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92 THEORY OF QUALITY [IV. § 47.
vary according as they occupy different positions in the partial-tone series, i.e. on their orders. Thus, a clang containing three tones may consist of (1), (2), (3), or of (1), (3), (5), or of (l), (7), (10), and so on, the quality varying in each instance.
3. The number and orders of the partial-tones present remaining the same, the quality will vary according to the relative degrees of loudness with which those tones speak. Thus, in the simplest case of a clang consisting of only (1) and (2), either tone may alter in intensity while the other remains constant, and so cause variation in the quality of the sound resulting from their combination.
It is clear that these three classes of variations are entirely independent of each other, that is to say, any two clangs may differ in the number, orders and relative intensities, of their constituent partial-tones. The variety of quality thus provided for is almost indefinitely great. In order to form some idea of its extent, let us see how many clangs of different quality, but of the same pitch, can be formed with the first six partial-tones, by variations of number and order only. We will indicate each group of tones by the corresponding figures inclosed in a bracket; thus e.g. (1, 3, 5) represents a clang consisting of the first, third and fifth tones.
All the possible groups, each of course contain-