Hearing the Irrational 69
chromatic genus, the latent problems hidden in the diatonic come forward sufficiently to
cause “ disruption ” : incipient instability and growing theoretical uncertainty. Only with the
enharmonic genus does this simmering instability disrupt both the diatonic and chromatic
genera and “ permit the creation of steps and leaps beyond all reason. ”^44 This, Vicentino
tells us, is the cause that should move us not only to call them “ irrational ratios ” but also,
by so naming them, to install them in mathematics and music jointly as having equal
existential force with the “ rational ratios ” we learned from arithmetic and the “ irrational
magnitudes ” from geometry. For Vicentino, music is the intermediate ground on which
arithmetic and geometry meet in such hybrid concepts as “ irrational ratios, ” shared between
mathematics and music.
The attitudes of Stifel, Cardano, and Vicentino about these mathematical issues reflect
their respective musical projects. As we saw, Stifel ’ s closest approach to affirming that
“ irrational numbers ” were “ real ” or “ true ” came in the context of his musical theorizing.
Yet this did not prove sufficient for him to maintain this position in the face of the actual
infinitude of fractional sums, the “ cloud of infinity, ” perhaps because his involvement
with music remained largely theoretical and restricted. Stifel ’ s main foray into practical
music was his thirty-two-strophe song to propagate Luther ’ s teachings, “ Johannes th ü t
uns schreiben ” (1522), based on the popular tune “ Bruder Veyt. ” Stifel ’ s composition led
him into a polemical war of song and countersong with the theologian Thomas Murner,
both always keeping this same melody for their new lyrics ( ♪ sound example 4.3).^45
Though Stifel ’ s song was very popular and went through many printings, even serving
as an important early example of the power of music that may have inspired Luther
himself, its melody was simple and derivative. Stifel merely provided new words to an
old tune; he had no vision of reforming the elements of music that would compare with
Vicentino ’ s ambitious project to (re)create a whole new genus of music. By comparison
with Stifel ’ s song, Cardano ’ s extant compositions are very ambitious, including a five-
voice perpetual canon and a tour de force of four simultaneous three-voice canons (twelve
voices in all).^46
Among Vicentino ’ s much larger output, his motet “ Musica prisca caput ” ( ♪ sound
example 4.4) dramatizes the emergence of the enharmonic genus to glorify his patron: its
first verse is in the diatonic genus, the second in the chromatic, while the final verse is in
the enharmonic, dramatically reserving the introduction of the diesis to produce a special
aura around the name of Cardinal Ippolito at the end of the motet. This motet ’ s pointed
delineation of all three genera provides yet another demonstration and justification of
Vicentino ’ s views to refute his critics and contest his condemnation. He tells us, as well,
that the whole d ’ Este family, including the cardinal and the prince of Ferrara, sang
this daring new music, quarter tones and all, “ with the most exceptional diligence. ”
Vicentino had evidently persuaded them that, in contrast to the public uses of diatonic
music “ in communal places for the benefit of coarse ears, ” such enharmonic music was
“ reserved ... to praise great personages and heroes for the benefit of refined ears amid the