Above All with the Greek Alphabet

Tonietti, Tito M.

doi:10.1007/978-3-0348-0672-5_2

Tito M. Tonietti⁵

Part of the book series: Science Networks. Historical Studies ((SNHS,volume 46))

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Abstract

I would like to begin with an argument which may be stated most clearly and most forcefully as follows:

Music was one of the primeval mathematical models for natural sciences in the West.

The other model described the movement of the stars in the sky, and a close relationship was postulated between the two: the music of the spheres.

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Notes

1.
Boyer 1990, p. 65.
2.
See Sect. 6.2.
3.
See below.
4.
Even if he is guilty of anachronism, in order to arrive more rapidly at the result, the reader inured by schooling to fractions will easily be able to calculate \(\frac{3} {2}: \frac{4} {3} = \frac{9} {8}\). However, the use of fractions in music had to await the age of John Wallis (1617–1703), Part II, Sect. 9.2. After all, the Greeks used the letters of their alphabet α, β, γ … to indicate numbers …
5.
Pitagorici 1958 and 1962. Boyer 1990, pp. 85–87. Cf. Centrone 1996, p. 84. The Pythagoreans are to be considered as adepts of a religious sect governed by prohibitions and rules, somewhat different from the mathematical community of today, which has other customs.
6.
Pitagorici 1958 and 1962. The adjective “harmonic” used for the relative mean, previously called “sub-contrary”, is attributed to him.
7.
Thomson 1973, p. 299.
8.
Thomson 1973, pp. 278, 281.
9.
Cooke 1997. Although Centrone 1996 is a good essay on the Pythagoreans, he too, unfortunately, underestimates music: he does not make any distinction between their concept of music and that of Aristoxenus . This limitation derives partly from the scanty consideration that he gives to the Aristotelian continuum as an essential element, by contrast, to understand the Pythagoreans. Without this, he is left with many doubts, pp. 69, 196 and 115–117. Cf. von Fritz 1940. Pitagorici 1958, 1962, and 1964.
10.
Plato 1994, pp. 25–27, 31–33, 61, 129–131.
11.
Plato 1994, p. 37.
12.
Plato 1994, p. 103.
13.
Heath 1963, p. 107.
14.
Pacioli 1509. See Sect. 6.4.
15.
In the pentagon, the diagonals intersect each other in this ratio.
16.
Weyl 1962.
17.
Heath 1963, p. 178. Fowler 1987, pp. 3–7.
18.
Plato 1999, pp. 117, 119, 125, 145, 149.
19.
Plato 1999, pp. 179, 181, 209, 187, 191, 195, 211.
20.
Plato 1999, pp. 457, 467, 469, 471.
21.
Plato 1999, p. 447.
22.
Plato 1999, pp. 471, 475, 477, 479, 481, 483.
23.
Plato 1999, pp. 491, 493.
24.
Plato 1999, pp. 497, 499, 501.
25.
Plato 1999, pp. 505, 507, 513, 506, 507.
26.
Plato 1999, pp. 511, 155.
27.
See above Sect. 2.2.
28.
Plato 1953, pp. 103–107.
29.
Plato 1953, pp. 128–130 and passim.
30.
We use the 1557 edition of Euclid , with the Greek text and the translation into Latin. An Italian translation is that of Bellissima 2003. Euclid 1557, p. 8 and 14; Bellissima 2003, p. 29. Zanoncelli 1990. Euclid 2007, pp. 677–776, 2360–2379 and 2525–2541.
31.
Euclid 1557, p. 10 and 16; Bellissima 2003, p. 37.
32.
Bellissima 2003. Euclid 2007, pp. 691–701.
33.
Tonietti 2000b.
34.
Hilbert 1899.
35.
Tonietti 1982a, 1983a, 1985a, 1990.
36.
Bellissima 2003, p. 31. Euclid 2007.
37.
Euclid 1956, pp. 349–350; Euclid 1970, pp. 146–150. Euclid 2007. Figure on every textbook.
38.
Euclid 1956, p. 354.
39.
Euclid 1956, p. 364.
40.
