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. . . it is nonsense to marvel
at the profusion
of melodies that can be drawn from the “seven tones”
of a scale, or the
“three tones” of a triad, or the twelve tones available
altogether. To
equalize corresponding tones of different octaves,
like d, d’, d’’, d’’’
etc., in a melodic sense is as nonsensical as it would be to equalize the
numbers 2,12,22,32 etc. From the melodic viewpoint the tone-gamut is
unlimited, though only a limited discernible sector is in practical use,
like the spectrum between the infrared and ultraviolet extremes. It is
unlimited like the series of numbers, though, like this, it renews itself
in periodic recurrence, the columns of tens in mathematics corresponding
to the columns of octaves in music. But the melodic line, composed of the
various absolute pitches (absolute according to their real vibration numbers)
is not concerned with the fact that by mere accident, and solely for practical
purposes, a certain periodic recurrence of names was agreed upon, just
a such a recurrence of names was agreed upon in the series of numbers.
We could just as well have given each number and each pitch a name of its
own, except that this way things would have been harder to learn and to
remember. Regardless of any names whatsoever for its single constituents,
it is the pitch-line, its curve or curves, its shape, its profile, its
ascensions and descensions which determine the character, the gesture
of the melody—the challenge of [one], the tenderness of [another], the
exuberance of [yet another].
Ernst Toch
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