Pairs of Interval Classes in Southeast Asian Tunings
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Construed non-numerically (Rahn 2011, 2012, 2013), the following normal-Forte-order formulations accurately model southeast Asian fixed-frequency tunings: sléndro 11111…, the ‘usual’ pentatonic 22323…, Thai pentatonic 11212, 5-tone pélog 11313; Thai ‘equiheptatonic’ 1111111…, diatonis/diatonic 1222122, and 7-tone pélog 1112112. In well-documented instances, two or more of these tunings appear in single pieces that have been realized in one or more cultural settings. In order to convey the consequences of such ‘translations’ from one tuning to another, seemingly distinct tuning, one can observe that since each tuning is ‘well-formed’ (Carey and Clampitt 1989), each maximizes the number of interval-pairs within particular generic-specific interval-classes. In ideal, mathematical terms, if d is the number of steps in a register, the number of such interval-pairs is d2(d-1)/2in ‘degenerate’ sléndro and Thai equiheptatonic, and d(d-1)(2d-1)/6 in the remaining, ‘non-degenerate’ tunings. The formulation outlined above identifies salient structural relationships between realizations of single instrumental pieces in otherwise contrasting tunings and between passages comprising ‘exchange tones’ (métabole) within individual pieces.
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