(§ 12.) "To avoid unnecessary labour, let us first learn what they can tell us, and see whether anything is to be added to it; retaining our own view on such subjects: namely this:—that those whose education we are to superintend—real philosophers—are never to learn any imperfect truths:—anything which does not tend to that point (exact and permanent truth) to which all our knowledge ought to tend, as we said concerning astronomy. Now those who cultivate music take a very different course from this. You may see them taking immense pains in measuring musical notes and intervals by the ear, as the astronomers measure the heavenly motions by the eye.
"Yes, says Glaucon, they apply their ears close to the instrument, as if they could catch the note by getting near to it, and talk of some kind of recurrences[337]. Some say they can distinguish an interval, and that this is the smallest possible interval, by which others are to be measured; while others say that the two notes are identical: both parties alike judging by the ear, not by the intellect.
"You mean, says Socrates, those fine musicians who torture their notes, and screw their pegs, and pinch their strings, and speak of the resulting sounds in grand terms of art. We will leave them, and address our inquiries to our other teachers, the Pythagoreans."
The expressions about the small interval in Glaucon's speech appear to me to refer to a curious question, which we know was discussed among the Greek mathematicians. If we take a keyed instrument, and ascend from a key note by two octaves and a third, (say from A1 to C3) we arrive at the same nominal note, as if we ascend four times by a fifth (A1 to E1, E1 to B2, B2 to F2, F2 to C3). Hence one party might call this the same note. But if the Octaves, Fifths, and Third be perfectly true intervals, the notes arrived at in the two ways will not be really the same. (In the one case, the note is ½ × ½ × ⅘; in the other ⅔ × ⅔ × ⅔ × ⅔; which are ⅕ and 16/81, or in the ratio of 81 to 80). This small interval by which the two notes really differ, the Greeks called a Comma, and it was the smallest musical interval which they recognized. Plato disdains to see anything important in this controversy; though the controversy itself is really a curious proof of his doctrine, that there is a mathematical truth in Harmony, higher than instrumental exactness can reach. He goes on to say:
"The musical teachers are defective in the same way as the astronomical. They do indeed seek numbers in the harmonic notes, which the ear perceives: but they do not ascend from them to the Problem, What are harmonic numbers and what are not, and what is the reason of each[338]?" "That", says Glaucon, "would be a sublime inquiry."
Have we in Harmonics, as in Astronomy, anything in the succeeding History of the Science which illustrates the tendency of Plato's thoughts, and the value of such a tendency?
It is plain that the tendency was of the same nature as that which induced Kepler to call his work on Astronomy Harmonice Mundi; and which led to many of the speculations of that work, in which harmonical are mixed with geometrical doctrines. And if we are disposed to judge severely of such speculations, as too fanciful for sound philosophy, we may recollect that Newton himself seems to have been willing to find an analogy between harmonic numbers and the different coloured spaces in the spectrum.
But I will say frankly, that I do not believe there really exists any harmonical relation in either of these cases. Nor can the problem proposed by Plato be considered as having been solved since his time, any further than the recurrence of vibrations, when their ratios are so simple, may be easily conceived as affecting the ear in a peculiar manner. The imperfection of musical scales, which the comma indicates, has not been removed; but we may say that, in the case of this problem, as in the other ultimate Platonic problems, the duplication of the cube and the quadrature of the circle, the impossibility of a solution has been already established. The problem of a perfect musical scale is impossible, because no power of 2 can be equal to a power of 3; and if we further take the multiplier 5, of course it also cannot bring about an exact equality. This impossibility of a perfect scale being recognized, the practical problem is what is the system of temperament which will make the scale best suited for musical purposes; and this problem has been very fully discussed by modern writers.