The Logarithmic Ear: Pietro Mengoli's Mathematics of Music

Summary

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. Here, I focus on the anatomical and mathematical basis of Mengoli's work.

Acknowledgements

I gratefully acknowledge the help of Noel Malcolm, Jackie Stedall, Jessica Wardhaugh, and the participants at the Cabinet of Natural History, Cambridge, as well as the two anonymous reviewers, all of whom commented on drafts of this material.

Notes

¹In particular, Pietro Mengoli, Geometriae speciosae elementa … (Bologna, 1659), and Maria Rosa Massa Esteve, ‘Mengoli on “quasi proportions”’, Historia Mathematica, 24 (1997), 257–80; eadem, ‘La théorie euclidienne des proportions dans les Geometriae speciosae elementa (1659) de Pietro Mengoli’, Revue d'Histoire des Sciences, 56 (2003), 457–74; eadem, ‘Algebra and geometry in Pietro Mengoli (1625–1686)’, Historia Mathematica, 33 (2006), 82–112. See also A. Natucci, ‘Mengoli, Pietro’, in Dictionary of Scientific Biography (New York, 1970–1990); A. Agostini, ‘L'opera matematica di Pietro Mengoli’, Archives internationales d'histoire des sciences, 3 (1950), 816–34.

²Pietro Mengoli, Speculationi di Musica (Bologna, 1670).

³P.J. Fricke, Moderne Ansätze in Mengolis Hörtheorie, in Musica scientiae collectanea, Festschrift K. G. Fellerer, edited by H. Hirschen (Cologne, 1973), 117–25; Paolo Gozza, ‘A mechanical account of hearing from the “Galilean school”: Pietro Mengoli's Speculationi di musica of 1670’, in The Second Sense: studies in hearing and musical judgement from antiquity to the seventeenth century, edited by Charles F. Burnett, Michael Fend, and Penelope Gouk (London, 1991), 115–33. A similar paper appeared in Italian as P. Gozza, ‘Atoms, “spiritus”, sounds: The “Speculations on music” (1670) of the Galilean Pietro Mengoli’, Nuncius: Istituto e museo di storia della scienza, 5 (1990), 75–98; I will refer only to the English paper.

⁴Gozza's paper also contains some worrying factual slips. He describes Mengoli's Elementa (note 1), as falling within ‘the Italian Renaissance tradition of classical geometry’ (116 note 3), although it contains a logarithmically based theory of the ratios of ratios and makes ample use of the algebra of Viète. He reduces the diversity of seventeenth-century theories of sound to a bald contrast between propagation and transmission (118), and regrettably oversimplifies Mengoli's own understanding of sound (123). It also seems stretching a point to assert that Beeckman and Descartes analysed musical perception ‘on the basis of a detailed anatomy and physiology of hearing’ (119).

⁵Pietro Mengoli, ‘Il Sig. Mengoli col seguente sicorso [sic: discorso?] provo che l'armonia della musica non è dissimile dell'armonia che unite formano le parti che costituiscono un bel sembiante’ [‘In the following discourse Signior Mengoli proves that musical harmony is not unlike the harmonies which unite together the parts which constitute a beautiful appearance’], in Amor tiranno, by Fantuzzi and Pellegrini (Bologna, 1649), 26–33 [not seen].

⁶Many of the Greek sources are collected in Andrew Barker (ed.), Greek Musical Writings II: harmonic and acoustic theory (Cambridge, 1989).

⁷Isaac Newton, Philosophiae naturalis principia mathematica (London, 1687), II, ch. 8. See also Brook Taylor, ‘De motu nervi’, Philosophical Transactions of the Royal Society of London, 28 (1713), 26–32; Joseph Sauveur, Collected Writings on Musical Acoustics (1700–1713), edited by Rudolf Rasch (Utrecht, 1984); Sigalia Dostrovsky, ‘The origins of vibration theory: the scientific revolution and the nature of music’ (Princeton, NJ: unpublished Ph.D. thesis, 1969); eadem, ‘Early vibration theory: physics and music in the seventeenth century’, Archive for the History of Exact Sciences, 14 (1974–1975), 169–218; and John T. Cannon and Sigalia Dostrovsky, The Evolution of Dynamics: vibration theory from 1687 to 1742 (New York, 1981).

⁸Examples are Nicolaus Mercator (in Bodleian MS Aubrey 25; London, Guildhall Library MS 51757 21; and Oxford, Christ Church MSS 1130 and D14); and Isaac Newton (in Cambridge University Library, Add. MS 4000, ff. 137r–43v and 104r–113v and Add MS 3958 (B) f. 31r).

