A

SYSTEM

OF

MECHANICAL PHILOSOPHY

...

WITH NOTES,

BY DAVID BREWSTER, LL. D.


394. It is in this delicate department of musical science that we think the great merit of Dr. Smith's work consists. We see that the deviation from perfect harmony is always accompanied with beats, and increases when they increase in frequency - whether it increases in the same proportion may be a question. We think that Dr. Smith's determination of the equality of imperfect harmony in his 13th proposition, includes every mathematical or physical circumstance that appears to have any concern in it. What relates immediately to our sensations is, as yet, and impenetrable secret. The theory of beats, as delivered by this author, affords very easy, though sometimes tedious, methods of measuring and insuring all the varieties which can obtain in the beating of imperfect consonances. It appears to us therefore very unjust to say, with a late writer in the Philosophical Transactions*, that this obscure volume has left the matter where it found it. The author has given use effective principles, although he may have been mistaken in the application ; which however we are far from affirming. Our limits will not allow us to give any account of that theory ; and indeed our chief aim in the present article is to give a method of temperament which requires no scientific knowledge of the subject. But we could not think of losing the opportunity of communicating, by the way, to unlearned persons, some more distinct notions of the scale of musical sounds, and of its foundation in nature, than scholars usually receive from the greater number of mere music masters. The acknowledged connection of the musical ratios with the pleasures of harmony and melody, has (we hope) been employed in an easy and not obscure manner ; and the phenomena which we have faithfully narrated, shew plainly that, by diminishing the rattling undulations of tempered concords, we are certain of improving the harmony of our instruments. We shall proceed therefore on this principle for the use of the mere performer, byt at the same time introducing some very simple deductions from Smith's theory, for which weexpect the thanks of all such readers as wish to see a little of the reasons on which they are to proceed.

* Dr. Thomas Young, to whom Dr. Robison here alludes, has published a reply to this, and a preciding part of the present article, in Nicholson's Journal, for August 1801. See also Dr. Young's Lectures on Natural Philosophy, vol. ii. p.607.

395. The experiment, of which we have just now given an account, shews that four consecutive fifths compose a greater interval than two octaves and a major third. Yet, in the construction of our musical instruments of fixed sounds, they must be considered as of equal extent ; since we have 7 half intervals in the Vth, and 12 in the octave, and four in the IIId, four Vths contain 28, and two octaves contain 24 ; and these, with the four which compose a IIId, make also 28. It is plain, therefore, that whatever we do with the IIIds, we must lessen the Vths. If therefore we keep the IIId perfect, we must lessen each of the Vths by 1[/4] of a comma ; for we learned, by the beating of the imperfect IIId c e, that the whole excess of the four Vths was a comma. Therefore the Vth c g must be flattened 1/4th of a comma. But how is this to be done with accuracy? Recollect the formula given a little ago, where the number of beats b in any number of seconds is In the present case q = 1, m = 3, N = 240 per second, and p = 4. Therefore the formula is = 2,25 in a second, or 9 beats in four seconds very nearly.

In like manner, the next Vth g d must be flattened 1/4th of a comma, by making it beat half as fast again, or 13 1/4 beats in four seconds (because in this Vth N = 360. But as this beating is rather too quick to be easily counted, it will be better to tune downwards the perfect octave g G, which will reduce N to 180 for the Vth G d. This will give use 1,68 per second, or 10 beats in 6 seconds very nearly.

There is another way of avoiding the employment of too quick beats. Instead of tuning the octave g G, make c G beat as often as c g. This is even more exactly an octave to g than can be estimated by a good ear. Dr. Smith has demonstrated, that when a note makes a minor concord with another note below it, and therefore a major concord with the octave to that note, it beats equally with both ; but if the major concord be below, it beats twice as fast with the octave above. Now in the present case, c g is a Vth, and c G a 4th. For the same reason c f would beat twice as fast as c F.

