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10. Mixed scales with three shi/ted centres. It is possible to arrange series of notes in such a manner that every successive note belongs to another centre. The result is ascale, which will be called a mixed scale because members of the various melodie subscales have been mixed up. The first example shown is the major scale of TARTINI, containingeight notes and supported by a perfectly symmetrical bass which shows three centres at distances of a perfect fifth . I. 11 '" ft d : .. 5 : I:dey:t:doh I I I r I r 1 46 MIXED SCALES The example shows, on aseparate stave, other scales which are suitable for use in accompanying parts.
The pairs of leading notes a and b Lj" f \~ and g (lah : dey = muy : soh = 20 : 21, or me: sey = tuy : ray = 20 : 21) have been met before. 13. Inversion, centres in t and e Lj" or t and d ~ (fah : tey 4: 7). The inversion in question produces the following notes: = f:g :a :c :eLj, f: aLj,: cLj,: dLj,: eLj, Because it would be a littIe troublesome to invent vocables for d Lj" c Lj" a Lj" it is permissible to shift part of the second group through the imperceptible interval of 224: 225, so as to get g ~ : b : c ~ : d ~ and vocalise with fa : soh : lah : doh : fey ta: süh : te: düh: rüh f : s : J : d ,&[ r r:J I t is important not to get too high a pitch with e L" (fey or rüh).
111, §§ 7 & 8). There are two subscales with centres j and aL (jah and tey) and two connecting leading notes e and b LP (me and dey) which have their centres in c and in d L (doh and mey). Example W will be best suited for soprano and strings. ACKNOWLEDGMENTS In conclusion I wish to express my indebtedness to several friends, to the late composer WILLEM PIJPER, to his pupil JAN VAN DIJK, to Professor DOUGLAS R. , for encouragement and much useful help. s : : d : p d : : : d d : : d : f : -~ d : f .