of scale and final was originally the product of Frankish and Italian music theory of the tenth and eleventh
centuries, in which an attempt was made to organize the chants of the Roman church according to the
categories of ancient Greek music theory, which was well known from treatises, even if practical
examples of ancient Greek music are virtually nonexistent. (As we shall see, the chants composed by later
Frankish musicians who had been trained according to this theory conform much more closely to our
accustomed idea of what a mode is.)
EX. 1-8 Differentiae of the first psalm toneWHY WE WILL NEVER KNOW HOW IT ALL BEGAN
Yet even if the ancient Greek catalogue of lyre tunings was conceptually foreign, hence irrelevant, to the
modal structure of Gregorian chant, the attempt to codify medieval modal theory according to Greek ideas
of order was not wholly misplaced. The Greek system and the Gregorian corpus did have one thing self-
evidently in common. They both employed what some scholars now call the “diatonic pitch set,” the field
of pitches and pitch relationships reducible to a specific arrangement of tones and semitones (“whole
steps” and “half steps”), of which the familiar major and minor scales are among the possible
representations.
When staff notation was introduced in the eleventh century, it made tacit yet explicit provision for that
arrangement. There is no way of telling the diatonic half steps (between B and C and between E and F)
from the whole steps on the basis of their appearance on the staff; from its very beginning, in other words,
the staff was “prejudiced” to accommodate the two different sizes of step-interval as musicians had from
time immemorial habitually “heard” and deployed them.
Thus there is no point in inquiring about the historical origins of the diatonic pitch set, our most
fundamental musical possession. We will never know them. We can do no better than the legends by
which the Greeks sought to explain the origins of their musical practice. In one of these, related by
Nicomachus in the second century CE, Pythagoras, the reputed inventor of music, heard beautiful sounds
coming unexpectedly out of a blacksmith’s shop. Weighing the anvils the smiths were striking, he
discovered the harmonic ratios governing the perfect (“Pythagorean”) consonances, as well as the whole
step. Laying these intervals out on a staff, and adding the two extra tones that are obtained when the
Pythagorean complex is transposed to begin on each of its own constituent pitches, we may arrive at a
primitive five-note (“pentatonic”) scale. Plugging the “gaps,” we find that we have “discovered” the half
steps (see Ex. 1-9a).