Harmonic Spirals

music theory

May 09, 2020

Let's create some harmonic spirals. We can create one by winding the number line like this:

  • you can think of this as a number line with a twist
  • starting at 1 (root pitch) in the center, we map one full rotation (=360 degrees) to the multiplication of 2 (octaves)
  • each line starts at an odd number representing the beginning of a new pitch color
  • the line extends to infinity, crossing the spiral at octaves of the color
  • all natural numbers will be part of a line
  • the whole image is a dense visualization of the harmonic series from the perspective of octaves
  • it gets even more accurate (but not as compact) if we use a logarithmic spiral
    • its radius gets two times bigger on every rotation.
    • its growth represents the behaviour of frequencies, as we go up the harmonic series.

3 Limit Spiral

Instead of the harmonic series, we can also display the ratios of any tuning system.

The following spiral displays stacked fourths, which can be seen as a 3 limit system.

Click to expand why I chose fourths instead of fifths

I picked fourths, as they grow slower in pitch, compared to fifths. This benefits our ears when listening to the non octave reduced pitches

Here is why stacked fourths are just a form of 3-limit:

(43)n=22n3n{(\frac{4}{3}})^n = 2^{2n} * 3^{-n}
  • The numbers are the powers of 4/3 (which is the ratio of a pure fourth)
  • The gray lines show where equal tempered notes would intersect the spiral
  • You can see that the further we go up, the error between equal temperament and stacked fourths gets bigger
  • In the settings, we can apply octave reduction to bring each 12 notes into one octave, creating a chromatic spiral
  • When octave reduced, we can see the "error" in the curvature of the (almost) octaves
  • If we check the tempered box, all fourths are comma corrected,
    • for example, the 12ths power iss added one full comma, the 24th power 2 etc (and all in between get logs of 2, see code for details)
  • The temperament leads to really complex ratios
    • We can also see a slight error of the temperment, I am not sure if this is a problem of Javascript or a problem of the algorithm
Unsolved: Comma Temperament vs Equal Temperament
  • We know that when applying fractions/factors of the comma (which is rational), all resulting tempered ratios remain rational.
  • In contrast, standard equal temperament is generated with multiples of the 12th root of 2, which is irrational
  • The question: What's the difference between "comma temperament" and standard equal temperament
  • There has to be some error in the "comma temperament" that I am not seeing right now

Further Reading

  • find unified way to generate those linear scales from generator interval + number of notes + number of notes per octave?!
    • e.g. generate(4/3, 25, 13) => stacks 24 fourths and packs each 13 fourths into one octave
    • or generate(3/2, 25, 12) => stacks 25 fifths and packs each 12 fifths into one octave
  • find way to animate between temperment and non temperment
  • find out why tempering Pythagorean tuning results in 12edo (or seems to)
    • at the first thought I cannot see where the irrationality comes from, as the comma is also just a fraction

Felix Roos 2022