Tuning Bass or Guitar to a Piano - using SuperAccurate Reference Tones

   When two pitches are equal, if they are sounded together, there will be zero beats
(beats being a slow vibrato that results when pitches are slightly detuned from each other.) Bass and guitar players (when not tuning using electronic devices!)  will typically get a pitch from the piano by playing in unison with  a chosen note,  tuning and playing simultaneously with the piano sounding.  They will tune the string until they hear a "zero beat rate" condition and assume that the string is now tuned to the piano.   From aural piano tuners, we know that making beat-rate speed comparisons is FAR MORE ACCURATE than attempting to find  this "zero beat-rate" condition.  Imagine, for example, trying to detect the sound of "1 beat per second", a very slow vibrato - it is nearly impossible to distinguish that from 0 beats per second, yet it represents a difference of 4 cents ( at A4)!  That is a huge error when you consider that a change 1/10 and even 1/100 of a cent  is  audible in the effect it has on the quality of intervals.  We can take a lesson from aural piano tuners and use a completely different approach that circumvents trying to hear an elusive zero beat-rate using REFERENCE TONES. 
    Imagine three notes    R, X and Y.   If the beat-rate between R and X   equals the beat-rate between
R and Y, then we can assume that X = Y ( they are tuned to the same pitch ).  If we change the pitch of R a slight bit, the 2 beat rates will both change but will remain equal.  The exact pitch of R ( Reference Tone) is not important as long as it's creating some beats that are slow enough to hear clearly.

(This will always be the case when x and y are both on the the same side ( # or b) of the perfect zero-beat point)
  R is the reference tone ( on the piano, a arbitrary note that we choose to use ).  X is the tone on the piano that we wish to tune to.  Y is the tone on the guitar or bass that we are tuning ( trying to accurately match the pitch to X).  Piano tuners use this system to verify that various intervals are narrow or wide from perfect by adjusting one beat rate to be slightly faster or slower than the other.  All intervals on pianos are a bit out of tune intentionally ( some are wide, some are narrow) to accommodate equal temperament.
  But I digress.   So how is this actually PRACTICAL?   Here is a real scenario involving tuning an electric bass using a test interval of a tenth, an interval with easy-to-hear beat rates on the piano. 
   R  :   Eb 2  on the piano  ( octave and 6th below middle C)
   X:    G3 on the piano  ( 4th below middle C)
  Y:   G harmonic on the G string of the bass ( octave above the open string)

GOAL is to perfectly match the G string harmonic to the piano G3
PROCEDURE:    1.   Play R and X together on the piano - listen to the speed of the beats.
               2. Play R and Y together ) piano and bass - listen to the speed of the beats.
     3.  Adjust Y until the beat rates are identical.

There you have it - the bass is matched to the piano probably to within 1/4 cent or even less.
   See my previous post about tuning and equal temperament as it pertains to guitar tuning.  http://puppydogmusic.blogspot.com/2011/04/piano-and-guitar-tuning.html

Beat rate comparisons are very useful when tuning 2 keyboards to each other to get quick accuracy.  Simply play an interval, like a major third, between the keyboards ( one note on one, one on the other) and judge the speed of the beats. Then reverse the roles of each keyboard using the same notes,  and adjust the pitch until both versions of the major third have equal beat-rates.