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My question is from my MUS 204 course, which is the following: How many cents comprise a quarter-tone in 15-EDO?

I believe I am missing one piece of knowledge to answering such a seemingly simple question. First, I understand a quarter-tone is 50 cents and second, I understand the frequency ratio in 15-EDO, correct me if I'm wrong, is 2^1/15 and a cent is 1/1200 of an octave.

I wanted to solve the problem similar to solving for the number of cents in just perfect fifth. (i.e., log of base 2 to the 3/2 = (number of cents)/1200, which is approx. 702 cents.) However, substituting 3/2 for 1/15 produces an incorrect number. (The correct number should be 80 cents.)

What correct steps must be taken to find the correct answer to the question above?

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    I find this question dubious. IMO there aren't any quarter tones in 15-edo. Oct 9, 2021 at 22:55
  • @leftaroundabout I answered on this basis
    – obscurans
    Oct 10, 2021 at 6:49

3 Answers 3

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Taking 12-EDO as a starting point...

  1. 12-EDO divides the octave into 12 semitones and, therefore, 24 quarter tones.
  2. The purpose of "cents" is to divide the octave using a linear scale rather than an exponential one.
  3. An octave being 1200 cents means that each 12-EDO quarter tone is 1200/24 = 50 cents.

15-EDO is to be treated analogously.

  1. 15-EDO divides the octave in 15 "semitones" and, therefore, 30 "quarter tones".
  2. It follows, then, that each 15-EDO quarter tone is 1200/30 = 40 cents.

As pointed out in the comments, the original textbook question is a dubious one. The concepts of "tone", "semitone", "quarter tone", and "perfect fifth" have specific meanings in 12-EDO that don't transfer — or, at least, don't necessarily transfer — to other EDO systems. It's misleading to speak of semitones and quarter tones, etc., in the context of 15-EDO.

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    I do not like it. By the same logic, a “quarter-tone in 22-edo” would be 27.3 cents, but that interval certainly shouldn't be called a quarter tone. Oct 9, 2021 at 22:53
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    @leftaroundabout It depends on what is considered a "tone" or "semitone" within the context of the particular tuning system. Given that the question originates in Music 204, I take it to be that they're trying to make a simple analogy with nice math rather than a generalized definition across EDOs. Frankly, I don't like the textbook question that prompted this, for exactly the reason you allude to: it's misleading to refer to 12-EDO concepts (semitone, quarter tone, perfect fifth) in the context of 15- or any other -EDO.
    – Aaron
    Oct 9, 2021 at 22:57
  • @leftaroundabout I've added a qualifying explanation to the answer. Would value your opinion on whether it offers an adequate caveat.
    – Aaron
    Oct 9, 2021 at 23:00
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    @PiedPiper The underlying frequencies, yes, but the math for cents themselves is linear, which I've understood to be the point of using cents.
    – Aaron
    Oct 10, 2021 at 15:01
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    @phoog that would make cent scale rather useless in comparing various temperaments or divisions. Oct 11, 2021 at 1:20
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substituting 3/2 for 1/15 produces an incorrect number

This is where you made a mistake. The formula to calculate number of cents between two frequencies f₁ and f₀ is:

1200·log₂(f₁/f₀)

For perfect fifth ratio of frequencies f₁/f₀ = 3/2 (in just intonation). However for a semitone in 15-EDO ratio of frequencies is not 1/15. It is 2¹⸍¹⁵.

Then the formulae are as follows: a semitone in 12-EDO has:

1200·log₂(2¹⸍¹²) = 1200·(1/12)·log₂(2) = 1200·(1/12) = 100 cents

Similarly, a semitone in 15-EDO is

1200·log₂(2¹⸍¹⁵) = 1200·(1/15)·log₂(2) = 1200·(1/15) = 80 cents

For a "quarter-tone", that is half of a semitone, in 15–EDO substitute 2¹⸍¹⁵ with 2¹⸍³⁰ to obtain 40 cents.

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    This is a useful answer, but just know, OP does indicate 2^1/15 for the semitone. You may want to revise your description of where OP misunderstood the math.
    – Aaron
    Oct 10, 2021 at 6:35
  • @Aaron OP wrote explicitly she "substituted 3/2 with 1/15" and that's a mistake. It might be a random error or maybe misunderstanding of where which numbers should go, but this is where discrepancy in calculations originates. Oct 10, 2021 at 6:53
  • Yes, I understand the error you're correcting. I just think you're explanation of what the error was isn't quite clear.
    – Aaron
    Oct 10, 2021 at 11:53
  • ...specifically, the second sentence of your explanation. IMO, that could be usefully clarified.
    – Aaron
    Oct 10, 2021 at 12:02
  • @Aaron I tried to clarify the answer. However I don't quite understand what is unclear in sentence "For a perfect fifth ratio of frequencies equals 3/2"? Oct 10, 2021 at 15:13
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TL;DR: This question is nonsensical.

Quarter tones simply don't exist in 12-EDO, since there is no way whatsoever to play one. They only exist in 24-EDO, and if you've added a ton of extra pitches in the middle, you are, clearly, not using 12-EDO in the first place.

Similarly, 15-EDO literally means one single step is 2^1/15, which comes to 80 cents exactly. That, and all multiples of it, are the only intervals that can even be played.

Whatever meaning you assign to a quarter tone, it's reasonable to assume it's somewhere around 50 cents, maybe a bit higher. By the plain English reading, 4 of these things should add up to a whole tone, 200 cents or so. 80 cents is just way too far off from 50 - it's more than half again as large!

A 31-cent error, for example, stands between the actual 7th harmonic, less 3 octaves, and the major second of 12-EDO. Nobody claims 12-EDO is remotely capable of representing this interval. It just doesn't exist here.

The answer is: in 15-EDO, there is no playable interval that is remotely usable as a "quarter tone".

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    This doesn't address the OP's issue all. If you could rephrase it in a more polite way, it could make a valid comment about semantics. Oct 10, 2021 at 7:19
  • Surely with guitar, trombone, violin, any extra pitches can be played.
    – Tim
    Oct 10, 2021 at 12:14

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