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The Experimental Approach

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Quantifying Music

Part of the book series: The University of Western Ontario Series in Philosophy of Science ((WONS,volume 23))

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Abstract

In the preceding chapter we investigated the results of the mathematical approach to the problem of consonance as defined by Zarlino. In the hands of Kepler this new approach led to a geometrical theory of harmony, while it induced Stevin to deny that the consonances are defined by the first few integers at all. However idiosyncratic the consequences were that Stevin was willing to draw from his argument against the supposed excellence of the first few integers, these do not diminish the basic soundness of his starting point. In terms of number alone, the argument was valid. It lost its relevance, however, from the moment the phenomenon of consonance was placed on a new, physical basis. In fact, a first step in this direction had already been taken some forty years earlier.

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Notes

  1. See, for instance, Dijksterhuis (1924), 179–190, and (1961), 269–271; a general overview is given by Drake in the D SB.

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  2. Palisca (1961), 105 sqq.; as he notes, the two letters have been reprinted in 1924/5 by Reiss, without any further comment. On Cipriano de Rore (a forerunner in his own right) see Reese (19592), 329–332.

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  3. Benedetti (1585), 283: “. . . cum nemo sit qui nesciat, quod quo longior est chorda, etiam tardius moveatur

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  4. Benedetti (1585), 283: “qui quidem numeri non absque mirabili analogia conveniunt invicem. Voluptas autem, quam auditui afferunt consonantiae fit, quia leniuntur sensus, quemadmodum contra, dolor qui à dissonantijs oritur, ab asperitate nascitur, id quod facile videre poteris cum conchordantur organorum fistulae.”

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  5. Palisca (1961), 109.

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  6. Walker has shown in his (1978), 31 that Palisca’s interpretation results from his mistranslating ‘analogia’ by ‘logic’ instead of ‘proportion’. Palisca’s other argument in favor of his interpretation is Benedetti’s announcement “Videamus igitur ordinem concursus percussionum terminorum ...” (283). ‘Ordinem’ might indeed mean ‘order’, but if Benedetti really had aimed at grading the consonances he would not have failed to say so three paragraphs later, where the alleged operation is in fact carried out, but turns out to serve quite another purpose (namely to show the ‘mirabilem analogiam’ discussed in the main text). So ‘ordinem’ is rather to be translated by ‘regularity’. Another example of Palisca’s modernizing Benedetti’s apparent meaning is on p. 109–110, where from Benedetti’s argument he extrapolates conclusions about the intervals with 7 that were indeed to be drawn later by Mersenne and Huygens (this book, Sections 3.5.3. and 6.2.3.), but of which there is no trace in Benedetti’s text (besides, the ratio 7: 5 corresponds with the augmented fourth rather than with the diminished fifth 10:7, as Palisca mistakenly believes).

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  7. No adequate history of early acoustics does yet exist. The statements made in the text are mainly based on some useful remarks in Lindsay (1966); Shapiro (1973), 134–143; Hunt (1978), Ch. 1, and Gouk (1982), Ch. 1.

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  8. GW 3, 141: “Copernicus divitiarum suarum ipse ignarus... “.

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  9. Bukofzer(1947),5–9;25–29.

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  10. Bukofzer (1947), 56/7; the opera has been recorded on Telefunken SAWT 9603/4-B.

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  11. Cp. Walker (1941/2), passim.

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  12. Walker (1978), 16.

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  13. Walker (1978), 15; from Palisca (1960), 63–9.

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  14. V. Galilei (1589), 92/3: “Gli intervalli musici, poi tanto sono naturali (com’io ho detto) quelli contenuti tra le parti del Senario, quanto gľaltri che son fuore di esse parti è tanto è naturale il Ditono contenuto dalla sesquiquarta quanto, quello che è contenuto dalla super 17 partiente 64. si come ancora tanto è naturale ľaccordare dell’ottava drento la dupla quanto è naturale il dissonare della settima drento la super 4. partiente quinta: & rompisi pur’il Zarlino la testa quanto vuole.” My translation differs somewhat from Palisca’s in his (1961), 122, where I found this particular passage.

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  15. Walker (1978), 17; reference to V. Galilei (1589), 117/8, and 124/5: “io subito confesserò che quello che noi hoggi cantiamo, convenga più che con altra Distribuitione con il medesimo Sintono di Tolomeo”.

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  16. Walker (1978), 20.

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  17. In Walker’s view Galilei’s point only lost its relevance when it was discovered that the overtones are the true causes of consonance. But they are not (cp. this book, p. 237). The first theory of consonance which made it clear that there is indeed an empirical basis underlying the simple ratios characteristic of the consonances was the coincidence theory. Galilei’s particular point of view was precisely what made him not hit upon this theory, just as we have seen to be the case with Stevin (this book, Section 2.3.8.).

