Any mathematicians among us?

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Any mathematicians among us?

2009-07-25 08:20 am

I would appreciate some help for my son-in-law please. We both figure there is a mathematical formula for the two problems I describe here. However, my two sons-in-law couldn't agree on a formula for a drumming problem.

Is there is a formula that will count the number of combinations of accented and unaccented notes in a bar of sixteenth notes where each sixteenth not may be either:
-Unaccented
-Accented

For example:
AUUU UUUU UUUU UUUU / UAUU UUUU UUUU UUUU / UUAU UUUU UUUU UUUU/ and so on all the way to AAAA AAAA AAAA AAAA / can be done manually, but there must be a formula.

Next, we'd like to know how the formula would change if a ghost note is added to the mix, so you now have three possible articulations for each note.

It's beyond me. Help please?

Re: Any mathematicians among us?

Reply #1 – 2009-07-25 09:52 am

2¹⁶ = 65,536
3¹⁶ = 43,046,721

Re: Any mathematicians among us?

Reply #2 – 2009-07-25 10:12 am

By way of explanation what Rick just said:

The first note can have 2 states (accented and unaccented), and thus creates 2 possibilities.
The second note, also having two states, doubles that number: 2 x 2 = 4 possibilities
The third note, in turn, doubles the number of possibilities of the two previous: 2 x 2 x 2 = 8

I'm sure you see the pattern emerging now...
The final 16th note: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = (2 to the power 16) = 65536

Add a three state possibility for every note, and every added note triples the number of possibilities:
3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 = more than you'll care to count.

Regards,
Alvatrus

Re: Any mathematicians among us?

Reply #3 – 2009-07-25 07:56 pm

Thank you both very much. This makes sense to me.