Number and sound
Octave, fifth and fourth are contained in the first four numbers. The lambdoma makes every ratio visible and audible at once.
According to tradition, Pythagoras discovered on a stretched string that pleasant intervals correspond to simple ratios of numbers. Divide the string in the middle and each half sounds an octave higher than the whole. Stop it at two thirds and you hear the fifth, at three quarters the fourth.
| Ratio | Interval | Example from C |
|---|---|---|
| 1 : 2 | Octave | C → c |
| 2 : 3 | Fifth | C → G |
| 3 : 4 | Fourth | C → F |
All three ratios need only the numbers one to four, exactly the numbers of the tetractys. For the Pythagoreans this was a strong argument that the world is ordered by number: the same figure that describes space also describes the harmony of tones.
The lambdoma
In the twentieth century the music theorist Hans Kayser took up this idea in his Harmonik. Its central tool is the lambdoma, a table in which every row has a denominator and every column a numerator. Each cell holds the ratio of numerator to denominator, and each ratio is also a tone.
Each cell plays the tone 220 Hz × numerator/denominator. Octave (blue), fifth (red) and fourth (green) are marked in colour. Highlighted is the diagonal on which every ratio equals 1. Fractions that can be reduced are grey.
A few patterns stand out immediately in the lambdoma:
- The diagonal from top left to bottom right contains nothing but ones: 1/1, 2/2, 3/3 … They all sound the same.
- Equal tones lie on straight lines through the top-left corner. 1/2, 2/4 and 3/6 give the same tone.
- Rows and columns mirror each other. A row holds the overtones above a fundamental, a column the undertones below it.
The name comes from the Greek letter lambda (Λ): write the numbers 1, 2, 4, 8 along one arm and 1, 3, 9, 27 along the other, and this shape appears. The scheme is already found in Plato’s Timaeus.
The rows and columns of the lambdoma are arranged like the two axes of a coordinate system. How closely it is tied to divisibility and the primes is followed up in the section Number theory.