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Ok, I am trying to figure out an equation that will allow me to get the number rhythm vales for a given unit of time (expressed in seconds)...

so my thinking was as follows:

Get the percentage of the tempo and scale it down:

(100 / 60.0 * tempo) * 0.01

... This will take a tempo of 30, and return 0.5... a tempo of 60 returns 1.0, a tempo of 120, 2.0, etc..

Then I need to inverse that so that 60 is actually 1.0, 30 is 2.0, and 120 is 0.5-- since a metronome marking of 30 should take 2x as long as 60... So I used exponents to do this:

(100 / 60.0 * tempo) * 0.01 / (tempo / 60.0) ^ 2

This gives me 1.0 for a tempo of 60, 2.0 for a tempo of 30, 0.5 for a tempo of 120... Exactly what I want.

So the last step now (and what I am struggling with), is to make this related to the rhythmic note value...

So, if I say quarter note = 120, that means there should be 2 quarter notes for every second...

The equation above will already give me 0.5 for a tempo of 120.. So it feels like I am half way there... But I still need to figure out how to get the highest note value... So if there are 2 quarter notes in a second, that should mean there are 4 8ths, 8 16th, 16 32nds, and 32 64ths...

so, if I traverse the list of rhythmic notes, dividing that 0.5 each time.. I should get where I want to be (which is to figure out what 0.5 converts to for 64ths, the fastest note duration)...

...

0.5 / 1 (quarter) = 0.5
0.5 / 2 (8th)     = 0.25
0.5 / 3 (16th)    = 0.166666

..... Ok already I know that is wrong, and I can see why... I don't want to be doing that.. I think what I want to be doing is:

0.5 / 2 (8th) = 0.25
0.5 / 2 / 2 (16th) = 0.125
0.5 / 2 / 2 / 2 (32nd) = 0.0625
0.5 / 2 / 2 / 2 / 2 (64th) = 0.03125

....... And that is where I am stuck.. What can I do to figure out this / 2 / 2 / 2 recursive business? Something tells me this is a logarithm since it's the inverse of an exponent.. but-- I don't quite get how to calculate Log base 2 against this...

Please help!

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I've seen your recent question on tempos and scaling - do you have any questions on my answer to the question here? –  Gerry Myerson Sep 20 '12 at 5:07
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1 Answer

A much simpler expression for your first display is "tempo divided by 60," and, for your second display, "60 divided by tempo."

Now for an $n$th note, you want "240 divided by (tempo times $n$)".

Thus, for a quarter, or 4th, note, you get 240 divided by (tempo times 4), which is 60 divided by tempo. For an eighth note, 240 divided by (tempo times 8), which is 30 divided by tempo. Sixteenth note, 240 divided by (tempo times 16), and that's 15 divided by tempo. And so on.

Going the other way, for a half note use $n=2$, for a whole note use $n=1$.

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