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4 HOPI SONGS
height or depth. The elements of the series are individual tones, and musical intervals are relations between these. Such relations may be conceived as themselves having relations in tone, combining intervals into orders. Excluding the limiting relation of identity or unison, an interval order may be defined as a set of intervals in which one of the tones, a or b, forming one of the set, forms with a tone c, different from either a or b, a second of the set in a certain sense; and one of these three different tones forms with a tone d, different from either a, b, or c, a third of the set in a certain sense: and so on until the set is exhausted.
The different tones conceived as the elements of an interval-order are known as the steps of the order. A step may be defined as a tone conceived as related to every one of a certain number of others, either directly by one of a certain set of intervals in a certain sense, or indirectly through others of the tones and by a certain sequence from the intervals, each in a certain sense.
It is through the conception of its steps that an interval-order becomes operative in interval combination. The viewing of notes as steps is the differentia of scalar music. A scale is a complex relation in tone with one or other element of which each note of certain music is identified by the maker.
In European music the names of the notes — a, b, c, do, re, mi — designate steps in one order of intervals called the diatonic scale. Every note of a European composition either has its diatonic name or is regarded as an accidental, the consciousness of scale being the invariable concomitant of musical production. The names of the accidentals also give them a place in the scale-system as new loci resulting from a changed conception of the step represented by a given tone, — or, as it is called, modulation from one key to another. Scale is even so familiar a fact as to be difficult to grasp. We fancy that music without it would hardly be music. Yet if the consciousness of scale be simply the embedding of each actual note in the same fancied complex of related notes, its presence is not essential to the combination of intervals into a work of art. Taking away the scale-consciousness takes away a norm