Zeuthen 1896; Cantor 1922; Smith 1923.
41.
Euclid 1956, pp. 355–356.
42.
Euclid 1956, p. 362.
43.
Euclid 1956, p. 97.
44.
Euclid 1956, pp. 349, 153–154.
45.
Euclid 1956, p. 153.
46.
Euclid 1956, II, p. 278.
47.
Euclid 1956, II, p. 412.
48.
Euclid 1956, II, p. 413.
49.
Euclid 1956, I, p. 155.
50.
Hilbert 1899.
51.
Boyer 1990, pp. 78, 99, 105–110.
52.
Zeuthen 1902, pp. 72–73.
53.
Zeuthen 1896, pp. 222–223.
54.
Euclid 1925, pp. 135–141.
55.
Euclid 1970, p. 147.
56.
Euclid 1916.
57.
Seidenberg 1960, p. 498.
58.
Seidenberg 1975.
59.
Fowler 1987, p. 21.
60.
Russo 1996, p. 73.
61.
Russo 1996, pp. 235–244.
62.
Russo 1996, p. 32. Mario Vegetti finds that Euclid ’s approach is used by Galen and Claudius Ptolemaeus as an axiomatic Platonic model. And yet, even this professor of ancient philosophy, though levelling out the procedure too much, realises that Euclidean rationality has to come to grips with Aristotle : “In the first place, the ontological obligation to consider the forms as transcendent, or at least as external to the empirical, disappears.” Vegetti 1983, p. 155.
63.
Euclid 1956, III, pp. 503–509.
64.
Euclid 1956, I, p. v.
65.
Tonietti 1982b, pp. 11–21.
66.
Winnington-Ingram 1970, p. 282.
67.
Aristoxenus 1954, p. 19.
68.
Aristoxenus 1954, pp. 20–21.
69.
Aristoxenus 1954, p. 22.
70.
Aristoxenus 1954, pp. 30–31.
71.
Aristoxenus 1954, p. 31.
72.
Aristoxenus 1954, pp. 24–25.
73.
Aristoxenus 1954, p. 32.
74.
Aristoxenus 1954, p. 47.
75.
Aristoxenus 1954, p. 48.
76.
Aristoxenus 1954, p. 79.
77.
Aristoxenus 1954, pp. 53–55.
78.
Winnington-Ingram 1970, p. 282. This writer shows the origin of her/his prejudices, because she/he immediately adds that “‘temperament’ would distort all the intervals of the scale (except the octave) and, significantly, the fifths and the fourths”. For her/him, the ‘correct’ intervals are, on the contrary, those of Pythagoras . See Part II, Sects. 11.1 and 11.3.
79.
See above Sect. 2.4.
80.
Aristoxenus 1954, p. 67.
81.
Aristoxenus 1954, p. 45.
82.
Tonietti 1991.
83.
Sambursky 1959, pp. 182–185.
84.
Boyer 1990, pp. 116–117. Thom 1980; Thom 2005; Tonietti 2002a.
85.
Aristotle 1982 [Metaphysics ] N5, 1093a, 1.
86.
Some followers of Aristoxenus have been listed and studied in Zanoncelli 1990. Aristoxenus remains one of the main sources regarding the Pythagorean sects for many scholars, who, however, curiously seem to avoid accurately the musical writings that are contrary to the Pythagorean scale. von Fritz 1940. Pitagorici 1964.
87.
Ptolemy 1682, pp. 1–3. We follow the edition of John Wallis , extracted from 11 Greek manuscripts compared together, with a parallel Latin translation: Armonicorum libri tres [Three books on harmony] . The famous Oxford professor so judged the Venetian edition of 1562 printed by Gogavino : “… versio … obscura fuerit & perplexa … a vero saepius aberraverit.” [“… the version is obscure and confused … it departs from the truth somewhat often.”]
88.
Ptolemy 1682, pp. 3–8.
89.
Ptolemy 1682, p. 8.
90.
Ptolemy 1682, pp. 13, 16–18 and 213.
91.
Ptolemy 1682, pp. 19 and 23–24.
92.
Ptolemy 1682, pp. 25–26. See Sect. 6.6.
93.
Ptolemy 1682, pp. 29–33.
94.
Ptolemy 1682, pp. 33–38; cfr. pp. 156–159.
95.
The brackets were added with the italics by Wallis . This enables us to measure the distance between the world of Ptolemy , where it was taken for granted that numbers were only those with a logos, rational and expressible, and the sixteenth century, when an equal existence and use would be granted also to non-expressible numbers, the irrationals.
