# The math of tempo and signature

Not sure where to put this topic, but in a sense it is related to the DAW so…
Is there a mathematical link between the tempo and the time signature? Say if you run 120bpm in 4/4 and you want a section to be 6/8, what would be the bpm to make it sound just as fast/slow? If the bpm is not set lower in this example it will sound faster.

Technically bpm is bpm, ie beats per minute and that’s your tempo. If instead you want for instance a measure in 4/4 to be just as long in 6/8, then try decreasing the tempo by 33%.

Firstly I’m after a mathematical model in general not just for the two signatures mentioned.

Secondly, 33% reduction would in the mentioned example give around 80bpm. That is much too slow in order to make this section sound as fast as the others. It is somewhere around 96bpm (if my ears are correct). But what is the exact correct bpm mathematically? (if there is such a thing).

Mmh, I have no idea what the mathematical description of a signature ‘feel’ is.

If you mean, you have step-entered some eighth-note triplets at 120 BPM in 4/4, then if you simply change the time signature, so that they would, technically, become straight 8th-notes in 6/8 (or, more correctly, 12/8), then no tempo change at all is needed for those notes to sound the same, but the metronome will sound wrong against them. You’d have to switch the MIDI track over to Linear Timebase (so that they won’t change during the next bit), then change the tempo from 120 BPM to 120 x 3/2 = 180 BPM.
But, as Strophoid said, the ratio is always 3:2 (or 2:3 in the “other” direction )… in 4/4, the “beat” is in quarter notes, but in compound time (6/8, 12/8 etc.) the beat is the dotted quarter-note (i.e. a quarter-note plus an eighth-note).

EXERCISE:
Start a metronome. Play six notes in a row, each on a tick of the metronome. Ask multiple different listeners to write in notation what they heard. Assume the listeners can not hear the metronome.
First listener writes two measures of quarter notes in 3/4 time.
Second listener writes one measure of eighth notes in 6/8 time.
Third listener writes three measure of half notes in 2/2 time.
Who is right?
They all are.
How are the tempos related?
They are the same.
A fourth listener writes one measure of triplet quarter notes in 4/4 time.
A fifth listener could write three measures of half notes in 4/4 time.
These listeners are also correct but they are hearing different base tempos.
The tempo is based on the perception of the listener and the time signature is just how they choose to notate what they hear. There is no hard mathamatical link and there are multiple ways to annotate the desired sound.

J.L.
p.s. I hope I got all those right.

Nope, that is NOT it.

I have a 120bpm 4/4 piece, followed by a 120bpm 6/8 piece. Lets say that every beat has a kick drum hit on it. That is 4 hits for the 4/4 bar and 6 hits for the 6/8 bar. With both running at 120bpm the 6/8 feels much faster. By trying and failing it seems that somwhere around 96bpm for the 6/8 part seem to match the 120bpm 4/4. Now they both are preceived (to me) to have the same “speed”. Since it is possible to find a matching bpm for the two signatures there must be a formula for it, no?

My point is that there is NO general mathematical model. Each specific case is different and based on the tempo, time signature, and (most importantly) what is being played.
To convert 120 bpm 4/4 triplets to 6/8 straight eighths is different than converting to straight quarters, triplets, or 5-tuplets.

By definition, a “beat” in the 120 bpm 4/4 piece is a quarter note. By definition, a “beat” in the 120 bpm 6/8 piece is an eighth note. When you say every beat has a kick drum in it do you mean that in the 4/4 piece every quarter note has a kick and in the 6/8 piece, every eighth note has a kick? If so, the kicks should be evenly spaced through the change in time signature. Of course, it depends on what actually is being played.
For example, If I am in 120 bpm 4/4 playing quarter notes and then start playing quarter triplets of course it sounds faster, but the tempo and time signature have not changed.
What are the NOTE value (eighth, quarter, sixteenth, etc.) of the kicks in your measures?

Yes. as Strophoid and Vic have already given. But not a general mathematical model that makes two different pieces of music in different time signatures sound the same speed.

Based on your discussion of the numerators in the time signature (which has nothing to do with tempo) it sounds like you are trying to get measures that are the same length rather than getting the same tempo or “beat”.
HTH
J.L.

