A complete view of Acoustical Science & its bearings on music, for musicians & music students.

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The Summation Tone, when present, renders a primary dissonance between Compound Tones harsher than it otherwise would be. Takefor example: let then and the Summation
Tone will be 80 + 90 = 170, a tone about midway between the 2nd partials, d (= 160) and r (= 180).
Summary. Beats are the source of all discord in music.
Starting with two simple tones in unison; if one of them be put slightly out of tune, slow beats will be heard, which are not very unpleasant, as long as they do not exceed one or two per second. On increasing the interval between the two tones, the beats gradually become more and more rapid, and at length form a harsh dissonance. If this interval be gradually increased, a point is finally reached, where all dissonance vanishes. The interval at which the dissonance just disappears, is termed the Beating Distance.
The harshness of any particular dissonance, depends partly upon
(1)  the rapidity of the beats, and partly upon
(2)  the region of Pitch in which the dissonance lies. Similarly, the Beating Distance for Simple Tones varies in
different parts of the realm of pitch, from a Tone at C2 = 1024 to a Major Third at C2 == 64.
Dissonance may arise between Simple Tones beyond Beating Distance, from the occurrence of Differentials.
A Compound Tone may be dissonant or harsh in itself, if it contain very high and loud partials.
Dissonance between Compound Tones may arise,
(1)  From beats between fundamentals,
(2)          ,,             ,,         the fundamental of one tone and an overtone of the other,
(3)  From beats between overtones only,
(4)  From beats due to Differential Tones,
(5)        Summation Tones.