1, 3, 5, 17, 257: Musical harmony

x1 the note itself, x2 the octave, x3 the fifth, x5 the third: here are the main harmonics behind the construction of the scale. According to the sequence of fifths: if F equals 1, C equals 3 (ou3 / 2, which is identical, knowing that a division or a multiplication by 2 in music does not change the note, but the octave), G is 9 (9/8), D is 27 (27/16), A is 27x3 = 81 (81/64), E is 81x3 = 243 (243/128), B is 243x3 = 729 (729/512). This sequence is in fact an approximation of the harmonic construction, the latter including the major thirds with the ratio x5. So there is a "jump" between D 27 and A 81, where A is worth 80, or 5x16 (therefore x5), so in third with the unit, F 1. Then we continue the sequence of fifths x3 A E B, i.e. E 5x3 = 15, and B 15x3 = 45. The harmonics x3 and x5 are therefore sufficient to build the 7 notes of the major scale, the white keys of the piano But insufficient for the black keys of the piano. If we continue we have F # = 45x3 = 135, this number is large .... by reducing it to the lower octaves we have 135/8 = 16.875, or almost 17 to 0.8%! This approximation, analogous to that between D 27 and A 80, is largely satisfactory for considering the number 17 as a new constructive harmonic of the scale. Continuing on this new harmonic 17, we have C # = 17x3 = 51, G # = 51x3 = 153, Eb = 153x3 = 459, and Bb in fifth with F # 17 being 17x5 = 85. By closing the loop, we get a very large approximation of the unit, because 85x3 = 255, or 256-1, 256 being a power of 2, so in octave with F 1. The last number of Fermat, 257, is also close to 256, so the octave found: in music, 1 = 256.