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Is it correct to state that any chords (or most of them), while playing, create Beating/Interference effect?

https://en.wikipedia.org/wiki/Beat_(acoustics)

Since semitones are not integer ratio, it should right? (example C major).

If thats true, why I don't perceive any "interference" playing and sustaining it?

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3 Answers 3

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Chords do cause beats. A basic technique of piano tuning is to count the beats when playing e.g. a major 3rd. But that's with two of the three strings that produce each note muted, a pretty pure sound.

Let all three strings ring, the beats are less noticeable. Not because they go away, but because the sound becomes more complex. Three sets of beats, not in phase with each other meld into a richer composite sound.

Same with other instruments. The more complex the tone, the more distant from a pure sine wave, the less apparent the beats will be. But they're there. You should be able to hear then in a dyad played on un-amplified guitar for instance.

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  • I think this question would be much more valuable if you could support it with an example, e.g. what exactly are the beats that can be heard in a major 3rd. Nov 8, 2021 at 17:07
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    You want an article on systems of intonation and the art of piano tuning? Big subject! But easily searched for. Nov 8, 2021 at 18:20
  • No, just one example. What are the beats that you can count when playing a major third on piano. Nov 8, 2021 at 18:35
  • @user1079505 it depends on which major third it is. Furthermore, the predicted beat frequency is unlikely to be found on any actual piano. I listened to a few professional studio recordings of world-class professional pianists, where presumably no expense was spared in tuning the instruments, to find an example for my answer, and the beating was not consistent. Keep in mind, in the treble register, there will be three strings tuned to each pitch, so in a major third there are 6 unisons and 9 major thirds sounding; each string relates to five others.
    – phoog
    Nov 8, 2021 at 21:07
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There are no semitones in a C major chord -- and some semitones are in fact integer ratios.

Two tones sounding simultaneously beat according to the difference of their frequencies. If you have one tone at 440 Hz and another at 441 Hz, you'll hear one beat per second. Raise the frequency of the higher tone to 442 Hz and you'll hear two beats per second.

Now an acoustically tuned A major triad comprising A4, C♯5, and E5 has the frequencies 440 Hz, 550 Hz, and 660 Hz. If there is interference, with what frequency should we expect to hear beating?

660 - 550 = 110
660 - 440 = 220
550 - 440 = 110

The beating is so fast that it sounds like another tone, or rather two other tones, one an octave below the root and the other two octaves below the root. These are known as difference tones. For more information, see the Wikipedia article on combination tones.

In equal temperament, no intervals are integer ratios aside from the octave. The interference in an equal-tempered triad is much harder to hear because the difference tones do not reinforce the chord. Instead of 550 Hz, C♯5 is 554.365 Hz, and instead of 660 Hz, E5 is 659.255 Hz. The difference tones are therefore

659.255 - 554.365 = 104.890
659.255 - 440.000 = 219.255
554.365 - 440.000 = 114.365

In the justly tuned case, the difference tones are precisely one or two octaves below the root of the triad. Here, however, they are not. Rather than being precisely at A2, the difference between E5 and C♯5 is closer to A♭2, but about 18 cents sharper than the equal-tempered A♭2. The difference between E5 and A4, instead of being precisely A3, is about 6 cents flat. Finally, the difference between C♯5 and A4, in contrast to the other two, is rather sharper than in just intonation, being a little less than 33 cents flat from B♭2. This means that the two difference tones that are precisely two octaves below the root in just intonation are instead 149 cents apart in equal temperament.

These difference tones are hard to hear because they don't reinforce each other, and they don't reinforce the harmonics of the actual tones. What one normally hears in an equal-tempered major triad, rather, is the beating caused by interference between the harmonics of the major third and of the other pitches. This is why even a piano that is in good tune will have a little vibrato in its major thirds. This can be heard very vividly by comparing the fourth-fret harmonic of an open guitar string with the corresponding fretted pitch on another string.

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  • I think the beating mentioned by Laurence comes from harmonics of the notes tuned in equal temperament. Nov 8, 2021 at 0:52
  • @user1079505 I was planning last night to do the calculations for an equal-tempered triad, but it was a bit too late. I'll do it now.
    – phoog
    Nov 8, 2021 at 8:21
  • "and some semitones are in fact integer ratios." I assume you are talking about e.g. 25/24 and 16/15 ratios in just intonation? Nov 8, 2021 at 11:44
  • @KarlKnechtel indeed.
    – phoog
    Nov 8, 2021 at 12:02
  • Aha. While that's true, I don't think it helps OP's understanding very much. I would expect that most people asking a question like this only have (conscious) exposure to 12TET (i.e, won't be aware of tuning differences between instruments, and may not have, broadly, considered the idea of pitches in between "the notes" as defined by e.g. a piano keyboard). Nov 8, 2021 at 12:09
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Mathematically the beat can be described in such a way: If you have a product of two (different) sine waves then the product of these can be written as a superposition of two frequencies, one having the sum of the frequencies and one having the difference.

Conversely a superposition of two sine waves of similar amplitude can be written as a product of one wave with the average frequency, and one with the deviation from the average frequency. Our brain hears this as twice the beat frequency since we cannot discern between positive and negative envelope.

So between any two notes you can get some beat. But let’s say we have a semitone at 440Hz and 440*2^(1/12), which is about 466Hz. Then the beat would have a frequency of about $26Hz$, which is nothing you ear can really perceive as beat. Meanwhile at 110Hz and 116.5Hz we have 6.5Hz, which you can perceive.

So this depends on the interval between the notes and on the frequency of the notes. The lower your notes are, the slower and thus more percievable the beat will get.

As soon as you have more than two notes you will not get as much of a regular beat, but more complex beats, which will even be harder to percieve as beat.

But even if you do not notice a beat, you still percieve it in a sense of "excitedness". A sound with little beat will be percieved as less excited and dynamic than a sound with a lot of beat.

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