Euler : E33







     Another of Euler's early books is his tentative theory of music, where he sets out initially in Ch. 1 to investigate both the generation of sounds in strings and tubes or pipes, not necessarly referring initially to musical instruments. Ch. 2 is initially concerned with philosophical matters, but gives some idea of how Euler thought. Later he considers some of the fundamentals of harmony.

References :

1. Mathematicians and Music

Author(s): R. C. Archibald

Source: The American Mathematical Monthly, Vol. 31, No. 1 (Jan., 1924), pp. 1-25.




The connections between sounds generally and their use in generating music in various instruments are discussed in a general manner.

Chapter I: Sound and Hearing.

A large number of ideas are presented in this introductory chapter, in which the various sources of sound known at the time are discussed, with reference to possible use in musical instruments. Some interesting comparisons are made between the similarities of sound waves in tubes and sounds produced by vibrating strings.

Chapter 2  : The Pleasures and Principles of Harmony.

In this chapter, after considering musical matters philosphically, Euler introduces a means of classifying the various ratios of notes found within an octave according to a numerical formula based on the possible subdivisions of the parts, resulting in pleasant sounds ;  he is largely successful in establishing a scale in this manner, but his scheme does not seem to have been adopted at the time.

Chapter 3  : Music in General.

In this chapter, Euler sets out his ideas about what music actually is ; he considers the link to speech and the written word, and to poetry in particular, which usually has a distinct rhythm set out in feet of some kind. This transfers to the musical equivalent in terms of beats and measures. Thus, there is a progression from everyday speech to verse to song and thence to the further abstraction of music of some form. This is the first characteristic of music, where high, low and intermediate pitches are used to produce a pleasing effect; the second characteristic is the timing of the particular sounds or notes, also coming from the poetic feet connection, whether they occur quickly or slowly, or in some mixture of the two, and whether the individual notes are of the standard whole time, half time, etc. Finally, the more difficult problem of giving a description of actual music, where these two characteristics are intermingled to provide a pleasing sound sensation is tackled.  As Euler states finally, this is the first time anyone has tried to give a scientific description of music. The work,  Musique Mathematique,  published in Paris 1865, uses Euler's work as a basis for a more in depth look at music ; it was written by musicians, who chose to remain nameless, and although they have successful in unraveling what Euler was saying about music, they do not seem to have understood his mathematical viewpoint, which they seem to have ignored, at least in this chapter.

Chapter 4  : Concerning Concords.

In this chapter, Euler sets out his ideas about using his special formula for generating concords and occasionally discords. This formula in generating concords goes far beyond the relatively simple ends of pleasing sounds, however it is relatively simple to understand, and some of the early history music by ancient Greeks is considered also. There was, and probably still is, an argument as to whether music is a science or an art, or a mixture of both.

Orders of Concords  : Orders of Concords.

This is an add on to Musique Mathematique, consisting of 4 sheets displaying the first 12 orders of concords considered by Euler.

Chapter 5  : The Succession of two Concords [or Consonants]

In this chapter, Euler sets out his ideas further by establishing the orders for the agreeability or charm of a succession of concords following his special formula.

Chapter 6  : Concerning Series of Concords.

In this chapter, Euler sets out his ideas further concerning the naming of the exponents of successive concords, the modes and indices, and provides us with an elementary example of how to do this numerically towards the end.

Chapter 7  : The Names Given to the Various Intervals.

Euler is rather critical about the historical names associated with the various interval, which had their beginning in antiquity with the Greeks. There are problems translating some of these, which are best left in the original. A comprehensive table provides a useful summary of these intervals in the octave as understood at the time.

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