§ 6. Graphic Representation of Consonance and Dissonance

Helmholtz has attempted to represent this result graphically, and from his work I copy, with some modification, the next two diagrams. He assumes, as already stated, the maximum dissonance to correspond to 33 beats per second; and he seeks to express different degrees of dissonance by lines of different lengths. The horizontal line c′ c″, Fig. 164, represents a range of the musical scale in which c″ is our middle C, with 528 vibrations, and c′ the lower octave of c″. The distance from any point of this line to the curve above it represents the dissonance corresponding to that point. The pitch here is supposed to ascend continuously, and not by jumps. Supposing, for example, two performers on the violin to start with the same note c′, and that, while one of them continues to sound that note, the other gradually and continuously shortens his string, thus gradually raising its pitch up to the octave c″. The effect upon the ear would be represented by the irregular curved line in Fig. 164. Soon after the unison, which is represented by contact at c′, is departed from, the curve suddenly rises, showing the dissonance here to be the sharpest of all. At c′, the curve approaches the straight line c′ c″, and this point corresponds to the major third. At f′ the approach, is still nearer, and this point corresponds to the fourth. At g′ the curve almost touches the straight line, indicating that at this point, which corresponds to the fifth, the dissonance almost vanishes. At a′ we have the major sixth; while at c″, where the one note is an octave above the other, the dissonance entirely vanishes. The e s′ and the a s′, of this diagram are the German names of a third and a flat sixth.

Fig. 164.

Maintaining the same fundamental note c′, and passing through the octave above c″, the various degrees of consonance and dissonance are those shown in Fig. 165. That is to say, beginning with the octave c′-c″, and gradually elevating the pitch of one of the strings till it reaches c″′, the octave of c″, the curved line represents the effect upon the ear. We see, from both these curves, that dissonance is the general rule, and that only at certain definite points does the dissonance vanish, or become so decidedly enfeebled as not to destroy the harmony. These points correspond to the places where the numbers expressing the ratio of the two rates of vibration are small whole numbers. It must be remembered that these curves are constructed on the supposition that the beats are the cause of the dissonance; and the agreement between calculation and experience sufficiently demonstrates the truth of the assumption.[77]

Fig. 165.

You have thus accompanied me to the verge of the Physical portion of the science of Acoustics, and through the æsthetic portion I have not the knowledge of music necessary to lead you. I will only add that, in comparing three or more sounds together, that is to say, in choosing them for chords, we are guided by the principles just mentioned. We choose sounds which are in harmony with the fundamental sound and with each other. In choosing a series of sounds for combination two by two, the simplicity alone of the ratios would lead us to fix on those expressed by the numbers 1, 5/4, 4/3, 3/2, 5/3, 2; these being the simplest ratios that we can have within an octave. But, when the notes represented by these ratios are sounded in succession, it is found that the intervals between 1 and 5/4, and between 5/3 and 2, are wider than the others, and require the interpolation of a note in each case. The notes chosen are such as form chords, not with the fundamental tone, but with the note 3/2 regarded as a fundamental tone. The ratios of these two notes with the fundamental are 9/8 and 15/8. Interpolating these, we have the eight notes of the natural or diatonic scale, expressed by the following names and ratios:

NamesC.D.E.F.G.A.B.C′.
Intervals1st.2d.3d.4th.5th.6th.7th.8th.
Rates of vibration 1,9/8,5/4,4/3,3/2,5/3,15/8,2.

Multiplying these ratios by 24, to avoid fractions, we obtain the following series of whole numbers, which express the relative rates of vibration of the notes of the diatonic scale: