<p>In music, it's common to hear the refrain that people like harmonics (notes played at frequencies that are rational multiples of one another) the most out of different combinations of notes (at least when the denominators of those rational numbers are low), which goes back to Pythagoras, and led to the development of just intonation, in which an instrument is designed to have its notes at these rational multiples of frequencies. This is because they produce less of a clashy sound, as waves of different frequencies add to and cancel each other out.</p>
<p>It is true that they produce a less clashy sound, however, I don't generally prefer that sound, and most instruments produce a varied enough sound that the difference is hard to hear anyway. All human scales which haven't been deliberately invented by neo-classical composers to be different, have notes organised nicely with frequency multiples right next to the powers of <span class="math-container">$2^\frac{1}{12}$</span> (if Pythagoras is to be believed, and they are not actually at these values) or, in the case of microtonal music found in a few cultures, <span class="math-container">$2^\frac{1}{24}$</span>. Pythagoras was working before the concept of fractional exponents, or even irrational numbers themselves, so it's not surprising he didn't consider them to be the basis of music.</p>
<p>The (somewhat) low-denominator approximations of these notes by Pythagoras don't follow any nice pattern like the powers of <span class="math-container">$2^\frac{1}{12}$</span>. The common scales and modes don't seem to favour low-denominator approximations among these numbers at all. The &quot;favourite&quot; relative frequencies of third and fifth don't have particularly simple denominators. People point out that (<span class="math-container">$10$</span> out of <span class="math-container">$12$</span>) of the powers of <span class="math-container">$2^\frac{1}{12}$</span> within an octave are weirdly close (with a hundredth) of nice rational numbers (with denominators below <span class="math-container">$15$</span> inclusive). But there are <span class="math-container">$72$</span> spaced-out numbers with denominators below <span class="math-container">$15$</span>, between <span class="math-container">$1$</span> and <span class="math-container">$2$</span> (excluding <span class="math-container">$2$</span> itself), so this doesn't necessarily seem <em>that</em> impressive, intuitively. And what of the nice rationals left out by the notes we like? Just intonation tuning makes anything in a non-standard key sound atrocious, and to my ear, listening to a piece of baroque music as it would originally have been played, it sounds atrocious in general. People say that it may sound bad because we're used to modern, equal temperament tuning.</p>
<p>But people from the time of just intonation would have been just as used to just intonation. So, as far as I can see, the hypothesis that we like just intonation is fabulously convoluted compared to liking a super nice set of numbers taken log <span class="math-container">$2$</span>, and there doesn't appear to be any evidence for it. Is it just psuedoscience? A reliable test would have to be blind and on non-musicians who haven't gotten on the bandwagon.</p>

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2023-04-02T03:56:58.483Z
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2023-04-02T04:14:32.327Z
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Is There Any Evidence that People Prefer Just Intonation
Thomas Anton
# Is There Any Evidence that People Prefer Just Intonation

(Asked by: username_0 on 2023-04-02)

In music, it's common to hear the refrain that people like harmonics (notes played at frequencies that are rational multiples of one another) the most out of different combinations of notes (at least when the denominators of those rational numbers are low), which goes back to Pythagoras, and led to the development of just intonation, in which an instrument is designed to have its notes at these rational multiples of frequencies. This is because they produce less of a clashy sound, as waves of different frequencies add to and cancel each other out.

It is true that they produce a less clashy sound, however, I don't generally prefer that sound, and most instruments produce a varied enough sound that the difference is hard to hear anyway. All human scales which haven't been deliberately invented by neo-classical composers to be different, have notes organised nicely with frequency multiples right next to the powers of $2^\frac{1}{12}$ (if Pythagoras is to be believed, and they are not actually at these values) or, in the case of microtonal music found in a few cultures, $2^\frac{1}{24}$. Pythagoras was working before the concept of fractional exponents, or even irrational numbers themselves, so it's not surprising he didn't consider them to be the basis of music.

The (somewhat) low-denominator approximations of these notes by Pythagoras don't follow any nice pattern like the powers of $2^\frac{1}{12}$. The common scales and modes don't seem to favour low-denominator approximations among these numbers at all. The "favourite" relative frequencies of third and fifth don't have particularly simple denominators. People point out that ($10$ out of $12$) of the powers of $2^\frac{1}{12}$ within an octave are weirdly close (with a hundredth) of nice rational numbers (with denominators below $15$ inclusive). But there are $72$ spaced-out numbers with denominators below $15$, between $1$ and $2$ (excluding $2$ itself), so this doesn't necessarily seem *that* impressive, intuitively. And what of the nice rationals left out by the notes we like? Just intonation tuning makes anything in a non-standard key sound atrocious, and to my ear, listening to a piece of baroque music as it would originally have been played, it sounds atrocious in general. People say that it may sound bad because we're used to modern, equal temperament tuning.

But people from the time of just intonation would have been just as used to just intonation. So, as far as I can see, the hypothesis that we like just intonation is fabulously convoluted compared to liking a super nice set of numbers taken log $2$, and there doesn't appear to be any evidence for it. Is it just psuedoscience? A reliable test would have to be blind and on non-musicians who haven't gotten on the bandwagon.

**Tags:** sensation, hearing, music, aesthetics

> **Comments:**
> **username_1**: Welcome Thomas. If this doesn't work as a question here for some reason, then try our sister site [music.se]. Remember to take their tour, read-up in their help-centre before posting.  
> **username_1**: Worth noting, there's also our [bioacoustics.se] site where you may be able to enquire about the rest of the animal kingdom if you wish.  

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## 0 answers

