1

I was doing some music theoryzing (circles of fifths and fourths) and found an interesting problem.

Suppose, we have $2$ sequences: A and B.

A

$a(i+1) = a(i) + 7 \pmod {12}$

$a(0) = 0$

As $7$ and $12$ are coprime, period of this sequence will be $12$:

| i    | 0 | 1 | 2 | 3 | 4 |  5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|------+---+---+---+---+---+----+---+---+---+---+----+----+----|
| a(i) | 0 | 7 | 2 | 9 | 4 | 11 | 6 | 1 | 8 | 3 | 10 |  5 | 0  |

B

$b(i+1) = b(i) + 5 \pmod {12}$

$b(0) = 0$

As $5$ and $12$ are coprime, period of this sequence will be $12$:

| i    | 0 | 1 |  2 | 3 | 4 | 5 | 6 |  7 | 8 | 9 | 10 | 11 | 12 |
|------+---+---+----+---+---+---+---+----+---+---+----+----+----|
| b(i) | 0 | 5 | 10 | 3 | 8 | 1 | 6 | 11 | 4 | 9 |  2 |  7 |  0 |

For some magic reason, if we read B backwards we get A:

$b(12) = a(0) = 0$

$b(11) = a(1) = 7$

$b(10) = a(2) = 2$

$b(9) = a(3) = 9$

...

$b(12 - i) = a(i)\; \forall i \in [0, 12], i \in \mathbb{N}$

Why does this happen? I am puzzled.

thanasissdr
  • 6,348
user4035
  • 351

1 Answers1

3

Observe that $7 + 5 = 12 = 0 \pmod{12}$, so $b(i) = b(i + 1) - 5 = b(i + 1) + 7 \pmod{12}$.

Dominik
  • 19,963
  • Sorry, I still don't understand the proof, can you give a more detailed explanation? Trying to prove myself now, using your hint. – user4035 Aug 23 '15 at 20:11
  • Just write down both your sequences. When you move in your first sequence from the left to the right, the value increases by $7 \pmod{12}$. In the second sequence, the value increases by $7 \pmod{12}$ if you move from the right to the left. – Dominik Aug 23 '15 at 20:16
  • "In the second sequence, the value increases by $7;(mod;12)$ if you move from the right to the left." - yeah, I see that. But why does this happen? I understand, that it's somehow related to the fact, that 7 + 5 =12. In your proof there is no a(i), so I don't see the relationship. Sorry. – user4035 Aug 23 '15 at 20:34
  • In my answer I've shown that $b(i) = b(i + 1) + 7 \pmod{12}$, that's why the value increases by $7$ if you move from the right to the left in the second sequence. – Dominik Aug 23 '15 at 20:38