Harmonic Spirals

tuning

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:

{(\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