[Smt-talk] scales and triads

[Dmitri Tymoczko]

On Aug 2, 2012, at 4:02 PM, 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. 

Interestingly, this all depends on the size of the particular chord or scale you are interested in.  For four-note chords, like major seventh chords, ascending-fifth motion is most efficiently realized with ascending voice leading -- such as (C, E, G, B)->(D, F#, G, B).  For five-note pentatonic scales, the best voice leading descends: (C, D, E, G, A)->(B, D, E, G, A).

For those interested in the geometry of music, it's a useful exercise to work out the formula relating the size of the chord to the size and direction of the minimal voice leading between Tx-related collections.

DT

Dmitri Tymoczko
Professor of Music
310 Woolworth Center
Princeton, NJ 08544-1007
(609) 258-4255 (ph), (609) 258-6793 (fax)
http://dmitri.tymoczko.com