groups and sets

[Erlich]
>a given consonant interval is always approximated by the same number of
>scale steps. This seems like an important grammatical feature without which
>the sense of the scale as a fixed melodic basis could be very difficult
>for a composer to convey. Stronger requirements such as Rothenberg
>propriety (which requires that a given number of scale steps always
>subtends an interval smaller than the smallest interval subtended by
>that number plus 1 scale steps) seem less useful as they exclude such
>common scales (such as the Pythagorean diatonic) from consideration.

I think agree, but isn't propriety a different measure than yours, rather
than a "stronger" one? That is, a scale could be proper without having a
1:1 relationship between its consonances and scale steps?

[Wolf]
>Hearing highly structured, non-tonal music in these temperaments may not
>reflect the modes of audition closest to the biases of the physiological
>system, but it does remind one ... that musical cognition, with training,
>can be a far richer resource than the ear alone.

Richer resource of what? The two are different things entirely. Lack of
acoustic pleasure does not prevent the delivery of beautiful symmetries,
no. And the World's most harmonious just scale will not deliever musical
symmetries if it is not designed and used as the resources of human memory
require.

>I can't help but note that I find it surpring that Mr. Erlich is framing
>his argument in this way. His own work in 22tet is within a system whose
>set and group properties are certainly more useful than the quality of its
>representation of ratios of small whole numbers.

I think I agree with this one. However, although I say that the "lack of
acoustic pleasure does not prevent the delivery of beautiful musical
symmetries", it may help. Mr. Erlich's ideas on consonance, as I
understand them, are designed to measure within what tolerance acoustic
pleasure may help set and group properties.

There is another, much finer tolerance, which measures only acoustic
pleasure. Thru the 7-limit, it about one cent for tones of the same timbre
and spatial location, and 4 cents for tones of different timbre or location.

Where Mr. Erlich and I differ is on the matter that having pure just
intonation should exclude desirable group properties. The 5-limit diatonic
scale works really well in just intonation, and I think that, despite the
more numerous and larger commas, his decatonic scales would as well. This
is because any sensitivity to mistuning in melody (making comma
adjustments) must be at least an order of magnitude rougher than the
acoustic pleasure tolerance of 1-4 cents.

Carl

For the record, I took Judith conrad statements as a humorous support of
my previous post. The Native Americans mentioned might though have
enjoyed hearing there music on the piano in the same absurd way I can
laugh at the version on In C performed by a Chinese Orch. with there
confusion over what semitone they are to play!
As to the MOS being formalized by Clough is erroneous in that Wilson
work predates the others! See Wilson Archive Below.
--
Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com