The math behind perfect 5th tuning

[Robert Scott]

     After reading all those posts about perfect 5th tuning, I got curious as
to how the math works out.  So here it goes.

     With perfect octave uniform tuning, (assuming zero inharmonicity),
the ratio of consecutive half-step fundamental frequencies is "r" where
"r" satisfies the equation:

    r ^ 12 = 2          (The symbol " ^ " means "raised to the power".)

Using logarithms, this solves to the familiar  r = 1.059463094.

     With perfect 5th uniform tuning, again assuming zero inharmonicity,
the ratio "r" satisfies the equation:

   r ^ 7  =  1.5

which solves to r = 1.059634023, which, as expected, is a little wider
than with perfect octave tuning.  So with this tuning, what does an octave
work out to?

   r ^ 12 = 2.003875474

which would make an octave wider than before by about 3.26 cents.  Now if
the 2nd partial inharmonicity just happened to be 3.26 cents, then you could
have zero-beat octaves at the same time as perfect 5th tuning.  But that assumes
that inharmonicity would not change the perfect 5th zero beat.  For that to
happen
you would need to have the 3rd partial inharmonicity equal to the 2nd partial
inharmonicity (i.e. both at 3.26 cents.)  - a neat trick if you could do it.

Getting back to the consequences of perfect 5th tuning, let's look at 4ths
and 3rds:

  4ths:   r ^ 5 = 1.335916983,   (sharp by 3.26 cents)
  major 3rds:  r ^ 4 = 1.260734323, (sharp by 14.4 cents)

Compare this with perfect octave tuning (r^12=2):

  4ths:  r ^ 5 = 1.334839854, (sharp by 1.90 cents)
  major 3rds:  r ^ 4 = 1.25992105, (sharp by 13.3 cents)

Of course, all this doesn't really apply exactly to pianos because perfect 5th
tuning is not exactly r ^ 7 = 1.5 due to inharmonicity.  But it ought to be
a pretty
good approximation.

   -Bob Scott
    Ann Arbor, Michigan