by Ray Tomes

Harmonics Theory: Patterns in the Harmonics

This blog series from FSC Science Director Ray Tomes will share 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.

Patterns in the Harmonics

For each harmonic H, the number of ways that it can be formed is called C(H). When C(H) is calculated out for higher numbers some very clear patterns emerge. For the patterns to emerge it is first necessary to see that there are two effects going on that cause things to race off in opposite directions. By balancing these two things we can see the beautiful music of the universe.

The first effect has been mentioned for non-linear systems, and that is that energy is lost to harmonics, mainly the low order harmonics. Such a pattern is shown in the figure below. In the theory we assume only that the power in each harmonic drops off as some inverse power of the harmonic number. It might be 1/H or 1/H^2 or 1/H^3 or even a non-integer number. In every case the result of going to higher order harmonics is a decrease in relative energy. But this recognises only one step in the process of making harmonics.

Chart: Relative Power for Harmonics for 1/h

The second effect is that numbers that are able to be factorised in a larger numbers of ways tend to be higher numbers rather than lower ones. This effect is in the opposite direction to the other effect but does not exactly cancel it out. Not all larger numbers have many ways of being factorised, but the larger the number the greater the possibility as shown in the figure below.

Chart: Strength of Harmonics 1 to 100

What we are really interested is, in any range what are the strongest harmonics which means the ones that can be factorised in the greatest number of ways for the size of the numbers. These will be the waves with the most energy for their size and which will presumably show up to our senses or on scientific instruments in preference to their weaker neighbours. The above two effects are combined below for the first 100 harmonics by showing C(H)/H as a function of H.

Chart: Relative Strength of Harmonics 1 to 100

Although 1/H works well in this range for detrending the rapidly rising C(H), it does not work so well for much larger H and C(H). This issue will be returned to later. For now, let us observe some of the patterns that are present in the harmonics from 1 to 100.

Firstly we see that multiples of 12 make stronger harmonics than other numbers and that very strong harmonics always have other strong harmonics at ratios of 2 above and below. By this is meant that if H is very strong then H/2 and H*2 will also be strong.

Next we may notice that there is a very strong musical pattern present. In particular, looking at the interval from H=48 to H=96 the very strongest harmonics are 48-60-72-96 which is a major chord in music (usually known as the ratios 4:5:6:8 resulting when cancelled by factors of 12). In addition the notes of the just intonation music scale are exactly represented in the harmonics 48-54-60-64-72-80-90-96. Additionally two extra notes labelled here as Mi-flat and Ti-flat are quite strong. These are needed to make dominant seventh chords.

Chart: Relative Strength of Harmonics 1 to 100 (with dominant seventh chords)

The reason that music and the strong harmonics are so closely related is that both are representations of frequencies which have as many small number ratios present as possible. As we look more deeply into these relationships we will see why past great scientists have referred to the music of the spheres. The underlying energetic frequencies of the universe have relationships that are everywhere musical, although as we go to higher numbers various different types of music are found and some of it would be rather foreign to our ears.