source file: mills2.txt
Date: Thu, 6 Jun 1996 15:21:56 -0700

Subject: Hindemith

From: PAULE <ACADIAN/ACADIAN/PAULE%Acadian@mcimail.com>

Gary wrote:

>   I seem to recall somebody saying the Paul Hindemith professed to teach
>composition in terms of the harmonic series, although (as far as I know)
>strictly within the 12TET system.

>   Anybody know any more on that?

Yes, Hindemith made it clear to his students that the major triad is 
patterned after the harmonic series, and many writers on orchestration show 
the "chord of nature" as a sort of ideal voicing: large intervals on the 
bottom, small ones on top.

Please read Partch's "Genesis of a Music" for an excellent historical survey 
of music theory as it relates to the harmonic series or just ratios.

Both Hindemith and Schoenberg conceived 12-tone composition in terms of the 
harmonic series, but while Schoenberg went too high up the harmonic series, 
Hindemith stayed down too low. Schoenberg could explain the whole chromatic 
scale as harmonic series above C, F, and G; however, in identifying C# with 
the 13th harmonic of F as well as the 11th harmonic of G, he was identifying 
it with two notes a semitone apart. Meanwhile, Hindemith expressed all 12TET 
intervals as 5-limit just ratios, and identified the strength of the 
relationship with the simplicity of the ratio:

Strongest
2/1 (octave)
3/2 (perf. fifth)
4/3 (perf. fourth)
5/3 (maj. sixth)
5/4 (maj. third)
6/5 (min. third)
8/5 (min. sixth)
9/5 (min. seventh)
9/8 (maj. second)
15/8 (maj. seventh)
16/15 (min. second)
45/32 (tritone)
Weakest

(He construed this sequence as sort of a "tone row" in his Ludus Tonalis.) 
So he is really constructing the chromatic scale by building major triads on 
C,  F,  G, Db, and Eb, and a minor triad on B: a rather arbitrary 
construction. He then assumes that the relationships he set forth will be 

heard as such without the conncting triads. I would argue that a tritone is 
never directly heard as a 45/32; it is almost as close to 7/5 as a minor 
sixth is to 8/5, so 45/32 is silly except in some kind of "extended 
referece" theory and then only in very specific situations. Other ratios he 
put forth are not going to be directly heard as such, or only in certain 
chords.


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