Middle C of the piano (or any instrument) has a vibration rate of about 260. Go up an octave from this and you double the number of vibrations per second; go down an octave and you halve the number of vibrations. Of any two notes that are an octave apart, the upper has twice the vibration rate of the lower. The whole range of audible notes, from 16 to 30,000 vibrations, thus amounts to about eleven octaves, of which music employs about eight octaves, finding little use for the upper and lower extremes of the [{230}] pitch series. The smallest step on the piano, called the "semitone", is one-twelfth of an octave; but it must not be supposed that this is the smallest difference that can be perceived. A large proportion of people can observe a difference of four vibrations, and keen ears a difference of less than one vibration; whereas the semitone, at middle C, is a step of about sixteen vibrations.
Mixture of different wave-lengths, which in light causes difference of saturation, may be said in sound to cause difference of purity. A "pure tone" is the sensation aroused by a stimulus consisting wholly of waves of the same length. Such a stimulus is almost unobtainable, because every sounding body gives off, along with its fundamental waves, other waves shorter than the fundamental and arousing tone sensations of higher pitch, called "overtones". A piano string which, vibrating as a whole, gives 260 vibrations per second (middle C), also vibrates at the same time in halves, thus giving 520 vibrations per second; in thirds, giving 780 per second; and in other smaller segments. The whole stimulus given off by middle C of the piano is thus a compound of fundamental and overtones; and the sensation aroused by this complex stimulus is not a "pure tone" but a blend of fundamental tone and overtones. By careful attention and training, we can "hear out" the separate overtones from the total blend; but ordinarily we take the blend as a unit (just as we take the taste of lemonade as a unit), and hear it simply as middle C of a particular quality, namely the piano quality. Another instrument will give a somewhat different combination of overtones in the stimulus, and that means a different quality of tone in our sensation. We do not ordinarily analyze these complex blends, but we distinguish one from another perfectly well, and thus can tell whether a piano or a cornet is playing. The difference between different instruments, which we have spoken of as a [{231}] difference in quality or purity of tone, is technically known as timbre; and the timbre of an instrument depends on the admixture of shorter waves with the fundamental vibration which gives the main pitch of a note.
Akin to the timbre of an instrument is the vowel produced by the human mouth in any particular position. Each vowel appears to consist, physically, of certain high notes produced by the resonance of the mouth cavity. In the position for "ah", the cavity gives a certain tone; in the position for "ee" it gives a higher tone. Meanwhile, the pitch of the voice, determined by the vibration of the vocal cords, may remain the same or vary in any way. The vowel tones differ from overtones in remaining the same without regard to the pitch of the fundamental tone that is being sung or spoken, whereas overtones move up or down along with their fundamental. The vowels, as auditory sensations, are excellent examples of blends, in that, though compounds, they usually remain unanalyzed and are taken simply as units. What has been said of the vowels applies also to the semi-vowels and continuing consonants, such as l, m, n, r, f, th, s and sh.
Other consonants are to be classed with the noises. Like a vowel, and like the timbre of an instrument, a noise is a blend of simple tones; but the fundamental tone in a noise-blend is not so preponderant as to give a clear pitch to the total sound, while the other tones present are often too brief or too unsteady to give a tonal effect.
Comparison of Sight and Hearing
The two senses of sight and hearing have many curious differences, and one of the most curious appears in mixing different wave-lengths. Compare the effect of throwing two colored lights together into the eye with the effect of [{232}] throwing two notes together into the ear. Two notes sounded together may give either a harmonious blend or a discord; now the discord is peculiar to the auditory realm; mixed colors never clash, though colors seen side by side may do so to a certain extent. A discord of tones is characterized by imperfect blending (something unknown in color mixing), and by roughness due to the presence of "beats" (another thing unknown in the sense of sight). Beats are caused by the interference between sound waves of slightly different vibration rate. If you tune two whistles one vibration apart and sound them together, you get a tone that swells once a second; tune them ten vibrations apart and you get ten swellings or beats per second, and the effect is rough and disagreeable.
Aside from discord, a tone blend is really not such a different sort of thing from a color blend. A chord, in which the component notes blend while they can still, by attention and training, be "heard out of the chord", is quite analogous with such color blends as orange, purple or bluish green. At the same time, there is a curious difference here. By analogy with color mixing, you would expect two notes, as C and E, when combined, to give the same sensation as the single intermediate note D. Nothing of the kind! Were it so, music would be very different from what it is, if indeed it were possible at all. But the real difference between the two senses at this point is better expressed by saying that D does not give the effect of a combination of C and E, or, in general, that no one note ever gives the effect of a combination or blend of notes higher and lower than itself. Homogeneous orange light gives the sensation of a blend of red and yellow; but there is nothing like this in the auditory sphere. In light, some wave-lengths give the effect of simple colors, as red and yellow; and other wave-lengths the effect of blends, as greenish yellow or bluish [{233}] green; but in sound, every wave-length gives a tone which seems just as elementary as any other.
There is nothing in auditory sensation to correspond to white, no simple sensation resulting from the combined action of all wave-lengths. Such a combination gives noise, but nothing that seems particularly simple. There is nothing auditory to correspond with black, for silence seems to be a genuine absence of sensation. There are no complementary tones like the complementary colors, no tones that destroy each other instead of blending. In a word, auditory sensation tallies with its stimulus much more closely than visual sensation does with its; and the main secret of this advantage of the sense of hearing is that it has a much larger number of elementary responses. Against the six elementary visual sensations are to be set auditory elements to the number of hundreds or thousands. From the fact that every distinguishable pitch gives a tone which seems as simple and unblended as any other, the conclusion would seem to be that each was an element; and this would mean thousands of elements. On the other hand, the fact that tones close together in pitch sound almost alike may mean that they have elements in common and are thus themselves compounds; but still there would undoubtedly be hundreds of elements.
Both sight and hearing are served by great armies of sense cells, but the two armies are organized on very different principles. In the retina, the sense cells are spread out in such a way that each is affected by light from one particular direction; and thus the retina gives excellent space information. But each retinal cell is affected by any light that happens to come from its particular direction. Every cone, in the central area of the retina, makes all the elementary visual responses and gives all the possible color sensations; so it is not strange that the number of visual [{234}] elements is small. On the other hand, the ear, having no sound lens, has no way of keeping separate the sounds from different directions (and accordingly gives only meager indications of the direction of sound); but its sense cells are so spread out as to be affected, some by sound of one wavelength, others by other wave-lengths. The different tones do not all come from the same sense cells. Some of the auditory cells give the low tones, others the medium tones, still others the high tones; and since there are thousands of cells, there may be thousands of elementary responses.