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  18. Van der Waerden (1979), 368/9.

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  19. V. Galilei (1589), 104: “ultimamente accertato con il mezzo dell’esperienza delle cose maestra”; translation taken from Palisca (1961), 128.

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  20. Walker (1978), 24.

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  21. Palisca (1961), 129–130.

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  22. Walker (1978), 17; 22/3.

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  23. The literature on Galileo is virtually boundless. A penetrating account of his work in mechanics and astronomy is given by Dijksterhuis (1961), Sections IV, 77–123 and 153–162. See also Drake in the DSB, and Drake (1978). In the following I quote the Discorsi from a 1730 English translation (Mathematical Discourses concerning Two New Sciences relating to Mechanicks and Local Motion) by Thomas Weston, facs. reprinted in Lindsay (1972); in the notes I give the Italian from the Ed. Naz., Vol. 8, and add references to the standard English translation by Drake (1974).

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  24. Math. Disc, 139; Drake (1974), 96;Ed. Naz. 8, 138: “... se io ancora da cosi facili e sensate esperienze trarrò ragioni di accidenti maravigliosi in materia de i suoni . . . “. See for a few prior remarks of Galileo’s on music and acoustics his Saggiatore: Ed. Naz. 6, 269; 280/1, and 349/350.

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  25. Math. Disc, 142; Drake (1974), 99; Ed. Naz. 8, 141: “. . . che impossibil cosa è il farlo muover sotto altro periodo che l’nico suo naturale”.

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  26. Math. Disc, 145; Drake (1974), 100; Ed. Naz. 8, 143: “. . . ed accadendo tal volta che ‘1 tuono del bicchiere salti un’ ottava più alto, nelľ istesso momento ho visto ciascheduna delle dette onde dividersi in due; accidente che molto chiaramente conclude, la forma dell’ ottava esser la dupla”.

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  27. Walker (1978), 29. The first reason seems to me invalid, as Galileo’s point here is not to count the number of waves, but to show that the octave splits each wave into two.

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  28. Math. Disc, 146; Drake (1974), 101; Ed. Naz., 143: “Tre sono le maniere con le quali noi possiamo inacutire il tuono a una corda: l’una è lo scorciarla; l’altra, il tenderla più, o vogliam dir tirarla; il terzo è l’assottigliarla.”

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  29. Math. Disc, 148; Drake (1974), 102;Ed. Naz. 8, 144/5: “L’invenzione fu del caso [...]. Raschiando con uno scarpello di ferro tagliente una piastra d’ottone per levarle alcune macchie, nel muovervi sopra lo scarpello con velocità, sentii una volta e due, tra moite strisciate, fischiare e uscirne un sibilo molto gagliardo e chiaro; e guardando sopra la piastra, veddi un lungo ordine di vergolette sottili, tra di loro parallele e per egualissimi intervalli l’una dall’ altra distanti.”

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  30. Math. Disc, 149; Drake (1974), 103; Ed. Naz. 8, 145: “quale veramente è la forma che si attribuisce alla diapente”.

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  31. Walker (1978), 30. [NB. In a letter dated 4th of February, 1983, Professor Walker suggested to me that his argument does not conclusively disprove the possibility of Galileo’s actually having carried out the experiment, as Galileo may have used for comparison a fifth from a harpsichord tuned in mean tone temperament.]

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  32. Math. Disc, 151; Drake (1974), 104; Ed. Naz. 8,146/7: “La molestia di queste [sc. dissonanze] nascerà, credo io, dalle discordi pulsazioni di due diversi tuoni che spropor-zionatamente colpeggiano sopra ‘1 nostro timpano, e crudissime saranno le dissonanze quando i tempi delle vibrazioni fussero incommensurabili [...]. Consonanti, e con duetto ricevute, saranno quelle coppie di suoni che verranno a percuotere con qualche ordine sopra ‘1 timpano; il qual ordine ricerca, prima, che le percosse fatte dentro all’ istesso tempo siano commensurabili di numero, accio che la cartilagine del timpano non abbia a star in un perpetuo tormento ď inflettersi in due diverse maniere per accon-sentire ed ubbidire alle sempre discordi battiture.”

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  33. Math. Disc, 151; Drake (1974), 104; Ed. Naz. 8, 147: “sarà dunque la prima e più grata consonanza l’ottava . . . “.

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  34. Math. Disc, 152; Drake (1974), 105; Ed. Naz. 8, 147: “tutte l’altre sono discordi e con molestia ricevute su ‘1 timpano, e giudicate dissonanti dall’ udito”.

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  35. Math. Disc, 155; Drake (1974), 101;Ed. Naz. 8, 149: “... fa una titillazione ed un solletico tale sopra la cartilagine del timpano, che temperando la dolcezza con uno spruzzo d’acrimonia, par che insieme soavemente baci e morda”.

