Notes to pages 63–70 291
- For the Jesuits ’ reliance on the certainty of mathematics, see Gatto 1994 , 17 – 64; Consentino, Homann, and
Luk á cs 1999 , 50 – 51; Dear 1995 , 39. In 1591, Robert Parsons denounced the English algebraist Thomas Harriott
as an Epicurean atheist and conjurer because of his mathematical atomism; see Neal 2002 , 25 – 27; Kargon 1966 ,
- See Tiella 1975 ; Rasch 2002 ; Wright 2002 ; Barbieri 2002. The split keys of Vicentino ’ s instrument were
not unique; other keyboards had earlier used this device. For sound examples, hear the CD attached to Cordes
- Cardano (1973 , 194 – 195) described Vicentino ’ s new instruments.
- Vicentino ’ s contemporaries were most familiar with just intonation (see box 4.2), in which the “ just diatonic
semitone ” (16:15 ≈ 1.067, 111.7 cents) was a common solution, but there were other competing possibilities,
such as the “ just chromatic semitone, ” sometimes called the “ minor semitone of the minor tone ” (25:24 ≈ 1.042,
70.7 cents). - Vicentino defined a minor diesis as “ one-half of the minor semitone ” and the major diesis as “ identical ” to
a minor semitone, but then we still have to divide that minor semitone exactly in half. See Vicentino 1996 , 59 – 62
(fol. 17r – 18v); Berger 1980 , 7 – 18. - Boethius cites “ Philolaus, a Pythagorean, ” who divided the tone unequally. See Boethius 1989 , 96 – 97; West
1992, 135 – 136. Boethius argues that the Pythagorean comma is the “ ultimate interval heard which can really be
perceived ” (Boethius 1989, 96 – 97). - See Plato, Republic 531a; Aristotle, Posterior Analytics 84b37 – 39; and Aristotle, Metaphysics 1016b18 – 24
(Aristotle 1984, 1:138, 2:1605), collected in Barker 1984 , 2:55, 70, 72, 135. See also Quintilianus 1983 , 81, 95. - Quintilianus 1983 , 84; Strunk and Treitler 1998 , 57.
- The word diesis originally meant a “ letting through, ” suggesting the performance practices of wind instru-
ments. See Aristotle, Politics 134a21; and Longinus, On the Sublime 39.2, in West 1992, 81 – 107, esp.
105 – 106. - For Aristoxenus ’ s references to the diesis, see Barker 1984 , 2:135, 140, 143, 145, 154, 165 – 166, 182,
- Ibid.,137.
- See Levin 2009 for a similar account of Aristoxenus. In Ptolemy ’ s view, Aristoxenus constructed the enhar-
monic genus by essentially assigning the unit of 6 to each diesis, in units where the tone is 24 units; see Barker
1984 , 2:384 – 391, 270. - Barker 1984 , 2:170n1. Socrates draws a square on the diagonal of a unit square in his conversation with the
slave boy in Plato ’ s Meno 84d – 85c. - Vicentino 1996 , 6, 12 (fol. 3r, 4v).
- For the role of Lodovico Fogliano ’ s geometric construction to divide intervals (on the model of figure 4.1 ),
see Pesic 2010. - Vicentino 1996 , 207 (fol. 66v).
- Zarlino 2011 ; Strunk and Treitler 1998 , 299. See also Mambella 2008.
- See Berger 1980 , 15 – 16.
- Vicentino 1996 , 207 (f.66v).
- See Oettinger 2003.
- These compositions can be found in Cardano 1973 , 139, 154 – 171.
- Vicentino 1996 , 33 (fol. 10v). For commentary on the musical details of his motets vis- à -vis his critics, see
li – lviii; recordings of this and his other motets are available on the CD accompanying Cordes 2007. - On the other hand, Vincenzo Galilei noted that enharmonic music “ was never sung without the instrument
named above [archicembalo] and if by misfortune one of the singers lost his way while singing, it was impossible
to put him back on the right track. ” After Vicentino ’ s death, “ it was practiced neither by his students nor by
anybody else ” ; Berger 1980, 73. - Cardano was critical of Vicentino ’ s scheme for tuning, which he found “ not unserviceable, but ... not entirely
accurate ” ; see Cardano 1973 , 194 – 195. Regarding Zarlino, see also Moyer 1992 , 202 – 225.