Perfect fourth

This is a second spin off of the future post “A search for interesting canons part 2” (part 1), (previous spin off) . Like the previous one this post is probably not interesting to you if you’re not into musical theory, but I’ll try to avoid jargon.  When searching for the best melody in canon it is necessary to define the rules of counterpoint, and for the most part I follow common practice rules. I do have 2 personal exceptions that I would like to document real quick, the first one is about the dissonance of the fourth.

Recap

In general 2 frequencies that share more harmonics (and conversely have a simpler ratio) sound better to the human ear. And the ideas of counterpoint adopt this with one exception: the perfect fourth. While in the case of the second and third their inverse frequency ratios (respectively the seventh and sixth) are treated the same, the perfect fourth is not treated identically to its inverse the perfect fifth. Specifically, it is treated as a dissonance when the interval occurs between the bass and another voice.

Reason for dissonance

The given reason for this is that with the interval of the fourth you can only deduce one consonant chord, with the top note of the interval as its root. Generally people prefer to hear the bottom note as the root. When you build a chord from the bottom note, the fourth is a dissonant note within that chord. The question is of whether the fourth is consonant is equivalent to asking whether you think you’re listening to one chord or another.

There might be a larger harmonic context that biases your interpretation of the chord. But at its basis you want to listen to whether you want the “fourth” to move downward to resolve an audible dissonance. Note that all of these are dissonant when following common practice counterpoint rules.
chords.png
The red ones have tension to my ear, the black ones sound consonant like F-major chords to me.

Conclusion

So rather than classify all perfect fourths as dissonant and disqualifying them, here are my two personal exceptions: The interval of an octave and a fourth can not be bare.
And: Any chord in the second inversion must be harmonically as sensible as the same chord interpreted as a root chord.

The bare octave+fourths sounding dissonant also makes sense considering the sparse overlap in their harmonic series compared to octave+third or octave+fifth.

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