Speculationi di Musica dedicate all’Eminentiss. e Reverendiss. Sig. Card. Azzolini da Pietro Mengoli, dottor dell’una, e l’altra legge, e di filosofia collegiato, Prior di S. Maddalena, e Publico Professor di scienze mecaniche nello Studio di Bologna

4to (105x140 mm). [24], 295, [1 blank] pp. and 2 folding tables inserted after H4 and Oo2. Errors in pagination in quire T and V. Collation: a-b⁶ A-Oo⁴. Half title: Musica speculativa del Mengoli. Contemporary cardboards (worn, rubbed and soiled, spine repaired). Title page soiled and with underlining, some damp staining which gets heavier near the end.

Rare first edition of this treatise, in which the Bolognese mathematician and priest Pietro Mengoli proposed a radical mathematical and physiological model for hearing. He argued that the soul perceives musical intervals through logarithmic abstraction of alternating air-particle frequencies hitting a double eardrum structure, departing from traditional Renaissance coincidence theories. Mengoli hypothesized an idiosyncratic anatomical setup featuring outer and inner eardrums to mechanically process incoming sound vibrations. Challenging Galileo’s coincidence theory, he asserted that perceived intervals arise from the difference in the alternating frequency with which air particles strike the eardrum rather than simultaneous pulse coincidences. Anticipating psychophysical principles, the soul calculates and abstracts intervals logarithmically - identifying a foundational “half-octave” (misura musica) as the baseline unit of auditory experience. Upholding traditional views, Mengoli maintained that intellectual judgment by the soul supersedes raw physical sensation in defining true musical consonance.

“In 1670, the Bolognese mathematician Pietro Mengoli published his Speculationi di musica, a highly original work attempting to found the mathematical study of music on the anatomy of the ear. His anatomy was idiosyncratic and his mathematics extraordinarily complex, and he proposed a unique double mechanism of hearing. He analysed in detail the supposed behaviour of the subtle part of the air inside the ear, and the patterns of strokes made on the eardrum by simultaneous sounds. Most strikingly, he divided the musical octave into a continuous set of regions which he colour-coded to show their effects on a listener. His work did not find its way into the mainstream of seventeenth-century mathematical studies of music, but when examined in its context it has the potential to shed light on that discipline, as well as being of considerable interest in its own right” (B. Wardhaugh, The Logarithmic Ear: Pietro Mengoli’s Mathematics of Music, in: “Annals of Science”, vol. 64, issue 3, 2007, pp. 327-348).

“According to Artusi, the sense of hearing could receive sounds as the matter of musical intervals and songs, but it could not judge how they relate with each other. He believed that using only hearing to perceive musical intervals would only result in confusion. Therefore, Artusi insisted on the necessity of intelligence for the clear and distinct judgment of music. Thus, the primacy of intelligence/reason over the sense of hearing was maintained in the tradition of music as a mathematical science. However, in the latter half of the seventeenth century, a theoretical work appeared that focused on the function of the sense of hearing: the Speculationi di musica (1670) by Pietro Mengoli […] Mengoli was active as a mathematician of the Galilean school, which promoted new natural sciences. However, when he was ordained in 1660, he began to waver between the religious doctrines of the Roman Catholic Church and the new scientific trends of the scientific revolution. After all, he represented Catholicism and opposed the Galilean doctrines (for instance, he supported the geocentric theory). Due to the fact that he developed his original metaphysics while attempting to reconcile the Catholic faith with the new scientific view of nature, the terms and logic in his works were often difficult to understand. Mengoli’s inclination toward speculation prevented his works from having much influence on future generations. It was not until the beginning of the twentieth century that his mathematical achievements were reevaluated. In addition to his works of pure mathematics (arithmetic and geometry), Mengoli wrote about ‘mixed mathematics’ (astronomy, optics, and music, etc.), in which natural phenomena were examined by means of pure mathematics. Speculationi is one of his works on ‘mixed mathematics’ and was written in Italian. It consists of the following sections: a dedication, preface, ‘Natural History of Music’ (Historia Naturale della Musica; a summary of the whole work), a ‘Protestation of the author’ (Protesta dell’autore; a declaration that the standpoint of the work is not against the doctrines of the Roman Catholic Church), and twenty-five speculations (Speculationi) - the main parts of the work. In 1673, three years after its publication, Speculationi was discussed in articles in ‘Philosophical Transactions’ of the Royal Society, and ‘Giornale de’ letterati’, a book review journal issued in Rome. Both articles mentioned that Speculationi contained new knowledge about the auditory organ, which reflected the prevalence of empiricism in those days. In the former article, after a short introduction of the work, the ‘Natural History of Music’ was wholly translated into English. Henry Oldenburg (1619-77), the first secretary-general of the Royal Society and the supposed author of this article, pointed out that Speculationi was ‘somewhat obscurely written’, indicating his difficulty understanding the work. In the latter article, the reviewer stated that Mengoli’s main purpose was to argue the nature of the soul (anima) rather than to teach music theory. This remark is particularly astute, because Speculationi deals exclusively with the recognition of musical intervals by the human soul, not with the practice of music […] Mengoli’s Speculationi is an unusual work on music. Although it deals with musical intervals and melodies, they are argued exclusively from the physiological aspect of musical hearing without notes. Its main purpose was to explicate how the soul recognizes the size of an interval when it is heard. Therefore, the work is a kind of epistemology that considers musical perception rather than music theory, as was pointed out by the article in ‘Giornale de’ letterati’. Certainly many of Mengoli’s theories appear very strange to us because they are deduced from his own speculations and do not involve any actual proof, however, his focus on the cognitive functions of the sense of hearing itself was significant. In this sense, we can regard Speculationi as a major milestone in the birth of aesthetics advocated by Alexander Gottlieb Baumgarten (1714-62) as a study of sensuous recognition and its proper objects: beauty and art” (O. Takaharu, Pietro Mengoli’s Theory of Perception of Musical Intervals: A Mathematical Approach to the Sense of Hearing in the Scientific Revolution, in: “Aesthetics”, no. 15, 2011, pp. 125-135).

