[Fwd: how is a piano tuned?]

[Danny Moore]

Avery,

You might want to mess with her a bit and tell her "A piano is tuned to the
12th root of 2" and see if she can figure it out.

Mathematically, the octave ratio is 2:1.  Assuming she's trying to tune an
equal temperament, the octaves must be in tune.  The thirds ratio is 5:4 (5/4
x 5/4 x 5/4 =125/64) whic is not equal to 2/1.  (To be equal, it would have to
128/64 or 5.04/4 cubed.)  To force the 3 contiguous 3rds to be equal, each
third must be expanded by 14/100 of a semi-tone (14 cents).  This action
forces the reciprocal of the major 3rd (the minor 6th) to be contracted by an
equal amount.  Same is true of 4ths:  Expand by 2 cents causes the reciprocal
(the 5th) to contract by 2 cents.  (For math students, 1 cent = 1/100 of a
semi-tone)

Solving mathematically, we are searching for the number that, multiplied by
itself 12 times = 2 otherwise known as the 12th root of 2.  That number is
1.059

No, I'm not a mathematician.  Jim taught us this the first day of college, but
Bartlett & I stayed anyway.

Danny

Avery Todd wrote:

> List,
>
>    A friend of mine forwarded this to me. Anyone want to answer this? :-)
> In this context, I don't think I could even if I tried.
>
> Avery
>
> >Avery, if you have time would you answer this girls question! I don't
> >think she would like my answer! :-)
> >
> >Thanks,
>
> >>Hi - I am a graduate student in Mathematics from the east coast.  In one
> >>of my mathematics classes there is a question which asks us how a piano
> >>is tuned.  So if you could please send me an email breifly describing
> >>the process I would be grateful.  Thank you very much in advance.
> >>
> >>Christine Palmer
> >>
> >>cpalmer@wpi.edu
>
> ___________________________
> Avery Todd, RPT
> Moores School of Music
> University of Houston
> Houston, TX 77204-4893
> 713-743-3226
> atodd@uh.edu
> http://www.uh.edu/music/