Sounds
Table of Contents
Aristotle has discussed accoustical questions.
I take pleasure in every kind of musical instrument and have paid considerable attention to harmony. But I have never been able to fully understand why some combinations of tones are more pleasing than others, or why certain combinations not only fail to please but are even highly offensive.
Then there is the old problem of two stretched strings in unison; when one of them is sounded, the other begins to vibrate and to emit its note; nor do I understand the different ratios of harmony [forme delle consonanze] and some other details.
Can one and the same pendulum really perform its vibrations, large, medium, and small, all in exactly the same time?
Aristotle has shown that the time of descent is the same along all chords, whatever the arcs which subtend them, as well along an arc of 180° (i. e., the whole diameter) as along one of 100°, 60°, 10°, 2°, ½°, or 4′. It is understood, of course, that these arcs all terminate at the lowest point of the circle, where it touches the horizontal plane.
If we consider descent along arcs instead of their chords then, provided these do not exceed 90°, experiment shows that they are all traversed in equal times.
But these times are greater for the chord than for the arc, an effect which is all the more remarkable because at first glance one would think just the opposite to be true. For since the terminal points of the two motions are the same and since the straight line included between these two points is the shortest distance between them, it would seem reasonable that motion along this line should be executed in the shortest time; but this is not the case, for the shortest time—and therefore the most rapid motion—is that employed along the arc of which this straight line is the chord.
As to the times of vibration of bodies suspended by threads of different lengths, they bear to each other the same proportion as the square roots of the lengths of the thread; or one might say the lengths are to each other as the squares of the times; so that if one wishes to make the vibration-time of one pendulum twice that of another, he must make its suspension four times as long. In like manner, if one pendulum has a suspension nine times as long as another, this second pendulum will execute three vibrations during each one of the first; from which it follows that the lengths of the suspending cords bear to each other the [inverse] ratio of the squares of the number of vibrations performed in the same time.
I can easily measure the length of a string whose upper end is attached at any height whatever even if this end were invisible and I could see only the lower extremity.
If I attach to the lower end of this string a rather heavy weight and give it a to-and-fro motion, and if I ask a friend to count a number of its vibrations, while I, during the same time-interval, count the number of vibrations of a pendulum which is exactly one cubit in length, then knowing the number of vibrations which each pendulum makes in the given interval of time one can determine the length of the string.
Suppose that my friend counts 20 vibrations of the long cord during the same time in which I count 240 of my string which is one cubit in length.
Taking the squares of the two numbers, 20 and 240, namely 400 and 57600, then the long string contains 57600 units of such length that my pendulum will contain 400 of them; and since the length of my string is one cubit, I shall divide 57600 by 400 and thus obtain 144. Accordingly I shall call the length of the string 144 cubits.
Nor will you miss it by as much as a hand’s breadth, especially if you observe a large number of vibrations.
Thousands of times I have observed vibrations in churches.
They have lamps suspended by long cords which had been inadvertently set into motion.
But the most which I could infer from these observations was that the view of those who think that such vibrations are maintained by the medium is highly improbable.
For, in that case, the air must needs have considerable judgment and little else to do but kill time by pushing to and fro a pendent weight with perfect regularity.
But I never dreamed of learning that one and the same body, when suspended from a string a hundred cubits long and pulled aside through an arc of 90° or even 1° or ½°, would employ the same time in passing through the least as through the largest of these arcs.
It still strikes me as somewhat unlikely.
Each pendulum has its own time of vibration so definite and determinate.
- It is not possible to make it move with any other period [altro periodo] than that which nature has given it.
For let any one take in his hand the cord to which the weight is attached and try, as much as he pleases, to increase or diminish the frequency [frequenza] of its vibrations.
It will be time wasted.
On the other hand, one can confer motion upon even a heavy pendulum which is at rest by simply blowing against it; by repeating these blasts with a frequency which is the same as that of the pendulum one can impart considerable motion.
Suppose that by the first puff we have displaced the pendulum from the vertical by, say, half an inch; then if, after the pendulum has returned and is about to begin the second vibration, we add a second puff, we shall impart additional motion.
And so on with other blasts provided they are applied at the right instant, and not when the pendulum is coming toward us since in this case the blast would impede rather than aid the motion.
Continuing thus with many impulses [impulsi] we impart to the pendulum such momentum [impeto] that a greater impulse [forza] than that of a single blast will be needed to stop it.
One man can give these impulses at the right instant as to ring a bell so large.
When six men seized the rope and tried to stop it they were lifted from the ground.
All of them were unable to counterbalance the momentum which a single man, by properly-timed pulls, had given it.
That is the wonderful phenomenon of the strings of the cittern [cetera] or of the spinet [cimbalo].
A vibrating string will set another string in motion and cause it to sound when the latter is in unison but even when it differs from the former by an octave or a fifth.
A string which has been struck begins to vibrate. It continues to do so as long as one hears the resonance.
These vibrations cause the immediately-surrounding air to vibrate and quiver.
Then these ripples in the air expand far into space.
They strike all the strings of the same instrument and even those of neighboring instruments.
That string which is tuned to unison with the one plucked can vibrate with the same frequency.
It acquires, at the first impulse, a slight oscillation.
After receiving 2, 3, 20, or more impulses, delivered at proper intervals, it finally accumulates a vibratory motion equal to that of the plucked string.
This is clearly shown by equality of amplitude in their vibrations.
This undulation expands through the air and sets into vibration the strings and any other body which has the same period as that of the plucked string.
Accordingly, if we attach to the side of an instrument small pieces of bristle or other flexible bodies, we shall observe that, when a spinet is sounded, only those pieces respond that have the same period as the string which has been struck.
The remaining pieces do not vibrate in response to this string, nor do the former pieces respond to any other tone.
If one bows the base string on a viola rather smartly and brings near it a goblet of fine, thin glass having the same tone [tuono] as that of the string, this goblet will vibrate and audibly resound.
That the undulations of the medium are widely dispersed about the sounding body is evinced by the fact that a glass of water may be made to emit a tone merely by the friction of the finger-tip upon the rim of the glass; for in this water is produced a series of regular waves.
The same phenomenon is observed to better advantage by fixing the base of the goblet upon the bottom of a rather large vessel of water filled nearly to the edge of the goblet; for if, as before, we sound the glass by friction of the finger, we shall see ripples spreading with the utmost regularity and with high speed to large distances about the glass. I have often remarked, in thus sounding a rather large glass nearly full of water, that at first the waves are spaced with great uniformity, and when, as sometimes happens, the tone of the glass jumps an octave higher I have noted that at this moment each of the aforesaid waves divides into two; a phenomenon which shows clearly that the ratio involved in the octave [forma dell’ ottava] is two.