Confusing problem, not sure how to define (repetition of musical pieces)

[Alfred]

If I have one musical phrase lasting 4 minutes and another lasting 6, when put together they should repeat every 12 minutes. If i introduce another phrase lasting 7 minutes, I'm guessing they will repeat every 84 minutes?

They question I'm trying solve is how to get (preferably) three phrases of reasonable lengths (around 2 - 15 minutes) that when put together will repeat every 15768000 minutes...

I understand these specifications may have been a bit ambitious, however any advise on how to go about solving this problem or even some pointers in the right direction would be very much appreciated!

Many thanks,

Alfred

 

[Dr.Peterson]

If I have one musical phrase lasting 4 minutes and another lasting 6, when put together they should repeat every 12 minutes. If i introduce another phrase lasting 7 minutes, I'm guessing they will repeat every 84 minutes?

They question I'm trying solve is how to get (preferably) three phrases of reasonable lengths (around 2 - 15 minutes) that when put together will repeat every 15768000 minutes...

You're looking for three small numbers whose Least Common Multiple (LCM) is 15,768,000 = 2^6 × 3^3 × 5^3 × 73. If you want whole numbers of minutes, at least one of them will have to be a multiple of 73 - not less than 15 minutes.

Maybe it can be done with everything in seconds. I'll let someone else try that, or prove that it also can't be done (as I suspect).

It appears that you are looking for a piece that takes exactly 30 years to play, if each year has exactly 365 days. But that means you have to ignore leap years. Is that what you intended?

 

[JeffM]

If I have one musical phrase lasting 4 minutes and another lasting 6, when put together they should repeat every 12 minutes. If i introduce another phrase lasting 7 minutes, I'm guessing they will repeat every 84 minutes?

They question I'm trying solve is how to get (preferably) three phrases of reasonable lengths (around 2 - 15 minutes) that when put together will repeat every 15768000 minutes...

I understand these specifications may have been a bit ambitious, however any advise on how to go about solving this problem or even some pointers in the right direction would be very much appreciated!

Many thanks,

Alfred

It seems to go beyond confusing to nonsensical.

You cannot combine a 4 minute phrase with a 6 minute phrase to repeat every 12 minutes.

6 + 6 = 12.

4 + 4 + 4 = 12.

4 + 6 + 4 = 14 and 6 + 4 + 6 = 16.

 

[Dr.Peterson]

It seems to go beyond confusing to nonsensical.

You cannot combine a 4 minute phrase with a 6 minute phrase to repeat every 12 minutes.

6 + 6 = 12.

4 + 4 + 4 = 12.

4 + 6 + 4 = 14 and 6 + 4 + 6 = 16.

I assume, from the OP's conclusions in the examples, that "combining" doesn't mean playing one after another, but perhaps three different instruments playing three different tunes (or parts) at once, each of which takes a different amount of time for one repetition. After 84 minutes, the 4-minute part will have been played 21 times, the 6-minute part 14 times, and the 7-minute part 12 times. It's the LCM we're looking for.

 

[JeffM]

I assume, from the OP's conclusions in the examples, that "combining" doesn't mean playing one after another, but perhaps three different instruments playing three different tunes (or parts) at once, each of which takes a different amount of time for one repetition. After 84 minutes, the 4-minute part will have been played 21 times, the 6-minute part 14 times, and the 7-minute part 12 times. It's the LCM we're looking for.

Well that makes sense: combining contrapuntally. But as you pointed out, 73 minutes is a long phrase. And even if you go to seconds, 73 remains a factor.

 

[Alfred]

You're looking for three small numbers whose Least Common Multiple (LCM) is 15,768,000 = 2^6 × 3^3 × 5^3 × 73. If you want whole numbers of minutes, at least one of them will have to be a multiple of 73 - not less than 15 minutes.

Maybe it can be done with everything in seconds. I'll let someone else try that, or prove that it also can't be done (as I suspect).

It appears that you are looking for a piece that takes exactly 30 years to play, if each year has exactly 365 days. But that means you have to ignore leap years. Is that what you intended?

Thanks for the replies,

I chose to ignore leap years in the end as I was only hoping to achieve a representation of 30 years.

I figured the most likely solution (if one did actually exist) could not consist of integers, but rather three numbers with enough decimal places so as to keep the phrases out of sync with one another just long enough to achieve the desired 15,768,000 minutes. However, where to begin finding an equation that would provide said numbers is somewhat beyond me...

Many thanks,

Alfred