Problems with JI

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 ... 

Why ... 

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