THE

MECHANICS' MAGAZINE ;

ON TUNING.

Sir, - Several of the first mathematicians, as well as musicians, have spent much time in endeavoring to discover the best manner of tuning instruments with fixed tones (piano-fortes, organs, &c.), though their endeavors have not, as yet, been attended with the desired success ; but, without branching into the depths of mathematical science, we will see how far the practice and theory of tuning agree together.

The system most in use, at present, for piano-fortes, is that known by the name of equal temperament ; which consists in dividing the imperfection of the scale or wolf, as it is generally termed, equally amongst the 12 major keys. This is open to many objections ; for, according to this system, there is not a single perfect third, fourth, or fifth in the whole instrument. Every third is sharp, and equally sharp ; every fourth sharp, and equally sharp ; and every fifth flat, and equally flat : therefore, not only is every major key rendered imperfect, but equally and similarly imperfect. This necessarily destroys that difference of character that should exist in a well-tuned instrument between the different major keys. And the minor keys are liable to the same defect, for the same reason. Thus is dull monotony substituted for pleasing and orderly variety ; and modulation from key to key loses, in a great measure, the very object of modulation, which is to relieve the ear, and cause us to return to the original key with increased pleasure ; arising from the systematic variety of the different keys through which we have successively passed. All those chords which nature has rendered perfect are made imperfect ; and, in those instruments where chords are made to sound a considerable time, as in organs, the imperfection of the equal temperament is most striking.

In opposition to this, we have unequal temperament ; but, as the equal temperament is fixed and unalterable, this comprises every possible variety of tuning. The tuners of the old school used the unequal temperament in their bearings, which were tuned by fifths and octaves ; and "The Wolf" was generally thrown into the key of E or A flat. The more modern tuners comprise their scale within the septave, and tune by fourths and fifths ; though we should more properly say, by thirds and sixths, as it is by those intervals that the scale is tried and regulated.

Many have considered that the fifth and octave are the only two perfect intervals, but the third and sixth should rather be considered so, for we may shaprten or flatten both fifth and octave very considerably, without affecting the ear ; but try them with the corresponding third or sixth, and the beating is very perceptible.

The mode of tuning by occasional fourths is one that will not suit the learner ; inasmuch as they do not strike the unpractised ear so well as fifths ; but experience has proved it to be best for practical purposes, especially if the scale be divided thus : - C-G, G-D, D-A, A-E, E-B, B-F sharp ; again, C-F, F-A sharp, A sharp-D sharp, D sharp-G sharp, Gsharp-Csharp, which completes the septave. It may be necessary to observe that the starting point is the middle C of the piano.

Earl Stanhope has constructed a very beautiful and scientific system of tuning, in theory, but extremely difficult in practice, and one that is very unsatisfactory to many ears ; and after all it is, perhaps, the better way to accommodate the temperament to the taste of those who prefer a particular key, which a good tuner always can do.

These few remarks are intended more for the practical tuner, than for the amateur or professional ; who, though they can point out the defect in that key which does not please them, yet cannot rectify it themselves...March 21, 1836.

ON TUNING - NEW MATHEMATICAL DIVISION OF THE SCALE.

Sir, - The following is a mathematical division of the scale, assuming the bass C as 30 inches : -

It is obtained thus : -

These multipliers are not arbitrary numbers, but as the second stage brings us beyond the first octave, we must double it to bring it within the octave ; and as the fourth stage brings us beyond the second octave, it must be twice doubled (or quadrupled), and so on of the rest.

A piano-forte tuned according to this scale would, I think, have a very pleasing effect ; but, independent of the impossibility of tuning to that exactness, piano-forte -makers, instead of doubling the length of string to produce the sound of the octave below, must necessarily use a thicker wire, or we should have piano-fortes as large as houses.

The following is Earl Stanhope's scale : -

C-C, perfect octave. C-G, perfect fifth. C-E, perfect third. E-B, perfect fifth. C-F, perfect fifth. F-B flat, perfect fifth. E-A flat, bi-equal third.* Aflat - C, bi-equal third. A flat - E flat, perfect fifth. Aflat - D flat, perfect fifth. D flat - G flat, perfect fifth. G-D, D-A, A-E, three tri-equal fifths.

These tri-equal fifths, though flat, are not of such a degree of flatness as to be offensive to the ear ; differing from a perfect fifth only 829,885 parts in 300,000,000 or 829885/3000000000. If this interval G-E, as in Kirnberger's method, be divided into one perfect fifth, and two equally flat fifths - such, for instance , as the perfect fifth G-D, and the equally flat fifths D-A and A-E ; then each of these two last fifths, by becoming too flat, is offensive to the ear. And if that same interval be divided into two perfect fifths, and one flat fifth, then this flat fifth so produced is still more offensive.

* A bi-equal third is thus obtained : - from one perfect octave deduct one perfect third, and divide the remainder into two equally sharp thirds.
A tri equal fifth is thus obtained : - divide the interval included by a perfect fifth from the key-note, and the second perfect octave above the perfect third from the same key-note, into three equally flat fifths, which are tri-equal fifths.

In tuning each key throughout the whole instrument, too much attention cannot be paid to the beatings, as that is by far the most accurate way of tuning by the ear. For, whenever a third, fourth, fifth, sixth, or octave, is quite perfect, there is no beating to be heard. But, on the contrary, when any of them are in any degree imperfect, though not distant from perfection, a beating is always audible. A very slow beating proves that the distance from perfection is not great ; but as the beating becomes more considerable, and, from the equality of the beatings, equal deviations may in like manner be correctly ascertained.

