We shall treat only of the diatonic scale, which is the basis of European music, and is approximated to as closely as is consistent with convenience of construction in key-board instruments, such as the piano, where the eight white notes beginning with C and ending with C an octave higher may be taken as representing the scale with C as the key-note.
The Diatonic Scale.
Now suppose we take G as the key-note and form its diatonic scale.
It was so early recognized as characteristic of Chopin that a magnificent example may be seen at the end of Schumann's little tone-portrait of him in the Carnaval: a very advanced Wagnerian passage on another principle constitutes the bulk of the development in the first movement of Beethoven's sonata Les Adieux; while even in the " Golden Age " of music, and within the limits of pure diatonic concord, the unexpectedness of many of Palestrina's chords is hardly less Wagnerian than the perfect smoothness of the melodic lines which combine to produce them.
But Zarlino uncompromisingly declared that the syntonous or intense diatonic scale was the only form that could reasonably be sung; and in proof of its perfection he exhibited the exact arrangement of its various diatonic intervals, to the fifth inclusive, in every part of the diapason or octave.
The brilliant success of Humperdinck's Hansel and Gretel, in which Wagnerian technique is applied to the diatonic style of nursery songs with a humorous accuracy undreamed of by Wagner's imitators, points a moral which would have charmed Wagner himself; but until the revival of some rudiments of musical common sense becomes widespread, there is little prospect of the influence of Wagner's harmonic style being productive of anything better than nonsense.
Claudius Ptolemy (130) rectified this error, and in the so-called syntonous or intense diatonic scale reduced the proportions of his tetrachord to s, iii, f, -i.
The frequency ratios in the diatonic scale are all expressible either as fractions, with i, 2, 3 or 5 as numerator and denominator, or as products of such fractions; and it may be shown that for a given note the numerator and denominator are smaller than any other numbers which would give us a note in the immediate neighbourhood.
If we write down the eight notes from G to g in the key of C, their frequency ratios to C, the frequency ratios required by the diatonic scale for G, we get the frequency ratios required in the last line: - We see that all but two notes coincidewith notes on the scale of C. But instead of A = we have n, and instead of f= 4 we have 4 b.
It is evident that for exact diatonic scales for even a limited number of key-notes, key-board instruments would have to be provided with a great number of separate strings or pipes, and the corresponding keys would be required.
The greater tone, lesser tone, and diatonic semitone of modern music. 2 Ptolemy set forth this system as one of eight possible forms of the diatonic scale.