O.K. One last try
No, that is wrong… you are comparing two different things…
In 4/4, a bar has 4 beats, each beat being a quarter-note (which is equivalent to two 8th-notes).
In 6/8, the bar has TWO beats, each being a dotted quarter-note (i.e. three 8th-notes). (that’s why I said the comparison should be made between 4/4 and 12/8, which has four dotted quarter-notes)
The “six” things you are referring to, in your description above, are in fact eighth-notes (that’s what the “8” is in 6/8 ), in two groups of three (three 8th-notes making a dotted quarter). You hear a big difference in the metronome, because in 4/4 it is beating out quarter notes (the lower “4” in 4/4).
You might hear it more clearly if you changed the 4/4 into 8/8 (nothing changes, except that the metronome plays twice as fast, and you now get eight “beats” to the same bar).
Without changing the tempo at all, the 8th-notes in 4/4 (or 8/8 ), are the same speed as the 8th-notes in 6/8… it’s just that it now has to play 6 of them in order to reach the end of the bar, instead of 8.

So if you want a bar of 4/4 to last the same time as a bar of 12/8, the 12/8 bar has to speed up by a factor of 3:2. (if you want 6/8, you should be comparing that with 2/4, not 4/4).

Ah yes of course, I see I had it wrong as well

The main steps for converting one tempo to its equivalent in a different time signature is as follows:
Define terms
#bpm = number of beats per measure, or the top number in the time signature
bpm1= original tempo
mepm1 = original number of measures per minute
mepm2 = new number of measures per minute
bpm2 = new tempo

Step 1: Convert the bottom number in the time signature so that the notes are the same. For example, if the original time signature is 4/4, and the new one is 6/8, convert the new one to 3/4. Now they are both using quarter notes.

Step 2: Find mepm1
mepm1 = bpm1/#bpm
Example mepm1 = 120/4 = 30 because 120 is the original tempo, and 4 is the number of beats per minute in the original time signature.

Step 3: Find mepm2
mepm2 = mepm1 + 1/#bpm2
Example mepm2 = 30 + 1/3 because 3 is the number of beats per measure in the new time
signature. (This is why we had to make them both use quarter notes in Step 1)

Step 4: Find the new tempo!
bpm2 = mepm2 * #bpm2
Example (30 + 1/3) * 3 = 91

So your new tempo should be 91bpm. This method should work for other conversions as well, as long as you complete Step 1 correctly.

I hope this makes sense!

Thx jesimusicgirl119!!!

However would you be so kind to do it the other way around as well?? Start with 6/8 in 91 bpm and convert to 4/4. I should end up with 120 bpm right? However after 4 hrs I cannot make the equation work!

Think I solved it.

Before you do the convertion described in step one (bottom number) you must make a note of if you are moving from a lower to a higher number or vice versa. The procedure described works for moving from lower to higher number. If doing it the other way around you need to deduct and not add, so the equation converting from 6/8 in 91 bpm to 4/4 is:

91-(1/3)*3=3X/4, making X=120

Holy wheel re-invention, batman!

The denominator isn’t really needed, and the ‘formula’ can be expressed simply:

Find the ratio of the old beat to the new, invert it, and multiply the tempo by its quotient.

The questions are:

• what rhythmic value does the beat have?
• what do you want it to become?

So for quarters to dotted quarters
the ratio of quarter to dotted quarter is 2:3, inverse is 3:2. Quotient is 1.5.

for dotted quarters to quarters
the ratio of dotted quarter to quarter is 3:2, inverse is 2:3. Quotient is .6666667.

The same process works for entire measures (or the duration of anything):
for a measure of 8ths to a longer measure of 8ths
ratio of 6 eighth notes to 8 eighth notes is is 3:4, inverse is 4:3. Quotient is 1.33333333.

The two methodes differ. Converting with one gives 91 bpm and yours 90. They cannot both be right. Listening to it the 91 seems to be correct, but they are so close I can’t say I am really able to hear the difference.

The way I showed you is simply an accepted standard- not something I came up with. It’s the same thing Vic and Strophoid explained above as well. It’s really not subjective.

Thanks for posting this. I think I get it. I’ve never seen “mepm” term before. I’d like to see more about this, perhaps with some examples. I’ll have to work with this to really understand it. Are there any further references on this method you’d suggest?

To my reading, this interesting and informative thread is really about Meter (which in Brazilian translates to Signature Track), which may be related to but is distinct from Tempo considerations, per se.

+1