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This is a very interesting video on shifting a tuning of a song in the middle of it for a quarter of a tone. It seems to be pretty well explained

"How to Modulate to G 1/2 Sharp (Jacob Collier-style)" by David Bruce Composer

youtube.com/watch?v=Xd54l8gfi7

@luka I wonder why the harmonics are off the reference frequency 🤔

@norwin you mean upper harmonics from the C key on piano don't match for example the fifth? isn't that because equal temperament is a mathematically divided octave (12-TET) and not based on harmonic series (naturally occuring resonances at half/third/quarter).

when a string in space is excited harmonics appear as ratios of rational numbers, - these results in frequencies that are slightly off from 12-TET

@luka Ah your explanation makes sense after reading up on en.wikipedia.org/wiki/Harmonic

I somhow had expected a fifth -> 1+7/12=1.59 to be in the harmonic series, and i suppose it is approximated by the 3rd overtone as 1*3=3 an octave above.

@luka semitones are additive, but overtones are multiplicative to the base frequency, thats the mismatch, if i got it right

This image was helpful
commons.wikimedia.org/wiki/Fil

@norwin yes this "multiplication" is the basis for ratios of harmonic series. from these just intonation is created - how 'naturally harmonically' strings resonate with fifths and fourths further on, and these are based on the fundamental 'base' frequency. That's why transposition is impossible with a single just intonation tuning - however with new systems (digital!) there could be (there is already for sure!) compositional method where tuning follows transpositions.

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