Commas formed by Cross-Products of 5-limit Patent Vals?

I have a simple question, but I am not sure how to go about answering it.

Suppose we took the set of all 5-limit patent vals and then took the cross product of every pair, yielding a 5-limit comma for each pair. Are there any 5-limit commas that cannot be produced by this process, no matter the pairs of vals you choose?

My intuition is to say yes (that there are some commas that can't be produced), but I don't know how to go about proving it. Does anyone have insight?

Ryan

As an example of an unobtainable comma, there is no patent val tempering
out 5/1 except for <0 0 0|. You'd need two nonzero patent vals tempering
out 5/1 to get that as a result.

-Mike

On Jun 2, 2012, at 3:18 PM, Ryan Avella <domeofatonement@yahoo.com> wrote:

I have a simple question, but I am not sure how to go about answering it.

Suppose we took the set of all 5-limit patent vals and then took the cross
product of every pair, yielding a 5-limit comma for each pair. Are there
any 5-limit commas that cannot be produced by this process, no matter the
pairs of vals you choose?

My intuition is to say yes (that there are some commas that can't be
produced), but I don't know how to go about proving it. Does anyone have
insight?

Ryan

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> As an example of an unobtainable comma, there is no patent val tempering
> out 5/1 except for <0 0 0|. You'd need two nonzero patent vals tempering
> out 5/1 to get that as a result.
>
> -Mike

Hmm, okay. So are there any others besides 2/1, 3/1 and 5/1? I guess 5/3 as well, since only 0p and 1p temper out 5/3.

Ryan

--- In tuning-math@yahoogroups.com, "Ryan Avella" <domeofatonement@...> wrote:
>
> I have a simple question, but I am not sure how to go about answering it.
>
> Suppose we took the set of all 5-limit patent vals and then took the cross product of every pair, yielding a 5-limit comma for each pair. Are there any 5-limit commas that cannot be produced by this process, no matter the pairs of vals you choose?
>
> My intuition is to say yes (that there are some commas that can't be produced), but I don't know how to go about proving it. Does anyone have insight?

Any such comma must have two patent vals which temper it out, rather than one or none. Hence 6/5, for instance, is tempered out only by the 2 patent val, 5/4 only by 1 and 3/2 by none.