Some tuners, in order to assist the fifths, have proposed to tune the octaves a little imperfect. The objections to this are obvious, for if we sharpen the octaves to assist the fifths, it injures the thirds; and if we flatten the octaves to assist the thirds, it injures the fifths. Such is the construction of the human ear, that we can hear a much greater deviation from perfection in the fifths than we can in the octaves, and a still greater deviation in thirds than either the fifths or octaves. Again, however small the deviation may be in a asingle octave, it becomes very sensible in two or three, and most offensive in six or seven.

We have been in the habit of considering the Wolf as an inherent imperfection in every instrrument that has exactly twelve fixed keys in each septave ; whereas, so far from being an imperfection, it is precisely the proper distribution of it that produces the charming variety of character between the different keys which is so essentially requisite in a well-tuned instrument...April 8, 1836.

ON TUNING PIANO-FORTES.

Sir. - Many, when they first commence learning to tune, ar contented to begin simply with tuning, properly so called, instead of what is technically called roughing up. This consists in taking the instrument rough from the stringer, and drawing it up until such time as it stands at concert pitch. It may be urged that a person who possesses a delicate ear will learn sooner of the first plan, in consequence of the ear not being vitiated by the discordant sounds that are the necessary attendants upon roughing up ; but such a person would be quite at a loss when he had to tune an instrument half a note, or even two half-notes (which is not uncommon) below concert pitch. From his comparatively bungling manner of proceeding, he would be three times as long over his work as one that has learned by roughing-up. In short, it is like learning to write elegantly before pot-hooks and hangers are acquired.

All this arises principally from not knowing the part in which the strain of the strings causes the pitch to fall, and to make the necessary allowance for it by drawing up that part above the pitch. If the bearings are comprised within the septave F-E, the pitch is found to fall from the included B, all the way up the treble. This is remedied by drawing that part up considerably above the pitch you are working upon, and, by the time you have finished the treble, it will have settled pretty well down to the perfect octave. Again, the first-named would probably make his pitch exact at starting, instead of allowing for the falling of it afterwards.

In horizontal grand and square pianoforrtes, this strain is very considerable, in cabinets not so great. The reason of this is, that in the latter the strain is in a perfectly vertical direction, and consequently, they stand longer in tune ; but in the two former it is all diagonal, and, indeed, it has been jocularly said, that the square piano-forte is so called, because there is nothing square about it!

The task of roughing-up is materially facilitated by stretching the strings with a well-known instrument called a rubber, made of wood, and, in most instances, covered with leather ; this is pressed downwards with considerable force upon the whole length of the string. It also has the advantage, where a astring is false, i.e. not perfectly round, of causing it to become more pure in its tone. Until of late years, piano-forte nakers were sadly bothered for wire. The best that could be procured then was the Berlin, or German wire, as it was generally called. But bad was that best ; it was only iron wire, and neither round, square, oval, nor any other shape. It was very scarce, and difficult to procure in time of was ; when Napoleon shut the foreign ports against us, to wit, it was a favour to get it at all at 10s. 6d. or 12s. per lb., and there has been the unexampled price paid for it of 25s. per lb. at a public sale. It was also very wasteful ; ring after ring having to be thrown aside in consequence of brittleness.

The proud boast was reserved for an Englishman* of overcoming these difficulties, and furnishing steel wire as near to perfection as any thing in this sub-lunary world. There had been countless trials and experiments made to give to steel such a temper as would fit it for music-wire ; but the patience of the English piano-forte makers had been nearly exhausted by their repeated disappointments, and it was some time before it came into general use. Now, not only is nothing else used in England, but at Paris, Vienna, Hamburgh, and even Berlin itself, the German wire has been completely beaten our of the market. This created a new era in piano-forte making ; for I think I may safely asset, that piano-fortes have been considerably more improved within the last ten or twelve years than during the previous thirty or forty. Independant of this, it has grown up more into a distinct trade per se ; formerly their shops were supplied with artisans from the joiners and cabinet-makers - now they are supplied with men regularly brought up to the business from their childhood...April 20, 1836.

* Wr. Webster, of Penns, near Birmingham.
He professes to send it out quite perfect, and will exchange any quantity from a quarter of an ounce to a quarter of a ton.

The different kinds of iron and steel vary very much in price. The following were the prices of most of the sorts in the London market, when the first edition of this work went to press. If I can obtain an accurate account of the present prices, I will give it in the appendix to this volume.

The high price of Swedish iron is owing to the great superiority of its quality when compared with common English iron ; and this is attributed to the circumstance of its being manufactured with wood charcoal, whereas most of the English iron is prepared by means of mineral coke. The Swedish iron, however, differs very much in its quality ; even one part of the same bar will often be of much greater value than the other. Formerly the wire-workers in Yorkshire used to go to Sheffield to by foreign iron, on account of the circumstance above mentioned : it was usual with them to cut the bars into two-feet lengths, and then select only such pieces as were fit for their purpose, leaving the other to be converted into steel. It is no uncommon thing to find a bar of foreign iron tough on one side and quite hard and brittle on the other. If such iron be exposed to the weather, this would soon become apparent by one side rusting and the other remaining quite clean, owing probably to the unequal distribution of carbon in its manufacture.

I have also been informed that we have no artist in England who can make iron-wire for musical instruments, and that all such wire is imported from Holland, and sold at very extravagant prices. The larger kind, such as is used in a piano-forte, cannot be bought for less than 8d. or 10d. per ounce.

(Samuel Parkes, Chemical Essays (1823))

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