Pietro Mengoli earned a degree in philosophy in 1650 and a degree in civil and canon law in 1653 from the University of Bologna, his hometown. He then began his studies in mathematics under the guidance of Bonaventura Cavalieri; after Cavalieri’s death (1647), he deepened his knowledge of geometry through a correspondence with G.A. Rocca of Reggio Emilia, who was also a student of Cavalieri and perhaps the only Italian mathematician of the time to have direct knowledge of R. Descartes’s Géométrie and F. Viète’s algebra. Even before graduating, in 1648-49, he obtained permission from the Bolognese Senate to give private arithmetic lessons at home for a very low salary. Beginning in 1650-51, the Senate assigned him a newly established lecture series on mechanics (lectura ad mechanicas), which he continued to teach until 1677-78. From the lecture outlines for the academic years 1654-55 (Series lectionum, Bologna, 1654) and 1656-57 (Lectiones mechanicas, Bologna, 1656), it is possible to identify the topics covered and the authors referenced (Archimedes, Pappus, Guldino, Galileo, Cavalieri, Baliani). From 1678 until his death, he held the chair of mathematics and astronomy that had previously been held by Cavalieri. The most productive period of his mathematical work was the decade from 1650 to 1660, during which he produced the works - all published in Bologna - Novae quadraturae arithmeticae (1650), Via regia ad mathematicas (1655), and Geometriae speciosae elementa (1659), which aroused the interest of European mathematicians (from J. Collins to J. Wallis, J. Barrow, J. Gregory, G.W. Leibniz, and J. Ozanam), but also earned him a reputation as a difficult, obscure, and confusing writer.

After these works, Mengoli abandoned the kind of theoretical research that had absorbed him during his youth. For a decade he wrote nothing, and when, in 1670, he resumed publishing, his interests and goals had changed considerably. What brought about such a decisive turning point in his life was his appointment, in 1660, as parish priest of the Church of St. Mary Magdalene on Via S. Donato and the associated title of prior. On the one hand, he was absorbed by his pastoral duties and the administrative tasks of the parish; on the other, his new spiritual responsibilities led him to reflect on the duties of a Catholic philosopher and to reevaluate the role of mathematics within his own encyclopedia of knowledge. Philosophical and religious questions concerning the relationship between reason and faith, revelation and science took center stage - issues that, especially after Galileo’s condemnation (1633), troubled Catholic scholars of natural philosophy and astronomy. Mathematics remained a central component of his work, but the emphasis shifted to its applications in the fields of music, astronomy, and chronology.

Le Speculationi di musica caught the attention of H. Oldenburg, secretary of the Royal Society, who - while lamenting the work’s obscure language - reviewed it in th ‘Philosophical Transactions’ (1674), where an English version of the preface, titled A Natural History of Music, also appeared. Oldenburg keenly grasped the original aspects of Mengoli’s treatise: his unique theory of sound, his rejection of the theory of consonance as a coincidence of sound vibrations - as proposed by Galileo in the Discorsi (1638) - and an intriguing physiology of the ear, based on the idea of the existence of a second eardrum in the inner ear. After centuries of obscurity, interest in Mengoli’s musical theories has been revived in recent years thanks to the studies of P. Gozza, who placed them within the context of the debates on sound and music that engaged seventeenth-century natural philosophers (P. Gozza, Music in Seventeenth-Century Natural Philosophy in Italy, in: ‘Nuncius. Annali di storia della scienza’, I, 1986, 2, pp. 33-36) and interpreted as an attempt to develop (and at the same time overturn) Galileo’s approach to music - on the one hand, by extending the study of consonance to the anatomical and physiological investigation of hearing, and on the other, by ‘explaining the pleasure of music through the action of the mind’ (P. Gozza, Atoms, ‘spiritus,’ Sounds: The ‘Speculationi di musica’ (1670) by the ‘Galilean’ Pietro Mengoli, in: ‘Nuncius. Annals of the History of Science’, V, 1990, 2, p. 97). Mengoli uses the reference to ‘sensible observation’ and ‘arithmetic and mechanical demonstrations’ for the ‘demonstration, apart from objects, of the immortality of the soul and its power to eternalize the temporal in the senses’ (ibid., p. 96).

The risk that his latest works of a philosophical-mathematical nature (Arithmetica rationalis, Bologna, 1674, and Arithmetica realis, Bologna, 1675) might arouse suspicions of heterodoxy in the Roman Inquisition prompted Mengoli, in 1675, to seek an influential patron in the person of Cardinal Leopoldo de’ Medici, after having already begun a correspondence the previous year with the powerful Medici librarian Antonio Magliabechi. Mengoli died in Bologna on June 7, 1686 (cf. M. Cavazza, Mengoli, Pietro, in: “Dizionario Biografico degli Italiani”, vol. 73, Rome, 2009, s.v.)