source file: mills2.txt
Date: Tue, 4 Mar 1997 08:21:30 -0800

Subject: Re: Schenker and tuning

From: kollos@cavehill.dnet.co.uk (Jonathan Walker)

Daniel Wolf wrote:
 
> Schenker himself wrote precious little that was useful about 
> intonation and - a contemporary of Schoenberg and Hauer - was active 
> in a Vienna that was solidly committed to equal temperament (listen to 
> any recording by Kolisch).

All that I've ever noticed of substance on intonation are the comments 
in Harmony (1906), where Schenker takes a Zarlinoesque approach, 
filtered through the harmonic series. He says that art must place 
restrictions on what nature offers, and so only the first six harmonics 
contribute to tonality -- whatever the vagaries of his explanation, his 
conception of at least the major scale seems to be firmly 5-limit. He 
criticises Riemann (as usual) for his utonal-type explanation of the 
minor -- he seems unconvinced primarily because such an explanation 
lacks an acoustical basis (this shows, as we might expect, that a purely 
mathematical explanation is not acceptable to him). He notes that the 
first minor triad available in the harmonic series is 10:12:15, and that 
the root is thus not identical with the fundamental. But he never quite  
settles on an explanation -- thus Daniel's phrase "a clouded major" 
seems quite fair (the tierce de Picardie also lurks in the shadows). 

As a further restriction upon natural resources, Schenker then says that 
art represents this essentially 5-limit system by the simpler 12TET. I'm 
not at all sure that this move is available to Schenker, but it 
undoubtedly places him within the decidedly pro-12TET Viennese milieu 
that Daniel mentions. 

> What would be very interesting would be to take one of Schenker's 
> graphic analyses and to construct a tuning for the piece based up 
> projecting the simplest rational interpretations at each stage of 
> prolongation and elaboration.

I've done some work on this. For example, two works of Beethoven would 
seem to defeat a 5-limit explanation even of the Background:

1. the funeral march of the Ab Sonata, op.26, which ends up a great 
diesis away from its tonic (repeating this in the reprise), i.e. 
(6/5)^4.

2. the song "In questa Tomba oscura", which ends up a lesser diesis away 
from its tonic, i.e. (5/4)^3

One other point of interest: in connection with the F major Prelude of 
Chopin's op.28, Schenker tentatively explains the Eb in the final chord 
as Chopin's "visionary" inclusion of the 7th harmonic (since the chord 
doesn't function as a dissonance to be resolved). Jonas, in his 
footnote, enthusiastically reinforces this explanation.

-- 
Jonathan Walker
Queen's University Belfast
mailto:kollos@cavehill.dnet.co.uk
http://www.music.qub.ac.uk/~walker/


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