Formulae

[Richard Moody]

----------
> From: JIMRPT@aol.com
> To: pianotech@byu.edu
> Subject: Formulae
> Date: Thursday, February 06, 1997 6:18 PM
>
>  List;
>
snip..
 >statements of equality are always relative to an implicit
> standard of tolerance."
> ergo:  2+5=7  TRUE
>           A440 x 2 = (2)A880  Mathmatically TRUE, Musically FALSE
(usually)
>           A440 x 2 = (2)A880 + 9cents  TRUE Musically?  (Generic S&S mod
'A')
>
> Besides aren't there 13 notes in a full octave?  :-)
> Jim Bryant (FL)

No there are only two.
When a pianist plays an octave, he/she can only tolerate 2 notes.  The same
with the listener. :))

It appears in the example above that an octave is given.  Also that in a
piano the frequency is not only doubled, but a little added because of
"stretch".   Stretch results because of the way the octave is tuned.  To
tune an octave from A440, the fundmental of A880 is tuned to the second
partial of A440.  Because of inharmonicity this partial may be sharp.
Suppose it is sharp by 9 cents as the example suggests.  Therefore given
the formula submitted by Niklas Eliasson (thank you 3x very much)  we have.

		F=f*2^(cents/1200)

Here F will equal the actual (in tune) frequency of the (theoretical)
fundamental of A880,, if the second partial of A440 or  f  is sharp by 9
cents.

on the calculator press these keys in the order given.  2  x^y  (  9  /
1200  ) *  880   =

And the result is  884.5866832365

Thus if the "generic" Steinway A really does have a second partial of A440
that is inharmonic by 9 cents sharp,  it would read  884.5866832365 cps on
a digital readout of a frequency meter  assuming the formula is correct and
the meter is accurate.
Thus for the octave A440  -  A880 to be beatless, the A880 would have to be
tuned to 884.5866832365  cps.  Hmmm   seems high doesn't it.

But it really doesn't matter what the numbers are, as long as the ear hears
the octave in question as beatless or nearly as possible.

We are at the point where the meaning of cents has to be nailed down.
The definition is 100 cents between each semitone. 1200 cents in an octave.
 If there are 100 cps between semitones, logic would assume 1 cent to equal
1 cps for this interval only.  If there were 50 cps  then 1 cent would
equal .5 cps.

Now it is my understanding that the SAT gives readings in cents, or perhaps
deviations from ideal, theoretical or desired pitches,, in cents.  The ear
hears deviations from desired pitch (but only from intervals) in beats.
Beats can easily be translated into  cps on paper.  I would like to get
cents translated to cps on paper. (ahem,, Jim ? : )  Why?  To be able to
tune different temperaments aurally. To caclulate synthesizer temperaments.
 To calculate scales, (using cps, how else?) and then compare the readings
of that scale from SAT's or other ETA's cent read outs.  To get a better
understanding of inharmoancity of piano strings. Exploring further the
theory that two pitches close to one another produce affects other than
beats or resultants and the nature of these affects, and or effects. etc.
etc.

Richard Moody