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Beyond Latin, French, English and German: The Invention of Symbolism

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And Yet It Is Heard

Part of the book series: Science Networks. Historical Studies ((SNHS,volume 47))

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Abstract

In 1628, Kepler had written that his adversary, Robert Fludd , had entered into a debate also with the Frenchman, Marin Mersenne (1588–1648), who had attacked him. In 1623, between the German and the Englishman, Mersenne had taken side with the first in his Questiones … in Genesim … [Investigations … on Genesis …], “holding tight to my hair” (that of the astronomer).

Le silence éternel de ces espaces infinis m’effraie.

Blaise Pascal , Pensées , [233].

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Notes

  1. 1.

    Kepler 1983, v. XI-1, pp. 471–472. Baumgardt 1951, pp. 181–182. Lenoble 1971, pp. 103–109 and 367–370. Amman 1967, pp. 210–219. Crombie 1981, p. 320.

  2. 2.

    Mersenne 1636. He dealt with harmony in several other books, written and published from 1623 to 1648. Lenoble 1971, pp. XII–XXV.

  3. 3.

    Mersenne 1636, [tr. 4] pp. 188–190. This long essay was composed of different Treatises on various topics, each numbered separately in the pages. I have assigned a number to the Treatises, for convenience in referring to them, in accordance with the order in which they were printed; details are in the Bibliography.

  4. 4.

    Mersenne 1636, [tr. 4] pp. 85–88.

  5. 5.

    Mersenne 1636, [tr. 4] pp. 82–90.

  6. 6.

    Mersenne 1636, [tr. 1] pp. 42–44.

  7. 7.

    Mersenne 1636, [tr. 4] pp. 97–99.

  8. 8.

    Mersenne 1636, [tr. 1] pp. 23–24. This was probably the first time that the term “frequency” had been used for the pitch of a sound. We shall see in Sect. 10.1 the idea of an unconscious counting in the soul, taken up by Leibniz .

  9. 9.

    Mersenne 1636, [tr. 1] “Seconde Observation” and p. 23; [tr. 4] p. 139. Crombie 1981, p. 319.

  10. 10.

    See Part I, Sects. 6.5 and 6.6.

  11. 11.

    Mersenne 1636, [tr. 1] pp. 4–10.

  12. 12.

    The analogy with waves in water would also be found in a page of Greek attributed to the Stoics, and in Boethius , but these were outside the Greek traditions for music which passed into Arabic culture; cf. Burnett 1991, p. 56. Here, Part I, Sects. 6.2 and 6.6.

  13. 13.

    Mersenne 1636, [tr. 4] p.112.

  14. 14.

    Mersenne 1636, [tr. 4] pp. 115, 132. In any case, he gave numerical rules about how to divide the neck of the lute in the equable temperament; [tr. 1] “Preface Generale” [v–vii]. He had still maintained the ‘Pythagorean’ essence of his initial writings about music, although in other aspects he had introduced modifications. Cf. Lenoble 1971, pp. 527–529.

  15. 15.

    Mersenne 1636, [tr. 4] p. 118. [tr. 1] “Preface generale” [v] and [xi]. Lenoble 1971, pp. 522–531.

  16. 16.

    Mersenne 1636, [tr. 4] pp. 133–137.

  17. 17.

    Mersenne 1636, [tr. 1] “Livre Second”.

  18. 18.

    Mersenne 1636, [tr. 4] pp. 251–253.

  19. 19.

    Mersenne 1636, [tr. 5] pp. 27–31. Sometimes, however, Mersenne used Kepler ’s measurements; Mersenne 1636, [tr. 1] pp. 104–105. Once, in 1627, he had appreciated the music of the heavenly spheres; now he rejected not only that of Fludd , but also that of Kepler . Cf. Lenoble 1971, pp. 367–370.

  20. 20.

    Mersenne 1636, [tr. 1], pp. 76, 167, 209–212; “Preface Generale” [i]; “Première Observation”, “Seconde Observation”. Lenoble 1971, pp. 190–199, 413, 555–560 and passim. Crombie 1981.

  21. 21.

    Mersenne 1636, [tr. 1] pp. 63–67. Mersenne criticised Maurolico (1611) for considering i proportional to r. Shea 1994, pp. 156–164.

