Calculating Beat Divisions Using Tempo and Time

Whether through modern subdividing delay pedals or sheer number crunching, here’s how to merge musical time with real time.

By Jon Chappell

The above delay pedals allow you to specify a musical subdivision as you tap a quater-note tempo. (Click any image to enlarge.)

Many modern delay pedals offer you the option of specifying the discrete repeats according to musical subdivisions. For example, if your pedal has a tap tempo function, all you have to is tap to the beat (which the pedal assumes is the quarter note), and the chosen mode (usually) does the work for you, spitting out perfect eighth notes, eighth-note triplets, or 16th notes.

A pedal with this ability (see photos above) is great to have on stage for certain kinds of music, especially dance and techno, where a throbbing but rhythmically related underpinning helps add a sense of activity to the music without overcomplicating the melodic harmonic content. If you can produce steady 16th notes by playing only four-to-the-bar quarters, think how much energy you’ll save!

They may seem miraculous, but all these modern marvel pedals are doing is some simple math and providing an onboard calculator. The convenience is that you only have to tap quarter notes, which is far easier than accurately tapping, say, a dotted-eighth rhythm in order to produce the famous “cascade” effect, as evidenced by Eddie Van Halen, Albert Lee, John Jorgenson, The Edge, and others. But the math itself is elementary school level stuff. You can do it in your head, or more quickly on any simple external calculator—even the one the comes with a “dumb” flip-phone. We’re talking about basic long division here.

THE BASICS: REAL TIME AND BEATS

Whenever converting back and forth between time (as is the language of delay pedals) and beat divisions (eighth notes, etc.), you use the value called “beats per minute” (BPM). This exists on metronomes, where, you’ll see “90 | Andante” (see Fig. 1). That’s just saying that there are 90 beats in a minute. That’s faster than one a second (which would be 60 BPM) but not as fast as two per second (120 BPM). In fact, it’s right in between one and two seconds at 1.5 beats per second.

But musicians don’t think of minutes or seconds, at least as they relate to beats. They think of BPM as fast and slow. For example, if the BPM setting on your metronome, drum machine or DAW’s conductor track, is too slow at 86 BPM, you crank it up to 90 or 92. BPM really only gets related to music when you bring in your digital delay. That’s when the math comes in handy.

BEATS ARE FROM MARS, DELAYS ARE FROM VENUS

There are many tricks and shortcuts for converting BPM to musical subdivisions, but here are two key numbers: 60,000 and 60. Either of those will convert a BPM to quarter notes. From there, you just perform the additional math to get to eighth notes (one half of a quarter), 16th notes (one quarter of a quarter), or 8th-note triplets (one third of a quarter note).

But how and why would you use 60 and 60,000? If you take 60,000 and divide it by the BPM, you get the quarter note rate in milliseconds. Milliseconds are the units that digital delays traffic in. So to take our example of 96 BPM, 60,000 / 90 = 625. That means quarter notes are coming 625 milliseconds apart—slightly faster than 1,000 ms, or one per second, but not quite as fast as two per second (500 ms).

The problem with 60,000 is that it’s a rather large number being divided by a small number. The second number is never going to be more than 208—the maximum tempo in normal music. Better to use 60. So to take the same example, 60/96 = .625. This is the same answer in seconds (not milliseconds) except with a decimal. However, if you ignore the decimal, you have 625. So 60,000 and 60 are related in that one answer gives you hundreds of milliseconds (666) and the other parts of a second (.666), but the second is more manageable, especially for calculators that limit you to just a few places.

Doing calculations on the fly can be a hassle, so you can create two types of cheat sheets that give you pre-determined answers. One way is simply to create a chart of all the basic metronome markings (there are only about 40 of them), fill in the appropriate slots, and carry it as a document in your laptop or smartphone. That’s the example shown in Fig. 1.

Fig. 1. You can make up a simple beat-division chart in milliseconds and save the document on your laptop or smartphone.

FORMULAS ON THE FLY

A better way is to create a spreadsheet that will do the calculations on the fly. For this, you enter a value in one cell that’s governed by a formula (which you’ve keyed in in advance). Once you enter the number—whether it’s a standard metronome marking like 96, 100, 120, or 132 or an unusual one like 121.25—the spreadsheet does the work for you. The formula works like this:

Cell A2 =

Apply formula for Cell B2: =60/(A2) * 1000

(Repeat for all Cells in Column B. Shortcut: shift-drag A1’s lower-right corner)

And Fig. 2 shows how it looks in Excel, a common spreadsheet program:

Fig. 2. Note that the formula is active in the formula field (indiated after "fx," and that "* 1000" (multiply by a thousand) gets rid of the decimal point.

The disadvantage with a spreadsheet is an obvious one: you have to know how to work a spreadsheet—and not just for creating sortable lists, which is what most people use them for. No, for this use you have to know how to format and enter a formula. But it’s really not hard, and once you get the protocol down, you can plug in any number. For example, Here are the formulas for deriving quarters, eighths, eighth-note triplets, and 16ths from a spreadsheet, based on 80 BPM in cell A2 above.

60/(A2) * 1000 = 750 (quarter notes, or beat for beats per minute)

60/(A2)/2 * 1000 = 375 (8th notes)

60/(A2)/3 * 1000 = 250 (8th-note triplets)

60/(A2)/4 * 1000 = 187.5 (16th notes)

THE BEST OF BOTH WORLDS

Perhaps the most ideal approach is to combine the fixed-document approach of Fig. 1 with the on-the-fly spreadsheet approach in Fig. 2. This is where you take the time to set up the formulas for the seven common subdivisions (quarter, quarter-note triplet, eighth, eighth-note triplet, 16th, 16th-note triplet, 32nd note) acroos the length of the columns. Then Enter in all 40 standard metronome markings, which you see in Figure 3.

Fig. 3. Standard metronome settings are limited to about 40 (39, to be exact), as found on the faceplate of this electronic metronome.

The cool part comes in the blank rows beneath your entries. The formula  still works, so you can plug in any number and the sheet will fill out  all the values across the seven columns.

CONCLUSIONS

Letting the spreadsheet fill in all the values in your matrix means no legwork on your part, other than plugging in six formulas across the top—simple, once you get the hang of “spreadsheet-speak,” and 40 rows of pre-designated BPMs (a no-brainer). But the best part is, you still have active, formula driven cells in Column A. Plus, you'll find you can use your newly cultivated spreadsheet chops in other applications, too--music and non-music related.