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I am searching for an equal temperament without just intervals.

If one exists, all intervals will as near as possible to just position and all notes will have a constant place.

If we use K to represent the frequency ratio of the semitone, the question is how big must K be? And how to determine the best K?

In standard 12-tone equal temperament, K = 21/12 ≈ 1.059463...

and in the equal temperament based on the just fifth, K ≈ (3/2)1/7 ≊ 1.059643....

The best K is between these two values. How can we find the best value for K?


Imagine how many different values K can have between 1.059643 and 1.059463! I exclude temperaments with more than 12 half tones.

Also, it's very important to have fixed places for ALL used notes. The violin like the piano also has fixed tones: the four open strings!

How to choose the best value for K for the semitone and keep only the unison perfect?


My actual idea is to try each of the 178 possible equal temperaments where the K for the half tone is between 1.059463 and 1.059643 (ET12).

If you can simulate any of them to tell me how it sounds for you? I would begin with K = 1.059464. If you decide to try, please begin with other end — K =1.059642!

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    I don't understand the question. Are you looking for an equal temperament that has more than 12 divisions of the octave? There is only one equal temperament where N=12; if there are twelve divisions and the size of the intervals is different from 100 cents (2^(1/12)) then they can't all be equal, or twelve of them will combine to form an interval that isn't an octave.
    – phoog
    Mar 20 at 18:45
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    Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking.
    – Community Bot
    Mar 20 at 18:58
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    @ElementsInSpace well 3-limit just intonation (a.k.a. Pythagorean tuning) gives a constant ratio between semitones but it doesn't close the circle of fifths, so there is (in theory) an infinite number of pitch classes (in practice, there is a wolf fifth, so unequal intervals). If we go to larger values of N then we can get very close to meantone tuning (which has good major thirds and slightly flat fifths) by using a subset of 31-TET and to 5-limit just (good major thirds and good fifths, but prone to pitch drift) with a subset of 53-TET. But these don't give equal sized semitones.
    – phoog
    Mar 20 at 21:02
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    @Tom K is the factor that relates the frequencies of adjacent semitones. 1.059463 is the twelfth root of 2 (rounded, of course). In tbe other hand, 1.059643 is the seventh root of 1.5. I hadn't realized the two numbers were different; now I understand the question!
    – phoog
    Mar 20 at 21:14
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    @phoog Yes, I got it also when I read the question, but I had the feeling it was not completely obvious ;).
    – Tom
    Mar 20 at 21:26

3 Answers 3

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In ET12 standard K=1.059463... and in the ET based on the fifth juste is 1.059643... The best K is between these two values. Any idea about?

Summary:

In brief, the cost of having a purer fifth is a less pure octave, and the consequences of having an impure octave are rather more significant than of having an impure fifth, so any value of K that isn't very close to the lower value will cause problems. On top of this, the impurity of the perfect fifth in standard 12-tone equal temperament is very slight.

Further discussion:

In 12-tone equal temperament, K≈1.059463 is the twelfth root of 2. That means that the pitch of each octave, 12 semitones above or below, is exactly twice or half.

The semitone based on the just fifth, K≈1.059643, is the seventh root of 1.5; if you use this as your semitone it means that the factor for an octave is 1.5^(12/7) or 2.003875. This means that if you tune the A above middle C to 440 Hz, the A below middle C will have a frequency of 219.5745 Hz. You can expect beating from this octave slightly slower than once a second -- enough to sound out of tune.

In cents, hundredths of a standard equal-tempered semitone, the octave is 3.35 cents too wide. By contrast, the standard equal tempered fifth is only 1.96 cents narrower than pure. If your ears are good enough to hear the difference, aren't they going to be that much more bothered by having the octave out of tune -- and by more than the fifth is in standard ET?

Our ears are more sensitive to the correct tuning of an octave than to the correct tuning of a fifth. What's worse, the octave differences would be cumulative. The high C of an organ's 4-foot stop is seven octaves above the low C of a 16-foot stop, so it would be seven times sharper. If you make the fifth exactly 701 cents, an octave is 1201.71 cents, but seven octaves is 8412 cents -- an eighth of a semitone wider than pure. The open low E of a contrabass is four octaves below the open E string of a violin. This would be nearly seven cents wider than pure.

In short, because we are so much more sensitive to out-of-tune octaves than to out-of-tune fifths, any K that is bigger than 2^(1/12) would still have to be much closer to that value than to 1.5^(1/7).

