**Music Theory: Tune Lengths, Part Two**

November 19, 2018

In the prequel, we explained the importance of the length of a song, and why jazz musicians should keep songs at a length of 3-4 minutes. Now we've explained why songs should be this long, the time is approaching that we explain how musicians keep their songs to that time (but some mathematics is coming first). We're going to go through everything step-by-step, and this post will cover the mathematics that is necessary if you're going to play a song and keep it to a certain length. It answers questions like, "If I'm going to play a 4-minute song at medium tempo, how many measures will there need to be in total?" But before we get to these more advanced mathematical questions, there are some basic rules that musicians can follow, shown below:

Some easy rules jazz musicians can go by when putting together a song are the following:

1. If you're playing at medium-tempo, any tune that is 32 measures in length (only one chorus) will take about one minute to play.

2. If you're playing a tune with a very fast tempo (like Cherokee), and it is 64 measures in length, it will take about a minute to play. Therefore, each chorus will take about one minute as well.

3. A 32-measure ballad usually takes approximately 2 minutes to play. Therefore, 16 measures at ballad tempo takes about a minute.

Now, here comes the mathematically-based rule for the length of a song (warning, this is about to get complicated. Take it slowly if you need to do so):

A song's total length (in minutes) and its total length in beats are directly related if the tempo is constant. For example, if a song's total length is 128 beats, we can assume that if the song** is 1 minute long, the tempo must be 128 beats per minute.

Then we will assign some letters to represent the values mentioned in the previous paragraph:

1. M stands for the length of the song (total) in minutes.

2. B stands for the length of the song (total) in beats.

3. T stands for tempo of this song. (The song's tempo must stay the same from start to finish for this to work in calculations.) It will be in beats per minute. (See this link for an explanation of what beats per minute are.)

Here is the equation we will use (notice that the letters below correspond to the ones we just used):

B ÷ M = T

In other words, beats divided by minutes equals the beats per minute (tempo).

So, let's use the formula. We'll say that M equals 4 (that is, 4 minutes) and that B equals 256 (that is, 256 beats per minute). So we will divide B by M (256 ÷ 4) to get 64 beats per minute as the answer. (Now, beats per minute might mean practically nothing to you even if you read the link; you should try this link, which leads to a website that compares beats per minute to rhythms that jazz musicians should know about. Also provided at the link are examples of tunes that can be played at those tempos.)

Now let's apply the formula to some tunes. Let's take "Satin Doll" as an example; let's say that someone plays it at 128 beats per minute (you can re-check the link if you want if you're not totally sure on how fast 128 beats per minute is), and we play the tune for a total of 4 minutes. How long will it be? Back to the formula. We need to insert the values that we just assigned to the equation:

B ÷ M = T

B ÷ 4 = 128

This will be a little awkward, but can easily be solved. Think about it like this: what can we divide by 4 to get 128? In other words, 4 multiplied by 128 is what? The answer is reached by calculating 128 x 4, and the answer is 512. So the answer to the "Satin Doll" problem is the following:

If we play "Satin Doll" at 128 beats per minute for a period of 4 minutes, the song will be a total length of 512 beats.

Now, to make this answer more friendly, let's convert the number of beats to measures. But here comes the hard part: what do we divide 512 by to get the answer we want? We need three different formulas:

1. If a song is in 3/4 time (e.g. "Bluesette" or "Waltz for Debby"), divide the number of beats (in the whole song) by 3 to get the number of measures.

2. If a song is in 4/4 time (e.g. "Summertime" or "Satin Doll"), divide the number of beats (in the whole song) by 4 to get the number of measures.

3. If a song is in 5/4 time (e.g. "Take Five"), divide the number of beats (in the whole song) by 5 to get the number of measures.

In the case of "Satin Doll", we would use the formula for 4/4 time to get the number of measures. So we divide the number of beats that we calculated earlier (512) and divide it by 4. The answer we get is 128 measures.

In other words, we have fond out that if we play "Satin Doll" at 128 beats per minute for a period of 4 minutes, the song will be a total length of 128 measures.

It's an interesting coincidence that 128 shows up twice there. It won't be like that this time: we're going to apply our formulas again, this time to a different song, and see what happens.

Let's pretend that we're going to get together a band and play "Tune Up". We're going to play for 2 minutes and the total number of beats will be ... ?

This is not too easy to figure out. So let's change the band's arrangements a little. We'll say that we're going to play for 2 minutes at 256 beats per minute. Let's get back to the formula:

B ÷ M = T

But this time, we're going to change the formula a little to our advantage. We already know what T is; it's 256, so it's not very smart to isolate it. It's best to isolate the variable we don't know: the number of beats, B.

So we will multiply both sides by M:

B ÷ M x M = T x M

You might think that the new equation is worse. In a way, you're right. On the left side of the equation, we're now dividing by M and then multiplying by M. That's redundant.

Which is exactly what we want.

By removing the redundant part of the equation/formula, it will become much shorter:

B = T x M

We've now got the formula we want for "Tune Up". We just need to add the 256 beats per minute and the 2 minutes to the equation, and we will know how many beats our song will be.

B = 256 x 2

B = 512

The answer we have reached is that "Tune Up" will be a total of 512 beats when we play it with the band using the arrangement. Now, let's use our beats-to-measures conversion formula:

If a song is in 4/4 time (e.g. "Summertime" or "Satin Doll"), divide the number of beats (in the whole song) by 4 to get the number of measures.

512 ÷ 4 = number of measures

128 = number of measures

The answer: there will be a total 128 measures in the band's arrangement for "Tune Up". (To keep it from getting too complicated, we'll assume that there is no introduction or ending in the arrangement.)

That's a lot of mathematics. It sets us up well to go deeper into this in a future post. In other words:

Will be continued

* Division symbol from rapidtables.com. Also, in this article, "song" means the whole song, while "tune" is used to refer to just one chorus.

** Correction: It originally said chorus, but song is the right word choice.