(adsbygoogle = window.adsbygoogle || []).push({}); Problem (in my words):

So, in music, the interval of an octave between two tones corresponds to doubling the frequency of the oscillation. In early Greek music, the interval of a fifth corresponded to multiplying the frequency by 3/2. With this definition of a fifth, prove that non of the tones you get by starting with a given one and going up by successive fifths can be equal to a tone you get by starting with that given tone and going up by successive octaves.

My shot at a proof:

From what I understand of the problem, I could just use modular arithmetic to prove this true.

Let's define an octave as adding 0mod2 to our original frequency. Likewise, lets define a fifth as adding 1mod2 to our original frequency.

Now, we can consider our 2 cases: one where the original frequency is an even number, and the other, where our frequency is an odd number:

Case 1: If the frequency is initially even, let's call it 0mod2. The next octave would thus be 0mod2 as well because 0mod2 + 0mod2 = 0mod2. The next fifth, however, would be 0mod2 +1mod2 = 1mod2. We quickly see that when our original frequency is even, that a fifth higher of that frequency is not the same as an octave higher.

Case 2: If the frequency is initially even, let's call it 1mod2. The next octave would thus be 1mod2, because 0mod2 +1mod2 = 1mod2. The next fifth would be 0mod2, because 1mod2 +1mod2 = 0mod2. Thus we see that going up a fifth is not equivalent to going up an octave.

Conclusion: We see that going up a fifth is thus not the same as going up an octave, and thus they can never be equal from going up in the same intervals.

I think I did it well, but I just need help making sure I've written the proof correctly and logically. If this isn't correct, please lead me in the right direction, but don't give me the answer. Thank you so much in advance for your help!!!! :)

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# Homework Help: Music and Math

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