Harmonics Theory, Part 10: Families of Harmonics

by Ray Tomes

Harmonics Theory: Families of 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.

Families of Harmonics

When several main line harmonics in a row are at ratios of 2 relative to each other, those families of harmonics do not stop just because they are not in the main line. For example, looking at 144 and 288 in the main line, we can trace a smooth curve back through 72, 36, 18 and 9 and forward through 576, 1152, 2304 and further on into the weaker harmonics jungle. These harmonics that are linked by ratios of 2 are called families of harmonics, and they can help us to see the fascinating structure present. In music all of the notes linked by ratios of 2 are given the same name, such as C, although they may be called C4, C5 and C6 so that we know which C is meant. The families are just like the musical notes, and adjacent families, which are different by only a ratio of 3, are like the most common musical modulations of adding or taking away a sharp or flat in music.

The first family goes 1, 2, 4, 8, 16, ... or just the powers of 2. Initially these are the strongest harmonics but at 8 the harmonics dip below the next family, their cousins, 3, 6, 12, 24, 48, 96, 192, ... which are all powers of 2 times 3. Again these are the strongest harmonics from 12 to 48 but then 96 falls below the line of the next family. This process keeps repeating with families that are powers of 2 times 3, 3^2, 3^3, 3^4 and so on. It can be seen that the peaks of these coloured parabolas occur between 12 and 24, then 144 and 288, and so on. This supports the view that a ratio of 3 happens for every 2.38 ratios of 2 because that also would fall between those same numbers.

However something else happens to our coloured lines as we move from left to right, the peaks of these parabolas are themselves making a parabola and as a result they stop being the highest harmonics of all. A new unmarked family of 720, 1440, 2880, 5760, ... takes over the lead. This family is of the form 2^n*3^2*5 and second cousin 5 has joined the fray.

The curved lines are getting a bit more crowded now, so the new families, which all have a factor of 5, have been added in darker colours. These new families again each make parabolas and the cousin families again extend this with a parabola of parabolas. At the right edge of the graph some of the strong harmonics are not members of any of the coloured families, but belong to a new cousins involving the ratio 7.

It is possible to make parabolas through harmonics that are linked by ratios of 3 rather than 2 also. An example would be 16, 48, 144, 432, ... and many others. Such groups fall away a little faster in each direction than the families based on ratios of 2. If we do the same thing with ratios of 5 or 7 then they fall away faster still. That is why repeated ratios of 2 and 3 happen in music but not with the higher primes.