I’ve been curious if errors introduced by tempered tuning necessary to fit a scale into a piano, might be corrected in some circumstances.
I imaging harmonizing singers and string quartets adjust their tuning to eliminate beat frequencies that the tempered piano scale might introduce. Don’t some guitar chords have perfect tuning?
It would be a fairly simple test to setup a controller for a listening test, that maintained the chord root note (or perhaps an inner note of the chord to minimize average shift) , while adding a bit of bend to the other notes for perfect fractional ratios.
I tried a little math exam of the notes in a C Major, from a MIDI notes table:
C major tempered freq.
C 261.63
E 329.63 (C x 5/4 = 327)
C-E tempered scale ratio 1.2599)
G 392.00 (C X 3/2 = 392.445)
(C-G tempered scale ratio 1.4983)
An academic paper examining scales noted:
The equal-tempered fifth consists of seven half-steps (700 cents), which is a completely acceptable approximation to a perfect fifth (702 cents). Similarly, the equal-tempered fourth consists of 5 half-steps (500 cents), which is a completely acceptable approximation to a perfect fourth (498 cents). Thirds are a different matter. The equal-tempered major-third is 400 cents, which is 14 cents sharp, and the equal-tempered minor-third is 300 cents, which is 15 cents flat.
Such corrections might be most noticeable playing pure tones on something like a Mellotron.