# 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