Euler and Music Musing Euler’s Identity

Euler’s identity e ^iπ + 1 = 0 is considered the most beautiful in the history of mathematics (Richard P. Feynmann, an American physicist who won the Nobel Prize in 1965 for his studies on quantum electrodynamics, is among the many that elected it “the most beautiful formula of all time”). Can this equation be represented by a system of sounds that correspond to it, albeit with the agility of an authorial contribution? Who was Euler really? Only a mathematician? Actually, he has also been involved in music for a long time, trying to establish in physical and mathematical terms which characteristics of music delight the soul and why. Without fear, therefore, he entered the difficult task of understanding good taste in music. Thanks to the collaboration with Chiara de Fabritiis, it was possible to interact with music and mathematics to take up in practice the opportunities suggested by Euler, applying them to his most famous equation. The encounter between music and mathematics achieves a translation not only into sounds of the five elements of identity (From a mathematical point of view, the elements that enter Euler’s identity are 5: e, π, i, 1, 0 (where 1 is the neutral element of the product and 0 is the neutral element of the sum), but also, as far as possible, of internal properties and relationships. The work is still in progress, although in advanced stages. Just as a whole world of mathematics is contained in this brief equation, so in the sounds that represent it there are centuries of history of music and related symbols and styles.

1. 1.

   Quadrivium, in Latin literally four ways, in medieval times, indicated, together with the Trivium, the scholastic training of the liberal arts, preparatory to the teaching of theology and philosophy. It included four disciplines attributed to the mathematical sphere: 1. arithmetic 2. geometry 3. astronomy 4. music. This subdivision is due to Marziano Capella, a late Latin philosopher (4th–5th century AD) who took care, among other things, of dividing all human knowledge into categories.

2. 2.

   Film of Mario Martone, 2014.

3. 3.

   See Giorgio Sanguinetti, Preface to the Italian edition of: La Musica nello stile Galante, R. O. Gjerdingen [12], p.11.

4. 4.

   The 25 March 1436 consecration of the Florence Cathedral, on the occasion of the completion of the dome built under the instructions of Filippo Brunelleschi.

5. 5.

   I. Harmonices Mundi Libri V, Lincii Austriae, sumptibus Godofredi Tampachii excudebat Ioannes Plancvs, 1619.

6. 6.

   Harmonie universelle, contenant la théorie et la pratique de la musique. Par F. Marin Mersenne [2], pp. 1636–1637.

7. 7.

   Abrégé de musique, par Monsieur Descartes, 1668.

8. 8.

   Musurgia universalis sive Ars magna consoni et dissoni in X. libros digesta. Par Athanasius Kircher [4], 1650.

9. 9.

   Caramuel Lobkovwitz [5], Juan 1606-1682, Primus calamus secundam partem metametricae exhibens.

10. 10.

    De Piero [8].

11. 11.

    Galeazzi Francesco [10], p. XXV. Euler’s quote refers to Tentamen… (op. Cit.) Chap. II paragraph 4.

12. 12.

    Here he means F. Marin Mersenne [2].

13. 13.

    Galeazzi [10], p.VI (translation by the author).

14. 14.

    See note 12.

15. 15.

    Schönberg writes about the dominant term: To tell the truth, the expression dominant for the fifth degree is not entirely correct, because this name suggests that this agreement “dominates” one or more others. (…) The name of “dominant” is usually justified by the affirmation that the first degree is introduced by the fifth, so that would be a consequence of this. (…) to follow means to obey but also to align, to come later: and if the tonic “follows” the dominant it is like when a king lets himself be preceded by his vassal, the master of ceremony and the quartermaster, so that they make the necessary preparations for the entry of the king who follows them: but the vassal is there for the king, and not vice versa. (Schönberg [11], pages 41–42, (translation from Italian by the author).

16. 16.

    Euler wrote: It is also absolutely necessary that a musical work resembles a prayer or a poem. As, in fact, in these it is not enough to join elegant words and phrases, but there must also be an orderly arrangement of the same things, and an appropriate distribution of the topics; so also in music there must be a similar principle. In fact, it is not very nice to have several consonances placed in series, even if individually they are quite pleasant, but it is necessary that the order be distinguished in these, just as if they were to express some prayer. In this problem it is especially useful to pay attention to the degree of ease or difficulty with which the order is perceived; and depending on how the established object requires, the joy and sadness will have to be changed, or now this, now that, will have to be increased or decreased. L. Euler, op. cit. p.87, (translation by the author).

17. 17.

    Dante: Paradiso Canto 1 versi 13/14.

18. 18.

    The first use of the symbol i to denote the imaginary unit is in a text of 1777, which Euler addressed to the Academy of Sciences of St. Petersburg and which was published posthumously in 1794 in one of the volumes of the Institutionum calculi integralis.

19. 19.

    Euler has been the first to use the letter e to denote Napier’s number in a short treatise, Meditatio in Experimenta explosion tormentorum nuper istituta (Reflection on experiments recently carried out on shooting with cannons) he wrote towards the end of 1727 or the beginning of 1728 (when he was 21 years old).

20. 20.

    Euler popularized this symbol by using it in the Introductio in Analysin Infinitorum of 1748 (previously he often used the letter p).

21. 21.

    Many thanks to Chiara de Fabritiis for the pleasant explanatory conversations.

Co-editor and Supervisor of the English translation Marianna Biadene.

Authors and Affiliations

1. Conservatorio “B. Marcello”, Venezia, Italy

   Davide Amodio

Corresponding author

Correspondence to Davide Amodio .

Editors and Affiliations

1. Sapienza University of Rome (retired) Rome, Italy, IVSLA - Istituto Veneto di Scienze, Lettereed Arti, Venice, Italy

   Michele Emmer

2. Department of Mathematics, University of Pisa, Pisa, Italy

   Marco Abate

Reprints and permissions

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

Policies and ethics