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 smithy, tested
The best-known version of the discovery is told by Nicomachus of Gerasa. Pythagoras passed a smithy and heard that the hammers together produced harmonious intervals. He weighed them, and their weights were as 12, 9, 8 and 6. This story was retold for two thousand years, including in the woodcut above.
But it is not true. The pitch of a hammer does not simply depend on its weight. And whoever hangs weights on strings is in for a surprise, which Vincenzo Galilei, Galileo’s father, described in 1589: for the octave one needs not twice but four times the weight, for the tension enters as a square. The simple ratios hold for the length of the string, not for everything that sounds. In this correction one sees how Pythagorean delight in number became physics: the numbers are right, but one must measure what they apply to (see The monochord).
Why these ratios
Galileo, Rameau and Euler already tried to explain why simple ratios in particular sound pleasant. The explanation that is still the most influential today was found in the nineteenth century. A vibrating string sounds not only its fundamental but at the same time overtones at twice, three times and four times the frequency, and so on. Hermann von Helmholtz showed in 1863: two tones in the ratio 1 : 2 or 2 : 3 share many overtones, and the rest lie far enough apart not to sound rough. With complicated ratios the overtones rub against each other. The first four partials of an ideal string, the fundamental and the first three overtones, stand exactly in the ratio 1 : 2 : 3 : 4, so the tetractys is in the physics of sound itself (see Everything is frequency).
How much of this is nature and how much habit is still being researched. A study in 2016 found among the Tsimane in the rainforest of Bolivia, who knew hardly any Western music, that they did not find consonances more pleasant than dissonances. The ratios are physical, the liking for them is probably also learned.
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 (teal) 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 numbers are already in Plato’s Timaeus, the lambda figure was drawn by later commentators, and Albert von Thimus extended it to the full table in the 19th century.
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.
Sources and further reading
Sources
- Plato, Timaeus 35b to 36b
- Nicomachus of Gerasa, Manual of Harmonics 6 (the smithy)
- Boethius, De institutione musica I 10 and 11
- Albert von Thimus, Die harmonikale Symbolik des Alterthums, 2 vols., Cologne 1868 and 1876
- Vincenzo Galilei, Discorso intorno all’opere di messer Gioseffo Zarlino, Florence 1589
- Hermann von Helmholtz, On the Sensations of Tone, Braunschweig 1863
- Josh H. McDermott et al., Indifference to dissonance in native Amazonians reveals cultural variation in music perception, Nature 535, 2016
Further reading
- Hans Kayser, Lehrbuch der Harmonik, Zurich 1950