Roots, Cents and Herz and ETD's

[Robert Scott]

Duncan,

I think I have your answer.  As Richard Brekne pointed out,
1.0005782715387 is not the 1200-th root of 2.  But your
spreadsheet seemed to say that it was.  So I got to thinking,
what is 1.0005782715387 and why did it seem to work out for you?
It turns out that 1.0005782715387 is exactly the 1199-th root of 2.
So I guess that your spreadsheet program has a counting error.
You may have left off the first or the last of the 1200 multiplications,
so that after 1199 multiplications you have exactly 2.0000.  Of
course the 100-th, 200-th, etc. multiplications will not match
the 12-note scale under this scenario.

Although you did not seem to suffer from any cumulative error in
the many multiplications, it would be more reliable to calculate
these very high powers of numbers near 1 by using logarithms:

    y          (y * log(x))
x     =   10

-Robert Scott
  Ypsilanti, Michigan


Duncan asked:

 >I filled this in in a spreadsheet, and had the 110 Hz multiplied 1200 times
 >by the number 1.0005782715387   in 1200 cells in a row.
 >220 came out to be the answer in the 1200th cell, NOT rounded off.
  ....
 >1200 row                 12row
 >A    110                   110
 >A#  116.4791983  116.5409404
 >B    123.4113571  123.4708253
 >C    130.7560774  130.8127827
 >C#  138.5379123  138.5913155
 >D    146.7828764  146.832384
 >D#  155.5185324  155.5634919
 >E     164.7740834  164.8137785
 >F     174.5804706  174.6141157
 >F#   184.9704764  184.9972114
 >G     195.9788344  195.997718
 >G#   207.6423453  207.6523488
 >A      220                   220
 >
   ...
 >My question is: Why don't these two rows exactly match ?