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If a whole tone is a 9:8 ratio between frequencies, and a major third is a 5:4 ratio between frequencies, and a major third is two whole tones, then to replicate two whole tones, wouldn't one just take a frequency f and do f*(9/8)*(9/8), which gives 81f/64, and that is not equal to 5f/4.. A bit confused. I got my ratios from 0:57 in this video:

Thanks so much!

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Pure intervals are small ratios. Stacking intervals does not necessarily end in small ratios again. So scales tend to be compromises. That's what "musical temperament" is about. Equal temperament is a compromise leading itself well to transposition. There are "meantone" temperaments that make several intervals pure and distribute others "equally" between them. Meantone temperaments usually focus on pure major thirds, while so-called "well-tempered" tunings try working from a set of pure fifths. Piano tuning is additionally made more problematic by the issue of "disharmonicity" that calls for "stretched octave" tuning.

Tuning systems and their compromises have been an issue since antiquity: the divergence of stacked pure intervals to (usually) full octaves is called the "Pythagorean comma".

The ratio of 2^1:12 has been the principal basis for scales for not all that long. Organs tend to be tuned differently since there the divergence from pure intervals tends to be particularly audible. Somewhat orthogonally, registers designated with 2⅔" are tuned in a pure interval in relation to the corresponding 8" pipe.

You'll find more than you ever wanted to know if you start looking...

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    The Pythagorean comma is specifically the difference between twelve stacked pure fifths and seven octaves. It's also very close in size to the difference between four stacked perfect fifths and two octaves stacked with a pure major third, which is one way of describing the syntonic comma.
    – phoog
    Nov 23, 2021 at 20:54
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Well actually, I answered my own question by waiting till the end of the video.. this is exactly why you can't tune a piano perfectly! In reality, a half tone is a ratio of 2^(1/12). And as a result, a major third, which is four half tones (two whole tones) is 2^(4/12) = 1.2599, which is very close to what we said initially of 5/4=1.25. Leaving this question and answer here in case I misunderstood and someone wants to correct

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    What do you mean by perfectly? What you have here is just that you cannot tune a piano in such a way that any interval is pure in the sense of just intonation. The actual reason why you in fact cannot tune a piano perfectly is not intonation, but inhamonicity (basically the overtones deviate from integer multiples of the bass frequency).
    – Lazy
    Nov 23, 2021 at 22:42
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In just intonation, a 5:4 major third is two whole tones, but there are two different sizes of whole tone. The first one is 9:8, but the second one is 10:9. The product of these is precisely 5:4.

In an answer, you wrote

a major third, which is four half tones (two whole tones) is 2^(4/12) = 1.2599, which is very close to what we said initially of 5/4=1.25.

It's not particularly close, but it's close enough for many people. It's also only one of several reasons why piano tuning is a compromise.

There's another major third, which is made up of two 9:8 whole tones, so it has a ratio of 81:64. Equal temperament is rather closer to this major third than it is to the 5:4 one, but this major third is even more out-of-tune with the fifth harmonic.

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You're sort of correct - you're describing equal temperament as though it were the only system rather than an arbitrary choice, but that's how equal temperament works.

Try this - a perfect fifth is 3*2 of your tonic, let's say A440. So you have E660. Do it again, you have B990. You can divide by two (an octave) at any time, to keep your number in the same octave you started in. So you have B#495. Do that twelve times in a row - you should get 440 again, meaning you've went around the circle of fifths and are back where you started.

You won't get 440, you'll be nearly a quarter of a semitone off. The various ways of dealing with that error answer your question in various ways. And continuing until you're in tune again won't work - you're looking for a power of 3 that's divisible by 2.

When classical music got more interested in changing keys, they started moving towards equal temperament. Bach's "well-tempered" was not equal temperament - the errors were distributed unequally, giving different keys different emotional qualities.Eventually the invention of, and difficulty of retuning, the piano, combined with the fashion for wide-ranging key changes in classical music, resulted in equal temperament as the accepted compromise.

Others have said that both major seconds are valid in different musical contexts, and have developed that - just intonation is their starting point, and Harry Partch's 43-tone system is one of the more extreme examples.

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  • A fifth above E is B, not F sharp.
    – phoog
    Nov 23, 2021 at 20:56
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    Yep. Fixed it. Sorry. But it doesn't change the principle.
    – KFW
    Nov 27, 2021 at 0:50
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    Yep. Fixed it. It doesn't change the theory, though, just leaves one step out.
    – KFW
    Nov 27, 2021 at 1:29

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