- 413 -Enders, Bernd / Stange-Elbe, Joachim (Hrsg.): Global Village - Global Brain - Global Music 
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Tonic base. Dually he considers the discharge 6-5 of the Subdominant agent into the Tonic associate. Both discharges are highly significant for key determination. The “DualDischarge”-Denotator of the Form PiMod12DualSteps expresses this duality:
“DualDischarge” : P iM od12DualSteps(((11,0),(8,7))
  • Function: Dominant - Subdominant. The initial inversion e711 maps the G-Major-Triad onto the F-Minor-Triad and vice versa (see DualFunction1), and it maps the G-Minor-Triad onto the F-Major-Triad and vice versa (see DualFunction3). Furthermore it exchanges the G-“Dominant-Seventh-Chord and the F-Minor-Chord with added Sixth (see DualFunction2).
    “DualFunction1” : P iM od12DualSets({7,11,2},{5,8,0})
 “DualFunction2” : P iM od12DualSets({7,11,2,5},{5,8,0,2})
 “DualFunction3” : P iM od12DualSets({7,10,2},{5,9,0})
  • Cadence: Authentic / Plagal. With respect to the functional dualism between Dominant and Subdominant and the modal dualism between the two Tonic variants as well as the two dominants there results a dualism between certain authentic and plagal cadences, namely those which differ modally in both chords:
    “DualCadence ” : PiM od DualSetSteps(
 (1{7,11,2},{012,4,7}),({5,8,0},{0,3,7}))
 “DualCadence ” : PiM od DualSetSteps(
 (2{7,11,2,5},12{0,4,7}),({5,8,0,2},{0,3,7}))
 “DualCadence ” : PiM od DualSetSteps(
 (3{11,2},{0,412,7}),({5,8},{0,3,7}))
 “DualCadence ” : PiM od DualSetSteps(
 (4{7,10,2},{012,3,7}),({1,5,9},{0,4,7}))
  • Semicadences: These are obtained as retrograde versions of the cadence steps. It is clear that the resulting denotators fit into the PiMod12DualSetSteps-Form. We omit the details. The inner symmetry of all four Dual-Forms makes clear, that it is not possible to formally decide upon what is left and what is right in Harrison’s table.
  • Alteration: Sharpen / Flatten. Harrison lists two corresponding pairs of alterations:
    • the flattening of the 7th scale degree in C-Major B ~> Bb together with the sharpening of the 6th scale degree Ab ~> A in C-Minor. In modulations these alterations produce, for example, a 4th scale degree in F-Major and a 2nd scale degree in G-Minor, respectively. The alteration-pair as well as both resulting Scales are explained by suitable denotators Alteration1 and DualAlteredScales1.


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