Notes to pages 16–25 287
- See Pesic 2003 , 5 – 21.
- Laertius 1972 , sec. 8:25. For Plato and the concept of zero, see Pesic 2004.
- Most interpretations of the Platonic dyad stem from the testimony of Aristotle, Metaphysics 987b23 – 35
(Aristotle 1984, 1561 – 1562); see Watson 1973. - Republic 526d – e (Plato 2006, 243); the identification of the Idea of the Good as the One has also been put
forward as a central tenet of the so-called unwritten teachings of Plato, as an esoteric continuation of the exoteric
discourse contained in his extant dialogues; see Klein 1992 ; Watson 1973. - Republic 527e (Plato 2006, 244).
- Ibid., 534d; 525d – e (Plato 2006, 242).
- For surreal numbers, see Knuth 1974 ; Conway 2001.
- See Pesic 2003 , 20. Adrastus (second century C. E.) is quoted from Barker 1984, 2:214; see Creese 2010,
5, 243. Psophos (noise) ironically inverts sophos (wise): the unwise merely make noise, as when Plato makes
fun of those who imitate “ the sound [ psophos ] of wind ” ( Republic 327a). - Successive musical intervals combine through multiplication because each successive interval acts to shorten
(or lengthen) the string representing the initial pitch. Thus, five successive rising 9:8 tones will lengthen the
initial string by (9:8)^5 ≈ 1.802 ... < 2:1, less than an octave; six tones overshoot the octave because (9:8)^6 ≈ 2.027
... > 2:1. For an overview of the problem of temperament, see Bibby 2004. - Republic 529a – d (Plato 2006, 246 – 247).
- Cited in Barker 1984 , 2:150. For a lively reconsideration and defense of Aristoxenus, see Levin 2009.
- See Barker 2000.
- See Ptolemy, Harmonics 99.1 – 111.15 (the end of the surviving text), in Barker 1984 , 2:378 – 391, and the
commentary in Barker 2000 , 158 – 191. - For the “ New Music, ” see West 1992 , 356 – 372; Csapo 2004. See Hagel 2010, 3–10, 17–28, 53–60, for the
common use of modulation. - See Jaeger 1969. Spitzer (1963, 10), argues that Democritus first joined together the four studies of the
quadrivium.
2 The Dream of Oresme - For an excellent treatment of the whole problem of the transmission of ancient music theory, see Mathiesen
1999 ; regarding the transmission of Boethius, see Mellon 2011. Concerning the period considered in this chapter
and following, see Carpenter 1955 , 24 – 27, 115 – 118, 313 – 315. For the earlier history of music in the quadrivium
see Libre 1969 , 175 – 191; Moyer 1992 , 11 – 35; and Vendrix 2008. - Zoubov 1961; Abdounar 2008.
- The earliest printed editions of Euclid in Latin date from 1482; vernacular translations were published in the
1560s and 1570s; see Grafton 1991 , 23 – 46. - Euclid himself had proved that there were an unlimited number of kinds (genera) of irrational magnitudes,
compared to the single genus of rational quantities; see Pesic 2003 , 18 – 21. - See Timaeus 39d and Campion 1994, 243 – 247. Hipparchus (second century B. C. E.) is credited with the
discovery that the location of the spring equinox would slowly precess around the zodiac from its initial position,
returning to its starting point after 26,000 solar years, the “ Great Year. ” Observationally, the “ north star ” and the
whole field of stars would change over that time as the north celestial pole traces out a slow circle: the present pole
star, Polaris, will give way to others but return to its place after 26,000 years. Plato ’ s “ Great Year ” seems to have
been a more general concept of cosmic recurrence based on the completion of the known cycles of the planets. - Oresme 1968a , 514 – 515, 340 – 341 For chorea and planets, see especially Wright 2001, 129 – 158; for the
religious context of chorea in the church, see also Mews 2009. For the carole , see Mullaly 2011, 47 – 48. - Oresme 1968a , 522 – 523; Oresme attributes this view to “ La Perpective de Witelo. ”
- Ibid. , 530 – 531. For Oresme ’ s relation to the Copernican view, see Blumenberg 1987 , 158 – 168.
- Oresme 1968a , 536 – 539.
- Oresme 1968b , 12.