Deriving the chromatic scale.

Introduction

I’m going to derive the chromatic scale mathematically as a function of the frequencies that are encountered in the circle of fifths.

In order to be able to evaluate this notebook, you’ll need to clone https://github.com/bwanab/music_prims and connect to it as a mix runtime.

Circle of fifths from first principles

The definition is the fifth above the fifth above the fifth infinitum.

Let’s start with the frequency of a low A = 110

a = 110

We know that the octave up is A 2 or 220. We might have learned in physics class that the first harmonic after the octave is the fifth which is A 1.5, thus the following function:

defmodule CircleFifthsFirstPrinciples do
  def fifth_above(a, b \\ 1.5) do
    v = a * b
    # here we're constraining the values to stay in the first octave
    if v > 225 do
      v / 2
    else
      v
      # in other words, we're wrapping around.
    end
  end

  def circle_of_fifths(a, f) do
    Stream.iterate(a, &f.(&1)) |> Enum.take(13)
  end
end

But, when we evaluate:

import CircleFifthsFirstPrinciples

circle_of_fifths(a, &fifth_above/1)

The last value that you can see is 223.00015 should be the same as the starting value times two (220), but it’s not. Why? I don’t know - physlcs. Sabine Hossenfelder has written a book (https://www.amazon.com/Lost-Math-Beauty-Physics-Astray/dp/0465094252) that argues the universe doesn’t owe us the beauty of small numbers. What we have above is just intonation(https://en.wikipedia.org/wiki/Just_intonation)--) where the notes are determined by small integer ratios. The way this was solved for centuries was to tune the octave to a true octave tuning which lead to one very noticeable off-pitch interval which was known as the wolf interval.

When one solves for the factor that gives the correct octave value it turns out to be 1.498307077 (https://en.wikipedia.org/wiki/Circle_of_fifths)..) This process is known as equal tempering. Basically, all intervals are minutely out of tune but resulting in no obviously out of pitch breaks.

defmodule BetterFifth do
  def better_fifth_above(a) do
    fifth_above(a, 1.498307077)
  end
end

import BetterFifth

circle_of_fifths(a, &better_fifth_above/1)

Now it looks right. The last value is well within the scope of human hearing to the double of the frequency of the root a.

What does the circle of fifths tell us? Once we’ve solved for the correct factor, we have a series of note frequencies such that each one is a fifth above the previous in exactly twelve steps to the octave up. If we sort those values we get the classic twelve note chromatic scale:

circle_of_fifths(110, &better_fifth_above/1) |> Enum.sort()

The chromatic scale

Given the circle of fifths, thus constrained, if we sort the values from low to high we now have the complete chromatic scale from A-110 to A-220. In other words, by computing circle of fifths, we’ve defined the twelve notes that are used for virtually all western music.

import MusicPrims

circle_of_fifths(110, &better_fifth_above/1)
|> Enum.sort()
|> Enum.zip(Enum.map(chromatic_scale({:A, 0}), &note_to_string(&1)))