(Ex. 3-1).
EX. 3-1 The four species of fifth and the “four finals”The ending notes of these four species-defining segments—D, E, F, and G—were dubbed “the four
finals” in Frankish tonal theory and named (in keeping with the Byzantine derivation of the mode system)
according to their Greek ordinal numbers: protus (first), deuterus (second), tritus (third), and tetrardus
(fourth) respectively. (The fifth A–E was considered a doubling, or transposition, of the first segment;
hence A was functionally equivalent to D as a final.) Full correspondence between the chant-
classification and the preexisting eightfold system of psalm tones was achieved by invoking the category
of ambitus, or range. Chants ending on each of the four finals were further broken down into two classes.
Those with the final at the bottom of their range were said to be in “authentic” tonalities or modes, while
those that extended lower than their finals, so that the final occurred in the middle of their range, were
called “plagal,” from the Greek plagios, a word derived directly from the vocabulary of the oktoechos,
where it referred to the four lower-lying scales.
Thus the four finals each governed two modes (protus authenticus, protus plagalis, deuterus
authenticus, and so on), for a total of eight, in exact accordance with the configuration (but only in vague
accordance with the content) of the eightfold system of psalm tones. In elaborating this system, the basic
fifth (or modal pentachord, from the Greek) whose diatonic species defined the final’s domain was
complemented with a fourth (or tetrachord) to complete the octave. (According to the terminology of the
day, the tetrachord was said to be conjunct—rather than disjunct—with the pentachord because its first
pitch coincided with the last one in the pentachord rather than occupying the next scale degree.) The
authentic scales were those in which the pentachord was placed below its conjunct tetrachord, so that the
final was the lowest note. In the plagal scales the tetrachord was placed below the pentachord, so that the
final came in mid-range. The result was a series of seven distinct octave species or scales with particular
orderings of the diatonic tones and semitones. There are only seven possible octave species but eight
modes; hence the last scale in Table 3-1 (tetrardus, plagal) has the same order of intervals as the first
(protus, authentic), but they are split differently into their component pentachord and tetrachord. Although
their octave species coincide, the modes do not, for they have different finals: D and G, respectively.
TABLE 3-1 Modes and Octave SpeciesIn Ex. 3-2 this table is translated into modern staff notation, giving the full array of so-called
“medieval church modes.” They will henceforth be numbered from one to eight, as they are in the later
Frankish treatises, and they will be given the Greek geographical names that the Frankish theorists