And first as to enharmonic scales, which are mentioned first, and seem to have been ancient, and regarded with high approbation.
It seems then that the enharmonic system would allow only of the following notes in an octave— where P means a note about half way between E and r, and Q one half way between 11 and a'. An edd scale truly for a.modern musician to look at; but, it may be, not incapable of pleasing effects to ears not accustomed to music in parts.
The chromatic scales come next in order, as follows :— To make something as like as we can to those scales, we should write down In modern musio The diatonic scales, Ptolemy allows, aro more agreeable to the ear, and his specimens are as follows : we shall now write the scale with the usual letters throughout.
These scales have all so far the diatonic character that they divide the tetrachord into two larger intervals followed by a smaller one : the scale of Didymus would have been exactly the modern untempered diatonic scale, if he had inverted the order of the two larger intervals in his second tetrachord. As to the other modes, the Dorians, &c., there is much confusion in Ptolemy respecting them, arising from the corruptness of the text, which Wallis has endeavoured to remedy. According to him, there are divisions of the octave, somewhat more fantastic than those which precede. In more recent times the idea has been started of their being simply different keys, or rather answering to different variations of the diatonic scale, by using intermediate semitones instead of some of the notes : it would be difficult, we think, to produce authority enough for this conjecture.
If it were true, as supposed, that the two octaves of the Greek scale, beginning, say with A, were minor, it would follow that Ptolemy, in his diatonic scales, exhibited the octave from a to a', as we have supposed. Accordingly, the principal mode of exhibiting the forma tion of the octave from two tetrachords and a tone would be the one we have taken, namely, On this point we shall only say that there never was, we believe, so strong a union of the three characters of scholar, mathematician, and musician, as was seen in Dr. Smith, the author of the Harmonics. lie
had studied the Greek scale attentively, and to him the first of these methods was a matter of cdurse. " The Greek musicians" (' Har monics,' 1749, p. 45), "after dividing an octave into two-fourths, with tho diazouctic or major tone in the middle Gefween them, and admitting many primes to the composition of musical ratios, subdivided the fourth into three intervals of various magnitudes placed in various orders, by which they distinguish their kinds of tetrachords." We do not, we confess, though admitting that it is exceedingly hard, and probably impossible, to reconcile the Greek writers with them selves and each other, find that sort of difficulty which Dr. Burney owned to, when he said that he neither understood those writers himself, nor had met with any one who did. He was a musician, and was looking out for an intelligible mode of arriving at and distributing the most agreeable concords, with a strong predetermination to arrive at musical truth or nothing. But the Greek writers were arithmeti cians, with as strong a determination to find natural foundations in integer numbers : they did not ask how to find sounds which would best suit the ear, but ho* to discover triplets of fractions which multiplied together should produce four-thirds of a unit. Pleased with the simplicity of the ratios which give the fourth, fifth, and octave, their efforts at musical improvement were confined to the attempt at discovering magic numbers to fill up the intervals. It was not until one of these philosophers had laboured at his abacus, and tasked his metaphysics to find d riori confirmation of some question in arith metic, that he strung his monochord and tried how his scale sounded : It would have been hard indeed if his ear had refused to sympathise with his brain. In all probability the musicians, whose object was simply to please, laughed at the arithmeticians, as Tycho BralaS did at Kepler, when the latter had discovered reason for the distances of the ' planets in the properties of solid bodies : they had motive enough, and, beyond all question, reason more than enough.