete' harmonic series.
Now it is interesting what the mathematics says about the difference between equal temperament and harmonic (pythagorean) temperament.
The dissonant out-of-phase frequency regarding the high e-string (compared to the low e-string) is called the syntonic comma (21,5 cent). Its mathematical representation as a frequency fraction is 81/80. The inverse gives the fractal decimal expansion
0,98765432098765432098...
the fraction 1/81 gives us
0,0123456790123456790123...
The first decimal expansion lacks the digit '1', whereas the second decimal expansion lacks the digit '8'. This is interesting, since it seems to show for me that within the tuning system of pythagorean temperament, one cannot build a scale with 7 notes that has the feature to be consistent in the sense that its initial value ('1') determines conistently the final value ('8').
As an analogy, one could say that such a system is inconsistent in the logical-formalistic sense that its starting premise does not lead for every iteration (cycle of fifths) to the same final result again. The result diverges for every iteration, so to speak.
So even here, in the world of music and its reference to the ability of the human ear to distinguish fine dissonances, diverging from the natural harmonic series, seems to imply that human consciousness can well distinguish the departure from consistency to inconsistency, the latter in the sense of in-phase or out-of-phase relationsships.
Maybe 'meaning' has also something to do with in-phase relationships and is somewhat the opposite to out-of-phase relationships? I a broad sense I would think so, at least for human relationships that must have a minimum of in-phase attributes for being able to properly exchange some information, emotions, thoughts etc. How to generalize the importance of in-phase relations to the material realm other than by the term 'coherence' - I don't now at the moment. I only know that the several interpretations of quantum mechanics are out-of-phase in the sense that they are incoherent to each other. Moreover, consistency does only arise on the macroscopic level when certain measurement results have been macroscopically fixed. Then there is some 'meaning' of what happened physically until measurement. Well, the old problem of how to unequivocally interpret QM.