⁹These and other mathematical and mechanical studies of music in late seventeenth-century England are examined more fully in Benjamin Wardhaugh, ‘Mathematical and mechanical studies of music in late seventeenth-century England’ (unpublished D.Phil. thesis, Oxford, 2006).

¹⁰Anon., ‘An Account of two Books’, P hilosophical Transactions of the Royal Society of London, 8 (1673), 6194–7002 (Review of Mengoli, Speculationi, at 6194–7000 [recte 6200]): at 6195.

¹¹All modern anatomical information is taken from Peter L. Williams and Roger Warwick (eds.), Gray's Anatomy (36th edition, Edinburgh, 1980), 1192–99.

¹²See in particular Hieronymus Fabricius ab Aquapendente, De visione, voce, auditu (Venice, 1600); Julius Casserius, De vocis auditusque organis historia (Ferrara, 1600–1601), reprinted in idem, Pentaesthesion (Frankfurt, 1610); and Helkiah Crooke, Microcosmographia (London, 1615); and see Wardhaugh (note 9), 134–76.

¹³Aristotle, De anima 420a; in Barker (note 6), 78.

¹⁴T. Willis, De anima brutorum quae hominis vitalis ac sensativa est, exercitationes duae (Oxford, 1672), trans. by S. Pordage as Two Discourses Concerning the Soul of Brutes (London, 1683), 74.

¹⁵Mengoli (note 2), iii.

¹⁶Gozza (note 3), 125, referring to Pietro Mengoli, La corrispondenza, edited by G. Baroncini and M. Cavazza (Florence, 1986), 54–55 (on Zani) and Memorie, imprese e ritratti de'Signori Accademici Gelati di Bologna (Bologna, 1672), 268–70 (on Manzi).

¹⁷Apart from a handful of more general medical works, the only exceptions I have found are Marcello Malpighi, De viscerum structura exercitatio anatomica. Accedit dissertatio eiusdem de polypo cordis (Bologna, 1666) and Cornelio Ghirardelli, Cefalogia fisonomica (Bologna, 1670), neither of which looks particularly promising as a source for Mengoli.

¹⁸Mengoli (note 2), 1–6, ‘Descrittione dell'Orecchio’.

¹⁹See note 14.

²⁰Examples are given in Wardhaugh (note 9), 155–75.

²¹Exceptions are the French anatomists Claude Perrault, Essais de physique, ou receuil de plusieurs traitez touchant les choses naturelles II: ‘De la bruit’ (Paris, 1680, 1688); and G.J. Duverney, Traité de l'organe de l'ouïe (Paris, 1683), trans. John Marshall as A Treatise of the Organ of Hearing (London, 1737).

²²Mengoli (note 2), ‘Historia Naturale Della Mvsica’ (unpaginated), and 6–14, ‘Descrittione del Suono’.

²³Mengoli 10: ‘l'aura è corpo, la cui proprietà è una larghissima indifferenza ad ogni termine: in conseguenza non hauerà proprietà alcuna di numero, e di proportione di parti, ne meno di luogo, ò sito, ò grauità’.

²⁴Mengoli 14: ‘l'aura è l'instromento immediato dell'anima, nel quale riceuendo, sente; e per lo quale operando, moue’.

²⁵Mengoli ‘Historia Naturale’ [i–iii].

²⁶Mengoli ‘Historia Naturale’, [v]: ‘le ariette affette dal suono entrano per la parte esterna dell'orecchio, l'vna dopo l'altra, e tutte per ordine vanno per le vie spirali, che iui si vedono, sino al fondo dell'orecchio; oue ciascuna tocca il timpano, e poi per altre vie spirali riesce fuori dell'orecchio, e dà il suo luogo ad altre ariette, che succedono à far l'istesso’.

²⁷Giovanni Battista Benedetti, Diversarum speculationum mathematicarum et phusicarum liber (Turin, 1585); see H. Floris Cohen, Quantifying Music: the science of music at the first stage of the scientific revolution, 1580–1650 (Dordrecht, 1984), 75.

²⁸On musical experiments in the seventeenth century, see Wardhaugh (note 9), 211–45.

²⁹See, for example, Ptolemy, Harmonics 15.12: Barker (note 6), 289 with note 64, although here the subjective experience of two notes’ being heard as one is used to distinguish one type of consonance from others rather than to characterize consonance in general.

³⁰See Cohen (note 27), 82–85.

³¹Musical experiments in the seventeenth century are documented in Wardhaugh (note 9), 211–45.

³²René Descartes, Musicae compendium (Utrecht, 1650); Marin Mersenne, Harmonie universelle … (Paris, 1636; facs. ed. Paris, 1963); Athanasius Kircher, Musurgia universalis, sive ars magna consoni et dissone (Rome, 1650).