  22. 22.

    Mersenne 1636, [tr. 1] pp. 67–71.

  23. 23.

    Mersenne 1636, [tr. 1] pp. 71–73.

  24. 24.

    Mersenne 1636, [tr. 1] pp. 74, 77–84.

  25. 25.

    Mersenne 1636, [tr. 1] pp. 84–156, p. 112.

  26. 26.

    Mersenne 1636, [tr. 1] pp. 135–137, “Preface Generale” [ii], 22, 167 and passim.

  27. 27.

    Mersenne 1636, [tr. 1] pp. 158, 162, 168. Cf. Bailhache 1993, Sect. 2.2.

  28. 28.

    Mersenne 1636, [tr. 1] pp. 169–170. In this connection, however, he had referred to Vincenzio Galilei in two previous books. Cohen 1984 , p. 183.

  29. 29.

    Mersenne 1636, [tr. 1] pp. 175, 179, 180, 185–187; [tr. 4] p. 216. [tr. 8] pp. 25–26.

  30. 30.

    Mersenne 1636, [tr. 3] p. 9.

  31. 31.

    Part I, Sects. 6.2, 6.5, 6.6.

  32. 32.

    Mersenne 1636, [tr. 4] pp. 123–126.

  33. 33.

    Mersenne 1636, [tr.1] p. 65. See above, note 21.

  34. 34.

    Maurolico 1611. Moscheo 1988, p. 341.

  35. 35.

    Tonietti 2006b.

  36. 36.

    Mersenne 1636, [tr. 3] pp. 107–148.

  37. 37.

    Mersenne 1636, [tr. 3] pp. 65–72. Cf. Lenoble 1971, pp. 514–521

  38. 38.

    Mersenne 1636, [tr. 3] p. 139.

  39. 39.

    Mersenne 1636, [tr. 4] pp. 245–251, 278–282 and 374.

  40. 40.

    Mersenne 1636, [tr. 4] pp. 200–201.

  41. 41.

    Mersenne 1636, [tr. 4] pp. 204, 209.

  42. 42.

    Part II, Sect. 8.3.

  43. 43.

    Mersenne 1636, [tr. 4] p. 293. Compare with Leibniz , who will make the same comparison between dissonances and sins, in Sect. 10.1.

  44. 44.

    Mersenne 1636, [tr. 4], pp. 290, 357–358, 143.

  45. 45.

    Mersenne 1636, [tr. 4] pp. 7–9.

  46. 46.

    Mersenne 1636, [tr.5] pp. 38 and 41.

  47. 47.

    Mersenne 1636, [tr. 5] pp. 45- - 45[bis]-46[bis]; the book presents page numbers that are repeated, and some are missing.

  48. 48.

    Part II, Sect. 8.2.

  49. 49.

    Mersenne 1636, [tr.5] pp. 48–49, 60–61.

  50. 50.

    Mersenne 1636, [tr. 5] pp. 331–335.

  51. 51.

    Mersenne 1636, [tr. 5] pp. 341–344.

  52. 52.

    Mersenne 1636, [tr. 5] pp. 208–211; [tr. 8] pp. 15–16. See Part II, Sect. 8.2.

  53. 53.

    See below, Part II Sects. 11.2 and 12.2.

  54. 54.

    Mersenne 1636, [tr. 5] pp. 362–363, [tr. 8] p. 20.

  55. 55.

    Mersenne 1636, [tr. 6] pp. 77–79.

  56. 56.

    Mersenne 1636, [tr. 7] p. 1.

  57. 57.

    Mersenne 1636, [tr. 7] pp. 17, 19.

  58. 58.

    Mersenne 1636, [tr. 7] pp. 23, 26; [tr. 5] pp. 227–228. Sachs 1996, p. 264.

  59. 59.

    Mersenne 1636, [tr. 7] pp. 26–46. Taking a ‘toise’ to be 1.7 m long, this would give a speed of 391 m/s, which is higher than the 331 m/s measured today. Cf. Lenoble 1971, pp. 485–486.

  60. 60.

    Mersenne 1636, [tr. 7] p. 49. Cf. Lenoble 1971, pp. 524–530 and 549–552. Amman n 1967, p. 23.

