XH 17: Paul Erlich's response

>This raises an interesting question: how _should_ we describe e-g-b-d'
>as a stable 20th-century sonority on a 12-tet organ or piano, say?
>Should we use "n-limit" terminology at all when the actual tuning in
>question doesn't really fit such a model, and more closely fits
>3-limit than 5-limit or 7-limit/9-limit?

Does the 12-tET major third, for example more closely fit the
3-(prime)-limit or the 5-(prime)-limit? Beyond 5:4 and 81:64, one can
use very complex ratios of either prime limit to approximate the 12-tET
arbitrarily closely, and your recent posting of a 5-(prime)-limit
approximation to 12-tET shows that you are at least partially aware of
this fact. (BTW, Helmholtz/Ellis have precedence over you in the
"Schulter Artusian 5-limit Just" tuning.) But perceptually, it is only
the simplest ratios which determine the character of an interval or
chord, even if the exact tuning is off by a sixth-tone or even a
third-tone. So the 3-(prime) limit description of the minor seventh
chord seems quite irrelevant despite its closeness, while the 5- and
7-(prime) limit descriptions both have some relevance, I feel, to the
way the 12-tET minor seventh chord is perceived, and the way it is often
performed on free-pitched instruments. In this context (namely, that of
describing individual sonorities), I believe the odd limit to be of more
utility. The odd limits of the aforementioned JI descriptions (whose
prime limits are 3, 5, and 7) are 81, 9, and 9, respectively. This gives
a much better idea as to the perceptual relevance of these descriptions
than the prime limit.