Space and time: the monochord in the lambdoma
One string, one movable bridge. On the monochord you can hear that pitch and string length are inversely related, like frequency and wavelength.
The monochord is the basic instrument of Pythagorean music theory: a single string over a sound box, with a movable bridge. If the bridge divides the string in the middle, each half sounds an octave higher than the whole string. At two thirds you hear the fifth, at three quarters the fourth.
The buttons set the bridge at the classical divisions and play first the whole string, then the shortened part. The slider allows any position.
Length and frequency
On the monochord a physical law can be heard directly: halve the length and the frequency doubles. String length and pitch are inversely proportional, just like the wavelength and frequency of any vibration.
In the lambdoma this corresponds to the two axes. One counts multiples (overtones, higher frequencies), the other parts (undertones, longer strings). Each cell connects both.
This relationship is more than acoustics. Wavelength is a quantity of space, frequency a quantity of time. If music in the lambdoma necessarily runs into this law, then it can also prompt reflections on the relation between space and time. And there are good reasons to complete these reflections into a fourness: the tetractys of energy, space, time and matter (see All is number, all is frequency).
Anyone who wants to go deeper will find extensive literature on the monochord, the lambdoma, harmonics and cymatics, the study of visible patterns of vibration.
An instrument for thinking
The monochord was never a concert instrument but a measuring device and a teaching aid. The oldest surviving text on it, the Division of the Canon, is attributed to Euclid. The Greeks called both the ruler and this instrument a “canon”, and that is where our word for a standard comes from. In the second century Ptolemy described in his Harmonics how to build and calibrate it. Boethius passed the knowledge on to the Latin Middle Ages in the sixth century, and around 1025 the monk Guido of Arezzo taught his choirboys to read new melodies off the monochord before they sang them.
What else the string reveals
Length is not the only thing that determines pitch. In 1636 the French friar and natural philosopher Marin Mersenne found the complete laws of the vibrating string. Frequency is inversely proportional to length, as the Pythagoreans heard. But it grows only with the square root of the tension and falls with the square root of its mass per unit length. Whoever doubles the tension therefore gets no octave but an interval between fourth and fifth. For the octave one needs four times the tension.
This is where the beautiful legend fails that Pythagoras found the intervals with blacksmiths’ hammers or with strings carrying weights in the ratio 1 : 2 (see Music: number in relation). On the string the simple ratios lie in the length, not in tension or weight, and with the hammer the pitch does not simply depend on the weight anyway. That is exactly what makes the monochord so valuable: it is the device on which the numbers really hold.
Something similar appears in two dimensions, though without simple pitch ratios. If one bows a metal plate, it divides like a string into vibrating fields and resting lines, and whole numbers determine the patterns (see Chladni figures).
Exercise
A monochord can be built in an hour. One needs a board about a metre long, two screws at the ends, a guitar string and a pencil as a bridge. Stretch the string, pluck it, and then mark the half, the thirds and the quarters with a tape measure. Put the pencil under the marks and you hear octave, fifth and fourth, and you hear at the same time that it is the numbers 1, 2, 3 and 4 that are sounding (see Number and sound).
Kayser’s drawing
Hans Kayser recorded the relation between lambdoma and monochord in a drawing of his own. The equal-tone lines of the lambdoma, the lines on which all cells have the same tone, match exactly the points at which the string is divided. What appears in the lambdoma as a table is, on the monochord, a sequence of bridge positions.
Sources and further reading
Sources
- Euclid (attributed), Sectio canonis (Division of the canon)
- Claudius Ptolemy, Harmonics I 8 and II 12
- Boethius, De institutione musica, book IV
- Guido of Arezzo, Micrologus, around 1026
- Marin Mersenne, Harmonie universelle, Paris 1636
- Robert Fludd, Utriusque cosmi historia, vol. 1, Oppenheim 1617
Further reading
- Hans Kayser, Lehrbuch der Harmonik (Textbook of harmonics), Zurich 1950