96.
(18:17) combined with (17:16) gives (18:16), which is equivalent to (9:8). Ptolemy 1682, pp. 39–48.
97.
Ptolemy 1682, pp. 49–50.
98.
Ptolemy 1682, pp. 61–62.
99.
Ptolemy 1682, pp. 66–78.
100.
Ptolemy 1682, pp. 78–79.
101.
Ptolemy 1682, pp. 79–85.
102.
Ptolemy 1682, pp. 89ff., 97, 156ff., passim, and 218.
103.
Ptolemy 1682, pp. 156–166.
104.
For example, Ptolemy 1682, pp. 97–98ff.
105.
Ptolemy 1682, p. 158.
106.
Ptolemy 1682, pp. 232, 236 and 238.
107.
Ptolemy 1682, pp. 239–248.
108.
Ptolemy 1682, pp. 249–258. See Part II, Sect. 8.3.
109.
Ptolemy 1682, pp. 260–273. Cf. Barker 2000. He showed that “Ptolemy understood very well what conditions must be met if experimental tests are to be fully rigorous, …”. However, concerning “… how far the treatise is faithful to the principles it advertises, … There are grounds for some scepticism here, …”. Therefore, in an independent way, my analysis does not side in Ptolemy ’s favour: because, with great probability, the Alexandrian did not test either the attunements of pipes, or Aristoxenus ’.
110.
Ptolemy (Tolomeo) 1985, pp. 60–63; translation corrected by me.
111.
See Sect. 5.4.
112.
Boyer 1990, p. 294.
113.
See Sect. 4.3.
114.
Boyer 1990, pp. 193–200.
115.
Peters 1990.
116.
Boyer 1990, pp. 94–96.
117.
Boyer 1990, pp. 105–110.
118.
Archimedes 1974, pp. 447–448.
119.
Napolitani 2001, pp. 21, 32–33, 36–37.
120.
Napolitani 2001, pp. 43–44.
121.
Archimedes 1960, II, pp. 478–479 and 484.
122.
Tonietti 1982a, 1988, 1990, 1992b; Napolitani 2001.
123.
Authier 1989, p. 107.
124.
Authier 1989.
125.
Archimedes 1960, pp. 467–473.
126.
Napolitani 2001, pp. 67–77.
127.
Boyer 1990, pp. 143–165; Napolitani 2001.
128.
Cf. Fano & Terracini 1957, pp. 356–360.
129.
Boyer 1990, pp. 166–184.
130.
Boyer 1990, pp. 201–204.
131.
Heron , Heiberg edition, IV, 162.
132.
Boyer 1990, pp. 211–215.
133.
Boyer 1990, pp. 210–211.
134.
Ben Miled 2002, pp. 351–352. See Chap. 5.
135.
Cf. Boyer 1990, though here at p. 209 the Italian translator turned her into a man.
136.
Boyer 1990, pp. 215–225.
137.
Napolitani 2001, p. 9.
138.
Lucretius I, 483–484; 1969, p. 32. The translations are mine, and Ron Packham ’s.
139.
Lucretius I, 268; 1969, p. 48.
140.
Lucretius I, 422–425; 1969, p. 28.
141.
Lucretius I, 699–700; 1969, p. 44.
142.
Lucretius I, 459–463; 1969, p. 30.
143.
Lucretius II, 126–132; 1969, p. 78.
144.
Lucretius II, 238–239; 1969, p. 84.
145.
Lucretius I, 827–829; 1969, p. 52.
146.
Lucretius I, 966–967; 1969, pp. 60–62.
147.
Lucretius I, 987; 1969, p. 62.
148.
Lucretius I, 1070–1071; 1969, p. 68.
149.
Lucretius II, 287; 1969, p. 88.
150.
Lucretius I, 1–2; 1969, p. 3.
151.
Lucretius II, 434–437; 1969, p. 96.
152.
Lucretius IV, 379–386; 1969, p. 232.
153.
Lucretius IV, 467–479; 1969, p. 236.
154.
Lucretius IV, 542–544; 1969, p. 240.
155.
Lucretius IV, 604–605 and 609; 1969, p. 244.
156.
Lucretius II, 410–413; 1969, p. 94.
157.
Lucretius II, 505–507; 1969, p. 100.
158.
Lucretius II, 618–620; 1969, p. 106.
159.
Lucretius II, 845; 1969, p. 120.
160.
Lucretius III, 117–160; 1969, pp. 150–152.
161.
Lucretius V, 1382–1383; 1969, p. 366.
162.
Lucretius V, 1399–1411; 1969, p. 368.
163.
Lucretius II, 874; 1969, p. 120.
164.
Lucretius II, 883–885; 1969, p. 122.
165.
Lucretius II, 930; 1969, p. 124.
166.
Lucretius II, 659–660; 1969, p. 108.
167.
Lucretius I, 78–79; 1969, p. 6.
168.
