Roots, Cents and Herz and ETD's

[Joe & Penny Goss]

Now there is the complex answer<G>
Joe Goss
imatunr@primenet.com
http://www.primenet.com/~imatunr/

----- Original Message -----
From: <larudee@pacbell.net>
To: <pianotech@ptg.org>
Sent: Sunday, March 11, 2001 9:37 AM
Subject: Re: Roots, Cents and Herz and ETD's


> 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.
>
>