FIG. 1-8 Illustration from a thirteenth-century manuscript of a famous music treatise by John of Cotton, now housed at the
Bavarian State Library in Munich, which shows Pythagoras in the blacksmith shop, measuring the harmonic consonances. The
inscriptions read, Per fabricam ferri mirum deus imprimit (“By means of a smithy God has imparted a wonder”) and Is
Pythagoras ut diversorum/per pondera malleorum/perpendebat secum quae sit concordia vocum (“It was this Pythagoras who, by
the weights of the various hammers, worked out the consonances for himself”). The lower panel shows a monochord, a more
“modern” device for tone measurement, and a harp, laterally strung like a lyre, which represents music’s power of ethos or
moral influence.
Another way of deducing the diatonic pitch set from properties of acoustic resonance is to generate it
by fifths radiating outward from a central tone. (If D is chosen for this demonstration the whole complex
may be represented on the staff without the use of accidentals.) A trace of this deduction survives in the
names of our scale degrees, “dominant” being the name of the tone produced by the first fifth “up,” and
subdominant (“underdominant”) being the name of the tone produced by the first fifth “down” (see Ex. 1-
9b).
But these deductions are all long after the fact and have nothing to do with history. They are
rationalizations, designed to show that our familiar musical system is “natural.” (Efforts to deduce the
diatonic pitch set from the so-called natural harmonics, or “overtones,” are especially ahistorical,
because the overtone series was not discovered and described until the eighteenth century.) Yet if the
immemorial diatonic pitch set is to be understood as “natural,” it must be understood in terms not only of
physical but of human nature. The historical evidence suggests that our diatonically apportioned musical
“space,” while grounded in acoustic resonance, may also be the product (or one of the possible products)
of a physiological predisposition governing “musical hearing,” that is, our discrimination of meaningful
pitch differences and pitch relationships.
Where actual musical practice is concerned, the relevant historical fact is that people have evidently
internalized the diatonic pitch set—carried it around in their heads as a means of organizing, receiving,
and reproducing meaningful sound patterns—as far back as what is as of now the very beginning of
recorded musical history, some three and a half millennia ago.
EX. 1-9A Deduction of the diatonic pitch set from the Pythagorean consonances