Lucretius V, 146–165; 1969, pp. 292–294.
169.
Lucretius V, 1183–1203; 1969, pp. 354–356; cf. Lucretius VI, 54; 1969, p. 376.
170.
Lucretius II, 1058–1063; 1969, p. 132.
171.
Lucretius II, 1070–1071; 1969, p. 134.
172.
Lucretius III, 163; 1969, p. 152.
173.
Lucretius III, 323–324 and 347–349; 1969, p. 162.
174.
Lucretius III, 440–441; 1969, p. 168. Cf. III, 554–557; 1969, p. 174 and III, 579; 1969, p. 176.
175.
Lucretius VI, 177–179; 1969, p. 382.
176.
Lucretius VI, 524–526; 1969, p. 404.
177.
Lucretius VI, 423–450; 1969, p. 398.
178.
Lucretius 1969, pp. 404–408 and 426–434.
179.
Lucretius V, 623–624; 1969, p. 320.
180.
Lucretius II, 720–722; 1969, p. 112.
181.
Lucretius VI, 788–790; 1969, p. 420.
182.
Lucretius VI, 981–983; 1969, p. 430.
183.
Lucretius V, 259–260 and 280; 1969, pp. 298 and 300.
184.
Lucretius VI, 931–935; 1969, p. 428.
185.
Lucretius VI, 1059–1060; 1969, p. 434.
186.
Lucretius I, 147; 1969, p. 60.
187.
Lucretius V, 67–69; 1969, p. 288.
188.
Lucretius V, 552–555; 1969, p. 316.
189.
Lucretius VI, 498–503; 1969, p. 402. Lucretius VI, 591–595; 1969, p. 408.
190.
Lucretius 1969, pp. 324, 352, 370.
191.
Lucretius V, 906–907; 1969, p. 426.
192.
Lucretius VI, 703–704; 1969, p. 414.
193.
Lucretius V, 419–431; 1969, pp. 308–310.
194.
Lucretius II, 217–220; 1969, p. 84.
195.
Lucretius II, 243–245; 1969, p. 84.
196.
Lucretius II, 292–293; 1969, p. 88.
197.
Lucretius II, 251–260; 1969, p. 86.
198.
Lucretius II, 172–174; 1969, p. 80.
199.
Lucretius 1969, p. 334.
200.
Lucretius V, 962; 1969, p. 340.
201.
Lucretius V, 175–178; 1969, p. 294.
202.
Lucretius V, 795–796; 1969, p. 330.
203.
Lucretius II, 1150; 1969, p. 138.
204.
Lucretius II, 1121; 1969, p. 136.
205.
Lucretius V, 826–836; 1969, p. 332.
206.
Lucretius V, 380–383; 1969, p. 306.
207.
Lucretius V, 999–1001; 1969, p. 342.
208.
Lucretius V, 1305–1307; 1969, p. 362.
209.
Lucretius V, 1423–1435; 1969, pp. 368–370.
210.
Lucretius V, 198–199; 1969, p. 296.
211.
Lucretius V, 95–109; 1969, p. 290.
212.
Serres 1980, p. 127.
213.
Serres 1980, pp. 159–163.
214.
Tonietti 2002a; Tonietti 1983b, pp. 279–280.
215.
Serres 1980, pp. 75–76.
216.
Serres 1980, p. 200.
217.
Schroedinger 1963.
218.
Forman 2002.
219.
Testi & contesti 1979–1983. Sivin 2005, p. 58. The context was described as “essential for understanding” also by Vegetti 1983, pp. 11–12ff.
220.
Euclide 1916, pp. 110–114.
221.
Euclid 1557.
222.
Lloyd 1978, p. 38. Vegetti 1979, pp. 61–69, 76, 93, 102.
223.
Lloyd 1978. Lloyd & Vallance 2001.
224.
Lloyd 1978, p. 38.
225.
Plato 1999, pp. 123, 133, 143, 471, 733, 781–782 … Vegetti 1979, pp. 20–22, 61, 101, 113, 121, 125, 132. Vegetti 1983, pp. 53, 59ff., 85, 94, 122.
226.
Sambursky 1959, pp. 13, 23–24, 37, 113, 124ff., 163ff., 228, 235. Lloyd 1978, pp. 105, 171, 173, 239–240, 264, 307.
227.
Sambursky 1959, pp. 39, 160–162. Vegetti 1979, pp. 59–62, 85–86. Vegetti 1983, p. 118.
228.
Vegetti 1983, pp. 30–31.
229.
Ovidio 1988, pp. 164–167.
230.
Sambursky 1959, pp. 205–208. The faith in progress, towards our physical sciences of the twentieth century, continually led Sambursky to make anachronistic comparisons between the ancient Greeks and us, taken as touchstones. Here, as regards poor Epicurus , who is presented in a contradictory manner, as if he had been afraid of religion, his judgement was: “… he abolishes any possibility of arriving at a comprehensive scientific conclusion.” and “… ‘scientific failure’ …”. Lloyd 1978, p. 169; Vegetti 1979, pp. 92, 94.
