Foundations

Ratios

Fractions as ratios, the intervals of the tetractys 2 : 1, 3 : 2 and 4 : 3, why twelve fifths never make exactly seven octaves, and why the division table contains every fraction exactly once as a new one.

For the Pythagoreans a fraction was not the result of a calculation but a ratio of two whole numbers, and they found the most important ratios in music. Divide a string in the ratio 1 : 2 and the octave sounds, at 2 : 3 the fifth, at 3 : 4 the fourth. All three lie in the numbers 1, 2, 3 and 4 of the tetractys. Theon of Smyrna writes in the second century, in substance, that all the consonances are contained in the tetractys. This chapter turns that into mathematics.

Fractions

So $\tfrac{2}{4} = \tfrac{1}{2}$, because $2 \cdot 2 = 4 \cdot 1$. The definition says when two ratios mean the same thing, without ever having to divide by anything.

The division table

Writing all fractions $\tfrac{n}{k}$ into a table, with the numerator $n$ in the column and the denominator $k$ in the row, every positive fraction occurs infinitely often: $\tfrac12 = \tfrac24 = \tfrac36 = \dots$. By Theorem 1, however, it stands there in lowest terms exactly once, and that is the field where it appears for the first time. The page Simplexes and primes in the division table colours these fields red. Following an idea of Georg Cantor, the same table is used to show that the fractions can be counted like the natural numbers: you run through it along diagonals and skip the fields not in lowest terms. In harmonics it is called the lambdoma.

How large is the share of fields in lowest terms? The probability that two random numbers are both divisible by a prime $p$ is $1/p^2$. So the probability that they have no common prime divisor is

The first equals sign follows from unique factorisation (Chapter 4), the second is Euler’s solution of the Basel problem from 1735, $\sum 1/n^2 = \pi^2/6$. The statement about the fields was shown by Peter Gustav Lejeune Dirichlet in 1849 and made more precise by Franz Mertens in 1874 and Ernesto Cesàro in 1881. The calculation above is the idea of the proof, not the proof itself, since “random natural number” first has to be made precise.

The intervals of the tetractys

The ratio of two tones is the ratio of their frequencies. Playing two intervals one after the other means multiplying their ratios.

With 2 : 1, 3 : 2, 4 : 3 and 9 : 8 the Pythagoreans had everything they needed to build a scale. But they came up against an obstacle that cannot be removed.

The Pythagorean comma

Stacking twelve fifths one on top of the other, you arrive almost exactly at seven octaves, but only almost:

The difference $\tfrac{3^{12}}{2^{19}} = \tfrac{531441}{524288} \approx 1.0136$ is called the Pythagorean comma. It is not an error of measurement and not an inaccuracy of tuning, but a theorem.

That is why every tuning system has to make a compromise. The equal temperament usual today spreads the comma over all twelve fifths and makes each a little too small. For the Pythagoreans it was perhaps the first encounter with a limit that lies in the numbers themselves. The second follows in the next chapter.

/-! Kapitel 8: Verhältnisse und das pythagoreische Komma -/

/-- Satz 3: Keine Anzahl reiner Quinten ergibt reine Oktaven,
    3^a ≠ 2^b für b ≥ 1. Eine Potenz von 3 ist ungerade,
    eine Potenz von 2 gerade. -/
theorem komma (a b : Nat) (hb : 1 ≤ b) : 3 ^ a ≠ 2 ^ b := by
  intro h
  have h3 : 3 ^ a % 2 = 1 := by
    rw [Nat.pow_mod]
    simp
  have h2 : 2 ^ b % 2 = 0 := by
    obtain ⟨c, rfl⟩ : ∃ c, b = c + 1 := ⟨b - 1, by omega⟩
    rw [Nat.pow_succ]
    simp
  omega

-- Das Komma: 3¹² gegen 2¹⁹, also 531441 gegen 524288
example : 3 ^ 12 = 531441 ∧ 2 ^ 19 = 524288 := by decide

Where this appears on the website

The pages of the Harmonics section build on these ratios, especially The lambdoma and Space and time: the monochord in the lambdoma. The connection between divisibility and sound described there rests on Theorem 1: fractions in lowest terms are exactly the new tones.