In the next place, the Vth d [a'] must be made to beat flat 15 times in 6 seconds.

In like manner, instead of tuning upward the Vth [a' e'], tune downward the octave a [a'], and then tune upward the Vth a e, and flatten it till it beat 15 times in 8 seconds.

If we take 15 seconds for the common period of all these beats, we shall have

The beats ofc g = 34.
G d = 25.
d [a' = 37 1/2.
a c = 28.

396. We shall now find c e to be a fine IIId, without any sensible beating ; and then we proceed in the same way, always tuning upward a perfect Vth ; and when this would lead too high, and therefore produce too quick beating, we should tune downward an octave. Do this till we reach b#, which should be the same with [c'], or a perfect octave above c. This will be a full proof of our accurate performance. But the best process of tuning is to stop when we get to g c. Then we tune Vths downward from c, and octaves upward when the Vths would lead us too low. Thus we get c F, F f, f bb, bb [b']b, [b']b eb, and thus complete the tuning of an octave. We take this method, instead of proceeding upwards to [b']# ; because those notes marked sharp or flat are, when tuned in this way, in the best relation to those with which they are most frequently used as IIIds.

397. This process of temperament will be greatly expedited by employing a little pendulum, made of a ball of about two ounces weight, sliding on a light deal rod, having at one end a pin hole through it. To prepare this rod, hang it upon a pin stuck into the wainscoating, and slide the ball downward, till it makes 20 vibrations in 15", by comparing it with a house clock. In this condition mark the rod at the upper edge of the ball. In like manner, adjust it for 24, 28, 32, 36, 40, 44, 48, vibrations, making marks for each, and dividing the spaces between them by the eye, noticing their gradual diminution. Then, having calculated the beats of the different Vths, set the ball at the mark suited to the particular concord, and temper the sound till the beats keep pace exactly with the pendulum.

398. But previous to all this, we must know the number of pulses made in a second by the C of our instrument. For this purpose we must learn the pulses of our tuning fork. To learn this, a harpsichord wire must be stretched by a weight till it be unison or octave below our fork ; then, by adding 1/40th of the weight to what is now appended, it will be tempered by a comma, and will beat, when it is sounded along with the fork ; and we must multiply the beats by 80 : The product is the number of pulses required. And hence we calculate the pulses fo the C of our instrument when it is tuned in perfect concord with the fork.

The usual concert pitch and the tuning forks are so nearly consonant to 240 pulses for C, that this process is scarcesly necessary, a quarter of a tone never occasioning the change of an entire beat in any of our numbers...

406. But we are forgetting the process of tuning, and have tuned three or four notes of our octave. We must tune the rest by considering their relation to notes already tuned. Thus, if g c makes 36 beats in 16 seconds, F c should make one third less, or about 24 in the same time : because N in the formula is now 160 instead of 240. Proceeding in this way, we shall tune the octave C [c''] most accurately as a system of mean tones with perfect IIIds, by making the notes beat as follows. A point [bold text] is put over the note that is to be tuned from the other, and a +, or a -, means that the concord is to be tempered sharp or flat. Thus, g is tuned from c,

Makec g beat-36 times in 16 seconds
G c+36
G d-27, i.e. 3/4ths of g c
c f-48
c [a']+60 times in 16 seconds
c e0, i.e. a perfect IIId
d f#0
c g#0
[a'] [c']#0
bb f downward-24, i.e. [2/3]ths of c g
bb [b']b0, i.e. a perfect octave
[b'] eb downward-43, i.e 5/4ths of c g
C [c']0, an octave.

Other processes may be followed, and perhaps some of them better thant the process here proposed. Thus bb and eb may be tuned as perfect IIIds to d and g downwards.

Also, as we proceed in tuning, we can prove the notes, by comparing them with other notes already tuned, &c. &c. &c.

