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Limits of Audibility. Feebleness of very Acute Sounds.
a sound, while the other maintains there is no sound at all. Thus while one person mentioned by Dr. Wollaston could but just hear a note four octaves above the middle E of the piano-forte, others have a distinct perception of sounds full two octaves higher. The chirp of the sparrow is about the former limit; the cry of the bat about one octave above it; and that of some insects probably more than another octave.
c. It is probable, however, that it is not alone the frequency of the vibrations which renders shrill sounds inaudible. There is no reason why an impulse, if strong enough singly to affect the ear, should lose its effect if repeated many thousand times in a second. On the contrary, such repetition would render the noise intolerable. But this is not the case with musical sounds; their individual impulses would, probably, be quite inaudible singly, and only impress by repetition. Now, as vibrating bodies have only a certain degree of elasticity, extreme frequency of vibrations can only take place when their dimensions are very minute; and consequently the excursions of their molecules from rest, and their absolute velocities, excessively minute also.
Thus, in proportion as sounds are more acute, their intensity, which depends wholly on the extent and force of their vibrations, diminishes. No doubt, if by any mechanism a hundred thousand hard blows per second could be regularly struck by a hammer on an anvil, at precisely equal intervals, they would be heard as a most deafening shriek; but in natural sounds the impulses lose in intensity more than they gain in number, and thus the sound grows feebler and feebler till it ceases to be heard.
105. The limits of the sense of hearing possibly, indeed probably, vary with different animals.
As there is nothing in the nature of the atmosphere to prevent the existence of vibrations incomparably more frequent than any of which we are conscious, we may imagine that animals like the Grylli, whose powers appear to commence nearly where ours terminate, may have the faculty of hearing still sharper sounds, which
Limits of the Sense of Hearing in Different Animals.
we do not know to exist; and that there may be other insects hearing nothing in common with us, but endued with a power of exciting, and a sense which perceives vibrations, of the same nature indeed as those which constitute our ordinary sounds, but so remote that the animals who perceive them may be said to possess another sense, agreeing with our own solely in the medium by which it is excited.
The same may, no doubt, be true of aquatic animals. The shrimp and the whale may have no sound in common. Spiders are said to hear the sound of music.
106. By the aid of the ascending and descending series of sounds in the natural scale thus obtained, pieces of music perfectly pleasing, both in point of harmony and melody, may be played; and they are said to be in the key of that which is assumed as the fundamental note of the scale, or whose vibrations are represented by 1.
If such a piece be analyzed, it will be found to consist entirely, or chiefly, of triple and quadruple combinations, or chords, such as the following:
First. The common or fundamental chord, or chord of the tonic, or the 1st, 3d, and 5th, (1, 3, 5,) or the 3d, 5th, and octave, (3, 5, 8,) sounded together.
Secondly. The chord of the dominant, or the notes (2, 5, 7,) sounded together.
Thirdly. The chord of the sub-dominant, or the combination (1, 4, 6.)
Fourthly. The false close, or the combination (1, 3, 6,) or (3, 6, 8.)
Fifthly. The discord of the 7th, or (2, 4, 5, 7.)
a. The common chord is the most harmonious and satisfactory chord in music, and when sounded the ear is satisfied, and requires
The Chord of the Tonic, the Dominant, and the Sub-Dominant. The False Close.
nothing further. It is, therefore, more frequently heard than any other, and its continual recurrence in a piece of music determines the key it is played in.
b. The fifth of the key-note is called, by reason of its near relation to the fundamental note, the dominant. The chord of the dominant is, then, the common chord of the dominant.
c. Note (4) is the sub-dominant, and its common chord is the chord of the sub-dominant.
d. The false close is the common chord of the note (6), only with a minor third instead of a major. The term false close arises from this, that a piece of music, frequently before its final termination, (which is always on the fundamental chord,) comes to a momentary close on this chord, which pleases only for a short time, but requires the strain to be taken up again and closed as usual to give full satisfaction.
é. The discord of the 7th consists of four notes; and is in fact the common chord of the dominant with the note immediately below it, or the seventh in order above it. The interval, however, between the notes (4) and (5), or between (5) and the octave of (4) next above it, is represented by the ratio,
or (taking 24 as the number of vibrations in a unit of time corresponding to note (1)) = 42. This interval, then, is less than the seventh of the diatonic scale, and is about half-way intermediate between the sixth and seventh of that scale. It is, therefore, called the flat seventh. (See article 108.)
This discord resolves itself into the chord (3, 5, 8); and unless that combination, or one equivalent to it, follows, the ear is not satisfied. The notes (4) and (5) are the essential ones of this discord, and the others are regarded as accompaniments. If played together, the ear requires that in the next chord (4) should descend or be succeeded by (3), while the note (7) is required to rise or be succeeded by (8.)
Modulation. Want of Intermediate Notes. Reduction of the Number of Notes.
107. With these chords and a few others, such as the chord of the 9th, whose essential notes are (1) and (2), or (1) and (9), may a great variety of music be played, but it would be found monotonous. The ear requires, in a long piece, a variety of key. The fundamental note occurs so often, that it seems to pervade the whole of the composition, and must therefore be changed. But this change of key, which is called modulation, is not possible without introducing other notes than those already enumerated.
In order to play equally well in all keys, every note must have others which differ from it by the intervals of a second, a third, a fourth, &c. But this is not the case in the diatonic scale. Thus the number of vibrations in a unit of time, corresponding to the major third of the note (2), is 27 × 33, nearly half way between those of note (4) and note (5). A new note would therefore have to be introduced, and similarly for other ratios or other
But this would require an enormous number of notes, and would render the generality of musical instruments too complicated. It becomes necessary, then, to consider how the number may be reduced, and what are the fewest notes that will answer.
108. The principle, on which the reduction of the number of notes is made, is, that, if two notes differ from each other by only one vibration in 80, the ear can hardly perceive the difference between them, and the substitution of one of them for the other will not be fatal to harmony.
a. The interval between two such notes is 8, and is called a
b. As an example of this principle, we will apply it to the re
Flats and Sharps. The Chromatic Scale.
search of the fifth of note (2). The number of vibrations corresponding to the fifth of this note is
27 × = 40,
which only differs by a comma from 40, the vibrations of note (6); and, therefore, note (6) may be used as the fifth of note (2), or as the dominant when (2) is the key-note.
109. Some new notes must, however, be interpolated; and, when any note is introduced between two others of the scale, it is denoted in music by the sign # sharp, or y flat.
Thus the major third of the note (2) is, as in art. 107, nearly half way between note (4) and note (5), and must, therefore, be interpolated. It is written either as (4) sharp, (4), or as (5) flat, (5) L.
110. The deficiencies of the diatonic scale are nearly supplied by the interpolation of a new note half-way between each of the larger intervals of the scale, thus
111. Musicians have long been at issue on the most advantageous method of executing this interpolation; and it is found necessary to depart from the pure and perfect diatonic scale, even in tuning the natural notes. To do so with the least offence to the ear is the object of a perfect system of temperament.
a. If, indeed, it were intended to give such a preference to the natural scale 1, 2, 3, 4, &c. as to make it perfect, to the sacrifice