286 Notes to pages 6–15
- See the exchange on Panofsky ’ s thesis, expressed (with some revisions) in Panofsky 1956a , followed by
Rosen 1956 and Panofsky 1956b. - See Drake 1970a , 43 – 62; 1970b , 1992.
- Drake and Galilei 2000 , 100 – 108 (143 – 150); see also Walker 1978 , 27 – 33; Cohen 1984 , 85 – 97.
- Drake 1970a , 53.
1 Music and the Makings of Ancient Science - I will restrict myself to Greek developments, but the foundations of Chinese and Indian natural philosophy
were also deeply involved with music. Indeed, evidence suggests influence flowing from India to Greece; though
the interrelations with China need further study, the centrality of music in all three traditions is no accident,
whatever may be the variety of historical influence and parallel invention. See Burkert 1972 , 471, 477; Sedlar
1980 ; Doshi 1985 ; Shankman and Durrant 2002. - Barker 1984 , 2:5, 28. Mart í nez 2012 critiques many myths about Pythagoras.
- Ibid. , 2:29. See also Hagel 2010, 143–151.
- Philolaus quoted from Barker 1984 , 2:36; Odyssey 5.248; Iliad 22.255. Spitzer 1963, 7 – 19, surveys Greek
treatments of cosmic harmony. - Barker 1984 , 2:38, 36. For “ unlimited ” and “ limiter, ” see also Burkert 1972, 251 – 269.
- Barker 1984 , 2:256 – 257; see also the helpful commentary in Nicomachus 1994 , 13 – 27, 83 – 97.
- Boethius 1989 , I.10.
- For a sensitive treatment of the significance of the fifth hammer, see Heller-Roazen 2011, who judges that
“ one can only speculate as to the reasons ” for its discord (17). - See Raasted 1979 ; Lloyd 1987 , 277 – 278. In Nicomachus ’ s version, Pythagoras went home to experiment
with strings tensed with different weights ( Barker 1984 , 2:257 – 258), which would also prove problematic. For
another account of sounding “ discs, ” see Barker 1984 , 2:30 – 31. - Creese 2010 gives the most complete account of this instrument and its theory, including a valuable discus-
sion of the smithy story (81 – 93). - For Ptolemy ( Harmonics 16.32), see Barker 1984 , 2:291; Burkert 1972 , 375 – 378. About 1580, Vincenzo
Galilei adduced experiments showing that the pitch of a string depends linearly on the square root of its tension
(rather linearly), but did not test the hammer story, which Mersenne (1972 , 166 – 171) finally rejected as “ very
false ” (1634); see Galilei 2003 , 326, 329, 339; Chua 2001. On the other hand, Walker (1978 , 23 – 26) argued that
Galilei himself could not have done the experiments he claimed to have performed. - Here following Porphyry ’ s Commentary , in Barker 1984 , 2:34 – 35.
- Philolaus and Aristotle cited in Barker 1984 , 2:38, 34; for further context, see Heath 1966, 47 – 48, 94 – 97,
115 – 116, 187 – 189. - Timaeus 37a.
- Laertius 1972 , sec. 8:25.
- Phaedrus 253b: “ convincing the boy they love and training [ rhythmizontes ] him to follow their god ’ s pattern. ”
Translations not otherwise identified are my own. - Republic 522c (Plato 2006, 237).
- The first ten Greek letters, followed by the accent ́ , denoted the first ten numbers: 1, 2, 3, 4, ... , 9 were
α ́ , β ́ , γ ́ , δ ́ , ... , θ ́ , while 10, 11, 12, ... were ι ́ , ι α ́ , ι β ́ , ... and 20 was κ ́. Thus, Greek numerals were more
alphabetic (and harder to use) than Roman. - Republic 526a (Plato 2006, 242).
- For arithmos , see Klein 1992 , 46 – 60; for rhythmos , see Pollitt 1972, 56 – 60.
- Iliad 23.239.
- This possibility goes beyond the suggestions of Tannery (1902 ) and Szab ó (1978 , 99 – 184) that “ all the
important terms of the theory of proportions have their origins in the theory of music ” ( Szab ó 1978, 170). See
also Borzacchini 2007.