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So equal temperament is a compromise to allow songs to be played without without having to retune them on the fly, to match the more consonant sounding just intonation intervals.

However, in a world where MIDI-based soft synthesizers with customizable tuning exist, it could be practical to optimise the tuning system to the played song, as the instrument can be tuned on a note-by-note basis.

I was thinking this could be algorithmically computed by shifting tuning towards just intonation based on if the note is used in multiple intervals or not. If it is not used in another context it would be completely tuned to its just intonation interval. e.g. the intro to Scar Tissue

An example of how it would work:

  1. MIDI file read into program to analyse things like song key, most frequent intervals, most played chords, etc.
  2. Program algorithmically computes tuning
  3. Program outputs tuning
  4. Tuning and song played back by synthesizer

Is there a known way to do this? Is just intonation the right 'ideal' to aim for? How do I ensure that the pitch does not drift?

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    This kind of thing has been explored, but mostly in the form of changing the tuning dynamically to make each chord as consonant as possible (dynamic/adaptive tuning/intonation) - would that be of interest to you too, or are you only interested in coming up with the fixed best compromise on a note-by-note basis for a given song? If the latter, why would the tuning drift? – topo Reinstate Monica Jan 19 at 10:53
  • Slightly off subject, but on a couple of my keyboards, they can be recalibrated to play in other tunings - just being one. But I have to tell them which key I'm going to play the piece in. Modulate and there's a problem. So 12tet, whilst being a compromise, seems the best solution, albeit not exactly in tune. But it's been working for many decades now. – Tim Jan 19 at 11:11
  • Might be of interest: music.stackexchange.com/questions/53218/… – Tom Jan 19 at 11:21
  • Why keep each note at the same pitch for the whole song? You could adjust notes as needed during the song, as musicians playing variable-pitch instruments do. – phoog Jan 19 at 22:11
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    matero: every temperament is a way of not having to retune "on the fly," since it is as you note not practical to do that with a keyboard instrument. The main distinguishing feature of equal temperament is that it sounds equally good (or equally bad) in every key. Meantone temperaments sound decidedly better in certain keys and decidedly worse in others, but they all allow you to play in the good keys without retuning, and that's something that you can't do with just intonation, unless you only use a subset of the chords of certain keys. – phoog Jan 19 at 22:19
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So equal temperament is a compromise to allow songs to be played without without having to retune them on the fly, to match the more consonant sounding just intonation intervals.

It's not only that - it also makes a given song sound more in tune when chords from different keys are used within the same song (whether due to modulation, chord borrowing, or whatever), and it makes all chords of the same type sound equally in tune. It might be that for many individual pieces of music, equal temperament (or something close to it) would still be judged to be the best tuning!

An example of how it would work:

MIDI file read into program to analyse things like song key, most frequent intervals, most played chords, etc.
Program algorithmically computes tuning
Program outputs tuning Tuning and song played back by synthesizer

Is there a known way to do this?

If you are thinking of writing that program, then yes - you could come up with your desired tuning and then use MIDI tuning standard to convey that tuning to an instrument that supports MTS. I'm not aware of a program that calculates the 'best' tuning for a piece - that might be a matter of opinion, depending on which chords you want to sound most in tune.

Is just intonation the right 'ideal' to aim for?

Not necessarily - Just intonation has its compromises too. As well as not working well for pieces that move between keys, It doesn't even make all the chords in a given key 'in tune' - So it doesn't necessarily represent 'pure' or 'perfect' tuning. This is one reason why adaptive/dynamic tuning has attracted more interest than just trying to 'statically' find the best tuning for a given piece.

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    Agreed. I'd just add, as a just intonation junkie, that there are kinds of music that don't aspire to making "all the chords in a given key 'in tune'". That doesn't mean that they are second-class citizens. – Scott Wallace Jan 20 at 15:12

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