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  36. Math. Disc, 156; Drake (1974), 101 ;Ed. Naz. 8,149: “la qual mistione di vibrazioni è quella che, fatta dalle corde, rende all’ udito l’ottava con la quinta in mezzo”.

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  37. Math. Disc, 156; Drake (1974), 107/8; Ed. Naz. 8, 150: “... allora la vista si confonde nell’ ordine disordinato di sregolata intrecciatura, e l’udito con noia riceve gli appulsi intemperati de i tremori dell’ aria, che senze ordine o regola vanno a ferire su ‘1 timpano”.

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  38. Costabel and Lerner (1973), Vol. 2, 210. Drake (1974), 107, notes that Galileo’s pupil Viviani had noticed the mistake, but apparently he corrected only the string lengths, without concluding that the experiment cannot have been performed in the way Galileo describes it.

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  39. Truesdell (1960), 36/7 reaches the same conclusion from the acoustical point of view.

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  40. Cp. Drake s.v. Benedetti in the DSB (p. 607/8).

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  41. Cp. on these matters Dostrovsky (1974/5), 181/2, and Gouk (1981), Ch. 1.

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  42. The two extreme positions in Koyre (1940) and Drake (1978); see for a more moderate view Dijksterhuis (1924), summarized in his (1961). See for an appraisal of the present state of the debate Segre (1980), passim.

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  43. Vogel (1955) analyzes extensively the role of the intervals containing 7 in the history of musical theory; cp. this book, Section 3.5.3. and 6.2.3.

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  44. DSB s.v. Mersenne, 317; 318.

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  45. Cp. Lenoble (1943), 524–531.

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  46. HU, Livre premier des consonances, prop. 26; 73. All references to Harmonie universelle are taken from the 1965 facs. ed. of the copy that contains Mersenne’s own handwritten marginal corrections and additions; the proposition numbers are as corrected by Mersenne.

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  47. A complete bibliography in Lenoble (1943), xi-xxxiv. Ludwig (1935), 18–32, briefly compares HU with the other works on music.

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  48. Namely, the Traité des instrumens in seven books (edited in an English translation by R. E. Chapman (1957)).

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  49. HU, prefatory matter, in between the Table des propositions and the Table des matieres’. “Ces pages vuides m’ont fait naistre l’occasion de donner un petit Abregé de la Musique Speculative, pour ceux qui n’ont pas loisir de lire nos Traittez tous entiers.”

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  50. Ibid.: “Le son n’est autre chose qu’un battement d’air, que l’oiiye apprehende lors qu’elle en est touchée.”

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  51. Ibid. : “Or ils representent le nombre & la comparaison de leurs batemens.”

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  52. Dostrovsky (1974/5), 185–188.

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  53. Dostrovsky (1974/5), 191–193, 209–217.

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  54. HU, Livre quatriesme des instrumens, prop. 9, coroll. 1; 211: “... le son de chaque chorde est d’autant plus harmonieux & agreable, qu’elle fait entendre un plus grand nombre de sons differens en mesme temps.. . “.

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  55. HU, Livre cinquiesme des instrumens à vent, prop. 12; 251: “D’où il est aysé de conclure que l’ordre des Consonances est naturel, & que la maniere dont nous contons en commençant par l’unité iusques au nombre de six, & au delà, est fondée dans la nature.”

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  56. HU, Livre sixiesme des orgues, prop. 28; 362/3; prop. 30; 366–8.

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  57. At times Mersenne employed another calculating rule than Galileo’s (which was to multiply the terms of the ratios of the intervals). However, his alternative methods always yield the same scale of degrees of consonance (within the compass of one octave, that is; the replicas make things more complicated).

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  58. HU, Livre premier des consonances, prop. 32, coroll. 2; 82. The distinction between ‘doux’ and ‘agreable’ is adopted for the first time in prop. 3; 11; but Mersenne is far from consistent in sustaining it, in accordance with his own vacillations.

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  59. HU, Livre premier des consonances, prop. 30; 77: “... il ne faut pas seulement iuger de la bonté des Consonances par la consideration des simples, mais il faut quant & quant considerer leurs repliques”.

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  60. HU, Livre premier des consonances, prop. 33; 82: “Pourquoy il n’y a que sept ou huit simples Consonances.”

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  61. HU, Livre premier des consonances, prop. 33; 86: “ ... car il n’y a nul nombre de mouvements ou de battements d’air, qui ne soient commensurables à tous autres nombres de mouvemens. C’est pourquoy ie m’estonne comme Kepler a osé apporter la comparaison des figures avec les Consonances, pour en tirer la raison de leur nombre & de leur bonté: ce qui seroit tolerable s’il se fust contenté de comparer lesdites figures aux Consonances & aux Dissonances par analogie, & par recreation ....”