  61. 61.

    Mersenne 1636, [tr. 7] pp. 47–50.

  62. 62.

    Mersenne 1636, [tr. 7] pp. 50–57.

  63. 63.

    See Part I, Sect. 5.3. Mersenne 1636, [tr. 7] pp. 54, 26, 64–68. Cf. Lenoble 1971, pp. 474–475, 554–555 and 588.

  64. 64.

    Mersenne 1636, [tr. 8] pp. 23–24, 26–27. He copied the last verse of Psalm 150 in Hebrew: “Let everything that has breath praise the Lord [Yah]. Praise the Lord [Yah]”. Thanks are due to Giovanni Mazzini for the translation. [tr. 3]. Cf. Lenoble 1971, pp. 96–109

  65. 65.

    Mersenne 1636, [tr. 1] “Abregé de la Musique speculative” V and p. 23; [tr. 5] pp. 361–362.

  66. 66.

    Mersenne 1636, [tr. 4], [tr. 5] and [tr. 6]. For example, Stillman Drake did not say anything about Vincenzio Galilei ’s new rules for pipes; Drake 1981.

  67. 67.

    Mersenne 1636, [tr. 2], [tr. 7] pp. 71–74, [tr. 4] pp. 331ff., [tr. 5] pp. 308ff. Lenoble 1971, pp. 581–594. Cf. Dear 2000.

  68. 68.

    Mersenne 1636, [tr. 4] p. 331f.

  69. 69.

    Fano & Terracini 1957, pp. 356–358.

  70. 70.

    Taton 1981.

  71. 71.

    Mersenne 1636, [tr. 7] pp. 61–64.

  72. 72.

    Fermat 1891, pp. 394–410.

  73. 73.

    Mahoney 1981.

  74. 74.

    Mersenne 1636, [tr. 5] p. [308bis]. Lenoble 1971, pp. 436–437.

  75. 75.

    See Part I, Sect. 3.4 and Fig. 3.10.

  76. 76.

    Boyer 1990, p. 422.

  77. 77.

    Lenoble 1971, pp. 430–435. Taton 1981a. See below, Sects. 9.3 and 10.1.

  78. 78.

    Pascal 1988, pp. 58, 110, 140, 154–159, 174–176.

  79. 79.

    Pascal 1988, pp. 57–58, passim. Today, in 2014, at least in some part of the world, those who exchange sex for money would no longer be called prostitutes and customers, but escorts and politicians. The problem of public morality is solved by words: Jesuit ability.

  80. 80.

    Pascal 2004, pp. 29, 67, 69, 365, 335, …passim.

  81. 81.

    The Greek philosopher Pyrrho of Elis, who died in 288 B.C., harboured doubts about everything, and taught that the truth could never be found. Consequently, some people called scepticism the Pyrrho nism, and the word remained common for a certain time.

  82. 82.

    If it is not true that n 2 = 2m 2, that is to say, \(\frac{n} {m} = \sqrt{2}\) with n and m whole numbers, then \(\sqrt{2}\) is irrational.

  83. 83.

    Pascal 2004, pp. 4–5, 576–579, 758, 344–345, 84–85, 360–361, 390–391.

  84. 84.

    Pascal 2004, pp. 34–35, 781, 236–237, 740–741, 718–719, …, passim.

  85. 85.

    Pascal 2004, pp. 52–57, 791–793.

  86. 86.

    Montaigne 1969, II p. 246; 2005, p. 771. Pascal 2004, p. 792. Pascal 2004, pp. 568–571 and 581.

  87. 87.

    Pascal 2004, pp. 85, 532–533, 93, 99, 171, 233, 266–269, 406–407, 512–517.

  88. 88.

    Pascal 2004, pp. 452–453, 893, 349, 894, 350–351, 355, 896.

  89. 89.

    Pascal 2004, pp. 100–101, 104–105, 150–151, 232–233, 564–565, 960.

  90. 90.

    Pascal 2004, pp. 132–133, 158–159, 160–167.

  91. 91.

