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are much worse than those of the tone, but also that these vary in themselves—the beats of thetone for example) not being so
harsh as those of thetone , nor those of thesemitone
, usually so discordant, as those of the semitone The small letters or asterisks in curved brackets show the positions of the Summation Tones generated by the fundamentals.
The facts thus summarized in fig. 80, will be found, on careful examination, to throw light on several fundamental phenomena in harmony relating to these intervals.
In the first place, it will be at once seen that with regard to Compound Tones such as those depicted, the Octave is the only perfectly consonant interval, that is, the only one absolutely free from roughness. Moreover, the student will readily perceive, that no roughness, except such as may be inherent in the tones themselves, can ever occur between two Compound Tones, at this interval, no matter what their constitution may be; for the higher of the two tones only adds to the lower one, elements which are already present.
The fact that the Octave is the only Interval devoid of all roughness, explains why this interval is the only one that can be used in all regions of the musical scale, on all instruments. Again the fact, that the Compound upper tone of the Octave, adds nothing new, but simply reinforces tones already present in the Compound lower tone, explains the similarity in effect of the two tones forming an Octave. We can thus understand, how it is that a company of men and women totally unskilled in music, and utterly unable to sing in Thirds, &c, yet experience no difficulty in singing together a tune in Octaves, and indeed when doing so usually consider themselves to be singing tones of the same pitch; in fact, such singing is called, even by musicians, unison singing.
Again, we see why a part in music for the Pianoforte, Harmonium, &c, may be doubled with impunity; for such addition adds nothing absolutely new; it simply reinforces the upper partials of tones already present, thus producing a brighter effect.
The Fifth as constituted in fig. 80 is not always a perfectly Consonant Interval, for as the figure shows the 3rd partial of the upper compound tone, dissonates with both the 4th and 5th of the lower one. The degree of roughness thus produced, will depend upon the intensity of these partials, and inasmuch as they are usually faint, the roughness will be but slight. Other things being