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I've been learning about the science behind music and came across the harmonic seventh interval as 7:4 ratio. Now we know that it's generally not used in just intonation as well as equal temperament. I was wondering why no one seems to miss it. Will C + E + G + harmonic seventh sound more pleasant than a normal C+E+G? Please note that I understand it doesn't fit in equal temperament. My question is more about why it is musically irrelevant.

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It's not "irrelevant" - for example Britten used in his serenade for tenor, horn and strings, which uses natural horn harmonics.

It was also used by organ builders as the "flat 21st" rank in mixture stops along with the just intonation 17th and 19th (i.e. just intonation E + G + 7th). The general effect is to add a "reedy" tone color to the fundamental pitch - you don't actually hear it as a 7th, unless it is out of tune or poorly adjusted.

The harmonic 7th chord is noticeably less dissonant than a tempered 7th chord, so it doesn't have the same "tension-creating" function, and the harmonic 7th doesn't have any tendency to "resolve" like the corresponding tempered 7th.

There is some dispute about whether or not the intonation of Barbershop singing uses harmonic sevenths - though that style certainly uses lots of "dominant 7th" chords.

Melodically, it's hard to tell the difference between the various sizes of 7ths and their inversion into various sizes of 2nds.

There are some sound examples at https://en.wikipedia.org/wiki/Harmonic_seventh and https://en.wikipedia.org/wiki/Harmonic_seventh_chord.

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