    Pascal ’s reasoning recalls that of lotteries, where by betting a small sum, or hardly anything, it is possible to win a lot. In the case of God, one would win Paradise for eternity. However, as in the case of lotteries, also Pascal ’s God becomes, in reality for common mortals (the non-elect), a tax: the most hateful one, because it is imposed on poverty and hope.

  92. 92.

    Pascal 2004, pp. 418–419, 522–523, 955, 524–533.

  93. 93.

    Pascal 2004, pp. 172–173.

  94. 94.

    Pascal 2004, pp. 49, 790, 74–75, 166–167, 835, 402–403, 466–467. Montaigne 1969, II p. 7, III p. 155; Montaigne 2005, pp. 430, 1252.

  95. 95.

    Pascal 2004, pp. 506–509, 902, 948.

  96. 96.

    See Part II, Sect. 8.2.

  97. 97.

    Pascal 2004, pp. 742–745, 799, 80–81, 364–365, 416–417, 716–717.

  98. 98.

    Pascal 2004, pp. 178–179, 578–581.

  99. 99.

    Pascal 2004, pp. 99, 427, 459, 465, 493, 944, 679, 999, 435, 923, 685, 1002.

  100. 100.

    Gracián 2008, pp. 88–89.

  101. 101.

    Pascal 2004, p. 1012.

  102. 102.

    Descartes 1966, X p. 141. Cf. Descartes 1979, p. 127.

  103. 103.

    Descartes 1966, X pp. 89–92, 96–98, 103–105 and 116. Cf. Descartes 1979, pp. 44, 73–75, 81–82, 24, 87–89, 101, 134.

  104. 104.

    Part I, Sect. 6.5. Descartes 1966, X p. 110. Also in the Traité de l’homme [Treatise on man]. Cf. Gouk 1991, p. 106.

  105. 105.

    Descartes 1966, X p. 115. Cf. Descartes 1979, pp. 132, 100, 134. This was the corpuscular model of Beeckman , from which he was to move away, to embrace sound waves. Descartes 1974, I pp. 104 and 109. See below in this section.

  106. 106.

    Descartes 1966, X pp. 108, 132–133 and 140–141. Cf. Descartes 1979, pp. 118–119, 126–127.

  107. 107.

    Descartes 1966, X p. 124. Cf. Descartes 1979, p. 109.

  108. 108.

    Maurolico 2000, Maurolico 201?.

  109. 109.

    Descartes 1966, X pp. 133 and 139. Cf. Descartes 1979, pp. 120 and 125. Evidently, he did not appreciate the intricate polyphonic games of the Flemish tradition.

  110. 110.

    Tonietti 2006b, pp. 157–158. Cf. Hooykaas 1981.

  111. 111.

    Descartes 1974, I pp. 286 and 288–289; 1976, IV pp. 678–683 and 722–725. Albert Ban adopted it; see below in this section. Mersenne 1946, III p. 458; 1955, IV pp. 51, 99, 143–145, 149–151; 1959, V pp. 570–571; 1970, XI pp. 81–82. Descartes 1979, p. 31.

  112. 112.

    Mersenne 1959, V p. 566; 1970, XI pp. 84 and 90.

  113. 113.

    Mersenne 1945, I pp. 201, 206, 256–259, 269–273. Descartes 1983, p. 567.

  114. 114.

    Mersenne 1636, [tr. 5] pp. 407–412.

  115. 115.

    Shea 1994, pp. 64–67.

  116. 116.

    Mersenne 1945, II pp. 276 and 286. See above, Sect. 8.1; see below in this section.

  117. 117.

    Mersenne 1945, II pp. 276 and 286. Descartes 1974, I p. 331.

  118. 118.

    Mersenne 1945, II pp. 282 and 293–294. Cf. Cohen 1984 , pp. 120–123.

  119. 119.

    Mersenne 1945, II p. 452. Cf. Cohen 1984 , pp. 127–139.

  120. 120.

    Mersenne 1946, III pp. 323–327, 414–415 and 418–419.

  121. 121.

    Mersenne 1636, [tr. 5] pp. 384–385 and 37–38. Mersenne 1946, III pp. 447–452; 1955, IV pp. 435–437; 1945, II pp. 514 and 519–520. See above, Table 9.2.

  122. 122.

