in the Musica enchiriadis itself suffice to prove, these simple devices were actually practiced in a
complex synthesis requiring considerable artistry—which, of course, is why a treatise needed to be
written about them in the first place. That artistic synthesis—not (as often assumed) mere parallel
doubling—was what the author of the treatise called organum.
FIG. 5-1 Polyphonic or organal settings of the sequence Nos qui vivimus (We the living) in Scolica enchiriadis, ca. 850.
The reason why parallel doubling is not acceptable without modification can be expressed in a single
word: tritones. If a given diatonic melody is doubled at a constant fourth or fifth below, then tritones will
emerge whenever the note B has to be doubled at the fourth or the note F at the fifth. Adjustment of the
doubling-voice (called the vox organalis) at these places produces a built-in discrepancy between its
mode species and that of the original chant (vox principalis). The result is “polytonality”—quite literally
so, given that parallel lines by definition never meet, and so two voices in strict parallel motion at any
interval except the octave will appear to end on different finals.
Consider Ex. 5-1b, the demonstration of “the symphonia of the diapente” (parallel doubling at the
perfect fifth) in the Scolica enchiriadis. The ending note of the vox organalis (G) contradicts the Dorian
final, and the B-natural in the vox principalis (a psalm tone) is answered in the vox organalis by a B-flat.
If the vox principalis is modified with a B-flat to agree with the vox organalis (and to smooth the contour
between its highest note and the F of its own medial cadence), then an E-flat (a note not present in the
normal diatonic system) must be introduced beneath it, which creates a new discrepancy between the
voices. (It can never be erased; if the vox principalis takes over the E-flat, the vox organalis will need an
A-flat, and so it will go on forever.)
EX. 5-1A Transcriptions of Scolica enchiriadis examples (Fig. 5-1), Double diapason