How many cents comprise a quarter-tone in 15-EDO?

Question

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?

Answer 1

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.

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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.

Answer 2

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.

Answer 3

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".

Answer 4

TBH I'm a musician and piano tuner, and not a mathematician. But my understanding is that equal temperment tuning on a piano reveals a mathematical discrepancy known as the Pythagorian Comma. It has to be said that merely understanding the mathematics of the equally tempered scale does not in and of itself make someone a good piano tuner!

As to quarter-tones, this is not a term used much by musicologists. Instead, the term micro-tones are used. The best examples I can think of where micro-tones are used would be either Arabic music, or Gamalan music (from, say, Bali, Sumatra, etc.).