Roots, Cents and Herz and ETD's

[larudee@pacbell.net]

Duncan,

There is nothing wrong with your calculations, but a logarithmic scale with 1200
divisions per unit doesn't fit into a logarithmic scale with 12 divisions per
unit.  Remember, you are applying the multiplier (either the 1200th or the 12gth
root of 2) to the result of the last calculation, NOT to the result of the 100th
or 200th or 300th calculation.  If you create a 1200 note ET scale and then
remove all but the notes divisible by 100, you end up NOT with a 12 note ET
scale but with a 12 note logarithmic scale, where the size of the interval
increases as you go up the scale.  The only place the two scales would match is
at a just fifth above the lower note, with the intervals (of 100 notes) below
becoming progressively narrower in the 1200 note scale than in the 12 note scale
and the intervals above progressively wider.

The 1200 note ET scale is therefore not equivalent to a 12 note ET scale divided
into cents.  In order to get the value of the cents in a 12 note ET scale you
have to first create the 12 note scale and then divide each note into 100 cents,
which is of course what we normally do.  You can of course divide each note into
logarithmic cents, but that would not be the same as a 1200 note ET scale.

Paul S. Larudee

"D.Martens" wrote:

> Hi,  apparently I wasn't clear enough in my last post,
> I'll rephrase the problem :
>
> Suppose you want to divide an octave in 1200 steps (cents), 100 for each,
> note, what should be the correct number if the first note is 110 Hz, and the
> last 220 Hz ? (the last ofcourse, is exactly double the amount of the first)
>
> Suppose you want to divide an octave in 12 steps, 1 for each  note, what
> should be the correct number if the first note is 110 Hz, and the last 220
> Hz ? (the last ofcourse, is exactly double the amount of the first)
>
> 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.
>
> I did the same for the twelfth root of two.
> again no surprise , the answer is 220 Hz in 12th cell in the other row,
> again NOT rounded off.
>
> I figured that if both numbers are exactly correct, they must have 12 places
> that exactly match: the 100th, the 200th,......the 12hunredth cell of the
> first row vs. the cells in the second row.
>
> 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
>
> As you can see in  the above table, differences are between 0.06 Hz and 0.01
> Hz, which is too much.
>
> 1.05946309436       for the 12
> 1.0005782715387   for the 1200
>
> My question is: Why don't these two rows exactly match ?
> Did I overlook something, or is it the computer's processor,
>
> Does anyone have a clue ?
>
> Duncan.