Calculating when metronomes at different tempos will sync up

I’m writing a piece where 5 players in 3 sections loop a certain bar halfway through. While 1 section loops the music at one tempo, the other two sections slowly phase out of tempo (akin to steve reich’s various “phase” pieces).

However unlike steve reich, the performers aren’t just speeding up to be a 16th off. Each player will be wearing headphones connected to a metronome. They will all start at the same tempo but when they phase, 2 other performers will gradually change the tempo for sections 2 & 3, making the music sound free but still have structure.

I want the piece to end with all 3 sections playing their last note in sync, I think that would have an AMAZING effect on the audience, everybody ending together after so much seemingly random music. My only problem is that it seems extremely difficult to calculate on which beat for each tempo the notes sync, especially since they change gradually. 

The tempos for each section are as follows:

(all are eight = tempo)

Section 1 = 121

Section 2 = 132

Section 3 = 110

When the phase begins, the 2 people who are manually changing the tempo do so at a rate of 1 click per measure in a 3/4 time signature until they reach the desired tempo. The starting tempo is 121 for each player, then sections 2+3 increase/decrease in tempo by 11 clicks over the course of 11 measures (in the tempo of section 1 (121)).

I may sound like I’m trying to do something impossible but if anybody has any tips or knows the math to figure this out that would be EXTREMELY helpful.

Also if anyone is confused by what I mean or needs any more details to figure it out just let me know!!

It is not easy for me to imagine what you want to do.
I think it is a work based on polytempo.
There is a fixed tempo section 121.
Another that starts at 132 and slows down.
Another at 110 that speeds up.
Until they are supposed to meet at 121.
Is it like that?
There aren't many examples of these things, as far as I know.
In the renaissance the canon in prolatio flourished, which you can explore.
Ligeti composed a piece called Peme symphonique, based on 100 metronomes. 
The only piece I know of that uses polytempo is by Charles Ives: the unanswered question.

4 hours ago, Luis Hernández said:

 

It is not easy for me to imagine what you want to do.
I think it is a work based on polytempo.
There is a fixed tempo section 121.
Another that starts at 132 and slows down.
Another at 110 that speeds up.
Until they are supposed to meet at 121.
Is it like that?
There aren't many examples of these things, as far as I know.
In the renaissance the canon in prolatio flourished, which you can explore.
Ligeti composed a piece called Peme symphonique, based on 100 metronomes. 
The only piece I know of that uses polytempo is by Charles Ives: the unanswered question.

 

 

I realize my explanation was a little confusing. I have 3 performers playing in unison at 121, looping one measure. then 2 of the players have their metronomes gradually changed to 132 and 110, 11 clicks from 121.

They each play in different tempos for the rest of the piece and i’m trying to see if there’s a way to calculate when all three tempos will hit a beat at the exact same time. I’m sure there’s some math I can use I’m just trying to see if anyone has tips.

Thank you for the recommendations, i’ll check them out ASAP!

If I understand correctly, you wish to find the beat where all 3 sections will align during/ after the tempo transition.

I have some idea right now, but may I know if 1 “click”=1 bar/ beat?

Here is my sketch:

There are 2 parts of this problem:

1) How much, after the transition section, will the 3 sections differ?

2) What is the first aligning beat after the transition section (i.e. first time all tempo=132, 121 and 110 appear at the same time)?

**********
Turns out (2) is an easier problem.

Let’s re-interpret the musical notation in a mathematical way. Tempo=132 is equivalent to saying each beat takes 60/132 seconds. Suppose it takes M beats at Tempo=132, P beats at Tempo=121 and N beats at Tempo=110 to reach the “aligning” beat. Then we have, the time to reach the aligning beat since the start of the first bar after the transition is

(60/132)M=(60/121)P=(60/110)N

So 55M=60P=66N

To find out the integers M, P, N that satisfy the equation, we need to find the HCF of (55,60,66). (yes sorry for the math) , which is 660.

Turns out M=12, P=11 and N=10 are the smallest integers that satisfy the equation… and that means at the 12, 11 and 10th beat at tempo=132, 121 and 110 respectively will all sections meet at the same beat, at the (60/11)-th second!

**************

The really tricky part is the first part of the problem - due to the transition section, player section is differed by some time.

3 hours ago, HoYin Cheung said:

If I understand correctly, you wish to find the beat where all 3 sections will align during/ after the tempo transition.

I have some idea right now, but may I know if 1 “click”=1 bar/ beat?

Here is my sketch:

There are 2 parts of this problem:

1) How much, after the transition section, will the 3 sections differ?

2) What is the first aligning beat after the transition section (i.e. first time all tempo=132, 121 and 110 appear at the same time)?

 

**********
Turns out (2) is an easier problem.

Let’s re-interpret the musical notation in a mathematical way. Tempo=132 is equivalent to saying each beat takes 60/132 seconds. Suppose it takes M beats at Tempo=132, P beats at Tempo=121 and N beats at Tempo=110 to reach the “aligning” beat. Then we have, the time to reach the aligning beat since the start of the first bar after the transition is

(60/132)M=(60/121)P=(60/110)N

So 55M=60P=66N

To find out the integers M, P, N that satisfy the equation, we need to find the HCF of (55,60,66). (yes sorry for the math) , which is 660.

Turns out M=12, P=11 and N=10 are the smallest integers that satisfy the equation… and that means at the 12, 11 and 10th beat at tempo=132, 121 and 110 respectively will all sections meet at the same beat, at the (60/11)-th second!

**************

The really tricky part is the first part of the problem - due to the transition section, player section is differed by some time.

 

yeah you get it, and by adding 1 click I meant increasing the tempo by 1.

The transition into 3 tempos is really the hard part that’ll take some more math, but thank you so much for your equation. the piece in mostly centered around the number 11 and occasionally 10 and 12 so the fact that those beats are where the parts sync up is perfect!

I’ll try it out a few different ways and see if I can figure out the math with the gradual “phasing” of tempos, but if I have to i’ll just change it to an immediate tempo change.