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10 HOPI SONGS
at first, to justify this assumption. In working with it, particular care must be taken lest the conception scale be substituted for the conception interval. Petitio principii more adroitly steals thejivery oi logic to serve unreason in than any other fallacy. On closer study of the melodies, even the legitimate assumption becomes doubtful. More probably these singers aim neither at diatonic intervals nor at any
\ other limited set, but at indefinitely varying approximations to the
/former, precise only as they are run into the mould of each individual
Aside from this probability and supposing a text settled on any assumption (other than that of the point to be proved), it is to be noted that any sequence of tone can be exhibited as moving within any scale if entire freedom as to frequence and position of changed notes be granted; since every note that does not fit can be given the latter standing. But while theoretically admissible, indefinite chromatic freedom is a practical impossibility. The explanation of the adiatonic features of these songs by accidentals will be disproved by showing that the implied interference with the scale is improbable. In the adiatonic sequences just cited, entitled through their repetition to a place in a corrected text of the melodies, this interference sometimes affects sev-
/ eral semitones, a degree of freedom hardly credible. Logical parsimony demands a simpler explanation of these songs than a chromatic structure rivaling that of modern European music.
It is to be noted in like manner that any melody restricted to intervals of a given scale can be exhibited as moving within that scale if entire freedom as to change of key be granted. For, if any given transition is impossible from the step with which its start is now identified, it will by hypothesis exist from some other step, and its start needs only to be identified with that; this change of scalar significance being a modulation. As before, the disproof of the explanation will consist in showing not the impossibility but the improbability of the changes assumed. The adiatonic sequences just cited are simply accounted for by the tendency of these singers to move toward or past coming notes, make such combinations as they please, and substitute other