RE: Re: Linear tunings, dual-purpose sonorities: beyond "odd-limits"

Margo Schulter wrote,

>As early as around 1030, Guido d'Arezzo lists acceptable intervals in
>polyphony as including not only stable unisons and fourths, but also
>the whole-tone (9:8), semiditone or minor third (32:27), and ditone or
>major third (81:64). In contrast, he excludes the minor second
>(256:243) and tritone (729:512).

As we've discussed before, the "odd-limit" concept for ranking consonance
only makes sense when the intervals involved in the comparison are
unambiguosly heard as the ratios which they are representing. Ratios of 729,
of 243, of 81, and even of 27 cannot realistically be placed in this
category. It's that "tolerance" thing we keep mentioning. Guido could be
said to be advocating, within the Pythagorean universe, a 9-limit standard
for polyphonic acceptability, with a tolerance of at least a syntonic comma.
Certainly vocal music of 1030, the combination of performance and auditory
uncertainties could easily add up to a tolerance for error greater than a
syntonic comma.

Same goes for Garlandia, de Liege, and Jacobus.

Vicentino's judgments are likely guided by an overarching heptatonic
(diatonic) melodic sense. Where intervals (such as 11:9 and 18:11) fall
between two diatonic intervals subtended by the same number of scale steps,
the categorical classification imprinted by a lifetime of musical experience
is clear and the perception less confused than that of intervals (such as
9:7, 12:7, and 7:6) which fall between intervals subtended by different
numbers of steps.