EX. 5-2A Disjunct tetrachord scale from Musica enchiriadisUnlike the octave system, with a scale constructed (as we saw in chapter 3) out of alternately conjunct
and disjunct tetrachords, the Musica enchiriadis scale is constructed entirely out of disjunct tetrachords.
Beginning with the familiar tetrachord of the four finals, d–e–f–g, you add a disjunct tetrachord below,
and thus obtain the B-flat in Example 5-1b. Add a disjunct tetrachord above and you get the B-natural as
part of the same scale. Add another tetrachord above that and F-sharp appears. Above that there will be a
C-sharp. (This much is actually demonstrated in the treatise. If one were to extend the scale at the bottom,
of course, one would keep adding flats, beginning with the E-flat hypothetically added to Ex. 5-1b.) The
result is a scale altogether without diminished fifths. An analogous hypothetical scale composed of
nothing but conjunct tetrachords would eliminate augmented fourths; it is not given in the treatise but can
be easily deduced: see Ex. 5-2b.
These scales produce perfect parallel counterpoint in theory but bear no relationship to normal oral
(that is, aural) practice. And that is why strict parallel doubling, though conceptually as simple as can be,
is literally utopian. It occurs nowhere in the “real world” of musical practice. Polyphonic music actually
composed according to the Musica enchiriadis scales (to quote a wry comment of Claude Debussy on a
piece the young Igor Stravinsky showed him in 1913) “is probably Plato’s ‘harmony of the eternal
spheres’ (but don’t ask me on which page); and, except on Sirius or Aldebaran or some other star, I do not
foresee performances... especially not on our more modest Earth.”^3
On our modest Earth, in other words, compromise with theory—that is, with imagined perfection—is
usually required. The author of the Scolica enchiriadis tacitly recognized this crucial point when
constructing an example to illustrate “the symphonia of the diatessaron” (parallel doubling at the perfect
fourth). The counterpoint in this case (Ex. 5-1c) has been “cooked,” precisely so as to avoid the
“polytonal” situation encountered in the case of fifths. The two lines end on the same final; that is to say,
they end on a unison. In order to meet, of course, they must stop being parallel. Instead, they approach the
final note in contrary motion. Such an approach is called an occursus, literally “a meeting.”
In order to smooth the way to the occursus (and also to avoid the B-flat from the Musica enchiriadis
scale, which would produce an augmented fourth against the E in the vox principalis), the vox organalis
behaves, in the second half of the example, like a drone—or like a sequence of drones. Instead of
following the contour of the vox principalis, the vox organalis hugs first the D and then the C, moving
from the one to the other when the opportunity presents itself to recover the correct symphonia (perfect
fourth) against a repeated note in the vox principalis. The voces organales above and below the vox
principalis in Ex. 5-1d, a composite organum simultaneously demonstrating octaves, fifths, and fourths,
behave similarly.
Curiously (and rather characteristically), the author of the Scolica enchiriadis does not actually
explain the modifications—the drones, the occursus—by which the purely conceptual idea of parallel
doubling is transformed into the actual practice of organum. Acknowledging that the case is not as
straightforward as the other examples, the author refers the discrepancy to “a certain natural law about
which we shall speak later” (but of course “we” never get around to it), meanwhile counseling the student