A Christmas Trinity III: Johann’s geometrical music.

Possibly the first mathematical law of nature was the Pythagorean discovery that the lengths of stretched strings that produce harmonious sounds when plucked are in simple numerical ratios to each other. The Pythagoreans knew the musical intervals unison 1:1, the fifth 2:3, the fourth 3:4 and the octave 1:2. All of these consonant intervals can be constructed with the first four integers, which had a special status in the numerological religious philosophy of the Pythagoreans and this alone was adequate as a metaphysical explanation for consonance for them and for the following centuries.

In the high Middle Ages European music was enriched by the discovery of polyphony and the consonance of the intervals major third 4:5, minor third 5:6, major sixth 3:5 and minor sixth 5:8. These discoveries caused problems for the Pythagorean music metaphysics as the integers 5, 6 and 8 did not have the special status of the first four integers and polyphony was not possible with the Pythagorean fifth as a fifth circle based on it leads to the so called Pythagorean comma that is if starting at say C you raise the musical tone in Pythagorean fifths 12 times you don’t arrive again at C but slightly removed from it. This means that a truly consonant scale can’t be constructed using Pythagorean intervals.

These problems led to a major discussion on music and its metaphysics at the beginning of the 17^th century with Vincenzo Galilei, Galileo’s father, defending the traditional Pythagorean standpoint and Gioseffo Zarlino arguing for a new metaphysics. Zarlino argued that all the intervals were contained in the number 6, which is the first perfect number, i.e. a number equal to the sum of its factors, furthermore God needed six days for the Creation, there are six planets Moon, Mercury, Venus, Mars, Jupiter and Saturn, six ‘natural offices’: size, colour, shape, interval, state and motion, six directions: up, down, forward, backwards, right and left, six faces on a cube and so on and so forth. This justification is of course purely arbitrary and one could produce a similar list for almost any number, also it fails to justify the 8 in the minor sixth. Zarlino deals with the latter problem in an Aristotelian manner by explaining that the minor sixth is only a potential consonance and not an actual one (Aristotle’s argument on infinity) and the consonance can be explained by the fact that 8 is twice 4. A lousy piece of ad hoc justification if ever there was one.

The problem of consonance and its mathematical basis became a major scientific theme in the 17^th century¹ with all of the major players making contributions, Stevin, Galileo, Mersenne, Descartes, Huygens etc. Here I just want to look at the solution offered by Johannes Kepler who was born on 27^th December 1571².

As I wrote in an earlier post Kepler believed that there were six planets because the planetary spheres were separated by the five Platonic solids, the basis for his belief that God is a divine geometer. Unfortunately although Kepler’s scheme gave a reasonable fit for the then known sizes of the planetary orbits it was not perfect and required fine tuning, which he supplied in the form of heavenly harmony. The Pythagoreans had propagated a theory of celestial harmony in which they claimed that distances of the planetary orbits stood in harmonious relation to each other and that in their journey through the heavens they created a music of the spheres that was only perceptible to the enlightened. Kepler took up this idea in his Harmonice Mundi and searched for harmonic relationships in various aspects of the planetary orbits, relative speeds at apogee or perigee etc. In the process of this search he famously stumbled across what is now called his third law of planetary motion, which is also known as the harmony law. In order to carry out this work Kepler first had to develop his own metaphysics of consonance and in fact he dedicated the whole of the third book (of five) of the Harmonice Mundi to this task.

Kepler sides with Zarlino and the moderns against Galilei and the Pythagoreans but unlike both parties he rejected a numerical metaphysics because he argued that numbers are abstracted from reality; in themselves they do not donate anything real. In line with his model for the solar system he proposed a geometrical solution because:

 

…the terms of the consonant intervals are continuous quantities, also the causes that distinguish them from the dissonances should be taken from the family of the continuous quantities, not from abstract numbers, as a discrete quantity.

 

Whereas Kepler’s solution for the number of planets was based on the polyhedra his solution for the riddle of the consonances was based on the polygons or rather those of the polygons that could be constructed in a circle using only a compass and a straight edge. He compiled the ratios between the number of sides of these polygons that spanned half of the circle or less and their residue or the whole number of sides and the ratio between the residue and the whole. So for the diameter (Kepler was the first mathematician to regard the diameter of the circle as a one sided polygon, a standard practice today when categorising plane figures) we have the ration 1:1 (residue) or unison and 1:2 (whole) or octave. The triangle delivers 1:2 (residue) or 1:3 (whole) both octave or 2:3 (residue to whole) fifth. This function up to the hexagon but the octagon produces problems, as it gives ratios of 1:7 and 7:8, which are dissonant. Kepler disposes of these by saying that the 7 is also produced by the heptagon, which is not constructible with straight edge and compass and so is not valid, more than somewhat ad hoc. For Kepler polygons greater than the octagon only produce multiples of intervals already generated and so do not need to be considered. The whole scheme broke down as Gauss discovered the methods of construction of such polygons as the 17-gon but by then Kepler’s geometrical music metaphysic was just a faint memory.

 

1) A full and detailed account of the development of mathematical music theory in the 17^th century can be found in Floris Cohen’s excellent Quantifying Music, D. Reidel, Dordretch/Boston/Lancaster, 1984 from which most of the above is stolen (um borrowed).

 

2) The Renaissance Mathematicus is suffering for a 24-hour time lag due to a time warp in the life of its author. Normal service will be resumed on New Years Eve.