Tuning Systems



Here is the tuning problem - a competition between different intervals.  Especially important are the lowest 3 most significant intervals in the harmonic series - The Octavethe Perfect 5th and the Major 3rd.  (remember that P4th is the inverse of the P5th).  In other words, if the Octave is tuned beatless and we try to tune the P5th beatless, it does’nt work out mathematically.  Also if you tune the M3rd beatless, it doesn’t work out with the Oct. or the P5th.

If you go around the “Circle of Beatless Perfect 5ths” you would think that you should end up with the original note - but you don’t - quite.  You end up 23.5 Cents (23.5/100ths of a half step) sharp.  We call this discrepancy the “Pythagorean Comma”.  (Yes, that’s the ancient Greek and he’s the one that first discovered it.)  

If all of the intervals in a chord are beatless, we call itJust Intonation”.  But the modern tuning approach (since J. S. Bach’s time) used by fixed instruments (like the piano) is to make all P5th’s a bit narrow so that in going around the Circle of 5ths, it comes out right.  We call that “Equal Temperament”.

Looking at the chart.

visualize the notes

exactly on the

“clock numbers”

and you end up with

Equal Temperament”.

In contrast, if you

visualize the notes on

the tick marks on the

OUTSIDE of the circle,

you end up with

beatless P5ths as we

want them in “Just

Intonation” where the

circle doesn’t come out. 

For instance, Bb going

counter-clockwise from

C to F to Bb doesn’t

match A# with the

beatless P5ths going

clockwise.  We’ve plotted going around the clockwise with the beatless P5ths as all sharps, whereas if you go around counter-clockwise, we’ve plotted them as flats.  With the beatless P5ths of Just Intonation, a Bb is different from an A# as are theoretically all the enharmonics (same pitch but different spelling) in Just Intonation.  However you can’t rely on this spelling in actual music notation, as arrangers and/or publishers are not consistent.  They often decide for practical reasons, to make it more understandable to the singer, not the theoretician.


Also we have plotted the beatless Major 3rd in Just Intonation from C to E which shows the discrepancy called the “Syntonic Comma”.  Notice that the Just Intonation M3rd is lower than the Equal Temperament M3rd or the Pythagorean M3rd.


In Barbershop singing, we want as many beatless Octaves, P5ths and M3rds as possible to reinforce each other and make our chords ring.    In other words, we want to sing in Just Intonation  (or “lock” the notes of the chord). They don’t ring if we use the “slightly-off-beatless” Equal Temperament.  But if we adjust the melody to Just Intonation, we will probably lose a bit in pitch level.  Thus we try to sing the melody in Equal Temperament to keep a level pitch and then tune the harmony notes to Just Intonation.  See Jim Richards‘ “Why Some Guys Go Flat” referenced on the next page.  As he explains, when singing harmony, you can’t necessarily keep the exact same common tone from one chord to a drastically different chord on the “Circle of 5ths” (i.e. when you jump from the I chord to the III chord).  Thus melody singers may need to think high for the M3rd’s (ET) and harmony singers may need to think lower (JI) for their M3rd to match the P5 & chord Root.

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