A "Just Intonation" issue with some tunings
Some very popular tunings ... utilize the root, 2nd, 5th and 6th notes together.
This sets up a "problem" when tuning the steel guitar using "Just Intonation"
Basically a tuning with the 1, 2, 5 & 6 notes in it ... cannot be made "beatless".
EX: E13/9 (lo to hi) E F# G# B C# E
The F# note can't be in tune with B and the C# at the same time ... somethings gonna beat ... .
In the key of E ... the following deviations for "Just Intonation" from "Equal Temperament" is as follows:
E = root = 0 cents
F# = 2nd = + 4 cents sharp
G# = 3rd = -14 cents flat
A = 4th = - 2 cents flat
B = 5th = + 2 cents sharp
C# = 6th = -16 cents flat
Heres the "problem"
F# is the fifth of B ... 5ths need to be +2 cents sharp of their root ... so tuning your F# to your B ... will put your F# ... + 4 cents sharp ...
Because the B is already +2 cents because its the 5th of E... so (+2) + (+2) = +4 ... exactly where it needs to be to fit in with the root E overtone series ...
So far so good ...
The problem is ... F# is the fourth of C# ... so if you tune your F# to the C # ... it will needs to be -2 cents flat of C# ... 4ths need to be - 2 cents flat of their root to fit in with its overtone series.
Since C# is already the sixth of E (-16 cents flat) ... that would put the F# ... -18 cents flat of its ET value.
There's the problem ... to get F# to "fit in" the key of E and agree with E's overtone series and the fifth B ... it needs to be +4 cents sharp.
But to "fit in" with the C# in this particular tuning ... it would have to be -18 cents flat.
Strumming E, F#, B and C# ... will produce a mess of nasty beats.
Most folks that use this E13/9 tuning ... tune the second (F#) to the fifth (B) and the sixth (C#) gets tuned to third (G#) ...
And the player just avoids hitting the F#, B and C# ... together.
Or you could just tune up "Equal Temperament" ... ha, ha