[Fwd: how is a piano tuned?]

[Howard S. Rosen]

A math major, I am sure, would be interested in knowing that the
frequencies of all the piano strings are in a geometric progression rather
than an arithmetic progression.(She *should* know the difference between
the two) The factor by which all piano string frequencies are multiplied to
arrive at the next frequency, is the 12th root of 2 (1.0594632 etc.) which
is explained in this manner:

Since all octaves are in the ratio of 2:1, we can designate every octave
note as a power of 2. The frequency then of all octave notes of the same
name would then be represented: 2 to the zero power, 2 to the single power,
2 to the second power, 2 to the third power, 2 to the fourth power etc. Any
octave note is designated by 1 full power unit higher than the previous
octave note. Since all of these octaves are divided into 12 equal parts
then the factor must then be 1 power unit divided by 12. Therefore, any
note of any octave will be multiplied by 1/12th of the octave, which is: 2
to the 1/12 power and she will know that mathematically it converts to the
12th root of 2.(this happens to be an irrational number, as the decimal
places are without limit. Maybe that's why tuning a piano is
impossible!@#$%)

I think that the problems of inharmonicity need not be explained. Perhaps
the above would suffice to interest a math major re: tuning a piano.


Howard S. Rosen, RPT
Boynton Beach, Florida
----------
> From: Danny Moore <danmoore@ih2000.net>
> To: pianotech@ptg.org
> Subject: Re: [Fwd: how is a piano tuned?]
> Date: Thursday, January 08, 1998 5:40 PM
> 
> 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/
> 
>