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Suppose I have the frequency $f$. According to just intonation, $\frac{3}{2} f$ is a perfect fifth.

Now compare the following:

$$ \left(\frac{3}{2}\right)^n \neq 2^m$$ for any integer $n,m$

But $$ \left(2^{\frac{7}{12}}\right)^{12} = 2^7 $$

Supposedly this difference explains why just intonation does not permit musical keys.

Can someone explain in mathematical terms why this is so? Why does just intonation based on ratios fail to permit keys but equal temperament $2^{\frac{k}{12}}$ permits them?

Given these two sets of numbers $$\lbrace 1,\frac{9}{8},\frac{5}{4},\frac{4}{3},\frac{3}{2},\frac{5}{3},\frac{15}{8}\rbrace$$ and $$\lbrace 1,2^{\frac{2}{12}},2^{\frac{4}{12}},2^{\frac{5}{12}},2^{\frac{7}{12}},2^{\frac{9}{12}},2^{\frac{11}{12}}\rbrace$$ what makes the former unable to form keys but the latter able to?

Stan Shunpike
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    The reason I understood why just intonation could not form keys was that an instrument tuned to a just intonation for a given key (call it C) could not reasonably reproduce a comparable scale in another key (call it D). The temperaments are empirically identified balances which produce acceptable scales starting from every key. I don't know if this relates to your ratios, however. – Cort Ammon Sep 11 '15 at 00:58
  • Maybe useful: http://math.stackexchange.com/questions/11669/mathematical-difference-between-white-and-black-notes-in-a-piano – Qiaochu Yuan Sep 11 '15 at 01:00
  • An immediate observation is that the frequency ratio of any two pitches a whole tone apart is the same in equal temperament, but not in just intonation. That is, $$2^{2/12}/1=2^{4/12}/2^{2/12}=2^{7/12}/2^{5/12}=2^{9/12}/2^{7/12}=2^{11/12}/2^{9/12},$$ but $$\frac{9/8}{1}=\frac{3/2}{4/3}=\frac{15/8}{5/3}\ne\frac{5/4}{9/8}=\frac{5/3}{3/2}.$$ More importantly, the fifth separating $1$ and $3/2$ is not the same as the fifth separating $9/8$ and $5/3$. Hence in the key with tonic $9/8$, the dominant sounds different than in the key with tonic $1$. – Will Orrick Apr 28 '17 at 06:02
  • @CortAmmon but nobody tunes instruments to just intonation. Tuning has always been a compromise, since before the invention of tonality (i.e., of "keys"). – phoog Feb 21 '22 at 16:54
  • @WillOrrick in the key with tonic $9\over8$ the tonic sounds different, because the tonic chord contains both $\hat1$ and $\hat5$ (see my answer for more detail). The thing that most people miss is that this makes "just intonation" fairly useless for tuning instruments. In general, performances can use just intonation only if the performers are able to use different frequencies for the same notes, so, for example, A might sometimes be 440 Hz and sometimes 445.5 Hz. – phoog Feb 21 '22 at 17:01

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Supposedly this difference explains why just intonation does not permit musical keys.

This supposition is incorrect. Unequal temperaments, including "just intonation," certainly allow for keys, as you can see from the couple of centuries' worth of tonal music written in various keys before equal temperament came into common use.

The more common (and more correct) statement about equal temperament is that it makes all of the keys "the same." This is because unequal temperaments cause different keys to have different intervallic relationships between analogous pitches.

As an example, using the notation $\hat1$ through $\hat7$ for the degrees of the diatonic scale (that is, $\hat1$, $\hat3$, and $\hat5$ represent C, E, and G in the key of C but D, F♯, and A in the key of D, and so on), suppose you base your tuning on A = 440 Hz using the usual just intonation intervals given for the major scale. This gives the following frequencies for your diatonic scale, in Hz:

$A = 220 \times 1 = 220$
$B = 220 \times \frac{9}{8} = 247.5$
$C\sharp = 220 \times \frac{5}{4} = 275$
$D = 220 \times \frac{4}{3} = 293.\bar3$
$E = 220 \times \frac{3}{2} = 330$
$F\sharp = 220 \times \frac{5}{3} = 366.\bar6$
$G\sharp = 220 \times \frac{15}{8} = 412.5$
$A = 220 \times 2 = 440$

Now the ratio between B and D is ${4\over3}\over{9\over8}$ or $32\over27$, while the ratio between C♯ and E is ${3\over2}\over{5\over4}$ or $6\over5$. In the key of G major, B and D are $\hat3$ and $\hat5$, whereas in A major, $\hat3$ and $\hat5$ are C♯ and E. Therefore, this tuning gives $\hat3$ and $\hat5$ a different intervallic relationship in different keys, Q. E. D.

In equal temperament, by contrast, these intervals, which both span three semitones, will have the same frequency ratio, namely $2^{(3/12)}$.

In fact, even the oft-repeated statement, "a keyboard tuned to just intonation is perfectly in tune in only one key" is false, because the keyboard isn't perfectly in tune even in that one key. The reason for this is that the frequency ratio for the perfect fifth between $\hat2$ and $\hat6$ is not the expected $3\over2$ but rather $40\over27$. In other words, if you tune a keyboard in C using these intervals, the perfect fifth in your D and D minor chords will sound like this instead of like this.

(The term "just intonation" is used here, as it often is, to mean "5-limit just intonation," which is to say that the ratios between frequencies have no prime factors greater than 5. The typical name for 3-limit just intonation is "Pythagorean," and it was mostly used before the development of tonal music requiring 12 pitch classes. Higher-limit just intonation is even less useful for tuning a standard 12-tone keyboard, because for example the minor seventh might sometimes need to be $9\over5$, sometimes $16\over9$, and sometimes $7\over4$.)

phoog
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