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?

  • 2
    $\begingroup$ 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. $\endgroup$ – Cort Ammon Sep 11 '15 at 0:58
  • $\begingroup$ Maybe useful: math.stackexchange.com/questions/11669/… $\endgroup$ – Qiaochu Yuan Sep 11 '15 at 1:00
  • $\begingroup$ 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 $1$. $\endgroup$ – Will Orrick Apr 28 '17 at 6:02

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