The playable palindrome
Reduce P(n) modulo 5 and read 1, 2, 3, 4 as the ratios 1:1, 2:1, 3:2, 4:3 of the tetractys and 0 as silence. The 51 values for n = 0 … 50 form a palindrome around a silent axis, with the fifth at the centre of each half. Press play to hear it in just intonation.
Counts
| period (25) | palindrome (51) | |
|---|---|---|
| silence | 9 = 3² | 19 |
| unison 1:1 | 4 | 8 |
| octave 2:1 | 4 | 8 |
| fifth 3:2 | 4 | 8 |
| fourth 4:3 | 4 | 8 |
Each period holds 9 silences and 16 = 4² consonances, 9 + 16 = 25 (echoing 3² + 4² = 5²). Its consonances multiply to 2⁸: exactly eight octaves, because a fifth and a fourth make an octave.
Piano roll
Click any column to hear that step. The solid line marks the silent axis at n = 25; dotted lines mark the pairs of fifths at n = 12, 13 and n = 37, 38. Played backwards, the piece is identical: it is its own retrograde.
In the OEIS: the values P(n) are A400796, and the digits H(n) = P(n) mod 5 played here are A400797, which records the period 25 and the mirror a(n) = a(25 − n).