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I've been doing some research into the harmonic series lately. What I've come to understand is that the lower the ratio between the harmonics, the more pleasant/consonant the interval is. So, an octave is consonant and so are perfect fifths. And if the harmonic ratio is very large, it is dissonant, like tritones and minor seconds. But the Perfect Fourth first appears as the 21st harmonic, which is very high. So why is the Perfect Fourth consonant?

I also realize that the perfect fourth and the perfect fifth are inversions of each other, so thus should have the same level of consonance, but how does this relate to harmonics?

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    It wasn't always considered as such a consonant. If you study classic harmony, you'll they were trying to avoid it (but it was used) – Shevliaskovic May 8 '17 at 20:19
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True that the 21st harmonic of a fundamental pitch is somewhere near 4 octaves plus a fourth. Somewhere near, as in 29 cents flat, at 52.71 half-steps out of 53.

More important for consonance, however, is that the 3rd and 4th harmonics of a fundamental pitch are almost exactly a fourth apart.   For the fourth between C4 and F4, that fundamental pitch is F2.

   Note   Pitch         Harmonics of F2  (pertinent)       Harmonics of C0  (misleading)

    F4   349.2 Hz             4      349.2 Hz                   21      343.4 Hz

    C4   261.6 Hz             3      261.9 Hz                   16      261.6 Hz

                              2      174.6 Hz                    8      180.8 Hz
                         fundamental  87.3 Hz  F2                4       65.4 Hz
                                                                 2       32.7 Hz
                                                            fundamental  16.4 Hz  C0

A C4–F4 fourth is extremely consonant, mathematically at least, for three reasons.

  1. One note is almost exactly an overtone of F2 while the other is utterly exact.

  2. The higher note has a very low harmonic number, at 4. The lower the better.   This is equivalent to having a simpler fraction for the pitch ratio.

  3. The fundamental pitch F2 is fairly high, at 87.3 Hz. The higher the better.   This often-overlooked factor helps explain why the same interval two octaves lower sounds much less consonant and is rarely used.

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The perfect fourth has a frequency ratio of 4/3. I don't know how you arrive at "19th harmonic" here. A perfect fifth has a frequency ratio of 3/2, and a perfect fourth is one octave up and a perfect fifth down again.

If you take a look at overtone series and "normalize" the octave (and remove previous ratios), the denominator will always be a power of 2 (because of normalizing the octave). You'll never get at 4/3, a perfect fourth, in that manner.

So you are doing something strange apparently. What is it?

  • I meant 21st harmonic, since that's when it first appears in the harmonic series. But the ratio of 4/3 makes more sense. Thanks. – akrs20 May 8 '17 at 23:06
  • Actually, as user40161 says, the ratio of 4/3 never appears in the harmonic sequence with the fundamental tone the lower. – Scott Wallace May 9 '17 at 8:46
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I wouldn't say that two intervals being inversions of each other will have the same level of consonance. A minor 9 is significantly more dissonant than a major 7, as one extreme example.

While the harmonic series seems to supply a decent correlation with consonance in the way that you have described, it can lead you astray. The way that I think of this is that if you look at the wave lengths of two intervals and see how frequently they align, you will get a better measure of harmonic consonance/dissonance. The octave is the most consonant interval at a 2:1 ratio (the upper note of the octave goes through two cycles for each one of the lower note), followed by the fifth with a 3:2 ration, then the fourth at 4:3.

An interesting way to conceptualize this is to look at intervals at different frequency ranges. I've mentioned that wavelength/frequency and how frequently those align between two notes is a valid measure of consonance. If you were to take an interval that is kind of in the middle, say a major third (5:4), play it around the middle of the piano, then play if down three octaves, you would find that the same interval sounds more dissonant further down the keyboard. The lower frequency notes have a longer wavelength and since we've established that how frequently two sound waves align determines consonance/dissonance, we can understand that the increased dissonance that we experience at a lower octave is the result of the longer wavelengths taking more time to align again as compared to the same interval in a higher register.

I'd also mention that in your quest to better understand consonance and dissonance, you should pay attention to relative levels of each based on context. Any note taken alone will not have a feeling of consonance or dissonance but when played in a series of notes, they will end up having noticeable dissonance. For instance, if you play an ascending major scale and stop on the 7th degree, you will find that the ear really wants to take the last step up to the octave, ie, the 7th degree of the scale is more dissonant than the others and desires resolution to the root, hence its name "leading tone". Similarly, you will notice that a major chord played by itself will sound nice and consonant but when played in a chord progression, different major chords have different levels of consonance or dissonance. For example, the dominant tonic relationship: if we are in the key of C and play a G major chord, it wants to resolve to C because it is dissonant within the context. You should find that this contextual dissonance outweighs the logic described above, eg, the G that wants to resolve to C will feel more dissonant than the C, even when the G is in a higher register (the above logic would dictate that G would be more consonant because it is the same group of intervals in a higher register).

I've heard a definition of dissonance that kind of breaks the usual concept and I can't entirely agree with. The idea was that dissonance is essentially the measure of how much a note or group of notes wants to resolve to another note or group of notes. I don't really take issue with this. However, using this logic, the least dissonant music would be atonal music since the idea of atonality is to avoid having any one note feel more like home than any other, so no note or group of notes has a strong pull to any other note or group of notes. This is the part I take issue with. Anyone listening to atonal music can tell you as objectively as a human is able that it is more dissonant than a I-IV-V-I progression. I only really mention this as another thing to consider.

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Despite being a mathematician, I feel that the idea that consonance is determined by simple frequency ratios is exaggerated. Does a fifth sound more consonant than a third? Play a melody with an accompaniment a third or a fifth away, which sounds nicer? A diminished fifth is famously dissonant but add a couple more notes to make a diminished chord and it does not sound dissonant.

Also, if it were that simple then any interval other than an octave would sound terrible on a well tempered instrument since all of the frequency ratios are irrational.

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