Harmony in Mathematics
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Music is the Mathematics of one who does not know that he is counting. Gottfried Wilhelm Leibniz (1646-1716)
When Pythagoras discovered that the major musical tones could be produced by shortening a string by simple whole number ratios, he had little doubt that eventually all of nature could be described and explained through the use of numbers - hence the motto of the secret society of which he was the founder and inspiration: "All is number". He was excited to discover that the octave was produced by the ratio 2 : 1, the major fourth 3 : 2, and the major fifth 4 : 3. The musical scale which Pythagoras deduced from these ratios was quite different from our modern scale (which is based upon a geometric sequence with common ratio \\( \\sqrt [12] 2 \\). He certainly applied his musical scale, among other things, to predict the relative positions of the sun and the planets from the earth (the harmony of the spheres), and to his theories of healing and medicine.
Pythagoras used the word harmony in a different sense than that which we use today, in which we describe a pleasant sounding combination of notes. To the Pythagoreans, armoniai described a well-ordered sequence of notes - a pattern or scale that was pleasing to the ear. Noone knows what this original scale was, but it seems likely that it was based upon the calculation which he himself named the harmonic mean.
Consider, for instance, the following ratios as one way of representing the fourth, fifth and octave notes of the scale:
C 1 : 1 F 4 : 3 G 3 : 2 C' 2 : 1 Then, by taking the harmonic mean of C and G it is possible to calculate a ratio for E; HM(C, E ) produces D, F and C' produce A, and A and C' can be used to find B. In this way, a scale emerges:
Note Harmonic
RatioHarmonic
DecimalWell-tempered
ScaleC 1 : 1 1.0 1.0 D 12 : 11 1.091 1.122 E 6 : 5 1.2 1.26 F 4 : 3 1.333 1.335 G 3 : 2 1.5 1.498 A 8 : 5 1.6 1.682 B 16 : 9 1.778 1.888 C' 2 : 1 2.0 2.0 Can you determine how each of the harmonic ratios was derived?
While this scale produces tones surprisingly close to the modern "well-tempered" scale, there are inconsistencies. (Using the Harmonic Mean method, for example, offers several options for computing parts of the scale - notes like A and B, in particular might be calculated in different ways). To go beyond a single octave, one would double each value to produce the next octave, and double each of those, in turn. The modern well-tempered scale is much more consistent across the range of notes (although might a scale based upon an arithmetic series also be consistent?)
Compare the sound of our modern (well-tempered) scale (based upon a geometric sequence) to that of one based upon the harmonic series of Pythagoras...
In the sense that the Pythagoreans were attempting to use Mathematics to describe and explain a universe which they did not understand, Mathematics itself could well be defined as a search for harmony - a search for patterns and relationships which will impose an order, an armoniai, upon an otherwise chaotic world.
A Picture of Harmony
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The Greeks knew no algebra as we know it today. Even their work with numbers was strongly visual - even now when we speak of numbers as "figures" we echo Pythagoras. Their Mathematics was strongly geometric in nature, a geometry largely bound by the Platonic restrictions of straight edge and compasses. But with these tools they could do a surprising amount of quite detailed mathematics - they could add, subtract, multiply and divide, using only these tools; after they overcame their initial distaste for irrationals, they were able to construct roots with relative ease.
It seems likely that their methods of calculating means, then, would also have been geometric. The most elegant of these constructions occurred within the semi-circle. Pappus of Alexandria, about 320 AD records the construction in which the radius CF gives the Arithmetic Mean, DF the Geometric Mean and EF the Harmonic Mean of AD and DB, as shown below.
Click anywhere on the first image showing a TI-Nspire construction to open an interactive version which you can control using HTML5 from Geometry Expressions, or watch the video of the GXWeb construction.
The Geometric Mean construction follows directly from the right-angled triangle in which an altitude is dropped to divide the base into lengths a and b. This may be proved by using Pythagoras' Theorem, but is far more quickly and elegantly achieved through the use of similar triangles. It is worth noting, too, that in this remarkable triangle, each of the three "uprights" rising from the base gives the Geometric Mean of the two rays extending from its foot. Thus in the figure above, not only is FB2 = OF.FD, but OB2 = FO.OD and BD2 = FD.DO. (The Greeks would have pictured these as squares equal in area to two rectangles!)
More elegant still was another of Pappus' constructions in which a tangent and a secant were taken from an external point, X, with the secant forming a diameter on the circle. In this case, if AD = a and DB = b, then AM(a, b) = DC, GM(a, b) = DE and HM(a, b) = DF.
Click anywhere on the first image showing a TI-Nspire construction to open an interactive version which you can control using HTML5 from Geometry Expressions, or watch the video of the GXWeb construction.
These representations provide immediate and convincing visual evidence for the important inequality, for a not equal to b:
AM(a, b) > GM(a, b) > HM(a, b)
Harmony in Perspective
