TUNING, the art of adjusting the several sounds of a musical instrument so as to make its scale appproach to correctness ; also that of putting two instruments, each of which has the parts of its scale in proper relative adjustment, into agreement with each other.

...The slight alterations which are made in order that any one of the twelve notes of the octave may be fit to be used as a key-note, without any shock to the ear, constitute the temperament of a scale ; the altered consonances are said to be tempered. Some writers call the interval from the false octave obtained byu the fifths to the true one, by the name of the wolf ... But by the term wolf other writers means the bad fifth which exists in the worst key, when the temperament is allowed to favour some keys at the expense of others. Simple as this little variation in the meaning of a term may be, it is worth while to notice it. A writer on tuning charges some of the pianoforte-makers of his time with utter ignorance of the scale, in stipulating with the tuners whom they employed that there should be no wolf. In all probability they only meant that no key should be worse than another, or that the temperament should be equal. This term wolf is said to be derived from the jarring of a bady-tuned consonance, supposed to resemble the distant howling of the animal ; we rather suspect it was so called because it was hunted from one part of the scale to another like a wild beast, in hopes of getting rid of it.

Two systems of temperament suggest themselves : the first, equal, in which the necessary defects of the scale are distributed equally thoughout it ; the second, unequal in which some mode is adopted of distributing the imperfection so as to make some keys feel it less than others. The most common practice of our day is to endeavor equal temperament. The two systems have their advocates, and the arguments for one and the other are as follows. In favor of equal temperament it is urged that all the keys are made equally good, and that in no one does the imperfection amout to a striking defect : also that in the orchestra there is little chance of any uniform temperament among the various instruments, if it be not this one. Against equal temperament it is urged that it takes away all distinctive character from the different keys, and leaves no one single key perfect. All these arguments have force, both for and against : for ourselves we consider those against equal temperament much the stronger. We have often felt that a pianoforte newly tuned has, with much correctness, a certain insipidity, which wears off as the effect of the tuning gradually diappears ; insomuch that the best phase of the instrument, to our ears, is exhibited during the period which precedes its becoming offensively out of tune. At this time the progress towards the state of being out of tune (for which there is no single word, maltonation would do very well) can only be called a change of the temperament ; andd the several keys begin to exhibit varieties of character which, until maltonation arrives, render the instrument more and more agreeable. But it must be remared with respect ot equal temperement, that it cannot be obtained in the ordinary way of tuning. The only way of obtaining a given temperament, equal or unequal, with certainty, is to take a monochord, and having calculated the proper lengths of the different strings, to form the successive notes on the monochord, and to tune the several notes of the instrument in unison with them. No tuner can get an equal temperament by trial : so that the question lies between the having all sorts of approximations to equal temperament, according to the propensities of different ears, or as many sorts of approximations to some other systems. Had the English nation been as musical as it is mechanical, a portable monochord, or a system of monochords, would have been invented, on which any system of temperament could have been readily laid down by rule, and thence transferred to the instrument.

The mode of proceeding by approximation to equal temperament is simply to tune the fifths a little too flat, and the following order of proceeding is the most usual, and has often been given. The first letter represents the note already tuned, the second the one which is to be tuned from it : a chord interposed in the parentheses represents the trial that should be made upon notes already in tune, in order to test the success of the operation as far as it has gone. The first step is to put C' in tune by the tuning-fork : -

C' ; C'C ; CG₁ ; G₁D ; DA ; AA₁ ; A₁E ; (CEG) EB ; (CEG, DGB) ; BB₁ ; B₁F# ; (DF#A) ; F#F₁# ; F₁#C# L (A₁C#E) ; C#G# ; (EG#B); C'F ; (FAC'); FA₁# ; (A₁#DF) ;A₁#A# ; A#D# ; (D#GA#) D#G₁# ; (G₁#CD#).

We have written all the semitones as sharps, whether tuned from above or from below. Of course, since the fifths are all to be a little too small in their intervals, the upper note must be flattened when tuned from below, and the lower note sharpened when tuned from above. In the preceding the octave CC' is completely tuned, and also the adjacent interval F₁C#. The rest of the instrument is then to be tuned by octaves. The thirds should all come out a little sharper than perfect, as the several trials are made : when this does not happen, some of the preceding fifths are not equal. The parts which are first tuned by fifths, and from which all the others are tuned by octaves, are called bearings.

We shall now show how, be means of the theory of the scale, to examine a system of temperament : the rest of this article is therefore only for those who have some mathematical knowledge of the scale. Everything will be expressed in mean semitones, and the following addition will be convenient. A major tone is 2.039100 mean semitones ; a minor tone is 1.824037 ; a diatonic semitone 1.117313 ; a comma .215063 ; the excess of twelve perfect fifths above seven octaves .234600, a little more than a comma, frequently called a comma ; the excess of an octave above three perfect thirds .410689. Various modes of dividing the octave have been proposed, that is, of creating imaginary subdivisions, by means of which to express the various intervals required. None is so convenient, in our opinion, as the expression by means of mean semitones and their fractions.