³³An example of the former is Francis North (Anon.), A Philosophical Essay of Musick, Directed to a Friend (London, 1677); an example of the latter is William Holder, A Treatise of the Natural Grounds, and Principles of Harmony (London, 1694).

³⁴See Jamie C. Kassler, The Beginnings of the Modern Philosophy of Music in England: Francis North's A Philosophical Essay of Musick (1677) with comments of Isaac Newton, Roger North and in the Philosophical Transactions (Aldershot, 2004).

³⁵See See Jamie C. Kassler, The Beginnings of the Modern Philosophy of Music in England: Francis North's A Philosophical Essay of Musick (1677) with comments of Isaac Newton, Roger North and in the Philosophical Transactions (Aldershot, 2004). 97.

³⁶Mengoli (note 2), 25–40, ‘Dell'vdire due Suoni insieme’.

³⁷Mengoli (note 2), 54: ‘senza l'vs o della ragione, solo col senso, e secondo il senso, l'huomo numera sino à qualche numero determinato’.

³⁸Mengoli (note 2), 54: ‘senza l'vso della ragione, solo col senso, e secondo il senso, l'huomo numera sino à qualche numero determinato’.

³⁹Mengoli (note 2), 54: ‘senza l'vso della ragione, solo col senso, e secondo il senso, l'huomo numera sino à qualche numero determinato’. 55.

⁴⁰Mengoli (note 2), 54: ‘senza l'vso della ragione, solo col senso, e secondo il senso, l'huomo numera sino à qualche numero determinato’. 56.

⁴¹Mengoli (note 2) 59–60 (chapter 11): ‘Quali interualli possa l'anima, secondo il senso, comprendere’ (thus, the table of contents page; the text has ‘Quali alternationi … ’).

⁴²Mengoli (note 2) 59–60 (chapter 11): ‘Quali interualli possa l'anima, secondo il senso, comprendere’ (thus, the table of contents page; the text has ‘Quali alternationi … ’). 99–100.

⁴³Mengoli (note 2) 59–60 (chapter 11): ‘Quali interualli possa l'anima, secondo il senso, comprendere’ (thus, the table of contents page; the text has ‘Quali alternationi … ’). 57–8.

⁴⁴Barker (note 6), 191 and references given there.

⁴⁵Cohen (note 27), 95.

⁴⁶Mengoli (note 2), 15–25, ‘Descrittione dell'Vdito’.

⁴⁷Mengoli (note 2), 15–25, ‘Descrittione dell'Vdito’. 18.

⁴⁸Mengoli (note 2), 15–25, ‘Descrittione dell'Vdito’. 19.

⁴⁹Mengoli (note 2), 15–25, ‘Descrittione dell'Vdito’. 20.

⁵⁰In an arithmetic progression, the difference between consecutive terms is constant (as 1, 2, 3, … ); in a harmonic progression, the difference between the reciprocals of consecutive terms is constant (as 1/2, 1/3, 1/4 … ).

⁵¹Mengoli (note 2), 22–23: ‘viene l'anima nell'organo dell'vdito à ritenere tutte le diuturnità innumerabili de'tremiti de'punti, che sono nell'aura giacente trà gli timpani dell'orecchio, e ad astraerle in vna somma, la quale io chiamo Logaritmo della ragione doppia’.

⁵²It appeared in Gregory of St Vincent, Opus geometricum (Antwerp, 1647).

⁵³Mengoli (note 2), 24: ‘inclinatione … à mouersi’.

⁵⁴Mengoli (note 2), 24: ‘inclinatione … à mouersi’.

⁵⁵Anon., ‘An Account of two Books’ (note 10), 6197.

⁵⁶For a discussion of the same confusion in the context of Newton's Principia, see Edith Sylla, ‘Compounding Ratios: Bradwardine, Oresme, and the first edition of Newton's Principia’ in Transformation and Tradition in the Sciences, essays in honor of I. Bernard Cohen, edited by Everett Mendelsohn (Cambridge, 1984), 11–43.

⁵⁷Mengoli (note 2), 41–53.

⁵⁸Mengoli (note 2), 41.

⁵⁹Mengoli (note 2), 42.

⁶⁰Mengoli (note 2), 44.

⁶¹Mengoli (note 2), 45.

⁶²Mengoli (note 2), 47: ‘Quanto vn'interuallo è minore del doppio, ha per equisoni gl'interualli altretanto minori dell'egualità, e del duplicato del doppio.’

