EFGABC', EAFAGBA If we chose to confound the intervals of a major and minor tone, WI should find the second of these (so it happens) correct, for the interval, of the original air are (in, minor tone; M, major tone ; a, semitone m 31 a 31 in (2 M + 2 m + s) (31+ In +s) (31+s) (3E+ s) a (31+a) 31, am those of the second are lim s 31m (2 M +2 irn + s) (31+ m +s) (in +8 (In + a) s +s) 31, which are undistinguishablo from each other, if D. and m be supposed (as is the fact) too nearly equal to make it wort] while to take account of their difference. But the third is slim 31 (211+ 2 s + m) (31+ rn + s) (31+ in) (31+ m)m (m + 31, which does no agree with either of the other two, nor can do so, except to an ea which cannot distinguish a from In or 31. To see what intermediat notes will be wanted, we must construct a diatonic scale on each of thy seven notes, which we shall now do, putting an equivalent to ever: note above C' or below C into the octave between C and CI, by balviR or doubling the fraction which expresses its vibrations. Moreover, w express the notes in tho diatonic scale on D by D, D, D, and and so on. Also let stand for an octave below stand for the note an octave below C, and so on, the rule that „,C and C,, are octaves when n: and n together mako nine. Al this is well known, if anything of the scale be practically understooc] What we have to do, for instance in forming the diatonic scale on F, i to take I, the representative of F in the diatonic scale of C, and multipl it successively by I, !, &c. Our scales then are as follows, puttin down under each note gained any note of the original diatonic scalm or any one of the scales previously formed, from which it differs insensibly little, removing each note an octave lower when necessary.
In order therefore to make an instrument which shall play in perfect tune in every one of these diatonic scales, we must have it capable of sounding the following notes, those of the original diatonic scale, or very near to them, being in parentheses, and requisite notes of nearly equal sound being written under one another.
n (Z In 1:)(1) Cl/ tt\iti (V) (2) in Y../ Wei /I \11) With this we might go on ad infinitum ; for it might be required to construct new diatonic scales upon every one of these new notes, which would introduce more new notes, on which again new diatonic scales might be produced, and so on. But since the original scale consists only of major and minor tones (nearly equal) and diatonic semitones (nearly half-tones), the now notes will very nearly divide the whole tones into equal parts, a circumstance of which advantage will presently be taken. In the mean time we proceed to explain, so far as it can be done, the distinction musicians draw between lam and sharps. Unfortunately we are unable to make writers on this subject agree with each other, or with themselves, as to the meaning of these words in an it-tampered scale. The conventions under which the names sharp and flat are used come easily enough, and temperament avowedly makes small adjustments and accommodations between the several notes, which cause the sharpened A to be practically the same as the flattened B, and so on. But what the clear and admitted distinction
of sharp and flat is previously to that adjustment, we wish we could evoke or provoke some musician to tell us. One word to those who write on the scale without much mathematical knowledge : get into a tempered scale as fast as you can, and keep there.
The nomenclature is regulated as follows. The notes A, B, C, D, E, F, 0 are preserved in every key ; so that if any key contain A and a note between A and 13, the latter is not called A.:—for then (A A:) A would occur twice in the scale—but Be, giving A Follow this rule in every one of the scales just given, and we shall find the following seta of notes in them severally :— That is to say, a diatonic scale on D, for instance, only keeps D, E, 0, A, B, or notes very near to them, of the diatonic scale, and requires the insertion of notes between F and 0 and C and D, which the avoidance of repetition of letters requires us, to denote by F.: and C:, and not by Ga or D. In the preceding keys then, we have five sharps mentioned (though really seven notes of the kind, two between C and D, two between F and 0, one between each of D and E, G and A, A and B) and one fiat (between A and B). Where are the other two sharp" I If we construct diatonic scales upon F: and C: we shall find with not precisely the same notes as before, but very near to them, excepting two notes which are new ; one between E and G (called E:), and one note between B and C (called =). But on which of the values of F: and C: in the table are these scales to be constructed, and why I Again, as to the Bate, if we construct diatonic scales on I3'), and on each new flat as it is successively Introduced, we shall find that our nomenclature gives ua new keys, ae follows :— But if we were actually to proceed to form this scale, beginning from l32 (V gained from the preceding process, we should find ourselves keeping very near the chromatic scale of sharps already obtained, so that the notes which appear in the preceding as remnants of the diatonic scale would really be close to the real notes. Let us see, for intanee, what the F would be in the key of Gy a fourth above F=le, is fourth above B b =II.
=p, A b., a fourth above El, D' a fourth above = 0,4, Dis G b, a fourth above Db = Ay.
F' (so called), a seventh above G x y F (so called) = F (really) = t.
Now from t the interval is only the comma, which we meet with so often elsewhere. But we should different values for the same flat in the different keys above, just as we have found different values for the same sharp in the preceding. To show, however, in how confused a state the natural chromatic scale has been left, we copy three scales, the first from Wallis (` Phil. Trans.,' No. 242, 1698), the second from La Borde(‘ Essai sur la Musique ancienne et moderne,' voL ii, p. 9, A.D. 1780), and the third from Montferrier (` Diet. des Math.; voL iii. p. 213, ex. 1840). All these writers omit the fiats, mentioning only the sharps :