...And how is this reduction meant to be read? Taken from p. 66 of David Neumeyer's The Music of Paul Hindemith

diagram showing phrase and section proportions in an unnamed Himdemith piece

This is the beginning of the section on analyzing melodies, and it totally blindsided me.

Note: I am not even sure this pertains to melody--it may be in reference to another section of the book?


The diagram itself does not directly relate to melody; rather, it's a map of the formal structure of the piece. It shows the major sections, the phrases comprising those sections, their lengths in measures, and how various divisions relate via the Golden Section (GS). Presumably melody will come in later, with Neumeyer showing how Hindemith composed his melodies to adhere to the structure mapped in the diagram.

For example, the A and B sections comprise 46 measures, and the A' section is 26 measures. 46/26 ~ 1.77, which is in the GS ballpark. This is illustrated by the bottommost bracket-curve pair. (At a guess, P.GS means "the Primary [longer] segment in the ratio", and S.GS means "the Secondary [shorter] segment".)

The topmost pairs of brackets illustrate similar relationships. The left-hand pair shows that the A section (17 measures) and the first phrase of the B section (12 measures) are in a ratio of 17/12 ~ 1.42. The right-hand pair shows the first two phrases of A' (19 measures) and the third phrase of A' (10 measures) as having ratio 19/10 ~ 1.9. If we squint, we see that these again are GS-ballpark ratios.

Each bracket-curve pair shows instances where Hindemith attempted to make his phrase structures reflect the GS.


)/2There is really not much relation. The Golden Section stuff came from an article in American Mathematical Monthly where some statistics on Mozart's works were presented. The development+recap was longer than the exposition by something like the Golden Section. The development+recap ratio to the entire length of a movement was much closer.

In a later article, the same authors found a trivial explanation for this effect. There is no relation to music. What happens is if one has two numbers A and B with B larger than A, then the ratio A+B to B is much closer to the Golden Section than is the ratio of B to A. It happens that in using the Fibonacci recurrence: F(i+1)=F(i)+F(i-1), the ratio of terms becomes very close to (Sqrt(5)+1)/2 regardless of F(0) and F(1) (not both zero).

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