    Mersenne 1946, III p. 586; [1959], V pp. 53–54 and 69–73.

  123. 123.

    Mersenne 1636, [tr. 8] pp. 21–22. Mersenne [1959], V pp. 615–617.

  124. 124.

    Descartes 1925, pp. 21 and 27.

  125. 125.

    Descartes 1925, pp. 38–39, 52–55.

  126. 126.

    Descartes 1925, pp. 63–65.

  127. 127.

    Descartes 1925, pp. 82, 84–85.

  128. 128.

    Descartes 1983, pp. 525, 531, 528–529. For Descartes ’ three dreams of 1619, see Shea 1994, pp. 125–127.

  129. 129.

    Descartes 1974, II p. 392. Descartes 1983, pp. 529–530, 546 and 569.

  130. 130.

    Descartes 1974, X pp. 154–155.

  131. 131.

    Descartes 1983, pp. 632–634, 664–665 and 682–685.

  132. 132.

    Shea 1994, pp. 156, 210, 223 e 250.

  133. 133.

    Mersenne 1962, VII pp. 231–232. Gingerich 1970, pp. 298 and 308.

  134. 134.

    Mersenne 1965, IX p. 193. Mahoney 1981a.

  135. 135.

    Descartes 1925, pp. 52–53.

  136. 136.

    Descartes 1967, XI p. 47.

  137. 137.

    Shea 1994, p. 19.

  138. 138.

    Descartes 1925, pp. 32, 36–37 and 44.

  139. 139.

    Descartes 1925, pp. 43 and 55–56. Descartes 1974, I p. 145.

  140. 140.

    Descartes 1925, pp. 55–56. Descartes 1975, III p. 274. Mersenne 1962, VII pp. 242–244. Tonietti 1983a; 1985a; 1985b; 1988; 1990.

  141. 141.

    Descartes 1925, pp. 37, 68, 35, 44.

  142. 142.

    Descartes 1925, pp. 70, 75, 77.

  143. 143.

    Descartes 1925, p. 93.

  144. 144.

    Mersenne 1962, VII p. 241.

  145. 145.

    Descartes 1925, pp. 28, 31 and 50. Descartes 1983, p. 216. Shea 1994, p. 20.

  146. 146.

    Descartes 1925, pp. 62–63, 79 and 86. Descartes 1974, I pp. 285–286; 1975, III pp. 258–259 and 270. Mersenne 1946, III p. 558. Shea 1994, p. 361.

  147. 147.

    Descartes 1964, VIII-1 pp. 53–54 and 89–90. Shea 1994, pp. 328–330.

  148. 148.

    Mersenne 1962, VII p. 241.

  149. 149.

    Descartes 1974, I p. 286. Crombie 1981. Shea 1994, pp. 345–346.

  150. 150.

    Descartes 1974, I pp. 303–306, 375 and 377; 1975, II pp. 380–405, 586; 1975, III pp. 249–250 and 257–258. Lenoble 1971, pp. 414–418, 420, 426, 582, 587 and 598. Descartes 1983, pp. 638ff. Shea 1994, pp. 381, 184, 186–187, 353 and 405–406.

  151. 151.

    Mersenne 1636, [tr. 1] pp. 157–158. Mersenne 1945, II pp. 231–236. Descartes 1974, I pp. 29 and 31–32. Cohen 1984 , pp. 187–201. Shea 1994, pp. 90–104.

  152. 152.

    Descartes 1974, I p. 142; 1975, II p. 699; 1975, III pp. 825–834. Cohen 1984 , pp. 119–120. Shea 1994, p. 85.

  153. 153.

    Maurolico 1575, pp. 150–151.

  154. 154.

    Descartes 1974, I pp. 24, 30, 110–111, 155–167, 172 and 177–178.

  155. 155.

    Descartes 1974, I pp. 94, 100–101 and 224–227.

  156. 156.

    Descartes 1974, I pp. 88, 108–109, 126 and 223.

  157. 157.

    Descartes 1975, III pp. 255, 261–262 and 825–834. Huygens 1889, II p. 547.

  158. 158.

    Descartes 1974, II pp. 388–389.

  159. 159.

    Cohen 1984, pp. 190–197. Shea 1994, pp. 82–106.

