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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?

Renée
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3 Answers3

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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.

Aaron
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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. – leftaroundabout Oct 09 '21 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 09 '21 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 09 '21 at 23:00
  • Very adequate IMO! – leftaroundabout Oct 09 '21 at 23:15
  • @Aaron The JI perfect fifth is 3:2, a concept that utterly transcends any tuning system. Semitones can be constructed as 16:15 (diatonic semitone, perfect fourth minus major third), 25:24 (chromatic semitone, major third minus minor third), and weirder combinations. Anything that claims to be called a "quarter tone" needs to be be somewhere half that, necessarily close to the 50 cent range. – obscurans Oct 10 '21 at 06:52
  • OP quotes the number 80 cents, which makes me think she already knows the information you provided here, and the question is on deriving the number of cents from frequencies. – user1079505 Oct 10 '21 at 07:21
  • Cents are not a linear division of a semitone (or of an octave). If they were linear, a quarter tone wouldn't be 50 cents. – PiedPiper Oct 10 '21 at 11:50
  • @PiedPiper Perhaps I used the wrong term. Cents allow for simple multiplication in that x semitones are, say, 50x cents. Is that not linear? – Aaron Oct 10 '21 at 12:00
  • @Aaron - Isn't a semitone [100 cents](https://en.wikipedia.org/wiki/Cent_(music))? – Dekkadeci Oct 10 '21 at 14:28
  • The underlying frequencies are exponential, just as they are with semitones and octaves. – PiedPiper Oct 10 '21 at 14:58
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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 '21 at 15:01
  • @Dekkadeci My error in the comment, yes. – Aaron Oct 10 '21 at 15:03
  • It seems to me that a cent ought to be defined as 100th of a division in whatever EDO system you're using. In that light, a quarter tone ought to be 62.5 15-EDO-cents – phoog Oct 10 '21 at 17:27
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    @phoog that would make cent scale rather useless in comparing various temperaments or divisions. – user1079505 Oct 11 '21 at 01:20
  • @user1079505 that's true, but there's no particularly compelling reason to measure all temperaments and divisions against 12-tone equal temperament. The main point, however, is that it makes less sense to adapt the definition of "semitone" to a different equal division of the octave while keeping "cent" related to 12-EDO, because "semitone" predates equal divisions of the octave altogether. Is a semitone in 31-EDO the 31st part of an octave? Of course not. – phoog Oct 11 '21 at 07:44
  • @phoog Yes, there is a reason, it's needed if you actually want to tune your instrument given way. Consider also, on this page: https://en.wikipedia.org/wiki/Equal_temperament intervals seem to be named based on their similarity to 12-EDO intervals. Calling a 15-EDO step a semitone seems to agree with this convention. – user1079505 Oct 11 '21 at 08:48
  • @user1079505 how does defining "cent" as 1/1200 of an octave instead of as 1/100 of an octave division help anyone tune a flute or a guitar to 15-EDO? It's not even possible to tune those instruments to 15-EDO. It's also not necessary to make reference to cents in order to tune any other instrument in 15-EDO; you can just calculate the frequencies directly. – phoog Oct 11 '21 at 08:57
  • Because 1/1200 fraction is already an established standard both for scientific research and in commercially available devices, like tuners. Also, as I mentioned, this lets you compare various tunings easily. Yes, cents are not _necessary_ for anything, if you prefer you can calculate frequencies for every purpose. And by the way, Adam Neely shows how to tune bass for 24-EDO, in a similar way you could get 15-EDO. https://www.youtube.com/watch?v=H4KIwA8O9LU – user1079505 Oct 11 '21 at 15:00
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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.

user1079505
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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 '21 at 06:35
  • @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. – user1079505 Oct 10 '21 at 06:53
  • 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 '21 at 11:53
  • ...specifically, the second sentence of your explanation. IMO, that could be usefully clarified. – Aaron Oct 10 '21 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"? – user1079505 Oct 10 '21 at 15:13
  • IMO it reads better now. – Aaron Oct 10 '21 at 16:44
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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".

obscurans
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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. – user1079505 Oct 10 '21 at 07:19
  • Surely with guitar, trombone, violin, any extra pitches can be played. – Tim Oct 10 '21 at 12:14