Verifying the Interval of a Fifth vs an Octave in Early Greek Music

[silvermane]

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, let's 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! :)

 

[LCKurtz]

I don't think modular arithmetic will work for you. For example, say you start with f =32. If you keep multiplying that by 3/2, after a few iterations what you get won't be an integer.

Just look at what happens to an original frequency f if you keep doubling it versus multiplying it by 3/2.

 

[Redbelly98]

In equal temperment, the ratio between a note and a half step above it would be
2^{1/12}.

A perfect fifth is 3 1/2 steps, or 7 semitones above a note, so the ratio for a perfect fifth would be
2^{7/12}.

Compare this to the ratio of 3/2 from the Greeks. If you raise both to the 12th power (which is the same as saying moving up 12 consecutive perfect 5ths, you'll see that they won't be the same.
(2^{7/12})^{12} \ne \left(\frac{3}{2}\right)^{12}

Moderator's note:
This doesn't seem to address the problem being asked. And if it did, it would quite probably violate our forum rules about giving out too much help on homework problems.

Another way to state the problem is: prove that going up by m fifths can never be equivalent to going up by n octaves, where m and n are integers.

 

[eumyang]

Actually, I think what I said does address the problem. It depends on how much music theory the OP knows, but I won't say anything further. It would help, however, if the OP could state the problem verbatim from the source.

 

[silvermane]

Moderator's note:
This doesn't seem to address the problem being asked. And if it did, it would quite probably violate our forum rules about giving out too much help on homework problems.

Another way to state the problem is: prove that going up by m fifths can never be equivalent to going up by n octaves, where m and n are integers.

Yes, before logging back in and seeing all these replies, I found out that modular arithmetic wouldn't work and did it just as you said. I also confirmed that I only needed to show the nth interval for a fifth wouldn't equal a kth interval increase of an octave:

(3/2)^n = 2^k

3^n = 2^(n+k)

Which is obviously not true for many reasons. Thank you everyone for all your help. I really appreciate all of you going out of your way to help me! :)

 

[silvermane]

Actually, I think what I said does address the problem. It depends on how much music theory the OP knows, but I won't say anything further. It would help, however, if the OP could state the problem verbatim from the source.


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The reason why it's in my words is because my professor is a little too wordy to understand the problem. I've cut out more than have of the "original problem", which only included historical facts about the octave and fifth. I felt that this information should be left out so that I can see what's going on, as well as those trying to help me. :)

 

[Redbelly98]

(3/2)^n = 2^k

3^n = 2^(n+k)

Nice. I myself wasn't sure where to go after the first equation. But of course, get it in terms of integers.