Let me try to allay some confusion associated with your question. "Standard tuning" refers to the
notes of the musical scale that are most conventionally used for tuning the open strings. It
does not specify the type of temperament used: that's something entirely different. These days, the standard tuning for a mandolin (and also a violin) is GDAE (low to high), and for a guitar, it's EADGBE (ditto). If you specify the octaves (like on a piano), then the mandolin is tuned to G3 D4 A4 E5. That G3 note is the G below middle C (C4), by the way.
The A4 note is usually tuned to a fixed frequency of 440Hz.
This is a different convention. This convention, called "concert pitch," does not depend on whether the tuning happens to be "standard" or something else, nor does it depend on the type of temperament used, either. It is an ISO standard, in fact (ISO16), and was adopted by the musical industry in 1926. Some folks still use other standard pitches, however, like A=432 Hz, or even A=444 Hz, or even A=466 Hz for old Baroque music.
Finally, the pitches of the other open notes on the mandolin (or violin, or whatever) are all derived from the reference frequency of the A4 string (A=440 Hz). It is these pitches that will depend on the temperament used. This is a third type of convention, if you will. For example: in a
just temperament, the high E string -- which is a perfect fifth above the A, will have a frequency that's 1.5 times higher, or E = 660 Hz. In
equal temperament (12TET), however, the frequency of the E string will be 1.498307 times higher, or E = 659.26 Hz. That's about 2 cents different! The other strings (G,D) will also wind up with slightly different frequencies, depending on the choice of temperament. (And there are more besides
just and
equal to choose from, by the way.)
YOUR TUNER TAKES CARE OF THIS. All tuners 'know' 12TET frequencies relative to A440 and tell you exactly what to use. Most tuners can also set the reference pitch to something other than A440, if you like, but they will still use 12TET! A few (like Peterson strobe tuners) offer special compromises (they call them "sweeteners") as alternatives to 12TET, but these only work in some circumstances -- read on.
For those who can follow the math:
That fifth in 12TET is exactly 2^(7/12) times higher than the root frequency, since a musical fifth is exactly seven musical half-steps above the root note, and each half-step is a factor of 2^(1/12) times higher in its frequency (i.e., it's higher by the twelfth root of two). Here the "^" symbol means exponentiation, and "/" means division. There are exactly 12 half-steps in a chromatic scale, so after raising a note by 12 half steps, we come to a note that's 2^(12/12) = 2^1 = 2 times higher: an octave. 12TET splits an octave,
multiplicatively speaking, into 12 equal, proportional parts. Each semitone is a fixed factor higher than the previous one.
Anyway, once the A4 note is set, the pitches of all the other mandolin strings are determined from this, based on (1) their assigned scale notes (that is, G,D, or E) and (2) the type of musical temperament used.
As you have seen, just temperament and equal temperament (12TET) assign DIFFERENT frequencies to the open strings.
Clearly, these are not compatible systems!
On a violin, the player compensates for pitch deviations associated with intially-just temperament (except for the open strings, which cannot be fixed) by adjusting their finger placement and vibrato. On a mandolin, you must use 12TET
because that's where the frets are positioned on the fingerboard.
If, on the mando, you mix just temperament for tuning the open strings with equal temperament for the fretted notes, then you're pretty much guaranteed to have to certain notes sound pretty far off, especially when the key isn't A or D major. 12TET minimizes these discrepancies as much as possible.
"Sweetened" tunings only exist successfully for playing in a few keys (usually just one, and maybe two), but
not in others. They make the other keys sound worse, in fact. CAUTION: There is no such thing as a "sweetened" tuning that works in all keys. Anyone who tells you that is selling snake oil.
A good example of "sweetening:" Many guitarists and most 5-string banjo players commonly drop the open B string note (2nd string) by a few cents from where their 12TET tuner suggests it should be. That's because the banjo is tuned to an
open G chord, and the B note will sound better
in that key. And the guitarists who do this are usually the same folks who play lots and lots of G chords in the root position with several open strings (e.g., bluegrass). This works well because the B note in 12TET is about 12 cents too high! In fact, it's among of the 'worst' compromises of the 12TET temperament. But if you do this, you'll need to play mostly in G. The banjo will sound pretty funky should you try to play out of E without a capo, for example. Even the keys of C and D will also sound pretty bad with a slightly dropped B string, to my ear. Finally, if you play a fretted B note on a different string from the open B (2nd string), the chances are that it will clash. Sweetenings only work sometimes. And sometimes they backfire.
This is a vast topic. To learn more, try starting with Wikipedia:
https://en.wikipedia.org/wiki/Equal_temperament#12TET
And please, folks,
trust your tuner! It probably "knows" more than you do about equal temperament (12TET)!!
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