Question about minimal comma pumps

I see I missed this message here:

http://launch.dir.groups.yahoo.com/group/tuning/message/97935

The question is - I don't get it. I get the concept of a temperament's
spectrum now, but what does the notation [2, s[1]] mean?

And when you say

"Call a routine which tests if an interval can be expressed in the group
generated by a list of intervals. If q can be so expressed, pass the list to an
output routine."

q referred to the regular temperament whose comma pump we're trying to
compute. So in this case, do you mean - if the spectrum of q expresses
the interval you want?

Secondly, might this approach perhaps be extended to enable you to
find comma pumps that start or end with a certain arbitrary chord? Say
in 12-tet, you want to go Cmaj -> C#dim -> Dm -> G7 -> Cmaj. In the
notation you used, this would be 1/1 -> 16/15 -> 16/15 -> 4/3 -> 2/3.
In 12-tet, you could replace that with

1/1 -> 6/5 -> 2/3 -> 6/5 -> 2/3 -> 6/5 -> 4/3

which brings you back to I. These are the famous "Coltrane Changes"
that Giant Steps is built off of. There are probably tons of awesome
ways to do this that involve the diaschismatic pun that I'm not even
seeing. Might there be a way to search for comma pumps like this?

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> I see I missed this message here:
>
> http://launch.dir.groups.yahoo.com/group/tuning/message/97935
>
> The question is - I don't get it. I get the concept of a temperament's
> spectrum now, but what does the notation [2, s[1]] mean?

It means start a list by including 2 and the lowest temperamental complexity interval on your spectrum list.

> And when you say
>
> "Call a routine which tests if an interval can be expressed in the group
> generated by a list of intervals. If q can be so expressed, pass the list to an
> output routine."
>
> q referred to the regular temperament whose comma pump we're trying to
> compute. So in this case, do you mean - if the spectrum of q expresses
> the interval you want?

No, is q expressible in this growing list of intervals we started by [2, s[1]].

> Secondly, might this approach perhaps be extended to enable you to
> find comma pumps that start or end with a certain arbitrary chord?

You can flesh out the numbers you get to make multipiers for defining chords. That's my approach with the examples I am generating, which if AKJ could somehow be located I could start putting up on the Xenwiki and then you could tell me how functional you think it all is.

> Might there be a way to search for comma pumps like this?

I wouldn't even know how to began without knowing what "like this" meant. However, the basic [5/4,5/4,5/4] pump could be broken up by noting that 5/4 = 3/2 * 5/6, which allows you to expand the basic pump out into the Giant Steps pump. There's an analogous extension you could do for 648/625 if you want to try to go Coltrane one better.