The 13-cycle as music
On the page about other primes, the sequence Qp(n) = ⌊n²/p⌋ mod p keeps the palindrome for every prime, and for p = 13 its counts make the triple 5² + 12² = 13². Here the same musical rule is applied: a residue H becomes the ratio H : (H − 1), so with 13 the cycle sounds every superparticular ratio from 2:1 down to 12:11, a much larger “tetractys”.
Counts in one period
What you hear
Each step sounds its interval over a fixed root. For p = 5 the ratios are the four consonances of the tetractys. For p = 13 they include the just major third 5:4, the minor third 6:5, the septimal intervals 7:6 and 8:7, and smaller and smaller steps down to 12:11, the kind of ratio (n + 1 : n) that Greek theorists called superparticular.
Every non-zero value occurs exactly p − 1 times per period and silence 2p − 1 times, so every interval of the set is heard equally often, and the whole period reads the same backwards.
The silence as p. With 5 the silence becomes 5:4, a consonance. With 13 it becomes 13:12 (138.6 cents), between a semitone and a whole tone and no consonance at all, and with 41 it becomes 41:40 (42.7 cents), smaller than a quarter tone. This is why the reading belongs to 5: see Why 5?
For p = 5 this floor sequence has the same counts as the Pentametric Cycle but is not identical to it; the Pentametric formula itself works only for 5 (see Other primes).