Formula for calculating the time of arpeggios (Digital Audio Workstation question)?

You've given the beats per minute as 98. You've given lots of other information. Unfortunately, none of it is helpful to answer your question specifically. Besides BPM, we need to know two other pieces of information:

1) What kind of note gets one beat? (It's represented by the bottom number in the time signature.)

2) What kind of note do you want to use for the arpeggio? (For example, an eighth note? A dotted eighth note? A triplet sixteenth note?)

Let's assume your time signature is 3/4 and you want your arpeggio to use regular 16th notes. These are important to the outcome and may not be what you want, but let's let them serve for the purpose of an example.

Step 1

Convert your beats per minute into seconds per beat. To do this, you divide the number of seconds per minute (60) by the number of beats per minute (98). That gives you about 0.612 seconds per beat. (Another way to look at it: determine the beats per second by dividing 98 by 60. You get 1.633. The reciprocal of beats per second is seconds per beat. The reciprocal of 1.633 is 0.612, so we've double checked our work!)

Step 2

The time signature of 3/4 says a quarter note gets one beat. So a quarter note occurs every 0.612 seconds. However, we want our arpeggio to play using sixteenth notes. Because 4 sixteenth notes make a quarter note, we'll divide 0.612 seconds by 4. That gives us 0.153, so a sixteenth note occurs every 0.153 seconds.

Step 3

A millisecond is one thousandth of a second. So, to convert 0.153 seconds into milliseconds, you'd need to multiply it by 1000. So your answer is 153 milliseconds.

Here's a formula you can use:

T = tempo in beats per minute

B = type of note that gets one beat (use ¼ for a quarter note)

N = type of note to use for the arpeggio

R = milliseconds for the arpeggio

R = (60 ÷ T) × (1 ÷ B) × N × 1000

Substituting the numbers from our example:

R = (60 ÷ 98) × (1 ÷ ¼) × ¹/₁₆ × 1000

Simplifying:

R = 0.612 × 4 × 0.0625 × 1000

Simplifying further:

R = 153

The formula will work for any time signature and any type of note. Here are some decimal values to use for N:

Whole note: 1

Half note: 0.5

Quarter note: 0.25

Eighth note: 0.125

Sixteenth note: 0.0625

Thirtysecond note: 0.03125

Dotted half note: 0.75

Dotted quarter note: 0.375

Dotted eighth note: 0.1875

Dotted sixteenth note: 0.09375

Half note triplet: 0.3333

Quarter note triplet: 0.1667

Eighth note triplet: 0.0833

Sixteenth note triplet: 0.0417

Note that this formula is useful not only for arpeggios, but for calculating milliseconds of delay when you want a digital delay to coincide with the beat of the music.