  160. 160.

    Mersenne 1945, II pp. 276 and 286. Cohen 1984 , pp. 185 and 286.

  161. 161.

    Cohen 1984, pp. 186–187, 286 and passim.

  162. 162.

    Mersenne 1945, II pp. 205–206. Cohen 1984 , pp. 116–161. Lenoble 1971, pp. 424, 427–429 and 434.

  163. 163.

    Cohen 1984, pp. 280, 148. H.F. Cohen attributed to him the idea that the pitch of sounds from pipes depended on their volumes, but without specifying the ratios. This philosopher of science considered the inverse proportion with the cube 1:23, followed by the Chinese and by Vincenzio Galilei , a “mistake”. On the contrary, he is the one who unfortunately perpetuated the same mistake committed by D. P. Walker , ignoring, like him, the modern manuals of acoustics, which justify the end-effect, or, worse still, cancelling the clear cultural differences between Chinese, Indians, Arabs and Europeans in music. Cohen 1984 , pp. 82–83 and 92–93. See above, Part I, Sects. 3.2 and 6.7.

  164. 164.

    Cohen 1984, pp. 279, 143.

  165. 165.

    Cohen 1984, p. 156.

  166. 166.

    Descartes 1983, p. 638f.

  167. 167.

    Scriba 1981.

  168. 168.

    Ptolemy 1682, pp. 283, 302, and 321.

  169. 169.

    Ptolemy 1682, p. 305.

  170. 170.

    Ptolemy 1682, pp. 306–307.

  171. 171.

    Ptolemy 1682, pp. 307–308 and 321.

  172. 172.

    Ptolemy 1682, pp. 308–309 and 318–320. As others had also begun to think, our mathematician realised that the strokes had to start together, otherwise the argument founded on coincidences would fall apart. Nevertheless, he maintained it undaunted, affirming, without giving any proof or justification, that if the strokes started without coinciding, after a while they would start coinciding again. Cf. Wardhaugh 2008, p. 160.

  173. 173.

    Scriba 1981, p. 152. Gouk 1989a, p. 159. Other interesting details about Wallis ’s musical endeavours can be found in Wardhaugh 2008, pp. 156–166–179. In fact, our famous mathematician maintained that “the ear might be a legitimate source of knowledge for what was none the less definitely a part of mathematics.” Unfortunately, it remains a mystery how he reconciled the judgement of the ear with his mathematics for music. Maybe, like his Ptolemy , he had faith in a superior a priori coherence?

    Thomas Salmon engaged in a dispute with a musician who relied only on the ear, and did not believe that music consisted of ratios. While he seemed to be distant from the real English musical environments, he cannot have been very close to the mathematical culture, either, because he had to ask Wallis for explanations about the calculation of ratios for music. Salmon made numerous attempts to adapt the tuning system of Ptolemy -Zarlino (which the English insist on calling the “just” intonation scale) to the requirements of modulations, proposing various interchangeable fingerboards. However, his conclusion was the same as we shall find further on in our story, that the equable temperament “may be allow’d by such Ears as are vitiated by long custome”.

  174. 174.

    Scriba 1981, p. 148. Boyer 1990, pp. 436–440.

  175. 175.

    Boyer 1990, pp. 348–358.

  176. 176.

    Boyer 1990, pp. 439 and 441. Kline 1972 , pp. 318, 251–252. See above, Sect. 6.5.

  177. 177.

    Scriba 1966.

  178. 178.

    Kline 1972 , p. 864.

  179. 179.

    Hooykaas 1981.

  180. 180.

    Gouk 1989a, pp. 157–158, 160–163; 1989b, pp. 188–190; 1991, pp. 99 and 113. Cf. Wardhaugh 2008, pp. 159 and passim. Resnick & Halliday 1961, pp. 414 and 431. See Part II, Sect. 10.2. For the scales proposed by John Pell (1611–1685), with 21 or 33 notes per octave, with the relative numbers, see Wardhaugh 2008, pp. 154–155.

  181. 181.

