66 Chapter 4
What follows, then, is not a digression into antiquity but an account of how ancient prob-
lems returned to life.
Boethius had not given the enharmonic diesis a precise ratio, perhaps because of its
very smallness.^31 Plato and Aristotle considered the diesis a kind of element, analogous
to a vowel or consonant; Aristoxenus also judged that “ the voice cannot distinctly
produce an interval even smaller than the smallest diesis, nor can the hearing detect
one. ”^32 The diesis, it was concluded, may be so small an interval that strict, secure
definition is elusive. This judgment may also reflect the material circumstances and
difficulties surrounding the production of this interval. Aristides Quintilianus (first
century c.e. ) noted that the enharmonic “ has gained approval by those most distin-
guished in music; but for the multitude, it is impossible. On this account, some gave
up melody by diesis because they assumed through their own weakness that the interval
was wholly unsingable. ”^33 Thus, even in ancient times the diesis involved discrimination
and virtuosity, as in the quarter-tonal “ bending ” of pitch possible on the aulos (a pipe
with finger holes and a reed mouthpiece, often played in pairs). Aristotle described the
aulos as “ orgiastic, ” its shrill wails often associated with Bacchic and Corybantic rites,
sending its hearers out of their minds ( figure 4.6 ).^34 Such associations would not mili-
tate toward fussiness in intonation, if indeed the diesis were an ecstatic “ bending ” of
a pitch not really to be measured by any ratio but only by the inspired frenzy of the
Dionysian virtuoso.
Though this aural interpretation goes against Pythagorean tradition and its ratios, its
ancient champion was Aristoxenus. Most of the information we have on the enharmonic
diesis comes from him, suggesting that this interval may have fit particularly well into
his thesis that discriminative hearing, rather than predetermined, fixed ratio, really deter-
mines musical intervals.^35 Aristoxenus stood at a critical point in the problem of sub-
dividing intervals, which (as we have seen) involves irrational magnitudes if the
divisions are to be strictly equal. In the face of this paradox, he opened the liberat -
ing possibility that we might weaken or abandon the demand that every interval be
strictly rational through his reliance on the sense of hearing, rather than on reason. He
himself never seems to remark that his line of argument would imply the possibility
of quantities that might bridge the rational and irrational; on the contrary, he explicitly
maintains a sharp “ division ... in respect of the differences between the rational and
the irrational. ”^36
By turning away from Pythagorean ratios, Aristoxenus took a crucial step toward treat-
ing an arbitrary musical quantity as a unit unto itself, apart from whether it is or is not
rational with respect to the initial integral units of string length.^37 His demonstrations recall
Euclid, who had shown in the Elements that magnitudes are rational or irrational only
relative to other magnitudes, not in any absolute sense. The diagonal of a square is incom-
mensurable with its side but may be perfectly commensurable with other lines (for instance,