231.
Vegetti 1979, p. 133.
232.
Lloyd 1978, pp. 223–225; Lloyd & Vallance 2001, p. 552. Vegett i 1979, pp. 111, 113, 125. Vegetti 1979, pp. 14, 23, 27, 33, 37–40. Vegetti 1983, pp. 116ff.
233.
Lloyd 1978, pp. 20, 120.
234.
Lloyd 1978, pp. 16, 119–120.
235.
Sambursky 1959, pp. 85 and 296.
236.
Lloyd 1978, pp. 219–220, 269, 278, 327.
237.
Lloyd 1978, p. 282.
238.
Sambursky 1959, pp. 197ff. Vegetti 1979, pp. 71–73, 90, 104–107, 110. Vegetti 1983, pp. 71ff., 170.
239.
Lloyd 1978, pp. 264–265, 308 and 323.
240.
See Chap. 5.
241.
Lloyd 1978, pp. 190–191. Vegetti 1983, pp. 113ff., 151ff., 162, 167–168.
242.
Bible , “Exodus” III, 14.
243.
Vegetti 1979, pp. 66, 73, 91, 94, 95, 116, 138, 142–143.
244.
Lloyd 1978, pp. 134, 303, 319. Vegetti 1983, p. 174. Samburky 1959, pp. 67ff.
245.
See above Sect. 2.7.
246.
Authier 1989, pp. 108, 116 and 123. Lloyd 1978, p. 194.
247.
See above Sect. 2.3.
248.
Lloyd 1978, pp. 47, 144–147, 240–248. Vegetti 1979, pp. 87, 95. Vegetti 1983, pp. 97ff.
249.
Plato 1994, pp. 142–143. Vegetti 1983, pp. 47–51.
250.
Lloyd 1978, pp. 287 and 302. Plato 1994, pp. 108–111. Vegetti 1979, pp. 105, 112, 120, 134.
251.
See Chap. 3.
252.
Euclid 1916, pp. xiii–xiv. Sambursky 1959, pp. 283 and 292, on the contrary, complained that all those war machines had not produced a “serious, multifarious technological development”.
253.
Authier 1989, p. 116.
254.
Lloyd 1978, pp. 216, 219 and 228. Sambursky 1959, pp. 45–46ff., wrote that musical harmony was “… the first example of the application of mathematics to a basic physical phenomenon”. Unfortunately, however, he added that the Pythagoreans had carried out “… authentic quantitative measurements, using wind instruments and instruments with strings of different lengths ….” This does not transpire from the completely different tradition that built up around them. Furthermore, if they had really done so, they would not have been able to maintain the ratios that were so dear to them; because reed-pipes and strings are tuned in accordance with different numbers, as will be seen in Sects. 3.2 and 6.7 below. It is clear that Sambursky does not seem to have had any direct experience with his ears, either.
255.
See Part II, Sect. 8.2. Vegetti 1979, p. 73.
256.
Sambursky 1959, pp. 55–56. Vegetti 1983, pp. 151ff., 156, 169ff., 175ff. Paul Tannery (1843–1904) did not contrast Aristoxenus sufficiently with the Pythagoreans and Platonics, putting them all together. But to the Frenchman should be recognized his great merit in attributing the correct role to music in the development of Greek mathematics. He went so far as to write: “… l’origine de la conception grecque de la mesure du rapport est essentiellement musicale, …” [“… the origin of the Greek idea of measuring the ratio is essentially musical, …”]; Tannery 1915 (1902), p. 73. Cf. Mathiesen 2004 who did not attribute Sectio Canonis to Euclid . Cf. Barker 2007 who believes that Sectio canonis is Euclid ’s.
257.
Vegetti 1979, pp. 108, 141, 119–121, 134.
258.
Vegetti 1979, pp. 43, 51. Vegetti 1983, p. 86.

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Tito M. Tonietti

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Tonietti, T.M. (2014). Above All with the Greek Alphabet. In: And Yet It Is Heard. Science Networks. Historical Studies, vol 46. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0672-5_2

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