    Gozza 1991, pp. 122, 126–129. The research of Paolo Gozza is developed, with some extra details, also by Benjamin Wardhaugh . In reality, this English scholar practically always concluded by raising bitter objections to Mengoli ’s theories. However, even Wardhaugh stated that the majority of theoreticians of music with mathematical criteria were incapable of explaining contemporary musical practices, because, on the contrary, these were based on the ear and on the equable temperament; Wardhaugh 2008, pp. 83–96.

    When also a professional musician like John Birchensha (c.1605–1681) attempted to present a theory of music to the fellows of the Royal Society, he remained distant from the practice followed by his colleagues: and even from his own practice. It seems that he used the equable temperament, whereas his theoretical scale had remained that of classical Pythagoreanism; Wardhaugh 2008, pp. 147–148 and 154–155.

  182. 182.

    Boyer 1990, pp. 426–427.

  183. 183.

    Drake 1970, pp. 483, 497 and 499. Drake 1981.

  184. 184.

    Descartes 1975, III pp. 756–758. Musical works by Constantijn Huygens were to be published, or re-published, at the end of the nineteenth century. Huygens 1888, I pp. 30 and 554; 1889, II pp. 547–548, 552, 555–556, 559 and 565. Shea 1994, pp. 336 and 412.

  185. 185.

    Mersenne 1960, VI p. 306. Descartes 1966, X p. 82.

  186. 186.

    Mersenne 1986, XVI p. 549–551.

  187. 187.

    Mersenne 1983, XV pp. 24–39. Huygens 1888, I pp. 30–31, 34–35, 87–89 and 83–86; 1889, II pp. 554–559. Lenoble 1971, p. 585. Shea 1994, p. 332.

  188. 188.

    Huygens 1888, I pp. 537–554.

  189. 189.

    Huygens 1888, I pp. 111, 135, 347 and 359. 1891, IV pp. 183–184.

  190. 190.

    Huygens 1888, I pp. 342, 349–350, 352, 356, 371–372, 508–509; 1890, III p. 228.

  191. 191.

    Huygens 1890, III pp. 199–200 and 252; Huygens 1899, VIII p. 241. Rasch 1992.

  192. 192.

    Tannery 1915 (1902) p. 71.

  193. 193.

    Huygens 1940, XX p. 11. On logarithms, see Wardhaugh 2008, pp. 40–42ff. On the musical manuscripts of Nicolaus Mercator , where among other things he divided the octave into 53 parts using logarithms, see Wardhaugh 2008, pp. 47–53 and 152–154. Wardhaugh ’s speculation about the hypothetical use made by Descartes of logarithms for his circular figures of the musical scales, does not sound very convincing.

  194. 194.

    Huygens 1890, III pp. 307–308; 1940, XX p. 12.

  195. 195.

    Huygens 1940, XX pp. 30–37.

  196. 196.

    Huygens 1940, XX pp. 43–60.

  197. 197.

    Huygens 1897, VII pp. 368–369.

  198. 198.

    Huygens 1940, XX pp. 68–69; 1901, IX p. 567.

  199. 199.

    Mersenne 1636, [tr. 4] p. 187. Huygens 1940, XX pp. 69–76.

  200. 200.

    In the edition, this escaped the notice of the editors of the page, by their own admission.

  201. 201.

    Huygens 1940, XX pp. 78–81.

  202. 202.

    Huygens 1940, XX p. 91. As this was clearly not a case of equal tones like those of Aristoxenus , the footnote on the Greek musician placed at this point by the editor of the edition appears to be inappropriate.

  203. 203.

    Huygens 1940, XX p. 104; 1937, XIX p. 377.

  204. 204.

    Huygens 1940, XX pp. 94–97. Aristoxenus 1954, p. 67.

  205. 205.

    Huygens 1940, XX pp, 98–102. If we take away the major third 5:4 from the fourth 4:3, instead of subtracting the Pythagorean ditone 81:64, what is left is a diesis of 16:15.

  206. 206.

    Huygens 1940, XX pp. 113–117.

  207. 207.

    Huygens 1940, XX pp. 120–122.

  208. 208.

    Huygens 1940, XX pp. 125–126, 129 and 133.

  209. 209.

    Huygens 1940, XX pp. 133–135. The difference from the equable temperament is explained in Cohen 1984 , p. 216.