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  62. HU, Livre premier des consonances, prop. 31; 87: “ ... nous pouvons neantmoins nous tenir à ce nombre, puisque la pratique y est conforme .. . “.

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  63. HU, Livre premier des consonances, prop. 33; 87/8: “Neantmoins toutes ces raisons ne me satisfont pas entierement, dautant que si le plaisir de la Musique commence par la consideration de l’esprit, qui est capable de contempler toutes sortes de raisons, il faudroit dire pourquoy les intervalles dissonans luy deplaisent dans les sons, puis qu’ils ne luy deplaisent pas dans les lignes, ny dans les figures ... .”

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  64. HU, Livre premier des consonances, prop. 33, coroll. 3; 88: “ . . . les sons peuvent apporter plus de lumiere à la Philosophie que nulle autre qualité; c’est pourquoy la science de la Musique ne doit pas estre negligee, quoy que les chants & les concerts fussent entierement abolis & defendus

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  65. HU, Livre premier des consonances, prop. 33, coroll. 4; 89: “Puisque le long exercise a coustume de rendre doux & facile ce qui sembloit auparavant rude & fascheux, ie ne doute nullement ques les intervalles dissonans, dont i’ay parlé dans cette proposition, à sçavoir les raisons de 7 à 6, & de 8 à 7, qui divisent la Quarte, ne puissent devenir agreables, si l’on s’accoustume à les oiiir & à les endurer, & que l’on en use comme il faut dans les recits & dans les concerts, afin ď emouvoir les passions, & pour plusieurs effets, dont la Musique ordinaire est privée.”

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  66. HU, Livre cinquiesme des instrumens á vent, prop. 13; 251–3.

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  67. HU, Livre quatriesme des instrumens, prop. 9; 208/9. Cp. Vogel (1955), 104/5.

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  68. HU, Livre premier des consonances, prop. 10, coroll. 3; 47: “ ... l’entendement suit toujours la iustesse des raisons & des proportions, quoy qu’il suffise d’en approcher pour satisfaire aux sens”.

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  69. HU, Livre premier de la voix, prop. 52; 81: “que l’air externe excite l’air interne de l’oreille, & qu’il imprime une emotion dans le nerf de l’oiiie, semblable à celle qu’il a receuë, & que l’esprit qui est tout dans chaque partie du corps, & consequemment dans ledit nerf, apperçoit aussi tost le mouvement des organes de l’oreille, & iuge par là les qualitez du mouvement du son, & des objects exterieurs qui le produisent: or l’on peut s’imaginer que l’esprit est comme un point indivisible & intellectuel, auquel toutes les impressions des sens aboutissent, comme toutes les lignes du cercle à leur centre, ou comme tous les filets d’une toille de l’araigne qui le filee & tissue . . . .” Cp. Crombie (1971), 301.

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  70. HU, Preface, & advertissement au lecteur, preceding the Livre premier des consonances; second page.

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  71. Cp. Barbour (1951), Ch. 8, esp. p. 191.

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  72. The five places in HU sue: Livre second des dissonances, prop. 11 ; 132; Livre premier des instrumens, prop. 14; 37/8 (calculation by Beaugrand); Livre sixiesme des orgues, prop. 16; 341–3; idem, prop. 38; 384/5 (Boulliau); Nouvelles observations physiques et mathematiques, obs. 8; 21 (Gallé).

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  73. HU, Livre premier des instrumens, prop. 15, coroll. 1; 41: “... la composition en sera beaucoup plus aysée, & plus agreable, & mille choses seront permises que plusieurs croyent estre deffenduës. . . “.

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  74. HU, Livre sixiesme des orgues, prop. 16; 342: “quoy qu’il soit meilleur de laisser les Tierces maieures iustes ... “. See also prop. 29; 366: “... i ‘aye monstre ailleurs qu’il les [i.e. les Quintes] faut affoiblir d’un quart de comma . . . “.

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  75. HU, Livre second des chants, prop. 7; 103: “Determiner s’il est possible de composer le meilleur chant de tous ceux qui se peuvent imaginer. ... “

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  76. Lenoble (1943), 525: “Evidemment [Mersenne] serait beaucoup plus à l’aise si l’art musical se réduisait à une algèbre des sons.”

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  1. Department of Social History of Science and Technology, Technical University Twente, Enschede, The Netherlands

    H. F. Cohen

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Cohen, H.F. (1984). The Experimental Approach. In: Quantifying Music. The University of Western Ontario Series in Philosophy of Science, vol 23. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-7686-4_3

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  • DOI: https://doi.org/10.1007/978-94-015-7686-4_3

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