Pair-of-Dimes lost

>In my recent post, "Xenharmonic?", I was using the word "paradigm" to mean a
>set of bounds placed over an entire field of human interest for a long
>period of time that's hard to change. This, it seems to me, is an
>embarrassment to civilization.

I'll definitely buy that. The compositional paradigms each tuning
provides are interesting, but there must always be a wide variety of
choices of those paradigms.


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Carl Lumma wrote

> 1. If we detune by hertz, a direct arithmatic variation in beating occurs,
> right?
>
> 2. Detune in cents and you say we get a "linear" change in beating?
> "Linear" and "geometric" are the same?

"Linear" and "geometric" are approximately the same when they're small
enough ...

Pitch (p) and frequency (f) are related as follows:

p = a*ln(f)/ln(2)

For cents, a=1200, but I'll set it to ln(2) for simplicity.

For a small change in frequency, df, we have:

p = ln(f+df)
dp = ln(f+df) - ln(f)
dp = ln[f(1+df/f)] - ln(f)
dp = ln(f) + ln(1+df/f) - ln(f)
dp = ln(1+df/f)

Now, it's a standard approximation that ln(1+x)=~x when x<<1.

For example:

(ln(441)-ln(440))*440=0.9989
(ln(442)-ln(440))*440=1.9955

It follows that dp = df/(f*ln(2)) oct

2 Hz at 440 is 2/440*1200/ln(2)=7.87 cents

So, detuning by a few cents gives approximately linear beating.
The approximation gets worse the larger the detuning. In fact,
you can approximate the discrepancy in cents as:

600/ln(2) * (df/f)^2 =~ ln(2)/2400 * dp^2

At 20 cents, this comes out as:

ln(2)/2400*20*20=0.12 cents

So, if you want accuracy to the nearest tenth of a cent, this
won't do. At 50 cents, it comes out as:

ln(2)/2400*50*50=0.72 cents

ln(2)/2400*50*100=1.44405662617

I think 1.4% is close enough, and you won't need much larger
detunings in dissonance calculations.


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