Re: JI as tuning by multitudes (integer ratios)

Margo Schulter wrote

>As you nicely sum up, this gives JI intervals, like integer fractions,
>a property of closure for operations like addition (multiplication of
>ratios) or subtraction (division of ratios).

Yes, that's it.

There's an interesting extra point here. If one just closes under multiplication
and division of ratios, and not addition / subtraction of ratios, then starting from
a finite set of ratios, you end up with a prime limit JI. . (using JI in your sense:
"27:16 is a JI interval, but not a `just' one in the sense of simplicity or direct
tuneability by ear.")

E.g. the set of all ratios that can be written using 2, 3, and 5 as the prime factors
will be 5-limit JI, and this system will have the property of closure,
while not having any 7-limit ratios.

Yet though closed, it still has fuzzy edges as it does have ratios that get arbitrarily close to
any other ratio you choose beyond the prime limit, such as 7/4 say.

If you try to remove all the fuzzy edges by adding limit points for all infinite sequences,
you will end up including all possible rational, irrational and transcendental numbers
in it.

The union of all the prime limit JIs will still have fuzzy edges because, for instance, it will
have all the rational approximations to the golden ratio:

e.g. the sequence of approximations from the Fibonacci sequence
1 1 2 3 5 8 13 21 34 55 89 144
->
1/1 2/1 3/2 5/3 8/5 13/8 21/13 34/21 55/34 89/55 144/89

Robert