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It should be observed, that, though such 3rd and 4th partials may be absent or weak in tones which are produced softly, they may become very prominent in those tones when sung or played loudly; consequently a Third which may be perfectly smooth and harmonious when softly played or sung, may become rough and unpleasant when more loudly produced: a remark which evidently applies to other intervals also.
The foregoing explains, why Thirds were not admitted to the rank of consonances, until comparatively recent times. For the compass of men's voices (in respect to which, the music among classical nations was chiefly developed) lies chiefly below middle C, and as we have just seen, Thirds in the lower parts of that compass are actually dissonant.
We have, in the above, also, the explanation of the rule in harmony which forbids close intervals between the tenor and the bass, when these parts are low in pitch.
To sum up the comparative smoothness of the Thirds: we find that these intervals may be almost or quite devoid of roughness when somewhat high in pitch, and may even excel the Fourth in smoothness under these circumstances, but that they rapidly deteriorate, as we descend below middle C.
For Compound Tones of such constitution as depicted in fig. 80, the Major Sixth seems decidedly equal, if not slightly superior, to the Fourth. As in the case of the latter interval, the 2nd partial of its upper tone dissonates with the 3rd partial of the lower, at the interval of a tone, but the roughness due to dissonances between the 4th and 5th partials in the latter interval is wanting in the former. As a set off to this advantage, however, we see that the Summation Tone in the Major Sixth when present, produces a tone dissonance with the 3rd partial of the lower tone.
On the other hand, the Minor Sixth is the worst interval we have yet studied. Its chief roughness is due to the semitone dissonance between the 2nd partial of the upper and the 3rd partial of the lower tone, which are usually pretty loud. A subsidiary roughness is seen above between the 3rd, 4th, 5th and 6th partials.
As an example of the Major Sixth,
take d = 384, then 1 = 640 2nd partial of 1 = 640 X 2 = 1280 3rd          d = 384 X 3 = 1152