Re: Re: Functional harmony

In the Riemann tradition, chords are notated by an elaborate system of
modified T, D, and S's, standing for the functions tonic, dominant and
subdominant. Diether de la Motte's _Harmonielehre_, a standard text in
German conservatories and universities, is an excellent introduction to this
tradition. The scale-step notation based on roman numeral I through VII
(sometimes with minor and major distinguished by small and capital letters)
is more closely associated with Vienna. The two last deep exponents of
scale-step harmonic theory were Schenker and Schoenberg, albeit with very
different perspectives and results.

Perhaps the most critical question for us in the tuning community is whether
the Riemannian traditions deals adequately with mediant tonal relations,
that is, harmonies related to the fixed functional harmonies by thirds.
Already with such scholars as Tanaka (before the turn of the 20th century),
tonal space was represented two dimensionally, in triangulated lattices.
Either of the assumed temperaments (meantone, 12tet) mapped the mediant
functions simultaneously onto functions related to the tonic by chains of
fifths, thus for many musicians the point may be somewhat moot.

Daniel Wolf