I had some time recently to play with rhythmic canons, which we have previously seen here.

In particular, I had been calculating the length of rhythmic canons mod 2 based on the motive {0,1,N}. It seems that Hélianthe and I have found a way to calculate the length for any N without actually having to calculate the entries of the canon. So, whereas before I would have to wait a couple of weeks to get the length for N>45, I could now get it in a matter of seconds.

And voilà ! The length of canons mod 2 for N up to 150.

We can see the specific cases studied by Hélianthe for and . There are a couple of interesting things as well: for example, what is going on for N=73, or N=85 ? (I have no idea…)

But the cool thing is that our method seems to work for any modulus , and not necessarily . Remember from this post that I began calculating the lengths for . Well, here it is for .

As in the case , we seem to have an interesting situation for , and .

In fact, I conjecture that for any different than 3, the length of the canon based on the motive {0,1,p^k} would be . For , as already discussed in the previous post, it would be .

Also, you’ll recall that Hélianthe found a very nice description of the entries of the canons mod 2 when , by organizing them in a table, and putting a 1 at the cell if the canon has an entry at (and else a zero). So, for example with , we get this table (white is 1, black is 0):

and for , we get this table:

and you can clearly see the recursivity between these tables.

For , the canons are not compact (i.e. there may be more than one entry of the motive {0,1,N} on a given beat), but we can still construct these tables, by putting in each cell the corresponding number of entries (between 0 and p-1).

So for example, with and , we get this table (black is still 0, white is 4):

and with :

(**Update**: ah, what the heck, let’s do it for as well :) :

Finally, for and , we get the following table:

and for , we get:

Exciting, isn’t it ? :)

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