We have directed to tune the two notes bb and eb bay taking the leading Vth downwards. We should have come at the same pipes in the character of a# and d# in the process of tuning upwards by Vths. But this would not have produced precisely the same sounds, althouhg, in our imperfect instruments, one key must serve for a# and bb. By tuning them as here directed, they are better fitted fo rthe place in which they will be most frequently employed in our usual modulations.

407. It may reasonably be asked, Why so much is sacrificed in order to preserve the IIIds perfect? Were they allowed to retain some part of the sharp temperament that is necessary for preserving the Vths perfect, we should perhaps improve the harmony. And since enlarging the Vth makes the tone greater, and therefore the limma mi fa much smaller, it will bring it nearer to the magnitude of a half tone ; and this will be better suited for its double service of the sharp of the note below, and the flat of the note above. Accordingly, such a temperament is in great repute, and indeed is generally practised, although the VIths and the subordinate chords of full harmony are evidently hurt by it. Even Dr. Smith recommends it as well suited to our defective instruments, and gives an extremely easy method of executing it by means of beats. His method is to make the Vth and IIId beat equally fast, along tiht the key, the Vth flat, and the third shap. He demonstrates (on another occasion) that concords beat equally fast with the same bass when their temperaments are inversely as the major terms of their perfect ratios. Therefore draw EG, and divide it into p, so that E p may be to p G as 3 to 5. Then draw C p, cutting gG in g', and EK in c' ; and this temperer will produce the temperament we want. It will be found, that E e' and G g' are each of them 32 of their respective scales.

Therefore make c g beat 32 times in 16 seconds

F a
Gc32
Gd24
Gb24, and tune b[b']
d [a']36, and tune a [a']
d f#36
a e27
a c#27
e [b']40 1/2, proving b [b']
e g#40 1/2
F c21 1/3, and tune F f
21 1/3, proving a
bb f28 1/2, and tune bb [b']b
eb [b']b38 [1/2] [...]

It may be proper to add to all these isntructions[] a caution about the manner of counting the clock while the tuner is counting the beats. If this is to continue for 16 seconds, let the person who counts the clock say one at the beat he begins with, and then telling them over to himself, let him say done instead of 17. Thus 16 intervals will elaps while the tuner is counting the beats. We he to begin to count at one, and stop when he hears sixteen, he would get the number of beats in 15 seconds only.

408. We do not hesitate to say, that this method of tuning by beats is incomparably more exact than by the mere judgment of the ear. We cannot mistake more than one beat. This mistake in the concord of the Vth mounts to no more than 1/100th of a comma ; and in the IIId it is only 1/180.

409. It may be objected that it is fit only for the organ and instruments of continued sounds, but will not do for the quickly perishing sounds of the harpsichord. True, it is the only method worthy of that noble instrument, and this alone is a title to high regard. But farther ; the accuracy attainable by it is much more exact, and more certain in its process, than any other. It does not proceed, by a random trial of flattened series of Vth, and a comparison with the resulting IIId, and a second trial, if the first be unsatisfactory. It says at once, let the Vth beat so many times in 16 seconds. Even in the second method, without counting, and merely by the equality of the beats of the Vth and IIId, the progress is easy. Both are tuned perfect. The Vth is then flattened a little, and the IIId sharpened ; - if the Vth beat faster than the IIId, alter it first.

All difficulty is obviated by the simple contrivance of a variable pendulum, already described. This may be made exact by any person that will take a little pains ; and when once made, will serve for every trial. When the ball is set to the proper number, and the pendulum set a swinging, we can come very near the truth by a very few trials.

N.B. In tuning a piano forte, which has always two strings to a key, we must never attempt tuning them both at once ; the back unison of both notes of the concord must be dampened, by sticking in a bit of soft paper behind it.

We hope that the instructions now given, and the application of them to two very respectable systems of temperament, are sufficient for enabling the attentive reader to put this method of tuning successfully in practice, and that he perceives the efficiency of it for attaining the desired end...


site index