by Ray Tomes

Harmonics Theory: Main Line Harmonics

This blog series from FSC Science Director Ray Tomes shares the fundamentals of physics in layman's terms, showing how present theory must inevitably lead to all waves losing energy and forming harmonically related waves. The end result is a very specific detailed structure that matches the observed universe and explains many previously mysterious observations. This series was previously published.

Detrended Strength of Harmonics vs. Log(10) Harmonic Number

Main Line Harmonics

The term "main line harmonics" is used for the harmonics shown in larger font in the above diagram. They are the series of harmonics starting from 1 that are always related to each other by a prime number ratio and which are the very strongest harmonics. There are some marginal cases to this definition as we might have chosen either 5760 or 8640 to lie between 2880 and 17280. Either way the ratios would be the primes 2, 3 or 3, 2; but 8640 was chosen as being a little stronger than 5760. Such a decision might be influenced by the formula for detrending the harmonics.

The purpose in highlighting these main line harmonics and the prime ratios between them is that the prime ratios contain a pattern which is almost regular. The early main line harmonics are 1, 2, 4, 12, 24, 48, 144, 288, 1440, 2880, 8640, 17280, 34560, 69120, 207360, 414720, ... and these have ratios between them of 2, 2, 3, 2, 2, 3, 2, 5, 2, 3, 2, 2, 2, 3, 2, ... and so on. It can be seen that there is one ratio of 3 for every two or three ratios of 2 and that the other primes occur much less often.

At one time I conjectured that the primes occurred with inverse frequency p*log(p) and because it it difficult to calculate very high harmonics this conjecture was used as a substitute for the actual calculations beyond the known limits. However with increasing computer speed and a few mathematical tricks it has been possible to calculate the harmonics out beyond 10^50 now and to find that the conjecture is not correct and gets gradually less accurate at higher order as the larger primes get further apart than expected. If this conjecture were correct, then a three would occur for every 2.38 twos, a five would occur for every 5.8 twos and a seven for every 9.8 twos. On that basis a 7 should be due to happen next and it does.

These proportions of 2s, 3s, 5s and 7s are quite consistent with musical practice. Pythagoras only used 2s and 3s in his musical scale, however Galilei, the father of the famous Galileo, showed that a major third is really 5/4 and not 81/64 as Pythagoras had assumed. Actually both of these ratios do occur but it is most often Galilei's ratio that happens in music. It has been expressed, for example by the mathematician Euler, that the ratio 7 is a bit too harsh to be used in musical ratios. However it is used in Indian music and in Blues music and should definitely be considered as a valid musical ratio, Part of the reason that it is not used more is not that it is naturally harsh but that it does not fit in with the present music notation which really is biased to the equitempered scale which only deals comfortably with ratios involving 2, 3 and 5. The proportional frequency of occurrence of 2, 3, 5 and 7 according to the harmonics theory gives a meaningful answer to the question that Euler asked concerning the concordance of sounds, a subject which has been studied by music theoreticians since Euler's time.