We're going to take a short break from the mathsy stuff, and consider some other things with regards to scales and music. The approach to scales I am taking is very much a western approach - although probably not unique to the west, it lacks things that are not present in the concept of scales in western music. It might be informative to look into what other cultures subsume into their concept of musical scales.
Many cultures consider scales not just a set of pitches, but a tuplet consisting of a set of pitches and a basic rhythm. Melodies or improvizations in a scale are expected to use that rhythm. The same pitch set can appear with another rhythm, but is then considered another scale. There's no guarantee that all rhythms are represented with all pitch sets. Examples of such systems are Indian ragas, Arabic maqamat and the ancient Greek scales.
Some Indian traditions also associate scales with time of day and other things - songs in some scales are supposed to be played in the morning, songs in others in the afternoon, etc.
In Europe, we tend to associate chords and progressions with keys, but since our keys are all identical copies that are transposed, the same chord progression can work in any key, provided it's been properly transposed. This is not necessarily the case in scales everywhere - even some renaissance instrument makers used temperaments that omitted parts of the gamut, and you can find some modern instruments even that lack several of the twelve tones usually expected - some harmonicas, for instance, or autoharps.
Some systems go further than that and approach their entire music's tuning like our harmonicas or autoharps - just pick one key, and don't bother making any other keys playable (duly note that the renaissance instrument makers that restricted the number of available keys still usually made their instruments have the full scales of C, F, G and a few others available). Indonesian Gamelan, to some extent, is an example of curtailing the available intervals in such a way. The Gamelan, however, further complicates things by having two entirely unrelated scales - one essentially five equal steps to the octave, the other close to a seven-note subset out of nine equal steps to the octave. Even then, variation between one village's gamelan and another may be drastic, and their tuning is thoroughly an art, and not an industrial standard.
Further, a scale may be somewhat flexible - lots of Chinese and Japanese music, as well as the American blues flex their tones' pitch considerable during the course of a song. This seems more common the smaller the size of the used tonal palette. Slightly comparable may be how the melodic minor is different when ascending from when it's descending. (In practice, the pitches that are altered - in Cminor, they'd be A and Ab, B and Bb - are not just altered when descending or ascending in actual compositions - it depends on chord choices, etc.)
Western music has been rather chord based for quite a while. Ancient Greek, Arab, Turkish and Persian music are rather more 'tetrachord'-based - a tetrachord is a span of subsequent tones , and the melody tends to weave melodies out of those four notes for quite a while, until switching to another tetrachord. From what I gather, they tend to switch tetrachords either so that the first scale and the last scale share the middle tetrachord - i.e. like going from GABC to CDEF to GABbC - CDEFGABc is C major, FGABbcdef is F major - CDEF is in both, or just switching between two tetrachords with the same end-points: GABC to GABbc. Playing around with such concepts (and extending them or retracting them), may provide some ways of making very 'culture-specific' music.
Unlike tetrachords, our chords tend to have the tones played simultaneously (although the practice of arpeggiation - sounding one chord tone at a time - can be played in a way that deviates from that), but they also form a very important melodic backbone; lots of melodies mostly consist of the tones of whichever chord is currently playing with the occasional tone outside of that set. We can of course come up with rather drastically different chords which might be used in similar ways.
Finally, other practices may be associated with different scales - working songs to ensure the right working rhythm, different scales for different religious uses, etc. There's a world of possibilities.
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