# A Dictionary of Music and Musicians/Beats

From volume 1 of the work.
​BEATS are a wavy throbbing effect produced by the sounding together of certain notes, and most noticeable in unisons and consonances, when not perfectly tuned to one another.
To explain their origin reference must be made to elementary facts in the science of sound. Sound is conveyed to our ears by the waves into which the air, or other medium, is thrown by the vibration of what is called the sounding body. These waves are proportionally relative to the rapidity of the vibrations of the note sounding, and therefore also to its pitch; they consist of alternate condensation and rarefaction, each vibration being considered (in England and Germany) to comprise both the compression and distension of the particles of the air analogous to the crest and trough of a wave of water. These are, as it were, opposite forces, and can be made to counteract each other if two waves be simultaneously produced which start at such a distance from each other that the condensation of one exactly corresponds to the rarefaction of the other. A very simple proof of this may be obtained by striking a large tuning-fork and holding it close to the ear, and turning it slowly round; when a particular point will be found on either side of the fork at which the sound ceases, although the fork continues to vibrate, because the two prongs are in such a position relative to the ear that their sound-waves in that direction mutually counterbalance one another.
Beats are produced by sound-waves which have such relations in size and rapidity, that at certain intervals they cross one another and, condensation and rarefaction being simultaneous for the moment, produce silence. For instance, if two notes which vibrate respectively 100 and 101 times in a second be sounded together, it is clear that the sound-waves of the latter will gain 1/100 on the former at each vibration, and half-way through the second will have gained so much that its condensation will exactly correspond with the rarefaction of the other note (or vice versa), and for the moment silence will result; and so for each second of time.
If the notes be further apart, as 100 to 102, the latter will gain twice as much in every vibration, and there will be two places where the waves counteract each other, and therefore two beats in each second. Hence the rule that the number of beats per second is equal to the difference between the rates of vibration of the notes.
It is found practically that it is not necessary for the waves to be exactly in opposition; for in the case of one note with 100 vibrations in a second and another with 103, though the three beats will be heard according to the rule above given, it is proved mathematically that there will be only one point at which the condensation and rarefaction are exactly simultaneous, and the other two extremes of opposition are not exact, though within 1/10000 of a second of coincidence.
In point of fact the sound will be lessened to a minimum up to the extreme of opposition in the position of the waves, and increased to the full power of the two sounds up to the perfect coincidence of the vibrations.
It will have been observed that the beats increase in number as the notes become more wide apart. According to Helmholtz they are most disagreeable when they number about 33 in a second, which is nearly the number produced by the sounding together of treble C and D♭. From hat point they become less and less harsh till with such an interval as treble C and E, which produces 128 beats in a second, there is no unpleasant sensation remaining.
Beats are of three kinds. The first and most commonly known is produced by the sounding together of two notes nearly in unison—to which the above description applies simply. They are associated with the name of the great violinist Tartini, for reasons concerning which a controversy has arisen, and which are too long to be here set down.
The second kind arises from the imperfect tuning of consonances—such as the third, fourth, fifth, sixth, or octave. Here the notes are too wide apart for the primary beats as described above to be noticeable But the primary beats are in this case thrown into groups or cycles, which produce the effect of beats. These were first investigated by Dr. Robert Smith, Master of Trinity Coll. Cambridge (died 1768), and are called after him.
The third kind, also due to the imperfect tuning of consonances, is that which has been most carefully investigated by Helmholtz, and is called by him the over-tone beat. It is produced exactly in the manner first described between the harmonics of one note and another fundamental note which is not in tune with the first, or between the harmonics of two fundamentals which are out of tune. ​For instance, if bass C be sounded with middle C, and the latter be slightly out of tune, middle C and the first harmonic of the lower C will be in the position of imperfectly tuned unisons, and beats will be produced. If C and G be sounded together, and the latter be out of tune, the second harmonic of the former and the first of the latter will clash in a similar manner, and beats will be produced between them. And so with other consonances.
The value of beats to organ-tuners is well known, as their disappearance when the notes are in tune is a much safer criterion of exactness than the musical sense unaided. Moreover it is possible to discover, by simple calculation of the number of beats in a second relative to the number of vibrations, the exact amount any note is out of tune with another.
[ C. H. H. P. ]