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 N=2^k and N=2^k+1. 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 p, and not necessarily p=2. Remember from this post that I began calculating the lengths for p=3. Well, here it is for p=5.


As in the case p=2, we seem to have an interesting situation for N=5^k, and N=5^k+1.

In fact, I conjecture that for any p different than 3, the length of the canon based on the motive {0,1,p^k} would be p^{2k}-1. For p=3, as already discussed in the previous post, it would be 3*(3^{2k}-1).

Also, you’ll recall that Hélianthe found a very nice description of the entries of the canons mod 2 when N=2^k, by organizing them in a N-1 \times N table, and putting a 1 at the cell (i,j) if the canon has an entry at N*i+j (and else a zero). So, for example with N=64, we get this table (white is 1, black is 0):


and for N=128, we get this table:


and you can clearly see the recursivity between these tables.

For p>2, 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 p=5 and N=25, we get this table (black is still 0, white is 4):


and with N=125:


(Update: ah, what the heck, let’s do it for N=625 as well :) :



Finally, for p=7 and N=49, we get the following table:


and for N=343, we get:


Exciting, isn’t it ? :)


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