Re: rothenberg and fifths

Carl,

>> I'm not positive, however, whether it's always the proper subset
>>of lowest cardnality in the system to do so, but that's the claim I was
>>making.
>
>I don't think so -- in 19-tET, a 10-tone MOS of generator 17...
>
> 0 1 3 5 7 9 11 13 15 17 19
>
>...is proper and exhausts the intervals of the temperament. Also the
>11-tone MOS of generator 12...
>
> 0 1 3 5 6 8 10 12 13 15 17 19
>
>...Both are smaller than the 12-note Yasser chain-of-fifths scale in
>19-tET.

Good point; the first scale you mention is also strictly proper, so there's
nothing particularly special about the chromatic scale in this respect. I
hadn't looked very closely at the problem.

>Perhaps the range of a "fifth" is better than most here, but many other
generators do produce proper scales with reasonable number of tones. And
>the fact that the fifth has been a historically popular generator isn't
>really explained its utility here, since a given culture usually uses only
>a single scale. Western music is overwhelmingly diatonic, for example.
>Why not use one of the proper MOSs of the minor third? And, as you pointed
>out, Rothenberg's model does not imply that improper scales are "bad".

Not "bad" but definitely "bad for a fugue." I'm not undertanding your
point on how the fact that a given culture sticks to a single scale applies
here. If certain scales work better than others given certain other
stylistic traits of the music, then certainly a stylistically coherent
music will often use only one scale, if it significantly better than others
in some respects. You're definitely correct, however, that the
characteristics I'm looking at are too weak to explain the use of the 5th
as a generator.

>I think we must attribute the popularity of the fifth to its strong position
>in the harmonic series, one way or another. Paul Erlich's idea of
>tetrachordality as a sort of 2nd-order octave equivalence is one good option.
>
>>You also responded to my point that this analysis might explain the use of
>>5ths in the generation of scales in many musical cultures, saying that this
>>fact is better explained by the concordance of the 3/2. The stumbling
>>block of this latter reasoning (this is not a point of my own invention) is
>>that such scales exist in cultures where harmony doesn't exist.
>
>Octaves exist in such cultures as well.

You got me there. I ought to stop slavishly repeating the arguments of
psychologists and think before I try to make a point.

>>I would agree: I have found that difference tones (a relatively weak
>>phenomenon of harmony) are clearly perceptable in certain melodic
>>situations, rapid sequences of high pitched tones.
>
>Yes, but I don't think we need difference tones to explain the 3/2 in
>melody.

I was unclear here. Difference tones were just an example of a weak
effect. Obviously coincidence of partials is a more relevant effect in
harmony which, one could argue, we might percieve between successive tones.

>>In any case, a stronger point is that many musical systems using fifth-
>>generated scales employ non-harmonic and rapid decay toned idiophones
>>such as xylophone-type instruments, where the consonance of the 3/2 is
>>irrelevant.
>
>I'm not so sure. And in fact, my position would be supported if one
>could find evidence that cultures using such instruments subject their
>fifths to greater mis-tuning than cultures using instruments with more
>harmonic timbres.

I believe you're probably right that people tend to tune xylophone-type
instruments further from pure in their fifths. I think this better
supports my point, however: if both the acoustical and Rothenberg-derived
considerations are relevant to differing degrees depending on the musical
situation, then where inharmonic timbres with short decays are used, the
Rothenberg-derived considerations will dominate and there will be no
particular tuning which is heard as pure. Maybe we actually are
unwittingly in agreement here: I would never want to claim that freq ratios
or the harmonic series are never or rarely musically relevant.

jason