are felt to be more of the nature of noise than musical tone, and this explains why they are so much less used than other instruments. Tuning-forks also produce very weak inharmonic tones, not only when struck, but, as Dr. Preyer has shown, when bowed strongly.
The use of upper partials is, then, to produce different qualities of tone, for without them, all instruments would seem alike. Thus Dr. Preyer found that for the Octave civ–cv (2048 to 4096 vibrations) many good observers were unable to distinguish the tones of forks from those of reeds, unless both were very loud. Moreover organ-builders have long been accustomed to obtain artificial qualities of tone by combining the Octave, Twelfth, Fifteenth, Seventeenth, etc. in the so-called compound stops (Sesquialtera, Mixture, Cornet). This was done not from any knowledge of the theory, but from a feeling that the quality of the single pipe was too poor for musical effect.
A still more important use of the upper partials is in distinguishing between consonance and dissonance. It was formerly supposed that the dissonance of two musical sounds depended solely on the complexity of the ratio between their prime tones. According to this view c′–f′♯ being as 45:32, would be dissonant even if there were no upper partials. Helmholtz has however shown that when c′ and f′♯ are struck together on any instrument whose tones are compound, the dissonance arises from the 3rd and 4th partials of c′ beating with the 2nd and 3rd of f′♯, thus (1):
and that the prime tones continue sounding without interruption. Hence when c′ and f′♯ are simple tones they give no beats, and in fact form as smooth a combination as c′ and f′. This theory has been carefully verified by Dr. Preyer. He used tuning-forks having from 1000 to 2000 vibrations per second; and by bowing them in such a manner as to get practically simple tones, he found that 5:7, 10:13, 14:17, and many like intervals were pronounced by musicians to be consonant. By stronger bowing the upper partial and resultant tones were brought out, and then these intervals were immediately felt to be dissonant. In the consonant intervals, on the other hand, the upper partials either coincide and give no beats, or are too far apart to beat roughly. Thus in the Fourth c′–f′ the affinity between the two notes depends on their possessing the same partial c′′′, and this relation is but slightly disturbed by the dissonance of g′′ and f′′ (see (2) above).
This theory also explains why such intervals as 11:13 are excluded from music. They are not consonant, for though they have a common partial it is high and feeble, and to get to it we have to pass over a mass of beating intervals. Nor are 11:13 connected by a series of consonant intervals as is the case with the dissonances in ordinary use. For example, C and F♯ are linked together thus, C–G–D–F♯, or thus, C–E–B–F♯.
Though the partial tones are generally heard simultaneously, they are sometimes separated by being made to traverse a considerable distance before reaching the ear. Regnault[1] found that when a compound tone is sent through a long tube, the prime is heard first, then the 2nd partial, then the 3rd, and so on. He also noted that the velocity of sound increases or diminishes with its intensity. Hence, as the lower partials are usually the louder, they arrive before the higher.
The word 'harmonics' was formerly (and is sometimes even now) used to mean partial tones. But a harmonic produced by touching a string at one of its nodes, or by increasing the force of wind in an organ pipe, is not a simple tone. If we touch the string at ⅓ of its length we quench the 1st, 2nd, 4th, 5th, 7th, etc. tones, but leave the 3rd, 6th, pth, I2th, etc. unchecked. Hence it is proposed by Mr. Ellis to limit the word 'harmonics' to its primary sense of a series of compound tones whose primes are as 1, 2, 3, 4, 5, etc., and to use the words 'partial tones' to mean the simple tones of which even a harmonic is composed.
PARTICIPANT (from the Lat. participare, to share in). One of the 'Regular Modulations' of the Ecclesiastical Modes. [See Modes, the Ecclesiastical; Modulations, Regular and Conceded.]
The Participant, though less significant, as a distinguishing feature of the Mode, than either the Final, the Dominant, or the Mediant, is of far greater importance than any of the Conceded Modulations. In the Authentic Modes, its normal position lies, either between the Final and the Mediant, or between the Mediant and the Dominant; with the proviso, that, should two notes intervene between the Mediant and Dominant, either of them may be used as the Participant, at will. In the Plagal Modes it is always the lowest note of the Scale, unless that note should be B or F; in which cases, C or G are substituted, in order to avoid the False Relation of Mi contra Fa: it is therefore always coincident, in name, with the Authentic Dominant, though it is not always found in the same Octave. In some cases, however, either Octave may be used indiscriminately as the Plagal Participant; and even the choice of some other note is sometimes accorded.
The following Table exhibits, at one view, the Participants of all the Modes in general use, both Authentic and Plagal.
| Mode | I. | G. | Mode | V. | G. | Mode | IX. | D. | ||
| " | II. | A2. A3. | " | VI. | C2. | " | X. | E2. E3. | ||
| " | III. | A. B. | " | VII. | A. | " | XIII. | D. | ||
| " | IV. | C. F. | " | VIII. | D2. | " | XIV. | G2. | ||
| 2 The lowest note of the Mode. 3 The highest note of the Mode. | ||||||||||
In some few of the Authentic Modes, and in
- ↑ Helmholtz, 'Sensations of Tone,' p. 721.
