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6                                            HOPI SONGS
On formal analysis the following appear the logical rights and duties of the two sides to the question: —
For scale. The coincidence of any music with the diatonic order of intervals is prima facie evidence that that order is the scale of the music; but the evidence is not complete without an explanation of its divergences therefrom. These may be accounted for in two opposite ways: either as inaccuracies of execution or complexities of conception. The musician may rudely grasp (or roughly deliver) the diatonic step he proposes; or he may change it by an accidental; or view it as a new step in uttering it, that is, go on in a new key.
Against scale. Any opposing theory of a music showing diatonic features must both explain its coincidences with diatonic form and show the insufficiency of mistake, alteration, and transposition to account for its divergences therefrom. The coincidences will be accounted for if proved characteristic of harmonic sequences, or the contribution of the observer. The divergences will be proved not to be mistakes if repeated by the same or different singers; and not to be accidentals or due to modulation if they imply complex changes of the pitch or conception of notes.
The evidence of the present notations bears strongly against the diatonic theory of this music on all four of these points. A measure of coincidence with the diatonic scale is implied in a predominant use of approximations to intervals of simplest ratio such as the phonograph here records. The diatonic form of the notations by ear proves in part the invention of the observer. The divergences of the songs from dia­tonic norms are in many cases shown to be intentional by repetition. In many cases also their explanation by the shifting of notes or of key demands an improbably complex use of the scale.
A measure of coincidence with the diatonic scale is implied in the predominant use of approximations to con­ sonant intervals in melody.
Any sequence of the seven consonant intervals, minor third, major third, fourth, fifth, minor sixth, major sixth, and octave, tends to move within, and hence to outline the diatonic scale. There are seven
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