I. Notes of observations on musical beats

Abstract

During the last three years I have been greatly occupied with observing and counting musical beats, for the purpose of discovering the cause and amount of error in Appunn’s reed tonometer, and of measuring the number of vibrations made in a second by tuning-forks and organs, as materials for my “ History of Musical Pitch.” The following are brief notes of some of the observations then made :— When two musical notes nearly but not accurately form a consonance, or are in unison, they beat. Under ordinary circumstances the number of beats in a second of a disturbed unison is equal to the difference of the number of double vibrations in a second made by each note. It is not so always, as will be shown later on. If x and y be the “ pitch ” or number of vibrations in a second, made by two musical tones, of which y is the sharper ; then, if my — nx =0, the tones form what I have termed a considence , that is, the n th partial of x falls on the same rank or seat as the m th partial of y . Considences are not always consonances, because other partials of the notes may beat roughly, as when m : n = 8 : 9 or 9 :10 or 15 :16, which are well known dissonances, but give appreciable considences. But if the pitch of either x or y be slightly altered, so that my — nx =± b , the two consident partials become what I have termed dissident , or placed on different ranks or seats, and b beats in a second are heard, being called “ sharp” when positive, that is when my > nx , and “ flat ” when negative, that is when my < nx . This includes the unison for which m = n . Hence all beats heard are beats of simple partial tones , however compound may be the tones which contain them. This agrees thoroughly with my observations. Tuning-forks are comparatively simple but always possess an audible second partial or octave, and sometimes higher partials still, capable of being so reinforced by resonance jars properly tuned to them, that beats can be separately obtained from them and counted. This is a matter of great importance in the construction of a tuning-fork tonometer. When the tone is very compound, as in the case of bass reeds (especially those of Appunn’s tonometer, furnished with a bellows giving, when properly managed, a perfectly steady blast for an indefinite length of time), beats can be obtained and counted from the 20th to the 30th and even the 40th partial, without any reinforce­ment by a resonance jar.