[Smt-talk] Smt-talk Digest, Vol 43, Issue 2

[Thomas Noll]

Dear Rick, dear Colleagues,
this is very interesting and invites further work. In addition to the inversions of the triads and the diatonic modes, one should perhaps also take into account the width-balances of the individual voices. The descending major second moves flatward, while the descending minor second moves sharpward. 
If we saturate the progression from a sixth-four-chord C-F-A to the root position C-E-G along the line of fifths we obtain a sharpward pentatonic mode shift from C-D-F-G-A  to C-D-E-G-A which corresponds to a falling voice leading in pitch height from F to E. [NB: This is in analogy with a diatonic mode shift, from C-Ionian to C-Lydian, which corresponds to a rising voice-leading in pitch height from F to F#]. The other voice leading from A to G succeeds through the gaps. It requires further thoughts to understand the nature of this situation. David Clampitt has done some work on the study of triadic inversions within the framework of height-width duality. In particular, he was able to formulate a modal refinement of the "Structure Implies Multiplicity"-Property from Clough & Myerson.  
Sincerely
Thomas Noll


On 02.08.2012, at 16:02, Richard Cohn wrote:

> In response to Thomas Noll's point (copied below): in the opening pages of Studies in the Origins of Harmonic Tonality, Dahlhaus traces a basic duality between chord-based (Riemann) and scale-based (Fétis) theories of tonality. This basic duality manifests as a directional duality in pitch space. From the scalar standpoint,  
> a transposition up by a perfect fifth is a rising motion: F "voice leads" to F# (in Tymoczko's generalized conception of voice leading). From the chordal standpoint, a transposition up by perfect fifth is a falling motion: C voice leads to B, and E voice leads to D. It is worth reflecting on why both transpositions involve right-ward motion on the Tonnetz. 
> 
> --Rick Cohn
> 
> 
> 
> Message: 1
> Date: Wed, 1 Aug 2012 15:08:37 +0200
> From: Thomas Noll <noll at cs.tu-berlin.de>
> To: Nicolas Mee?s <nicolas.meeus at paris-sorbonne.fr>
> Cc: smt-talk at lists.societymusictheory.org
> Subject: Re: [Smt-talk] Gravity (Was: Car names)
> Message-ID: <7EAEB86B-3481-4552-A715-2F0683FA009E at cs.tu-berlin.de>
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> 
> If we assume a directional markedness in the pitch height dimension (e.g. downwards being unmarked) we might assume an analogous markedness along the line of fifths (e.g. flatward being unmarked). Such an assumption implies an interesting question: How do the two kinds of markedness interrelate? Jacques Handschin argues in favor of an affinity between ascending pitch height and sharpward oriented fifths. That same type of affinity would then hold between descending pitch and flatward oriented fifths. This affinity is contrapuntally supported by the ultimate progression between tenor and bass in the cadence (as well as in the Ursatz). But for modal tone relations Handschin's assumption might nevertheless be wrong. There are good mathematical reasons to postulate that the combination between ascending pitch and flatward oriented fifths is the unmarked one.
> Sincerely
> Thomas Noll
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Thomas Noll
http://user.cs.tu-berlin.de/~noll
noll at cs.tu-berlin.de
Escola Superior de Musica de Catalunya, Barcelona 
Departament de Teoria i Composició 

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