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I'm thinking of bass, viola, violin, and cello, though my question could apply to any instrument tuned in stacked fourths/fifths. Typically the strings are tuned using the 3rd and 4th harmonics so that each string is a just perfect fourth/fifth from the next. However, this leads to the bottom string being technically a Pythagorean minor third / major sixth to the top string. To use bass as an example, the ratio between the E and G strings would be (4^4/3^4) = 2*(32/27). However a 5-limit minor tenth would be the ratio 12/5 = 6/5.

I've discovered that it's possible to tune this 12/5 ratio by simultaneously playing the 3rd and 4th harmonics of the E string compared against the 5th harmonic of the G string. I am aware that any such tuning would lead to a wolf fifth/fourth between open strings. Despite this, have any prominent string players tuned in this manner or similar (such as, to use bass again as an example, tuning the A string to match with the E string rather than the D string)?

My current suspicions lie in flageolet passages. For example, at the end of the 2nd movement of the Koussevitsky concerto for double bass, Koussevitsky plays on his recording with harmonics. Playing through this passage, the high E on the A string forms an odd (to my ear) interval against the high F# on the D string; but when I tune my A string to form a 5:9 ratio against the G string, the melodic interval sounds more in tune. This suggests to me that other players might be tuning their strings in the manner I described to compensate for this.

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    I think you've miscalculated your intervals. A twelfth is a fifth plus an octave, and the fifth is a perfect interval, so there is no such thing as a minor twelfth or a major twelfth. Similarly, an eighteenth is a fourth plus two octaves, so there is only a perfect eighteenth, not a major or minor eighteenth. The interval between the outermost of four strings tuned in fifths is a minor seventeenth. – phoog Jul 4 '20 at 20:47
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    @phoog I edited it. Isn't a stack of three fifths (i.e. CG+GD+DA) equal to M6+P8 = M13? – awe lotta Jul 5 '20 at 13:32
  • Yes, that is a major 13th. I was confused and in a hurry. I was thinking that the interval is equal to two octaves plus a minor third, but it's actually equal to two octaves minus a minor third. – phoog Jul 6 '20 at 20:49
  • I don't know that much about non-fretted string instruments, but if you want to get a different temperament, couldn't you leave the tuning as it is and play your notes a bit sharper or flatter? I have heard of some chamber musicians who deliberately play a few cents sharp/flat to get closer to the Pythagorean perfect intervals with the notes that other instruments are playing. – mkorman Aug 25 '20 at 13:10
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Most players tune their instruments to a single note and then tune the fifths by ear. I don't know whether anyone tempers the fifths by ear, but if they do it is probably not by as much as would be needed to create a just major 13th. This would mean tuning a violin's E string to 657.27 Hz, for example, which would beat nearly three times a second against the acoustical fifth above A (at 440 Hz) of 660 Hz.

However, players who are playing with keyboard instruments that aren't in equal temperament typically tune each string to the keyboard. This is commonplace in performances of early music with period instruments. In such cases the outer strings may be a just major thirteenth from one another; for example, in the Werckmeister temperaments, most of the fifths are acoustically pure, but the tempered fifths include most of those that correspond to open strings. For example, of the four temperaments covered in the Wikipedia article, every one has a narrow fifth between D and A, which affects violins, violas, cellos, and basses.

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I know that some string players use baroque tuning for certain pieces in an effort to more authentically present the music. Some pieces are easier to play with alternative tunings, and some more modern pieces will even specifically call for certain unusual tunings. Basses specifically will tune their lowest string even lower than standard to be able to reach notes outside their typical range if the score calls for it.

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