Enders, Bernd / Stange-Elbe, Joachim (Hrsg.): Global Village - Global Brain - Global Music

the sharpening of the 4th scale degree F

F_# in C-Major together with the flattening of the 2nd scale degree D

D_b in C-Minor. In modulations these alterations produce, for example, a 7th scale degree in G-Major and a 6th scale degree in F-Minor, respectively. Again, the alteration-pair as well as both resulting scales are explained by suitable denotators Alteration₂ and DualAlteredScales₂.

“Alteration1” : PiM od12DualSteps((8,9),(11,10)) “Alteration2” : PiM od12DualSteps((5,6),(2,1)) “DualAlteredScales1” : PiM od12DualSets( {0,2,4,5,7,9,10},{0,2,3,5,7,9,10}) “DualAlteredScales2” : PiM od12DualSets( {0,2,4,6,7,9,11},{0,1,3,5,7,8,10})

Finally, we inspect two mutually dual sequences discussed in Harrison’s book:

Sequence 1 (Bach)

Figure 2:

Sequence from Bach’s G-Minor Fantasy, BWV 542 mm. 31-34 as simplified to Harrison ([4], p. 33

We consider four Denotators of the Form PiMod₁₂Set denoting the 3rd and the 2nd chord of this sequence (a minor tonic third t and a Major Dominant D with respect to C-Tonality), as well as the union of 3rd and 4rd chord (the “sharpening” set #) and the union of the 2nd and 3rd chord (the authentic cadence set A)

“t” : PiM od12Set({0,3}) “D ” : PiM od12Set({7,2,11}) “# ” : PiM od12Set({0,3,4,7}) “A ” : PiM od12Set({7,2,11,0,3})

The entire sequence can be modeled as a global composition by gluing several copies of these four charts. In fact, it is the Colimit of the following diagram of Denotators(!):

PICT