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are used as nearly synonymous. It will be convenient to restrict the meaning of the latter so that it shall denote a sound which does not admit of resolution into simpler elements. A single sound of determinate pitch will accordingly, in what follows, be called a tone, or simple tone. For a compound sound the word clang will be a serviceable term. The series of elementary sounds into which a clang can be resolved we shall call its partial-tones, sometimes distinguishing among these, the lowest, or fundamental tone, from the others, or overtones, of the clang. This nomenclature is the direct adaptation of the German terms employed by Helmholtz. Its introduction is due to Professor Tyndall.

47. This long discussion has paved the way for the complete explanation of musical quality which is contained in the following proposition. The quality of a clang depends on the number, orders and relative intensities of the partial-tones into which it can be resolved. We have here three different causes to which variations in the quality of composite sounds are assigned.

1. A clang may contain only two or three, or it may contain half-a-dozen, or even as many as fifteen or twenty, well developed partial-tones.

2. The number of partial-tones present remaining the same, the quality of the resulting sound will

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 (1), (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 by the corresponding figures inclosed in a bracket; thus e.g. (1, 3, 5) represents a clang consisting of the first, third and fifth partial-tones.

All the possible groups, each of course contain

ing the same fundamental tone, are given in the following enumeration:

Two at a time:

(1, 2), (1, 3), (1, 4), (1, 5), (1, 6).

Total 5.

Three at a time:

(1, 2, 3), (1, 2, 4), (1, 2, 5), (1, 2, 6), (1, 3, 4), (1, 3, 5), (1, 3, 6), (1, 4, 5), (1, 4, 6), (1, 5, 6). Total 10.

Four at a time:

(1, 2, 3, 4), (1, 2, 3, 5), (1, 2, 3, 6),

(1, 2, 4, 5), (1, 2, 4, 6), (1, 2, 5, 6),

(1, 3, 4, 5), (1, 3, 4, 6), (1, 3, 5, 6), (1, 4, 5, Total 10.

Five at a time:

(1, 2, 3, 4, 5), (1, 2, 3, 4, 6), (1, 2, 3, 5, 6),
(1, 2, 4, 5, 6), (1, 3, 4, 5, 6).

Total 5.

Six at a time:

Total 1.

(1, 2, 3, 4, 5, 6).

The whole number of groups is 31, or if we allow the fundamental-tone (1) to count by itself as a sound of separate quality, 32. Let us next examine how many clangs of different quality can be obtained from a single combination of three fixed partial-tones by variations of intensity only, supposing that each

tone is capable of but two degrees of loudness. Representing one of these by f, and the other by p, we indicate, e.g., by (ƒ, p, p) a clang in which the fundamental tone is sounded forte, and the two overtones piano. The different cases which present themselves are the following:

(ƒ, ƒ, ƒ), (ƒ, p, ƒ),

(p,ƒ, ƒ), (p, p, ƒ), (ƒ,ƒ, p), (ƒ, p, p), (p, ƒ, p), (p, p, p)

or seven in all, since (p, p, p) has the same quality as (f,f,f). The number of cases increases very rapidly as we take more partial-tones together. Thus a clang of four tones will produce 15 sounds of different quality; one of five tones 31; one of six tones 63; by variations of intensity only. Altogether we could form, with six partial-tones, each susceptible of only two different degrees of intensity, upwards of four hundred clangs of distinct quality, all having the same fundamental tone. The supposition above made utterly understates, however, the possible variety of quality dependent only on changes of relative intensity. A very slight increase or diminution of loudness, on the part of a single constituent tone, is enough to produce a sensible change of quality in the clang. We should be still far below the mark if we allowed each partial-tone four different degrees of intensity, though even this supposition would bring us more than eight thousand separate

cases. Since many more variations of intensity are practically efficacious, and also since the disposable partial-tones need by no means be limited to the first six, the above calculation will probably suffice to convince the reader that the varieties of quality which the theory we are engaged upon is capable of accounting for are almost indefinitely numerous. This is, in fact, no more than we have a right to demand of the theory, when we reflect on the fine shades of quality which the ear is able to distinguish. No two instruments of the same class are exactly alike in this respect. For instance, grand pianofortes by Broadwood and by Erard exhibit unmistakable differences, which we describe as 'Broadwood tone' and Erard tone.' Less marked, but still perfectly recognisable, differences exist between individual instruments of the same class and maker, and even between consecutive notes of the same instrument. To these we have to add the variations in quality due to the manner in which the performer handles his instrument. On the pianoforte the kinds of tone elicited by a dull slamming touch, and by a lively elastic one, are clearly distinguishable. With other instruments such distinctions are much more marked. On the violin we perceive endless gradations of quality, from the rasping scrape of the beginner up to the smooth and superb tone of a Joachim. A

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