Recently I was wondering why high pitched instruments seem to have more intonation difficulties than lower pitched instruments. Since this is a common observation, I figured there was probably some reason for it which could be described in simple physics. After a little thought I came up with the following idea, which I believe is a well known phenomena.

BACKGROUND

We humans hear frequency logarithmically to the base 2. That means each higher octave is a doubling of the frequency. For example, the difference between 20 Hz and 40 Hz, sounds the same to the human ear as the difference between 4000 Hz and 8000 Hz. Even though the latter 2 notes are much further apart in absolute terms, both are exactly one octave apart to human perception.

When 2 different frequencies sound, we hear the net resulting waveform of one superimposed on the other. If they are far apart then we perceive them as 2 distinct tones. But if they are close enough together then we perceive them as a single tone with "beat" pulses in it. For example, if we hear A=440 and A=443 at the same time, then we hear an "A" that pulses 3 times per second. The rate of the pulsing is the difference between the two tones.

How close together do they have to be to create audible beats? Most references say within something like 10 Hz. Beyond that the beats are perceived differently as a dissonant noise. When the tones are even further apart than that, the difference in frequency itself becomes an audible frequency and may be perceived as a separate tone. This is called a difference tone, and I may eventually extend this example to include that. But for now it's beyond the scope of this example. Here we focus on the audibility of beats.

INTONATION AUDIBILITY

So the question is, when does intonation error become perceptible in music?
I believe that with most people, it becomes perceptible only when the notes are longer than the beats. That is, when any individual note contains more than 1 beat.
In other words, if each individual note ends quickly so that a beat never pulses, then there are no beats and we perceive the notes as "in tune".
This means the perception of intonation is a bit more complex than simply hearing that tones have different frequencies. It would depend not only on how far apart the tones are, but also what octave they are in, and also how fast the notes are moving.

This example demonstrate this idea over a 2 octave range: at 500 Hz, at 1000 Hz, and at 2000 Hz.
We start with a difference of 2 Hz at 500, which becomes 4 Hz at 1000 and 8 Hz at 2000.

  • First we play each tone separately. The difference is so small (about 6.7% of one half step) that to most listeners they will sound identical when heard separately.
  • Then we play both tones together at the same time. The difference becomes obvious because the tone pulses with beats.
  • Then we play both tones together at the same time. But instead of one long tone, we play a series of short tones at a rate of 4 per second (240 per minute).
    • In all the tests, the short tones are at the same rate - 4 per second.

    FIRST OCTAVE

  • Tone: 500 Hz.
  • Tone: 502 Hz
  • Tone: Both tones together, sustained.
  • Tone: Both tones together, short notes

    • SECOND OCTAVE

  • Tone: 1000 Hz.
  • Tone: 1004 Hz
  • Tone: Both tones together, sustained.
  • Tone: Both tones together, short notes

    • THIRD OCTAVE

  • Tone: 2000 Hz.
  • Tone: 2008 Hz
  • Tone: Both tones together, sustained.
  • Tone: Both tones together, short notes

    • I make the following observations:

  • At 500 Hz, it's impossible to tell that the short notes are not in tune. As you can both hear and see in the waveform, there are no beats. Each note ends before a beat occurs.
  • At 1000 Hz, it's possible to tell the notes are out of tune, but it may require a trained listener. Each note has between 1 and 1.5 beats - the beat begins to rise just before the note ends.
  • At 2000 Hz, it's obvious the short notes are out of tune. Each note has a full swing beat plus a smidgen more.

    • Some predictions this experiment suggests, but doesn't scientifically prove:

  • With exactly one beat per note, intonation error is perceived as dynamics - each note getting quieter due to the beat's destructive superposition.
  • Well trained listeners can detect intonation error between 1 and 2 beats per note, by hearing the pulse just begin to rise before the note ends.
  • 2 or more beats per note is easily audible to most untrained listeners.
  • Intonation differences are more audible in higher octaves because the same intonation difference beats faster, so it more likely to beat faster than the notes.
  • Most listeners subjectively perceive a faster pulsing beat to be more obvious and dissonant compared to a slower pulsing beat. Thus even if lower pitched instruments are out of tune, and they play sustained notes long enough for it to be audible, most listeners would describe the resulting sound as less "bad" sounding.

    • The net result is that an intonation error that is inaudible at 500 Hz, becomes audible one octave higher and obvious two octaves higher. Thus, intonation does indeed get more demanding as pitch rises because the human ear becomes more sensitive to intonation differences at high frequencies. When piccolo, flute and violin (and to some extent oboe) players are off in their intonation, the results will be more easily heard by most listeners, and it will be perceived as more "bad" sounding or dissonant compared to the same intonation error in the lower pitched instruments.

    Of course all musicians should strive to play in tune. The point is not that the the lower instruments don't have to be in tune, but that the results of intonation problems in higher pitched instruments are more apparent and perhaps more dissonant to the listener.

    NOTE: I don't know whether this experiment has anything to do with absolute pitch perception. Perhaps those with absolute pitch can hear the two tones are out of tune even when the notes are quicker than the beats. Then again, perhaps not - it's an interesting question. Several people have listened to these samples and reported similar perceptions to my own. But none of them have had (or claimed to have) absolute pitch perception.