Those who are acquainted with the mathematical theory of the scale know that all the concords cannot be made perfect : others map learn it in the following way :—Suppose twelve perfect fifths to Is tuned upwards from c, and give the results the names of the note which they would represent in the scale of twelve semitones, and is the nomenclature of sharps, If the scale were perfect. We should the] have C, 0, Al, Y.', IS*, Fi:, G':, D':, A':, 11':.
Now since le: the same note as c', it appears that twelve perfec ftha should be tho same as seven octaves; and if we pass to the ctave below, as soon as we get out of the octave beginning with c, we hould have c, a, a, A, E, &e., the last being c'. It will be found, how ver, that this is not true, but that the note obtained from twelve scents by fifths intermixed with six descents by octaves is sharper hen 0. Again, if the perfect thirds be tuned from c, we should have E, o:, ; but it will be found that c' obtained in this way is too lat. The octave derived from the fifths vibrates 223 timed where it hould vibrato 220 ; and that derived from the thirds vibrates 125 imes where it should vibrate 128. The slight alterations which are uade in order that any one of the twelve notes of the octave may be it to be used as a key-note, without any shock to the ear, constitute he temperament of a scale ; the altered consonaucea are said to be empered. Some writers call the interval from the false octavo obtained By the fifths to the true one, by the name of the wolf; and using ho word in this sense, Lord Stanhope contends (and justly) for five mires, one from the fifths, and one from each of the seta of thirds mginning with c, c:, n, n;;; and three more might have been got from ho minor thirds. But by the term wolf other writers mean the bad fifth which exists in the worst key, when the temperament is allowed .o favour some keys at the expense of others. Simple as this little ..ariation in the meaning of a term may be, it is worth while to notice L A writer on tuning charges some of the pianoforte-makers of his Imo with utter ignorance of the scale, in stipulating with the tuners a-hom they employed that there should be no wolf. In all probability :hey only meant that no key should be worse than another, or that the :emperament should be equal. This term wolf is said to be derived from the jarring of a badly-tuned consonance, *supposed to resemble the Extant 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 throughout it ; the second, unequal, in which some mode is adopted of distributing the imperfection so as to make some keys fool it less than others. The most common practice of our day is to endeavour at
equal temperament. The two systems have their advocates, and the . arguments for one and the other are as follows. In favour of equal temperament it is urged that all the keys are made equally good, and that in no cue does the imperfection amount to a striking defect : also that in the orchestra there is little chance of any uniform tempera ment among the various instruments, if it be not this one. Against equal temperament it is urged that it takca 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. Wo have often felt that a pianoforte newly tuned has, with much correct ness, a certain insipidity, which wears off as the effect of the tuning gradually disappears ; insomuch that the best phase of the instrument, to our ears, is exhibited dining 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 ; and the several keys begin to exhibit varieties of character which, until maltonation arrives, render the instrument more and more agreeable. But it must be remarked with respect to equal temperament, that it cannot be obtained in the ordinary way of tuning. The only way, unless beats be used, of obtaiuing 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 iu 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. Ilad the English nation been as musical as it is mechanical, a portable mono chord, or system of monochords, would have been invented, on which any given 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 representa the note already tuned, the second the one which is to be tuned from it : a chord interposed in parentheses represents the trial that should be made upon notes already in tuue, in order to test the success of the operation as far as it has gone. Tho first step is to put c' in tune by the tuning-fork