  210. 210.

    Huygens 1905, X pp. 169–174.

  211. 211.

    Huygens 1890, III pp. 307–308; 1940, XX pp. 160, 168–169. Mersenne 1636, [tr. 1] p. [vi]; [tr. 4] p. 132. See above, Sect. 9.1, Table 9.1.

  212. 212.

    Huygens 1895, VI pp. 473, 484 and 515.

  213. 213.

    Huygens 1940, XX pp. 161, 159 and 162.

  214. 214.

    Huygens 1940, XX pp. 171–173. See Sect. 8.1.

  215. 215.

    Huygens 1940, XX pp. 559–561; 1937, XIX p. 356.

  216. 216.

    Huygens 1937, XIX pp. 364–373.

  217. 217.

    Huygens 1937, XIX pp. 375–376.

  218. 218.

    Huygens 1937, XIX p. 351. Resnick & Halliday 1961, p. 403. Huygens 1950, XXII p. 16.

  219. 219.

    Huygens 1934, XVIII pp. 489–492.

  220. 220.

    Huygens 1944, XXI pp. 751–755.

  221. 221.

    Remember Sect. 9.2. On Ban, see above all Rasch 1992, pp. 192–196.

  222. 222.

    Huygens 1950, XXII p. 53. On the death of Ban, see Huygens 1889, II p. 547.

  223. 223.

    Huygens 1889, II p. 510; 1944, XXI p. 788.

  224. 224.

    Huygens 1944, XXI pp. 539–559.

  225. 225.

    Huygens 1893, V pp. 63, 98–105 and 346; 1895, VI pp. 485, 500 and 518; 1897, VII pp. 313–314; 1899, VIII pp. 239 and 241; 1901, IX pp. 20, 230 and 568; 1905, X p. 598. Boyer 1990, pp. 431–436. Cf. Bos 1981, where music is unfortunately ignored.

  226. 226.

    Huygens 1888, I p. 88; 1891, IV pp. 183–190, 334, 341 and 356; 1897, VII pp. 108–110 and 122–123.

  227. 227.

    Huygens 1888, I p. 17; 1950, XXII p. 411.

  228. 228.

    Huygens 1888, I p. 17; 1893, V pp. 63–64; 1899, VIII p. 322; 1950, XXII pp. 405–406.

  229. 229.

    Newton 1687 (1972), Sectio II liber secundus.

  230. 230.

    Huygens 1901, IX pp. 258, 366, 471–475, 497–520; 1950, XXII p. 153.

  231. 231.

    Huygens 1901, IX pp. 569–576; 1905, X pp. 224–225, 298–299.

  232. 232.

    Bos 1981, pp. 608–609.

  233. 233.

    Huygens 1901, IX pp. 258, 367–368, 472, 522–524.

  234. 234.

    Huygens 1901, IX p. 522.

  235. 235.

    Huygens 1901, IX p. 525. Dijksterhuis 1971, pp. 492–509 and 612–620. Bos 1981, pp. 609–610, 602–603.

  236. 236.

    Bos 1981.

  237. 237.

    Bos 1981, pp. 607–608.

  238. 238.

    Cohen 1984, pp. 206–230. What would be appreciable here is the desire to measure the theory of music against the specific tastes of Huygens; if it were not for the event that he had written, aroused by his historiographic hypothesis: “… of a Lully not a single word”, p. 225. With reference to the Franco-Florentine composer, an expression of praise above has been found about him: “belle opere [fine works]”. Huygens 1899, VIII p. 24. See above all Rasch 1992, pp. 200–201. Rasch 1992, pp. 185–210.

  239. 239.

    Huygens 1905, X p. 651.

  240. 240.

    Rasch 1992, p. 203.

  241. 241.

    Mikami 1913. Li & Du 1987. Jami 1988b.

  242. 242.

    Martzloff 1981a; 1981b; 1988; 1993. 1981b, pp. 223, 327.

  243. 243.

    Needham 1959, p. 168.

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    Tito M. Tonietti

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Tonietti, T.M. (2014). Beyond Latin, French, English and German: The Invention of Symbolism. In: And Yet It Is Heard. Science Networks. Historical Studies, vol 47. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0675-6_3

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