⁶³Pietro Mengoli, Geometriae speciosae elementa: … Quartum de rationibus logarithmicis. Quintum de proprijs rationum logarithmis … (Bologna, 1659).

⁶⁴Massa Esteve, ‘La théorie euclidienne des proportions’ (note 1). See also A. Agostini, ‘La teoria dei logaritmi da Mengoli a Eulero’, Periodico di matematiche, s. IV, 2 (1922) (I have not seen this article).

⁶⁵Wardhaugh (note 9), 80–109, discusses other musical uses of logarithms in the seventeenth century, including those by Nicolaus Mercator and Isaac Newton.

⁶⁶The others are lucidly detailed in Bradley Lehman, ‘Bach's Extraordinary Temperament: our Rosetta Stone’, Early Music, 33 (2005), 3–23 (part 1), at 3–5.

⁶⁷For example, Descartes asserted that the ear cannot distinguish smaller differences of pitch than a syntonic comma (81:80, about 22% of a semitone) ‘without effort’: Kent says that ‘this is either a rather naive excuse for remaining within Zarlino's senario or evidence that Mersenne was correct in claiming that Descartes had a “poor ear”.’ René Descartes, René Descartes Compendium of Music (Compendium Musicae), trans. Walter Robert, intro. and notes by Charles Kent (American Institute of Musicology, 1961), 17 and note.

⁶⁸For example, British Library, MS Harleian 4160, ff. 1–41: Anon., ‘Musical observations and experiments in musical sounds belonging to the Theoric part of music’ (c.1698), especially ff. 35r–35v ‘The most exact way for the tuning of an Organ Harpsechord virginal or Espineta’; and Oxford, Bodleian Library, MS Mus. Sch. d375*, ff. 32r–40r: Anon. [Thomas Salmon?], ‘The Use of the Musical Canon’. See also Mark Lindley, Lutes, Viols and Temperaments (Cambridge, 1984), passim.

⁶⁹Robert Dowland, Variety of Lute-Lessons (London, 1610), 16.

⁷⁰This spread is described in Wardhaugh (note 9), 273–317; the argument which follows is documented more fully in ibid., 178–89.

⁷¹Two important ancient loci for the harmonious soul are Plato, Timaeus, 35a–36b, 47d and Aristotle, De anima, 407b27–408a33; on neoplatonic and magical uses of the idea, see Gary Tomlinson, Music in Renaissance Magic: towards a historiography of others (Chicago, 1993); and Penelope M. Gouk, Music, Science and Natural Magic in Seventeenth-Century England (New Haven, CT, 1999), passim; on orthodox theological uses see for example Peter Dear, Mersenne and the Learning of the Schools (Ithaca, NY, 1988), especially 96–116, 139–69.

⁷²J. Bruce Brackenridge and Mary Ann Rossi, ‘Johannes Kepler's On the More Certain Fundamentals of Astrology. Prague 1601’, Proceedings of the American Philosophical Society, 123 (1979), 85–116: see thesis 43, p. 139.

⁷³Robert Hooke, ‘An Hypothetical Explication of Memory; how the organs made use of by the mind in its operation may be mechanically understood’ (Waller's title), in ‘Lectures of Light’ VII, in idem, The Posthumous Works of Robert Hooke, edited by Richard Waller (London, 1705), 138–48.

⁷⁴See, for example, Roger North, Roger North's Cursory Notes of Musicke (c.1698–c.1703): A Physical, Psychological and Critical Theory Edited with Introduction, Notes and Appendices, edited by Mary Chan and Jamie C. Kassler (‘North papers 1’) (Kensington, NSW, 1986), passim.

⁷⁵The most systematic attempt to create an experimental science of music in the seventeenth century was perhaps that of Thomas Salmon; see in particular Thomas Salmon, A Proposal to Perform Musick, in Perfect and Mathematical Proportions (London, 1688); and idem, ‘The Theory of Musi ck reduced to Arithmetical and Geometrical Proportions’, Philosophical Transactions of the Royal Society of London, 24 (1705), 2072–77, 2069 [misnumbered].

⁷⁶Mengoli (note 2), 60–61: ‘Del partire secondo il senso’.

⁷⁷Mengoli (note 2), 64–65: errori ‘inosseruabil[i], secondo il senso … gradeuoli all'anima … à pena tollerabili … troppo euidenti, ed intollerabili’.

⁷⁸Mengoli (note 2), 64–65: errori ‘inosseruabil[i], secondo il senso … gradeuoli all'anima … à pena tollerabili … troppo euidenti, ed intollerabili’. 111–5.

⁷⁹C. Palisca, Humanism in Italian Renaissance Musical Thought (New Haven, CT, 1985); A.E. Moyer, Musica Scientia: Musical Scholarship in the Italian Renaissance (Ithaca, NY, 1992).