...We are for variety in the several keys, and against equal temperament ; but we do no like variety without law. We do not like, for example, to find the greatest temperament in one key, and the least in an adjacent key, as that of the dominant or subdominant. Suppose then we ask what can be done towards an ascending and descending temperament, say from the key of C, shall increase through the keys of C, G, D, A, E, B, and diminish through those of F', C', G', D', A', F. And as a first step, let the increments and decrements of the temperaments of the fifths be equal, or let C = m, G = 2m, D = 3m, A = 4m, E = 5m, B = 6m, F' = 7m, C' = 6m, G' = 5m, D' = 4m, A' = 3m, F = 2m. Here, as far as the fifths are concerned, the effect of modulation into the dominant or subdominant keys is the same everywhere, as much as in equal temperament. And, from the first rule, we have 48m = .2346, or m = .0048875, and the greatest temperament of a fifth is seven times this, or .034. Now if we compute the temperaments of the thirds, major and minor, from the fourth and fifth rules, we may exhibit the temperaments of all the keys, as follows : -

The three columns contain the temperaments of the minor third, major third, and fifth. The effects of modulation into adjacent keys are everywhere small, nowhere amounting to more than about the tenth of a comma, in alteration of temperament ; while the fifths are in different keys so differently tempered, that is in C that interval may be called perfect, while in F# there is nearly twice as much temperament as in the equal semitone system. There is then variety without sudden change. In the system of equal semitones, the temperament of the minor third, major third, and fifth, are always

-.156   +.137   -.020

Now to form the scale in this system. Proceeding by the table given above, of which we take a few steps as an example, we have

* Throw out the twelves as fast as they arise.

Proceeding in this way, we find for the intervals of the several semitones from the key note, expressing in mean semitones, the following table :

To carry this or any other system strictly into practice without comparisons with the monochord, or the use of beats, presently described, would be impossible ; but the following might be suggested as an approximation. In tuning by fifths, let the intervals CG and FC be made perfect, or all but perfect ; let there be greater temperament in GD, DA and D#A#, A#F ; and most of all, decidedly in the remaining intervals.

The system of temperament is sometimes described by giving the number of vibrations made by the several semitones, or numbers proportional to them. It is easy enough to deduce the number of mean semitones in each interval from such data, either by the common tables of logarithms or by that given in SCALE.

First, by the common table of logarithms. From the logarithm of the number answering to the higher note, subtract that answering to the lower ; from the result take its three-hundredth part, and multiply the remainder by 40. The product is the number of mean semitones in the interval, with an excess of very little more than the thousandth of a mean semitone in an octave. For example, to find the intervals, in means semitones, of a fifth and of a comma ; in the former of which the lower note makes two vibrations while the higher makes three, and in the latter 80, while the higher makes 81 :

Next, by the table in SCALE (p.506). If the nubmers be in the table, simply subract the logarithm of the lower number from that of the higher, and the result is the answer required, within about the hundredth of a mean semitone. But if the numbers be not in the table, divide both by any number which will being them within the table, accurately or approximately, and then subtract as before ; interpolation may of coursse be employed, but if the skill of the computer does not reach so far, he must be content with a less accurate result, or must use the common table, in the manner just explained. For instance, one note makes 4622 vibrations, while another makes 5033 ; required the interval between them. Divide both by 30, which gives 154.1 and 167.8 ; if without interpolation, say 154 and 168. Opposite to 168 is 88.80, and opposite to 154 is 87.20, differing by 1.5, or a mean semitone and a half. The interpolated logarithms are 88.08 and 87.21, differing by 1.47. The more accurate result of the former rule is 1.4752.

The tuning of a piano-forte is generally done by ear, but in that of an organ, recourse is had to the beats which imperfect consonances always give. In the tempereament of this last-named king of instruments, less liberty is allowable that in that of the stringed instreuments : for not only do the beats become unpleasantly frequent when a consonance is too imperfect, but the imperfection of the consonance itself is more perceptible when notes are held, as in the organ, than when they die rapidly away, as in the piano-forte. These beats are described in ACOUSTICS (p.97) and when the lower note is known, and also its number of vibrations, the number of beats which are made in a given time, as ten seconds, a minute, or any other which is convenient, can be calculated from the known imperfection of the consonance, and the number of vibrations of the lower note. Theoretically speaking, it makes some little differennce whether the consonance is tempered sharp of flat, but not to an extent which it is worth while to consider. The rule for determining beats is as follows : let the lower note of the (perfect) consonance make n vibrations, while the upper note makes m, the fraction m ÷ n being in its lowest terms, and let N and M be the actual numbers of vibrations in the lower and higher notes, per second : then mN = nM. Let μ be thre fraction of a mean semitone by which the consonance is tempered ; then the number of beats in a minute is found by taking the fraction μ of the production of mN or nM, multiplying by 1109 and dividing by 320, or by 4, 8, and 10. The algebraical formula is,

[1109/320] μmN or [1109/320] μnM ([1109/320] = 3.4656)

For example, let the note C' make 512 vibrations, it is required to find the number of beats per minute in the consonance C'G', when tempered as in the system of mean semitones. Here N = 512, m = 3, mN = 1536. The exact fifth is 7.01955 means semitones, whence the fraction is .01955, since the tempered fifth has seven semitones exactly. Multiply .01955 by 1536, which gives 30.0288 ; multiply by 1109 and divide by 320, which gives 104.07 (say 104) beats in a minute.

Tables for facilitating the calculation might easily be made, but it is hardly worth while to insert them here. The beats are usually, we believe, sumply counted with a watch, but it would be both convenient and exact to have some such machine as Dr. Smith recommended, a pendulum which could be easily altered in vibration, and first adjusted to move exactly with the beats : the pendulum might then be subsequently compared with the watch. Without such a contrivance it is very difficult to tune the piano-forte by beats, since they do not last long enough in sufficient intensity : with it the last named instrument might easily be tuned on any system of temperament, and those who practice the art would have the advantage in hearing different systems, knowing at the same time what those systems are. At present the organ-builder is the only tuner who makes any approach to science ; all the rest judge only by the ear, which may vary from time to time, or even with the state of the body, or the weather. We have as many reasons for thinking that the ear alone is a valuable judge in so nice a matter as temperament.[(Professor De Morgan)]

Notes and Queries : Organ tuning (1857)

Augustus De Morgan, A Budget of Paradoxes (1911)

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