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. [Illustration: 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] [Illustration: FIG. 165.]