(I deliberately used the organ as the example keyboard because pianos are actually tuned using "stretched" octaves to account for the fact that their overtones aren't precisely harmonic. The degree of inharmonicity is greater in the upper and lower parts of the range, so the degree of stretching isn't constant, and in any event the point of the stretching is to make the octaves sound like they are tuned justly, not to make them sound wide.)

It is also perhaps worth mentioning that for a few centuries from the middle of the Renaissance until the advent of equal or near-equal temperament, keyboards were tuned with many of their fifths significantly flatter than the fifth of equal temperament, with the occasional fifth significantly wider, because there was an additional desire to have the major third closer to acoustically pure -- 13.7 cents narrower than standard equal temperament. As it became increasingly desirable to be able to play in any key, it became apparent that the fifth would have to take precedence over the third because we are more sensitive to out of tune fifths than out of tune thirds. For the same reason, the octave has precedence over the fifth. It's difficult to imagine that changing this could be an improvement.

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  • Is it something about beating that makes out of tune octaves worse than fifths which is worse than out of tune thirds, etc? I expect out of tune unisons would be worse than octaves. I suppose it has something to do with the interval ratio being closest to 1:1. Those intervals closer to 1:1 being out of tune make worse beating than those further away from 1:1? Mar 26 at 13:32
  • I believe it's basically along those lines, yes. More precisely, it's about the distance between the two fundamentals and the coinciding overtones. For the unison, it's the two fundamentals, obviously; for the octave, the first overtone of the lower pitch coincides with the fundamental of the second. For the fifth, the coinciding overtone is a twelfth above the lower pitch, and for the third it is a seventeenth above. I think octave equivalence must also play a role.
    – phoog
    Mar 26 at 17:31
  • Thanks @phoog. Your answer is very well written and explained. I don't know how to do this math, but you explained the concepts in a way that was perfectly clear to me. Mar 28 at 17:01
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K=1.059463 was chosen as it's the 12th root of 2. On a 12-tone scale, that ensures that an octave is exactly a ratio of 2:1. Any other value of K would leave you with octaves that aren't.

Assuming you want a 12-tone scale, then the choice is either go for 12TET with K=1.059463, or forget equal temperament and go for something like Just Intonation or Pythagorean Intonation. Those mostly have nicer sounding intervals, but always produce wolf intervals somewhere in the scale.

If you increase the number of notes to n-TET, then for sufficiently large n you can get as close to any intervals as you like. But it becomes impractical to make an instrument capable of doing that. If you try to eliminate some notes to make things more practical, then the wolf intervals come back.

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    Just and Pythagorean are basically useless for harmonic music on a 12-tone keyboard. For music using a limited set of keys (early baroque and earlier), you will want some sort of meantone temperament. In the middle baroque and later, you have other temperaments that increasingly made more keys useful. But it is always some sort of temperament.
    – phoog
    Mar 20 at 20:59
  • Well, with some caveats. Pythagorean tuning is perfect for music that treats the major third as a dissonance. Just intonation is fine if your piece avoids the narrow fifth between the 9:8 major second and the 5:3 major sixth and if you don't mind retuning frequently as you move from one piece to the next.
    – phoog
    Mar 21 at 11:03
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    Yes, unequal temperaments always have some intervals worse than others — but they don't all have a ‘wolf’, nor even one anywhere near as bad.  (Some of the more sophisticated temperaments, such Werckmeister's or Valotti, have all intervals and keys working well, and even meantone tunings are pretty usable.)
    – gidds
    Mar 21 at 16:35
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I'm not sure I understand the question. All equal temperaments with K intervals use the 2^(1/K) as a step; it's the only solution to the problem.

You could use more than 12 intervals in an octave but that doesn't give the same scales as used in Western (or Persian or Arabic or Turkish....) music. Equal nineteenth roots of two give a fair scale. Others have been chosen. A division into 53 equal tones has been known since about 300 BC in China and has been rediscovered now and then.

Another possibility is Wendy Carlos' temperaments; these are equally based but without octave equivalence.

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  • "You could use more than 12 intervals in an octave but that doesn't give the same scales as used in Western ... music": you can use a subset of such a system as your European scale, however.
    – phoog
    Mar 20 at 21:10
  • "Equal nineteenth roots of two give a fair scale" - what exactly do you mean by "fair" here? Mar 21 at 3:56
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    It's not accurate enough for fine-tuning to "just" (if that's the point) nor is it wide enough to imitate half-steps. A 53-step scale (53 roots of two) hits things better. Unusual tunings are interesting in themselves, but most don't fit "normal" music better than 12-tet.
    – ttw
    Mar 21 at 4:04

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