Hindemith

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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