The math behind perfect 5th tuning

[Richard Moody]

----------
> From: Robert Scott <rscott@wwnet.com>
> To: pianotech@byu.edu
> Subject: The math behind perfect 5th tuning
> Date: Tuesday, February 04, 1997 7:50 AM

Robert writes
>      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".)
>



	This is probably why it requires super effart to get  even a C in math
courses.  My reasoning  doesn't always seem to jibe with the mathminded.
Some questions I ask they look at me like I would look at someone who can't
hear beats.  Or like the choir master who encounters the son of the patron
who can't match pitch.

	Anyhow.... To this mind (I know that's a matter of opinion), the problem
seems to be that 12 notes have be placed in an octave. You start with one
note  say   n and end up with 2n.  You have to multiply that n by the same
ratio    r      12 times to end up with 2n.  What is    r    ?

Does this work??      n times a ratio twelve times equals two times n.

Or symbolically   n(r^12) = 2n             hmmmmm   the two  n's   cancel


	r^12 = 2      or r =  the twelth root of 2

Is this right ???

> 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.


One more question.   how did you come up with cents?   Is    r    the ratio
of a fifth larger than r the ratio of an octave   by 3.26 percent  ??    Or
is 2.003875474 larger than 2.00000000 by 3.6 %  ?   I could try it on the
calculator but I gotta go.    I am wondering if the question  is correct
though?

Richard Notthereyetinmath  Moody



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
>