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. Further notes are added to the chapter links

References :

1. Mathematicians and Music :Author(s): R. C. Archibald

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

2. An interesting book : The Theory of Harmonics by John Keeble, circa 1785 sets out the history of the development for ancient Greece times. It is available as a pdf download on the web, or can be bought quite cheaply.

3. Tuning and Temperament by J. Murrey Barbour publ. by Dover is the bible for this sort of discussion, though it assumes an advanced level of understanding.

4. Ancient Greek Music by M.L. West publ. by Oxford is more readable, and supposes to take us from basics up to a fuller understanding of a very complex study.

5. It is not a bad idea to have a suitable introductory physics text available that has a chapter on music. Such a one is a rather elderly text in my possesion  Analytical Experimental Physics, by Ference, Lemon, & Stephenson, pub. by Uni. of Chicago, 1956. Chapter 33.

6. Charles Samuel Smith produced a translation of this work for his Ph.D. dissertation at Indiana University in June 1960. It is available as a microfilm from some university libraries in the U.S.



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.

                                                         Chapter 8  : Concerning Musical Modes.

Euler sets about the difficult task of describing the various tones used in constructing a scale of notes in suitable ratios of pitch or frequency in an octave, which is pleasing to the listener. He does this by looking at the exponent 2m3p5q for successive values of m, p and q; in all 17 such modes are considered, dating from the earliest in ancient times, arriving at the diatonic, chromatic and enharmonous modes.


Chapter 9  : Concerning Kinds of Diatonic-Chromatic scales.

Euler produces a mathematical generalisation of these scales, which he then adjusts as a 'well-tempered' scale to be the best available.

Chapter 10  : Concerning Kinds of Diatonic-Chromatic scales.

Euler adds a few more scales to his grand total, but the exercise has become more mathematical as the divisions between the notes on the scales are barely perceptible.

Chapter 11  : Consonances in the  Diatonic-Chromatic Genus.

Euler sets out an extensive listing in note notation of the various consonances or chords that arise from this scale.


                                                 Chapter 12 : Concerning the modes and systems within the Diatonic-Chromatic Genus


                                                   Chapter 13 : Concerning the Method of Composition in a given Mode and System


                                                    Chapter 14 : Concerning the Interchange of  Modes and Systems.



Ian Bruce. May 3rd , 2019 latest revision. Copyright : I reserve the right to publish this translated work in book form. However, if you are a student, teacher, or just someone with an interest, you can copy part or all of the work for legitimate personal or educational uses. Please feel free to contact me if you wish, especially if you have any relevant comments or concerns about this work.