⁸⁰See S. Schaffer, ‘Golden means: the Guinea trade’ in Instruments, Travel and Science: Itineraries of Precision from the Seventeenth to the Twentieth Century, edited by Marie-Noëlle Bourguet, Christian Licoppe and H. Otto Sibum (London, 2002), 20–50, citing Robert Boyle, Medicina Hydrostatica: or, Hydrostaticks Applyed to the Materia Medica (London, 1690).

⁸¹Barker (note 6), 119–84; Michael Fend, ‘The Changing Functions of Senso and Ragione in Italian Music Theory of the Late Sixteenth Century’, in Burnett, Fend, and Gouk (note 3), 199–221.

⁸²Anon., ‘An Account of two Books’ (note 10), 7000.

⁸³Henry Oldenburg, Correspondence, edited by A Rupert Hall and Marie Boas Hall (Madison, WI, 1965–1977) X, 6–8: 7 June 1673.

⁸⁴Mengoli (note 2), 67–91: ‘Della perfettione de gl'interualli, e sua misura’.

⁸⁵Mengoli (note 2), 67–91: ‘Della perfettione de gl'interualli, e sua misura’. 67–68.

⁸⁶Mengoli (note 2), 74–75: ‘sono come 100 à 1000, in ragione delle istessa perfettione participata di 5 à 4 … la perfettione propria naturale delle perfetta, alla perfettione participata della sua imperfetta, hà vna ragione infinita … ’.

⁸⁷Mengoli (note 2), 92–99: ‘De gli errori nelle Alternationi, secondo il senso’.

⁸⁸Mengoli (note 2), 75–88, esp. 88: ‘La perfettione dell'alternatione della egualità, alla perfettione dell'alternatione di ciascuna disegualità della prima, ò della seconda sorte, è altretanto molteplice, quanto è il minimo commun diuido de i numeri della proposta disegualità.’

⁸⁹Mengoli (note 2), 91: ‘Le perfettioni naturali delle alternationi di due proposte disegualità della prima, ò seconda sorte, sono reciprocamente, come i communi diuidui minimi de i numeri delle proposte disegualità’.

⁹⁰Mengoli (note 2), 116–34: ‘De i veri numeri de'suoni, e di varie proprietà, che ne riportano gl'interualli’.

⁹¹Mengoli (note 2), 116–34: ‘De i veri numeri de'suoni, e di varie proprietà, che ne riportano gl'interualli’. 118–19.

⁹²Mengoli (note 2), 116–34: ‘De i veri numeri de'suoni, e di varie proprietà, che ne riportano gl'interualli’. 120–21.

⁹³North (note 74), pp. 81–82 of MS.

⁹⁴Mengol i (note 2), 137.

⁹⁵Mengoli (note 2), 135–39: ‘Trà quali veri numeri de'suoni la specie di ciascun'interuallo sia pi[ugrave] perfetta’ (‘de'suoni’ is missing from this heading in the table of contents.)

⁹⁶Mengoli (note 2), 140–43: ‘Dell'attentione Musica attiua dell'anima’.

⁹⁷Mengoli (note 2), 140–43: ‘Dell'attentione Musica attiua dell'anima’. 141.

⁹⁸Mengoli (note 2), 140–43: ‘Dell'attentione Musica attiua dell'anima’. 143–62.

⁹⁹Robert Hooke, ‘A Curious Dissertation concerning the Causes of the Power & Effects of Music’, London, Royal Society Library MS Classified Papers II no. 1, transcribed in Penelope M. Gouk, ‘The Role of Acoustics and Music Theory in the Scientific Work of Robert Hooke’, Annals of Science, 37 (1980), 573–605; at 601: the tympanum ‘can be soe tuned … or stretch'd, that it becomes harmonicall or unison to whatsoever sound is heard.’ See 591–92 for Gouk's tentative dating of this manuscript.

¹⁰⁰Noel Malcolm, ‘The Library of Henry Oldenburg’, Electronic British Library Journal (2005), article 7, pp. 10, 45.

¹⁰¹Mengoli (note 2), 173–225: ‘Della Modulatione’ and 225–36: ‘De gli Accordi di pi[ugrave] suoni’.

¹⁰²Mengoli (note 2), 240–60: ‘Delle Passioni dell'anima’; table at 258–60.

¹⁰³A broadly similar seventeenth-century example is in Cambridge University Library, Add. MS 4000, ff. 104r–113v: Isaac Newton, musical calculations (at 110r–112r) and ff. 137r–143v: idem, essay ‘Of Musick’ (at 138r–139r).