Page:Polynesian Mythology by George Grey (polynesianmythol00greyuoft).djvu/363
presented nearly by 120 being the lowest note; then 116.60:113.39.
I shall not trouble the reader with chronological or scholastic differences; the diesis of Archytas being given by Vincent as , that of Eratosthenes as 117:114, for keen indeed must be the ear that could discern between and (except in harmony); much more difficult still would it be to discover a difference between 116.60:113.39 and or .
If any wish to examine this matter more closely, they can consult the Treatises on Harmonics. Mr. Vincent has calculated these differences by logarithms to the 60 root of 2.
My point is, to prove that the ancients did possess and practise a modulation which contained much less intervals than ours, and that such, or an approach to such, modulation (though probably but imperfect) is still retained among some people, and that the principles on which the Greeks founded their enharmonic genus, still survive in natural song, though I will not be bold enough to assert that sometimes these songs may not change into one of the chromatic χϱόαι, which, for want of practice, I might not be able to decide. One thing, however, is certain, that, as Aristoxenus tells us, no perfect ear could modulate more than two dieses at a time (and then there was a "saltus" or interval of two tones), and as the New Zealand songs frequently exhibit more than two close intervals together, it is more than probable that many of these songs are a chromatic, represented by 120, 114, 108, or 120, 112½, 108; but it will not be worth while for the present purpose to discuss this nicety, as all we want is a practical approximation.