Is there an "algorithm" for the time duration of a repeating rhythm?
Is there a way to determine the amount of additional rest time required after a rhythmic figure in order to ensure that the entire phrase loops seamlessly within a defined time signature? I want to measure the amount of time in seconds.
For example, I used a stopwatch to tap a rhythm. I measured how much time separated each tap (and I assumed the taps themselves don't consume any time). I did this twice: the first time, I included a rest at the end of the phrase. The second time, I didn't include a rest at the end of the phrase. The following numbers give the duration (in seconds) between successive taps. For example, the first item, "1. (tap) 0.274," indicates that the amount of time between the first tap and the second tap is 0.274 seconds. Here's scenario 1, which includes the final rest at the end of the phrase:
(tap) 0.274 s
(tap) 0.257 s
(tap) 0.262 s
(tap) 0.249 s
(tap) 0.468 s
(tap) 0.504 s
(tap) 0.237 s
(tap) 0.251 s
(tap) 0.466 s
(tap) 0.978 s
There were 10 taps in this rhythm, as shown. After completing duration no. 10, the rhythm would start again, so that the 11th tap would be the start of the second loop. The total time for this phrase is 3.946 seconds. The number of taps per minute is 167.26.
However, if I were to remove the final 0.978 s duration after the 10th tap, (scenario 2), then the total time for the phrase is 2.968 s, and the number of taps per minute is 202.16. In this second instance, the loop would rush into the beginning and would eventually sound like another rhythm. In other words, the loop was too short when I skipped this final 0.978 s duration.
I am trying to understand how would someone take scenario 2 and determine from the values that, in order to have a seamless loop, they would need an extra duration of 0.978 seconds at the end. How can this be determined? This is a 4/4 rhythm.
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General Method
There is a way to figure it out. First, let's establish the fundamental unit of your rhythm. 1, 2, 3, 4, 7, and 8 all have the duration of one fundamental unit. This fundamental unit is called a "beat." So duration 5, duration 6, and duration 9 all last two beats each. Duration 10 lasts four beats.
Using this as the fundamental unit, one beat lasts 0.25±0.01 s. (±0.01 s is the standard deviation based on the fluctuation in your counting of the fundamental unit.) If we have 0.25s/beat, we can take the reciprocal to get:
So your pattern has a tempo of 240 bpm. Since we're working in 4/4 time, each beat is a quarter note (that's what the denominator of 4/4 indicates). Here's your rhythm:
Using this example as a guide, here's the general approach:
determine the key signature (X/Y, e.g., 3/4)
determine the duration of one beat (Y represents the duration of a beat; normally we refer to the duration as a quarter note or eighth note, but you're referring to it as an amount of time in seconds)
multiply the beat duration (Y) by the number of beats (X) to get the time duration of a single measure--the duration that you must repeat/loop
optional: multiply the value from step 3 by 4, because music is often written in 4-bar phrases
First Example
For example, in your case, we would have:
the key signature is 4/4
one beat lasts for a time duration of 0.25 s
multiplying 0.25 s by 4, we get 1.0 s as the time duration of one measure
optional: multiplying the value from step 3 (1.0 s) by 4 gives 4.0 s
And so, in your example, each phrase that you loop must be 4.0 s long. (This assumes you'll be using a 4-bar phrase, which is true in the case of the example you provided.) When we add up the duration of 1-10, we get a total of 3.946 s, which almost perfectly matches are ideal value of 4.0 s. In the second case you describe, the time is only 2.968 s, which means you need to add a final 1.032 s of "rest" to bring the total phrase up to 4.0 s (2.968 s + 1.032 s = 4.000 s.) This additional "rest" duration of 1.032 s is almost exactly the same as the value you found to work: 0.978 s. The difference is simply a result of imprecision.
Second Example
Here's a second way to do the calculation. Let's say you have these values, and we're working in a 4/4 time signature again:
(tap) 0.173 s ? 0.2 s
(tap) 0.182 s ? 0.2 s
(tap) 0.390 s ? 0.4 s
(tap) 0.599 s ? 0.6 s
(tap) 0.397 s ? 0.4 s
(tap) 0.186 s ? 0.2 s
(tap) 0.425 s ? 0.4 s
You can see I've rounded the values. The reason is that it's extremely reasonable for the taps to be off by ~0.03 s. All of the original values above are all within that tolerance, and so it's fine to standardize the numbers by rounding to the nearest tenth of a second.
As we can see from the list above, the smallest unit (the duration of 1 beat) is 0.2 s. Let's convert each time (above) into a number of beats. We divide every time value by the duration of 1 beat to get the number of beats that each tap counts for:
(tap) 0.2 s / 0.2 s = 1 beat(s)
(tap) 0.2 s / 0.2 s = 1 beat(s)
(tap) 0.4 s / 0.2 s = 2 beat(s)
(tap) 0.6 s / 0.2 s = 3 beat(s)
(tap) 0.4 s / 0.2 s = 2 beat(s)
(tap) 0.2 s / 0.2 s = 1 beat(s)
(tap) 0.4 s / 0.2 s = 2 beat(s)
Adding up all of the beats gives a total of 12 beats. In 4/4 time, one measure contains 4 quarter-note beats, or four 0.2-s-long beats. So 12 beats would be 3 measures, and 16 beats would be 4 measures. To extend the current phrase to a 4-bar phrase, you need to add 4 more beats, each of which would last 0.2 s. So the time you need to add is 4 x 0.2 s = 0.8 s. This might sound even more natural. Here's what that could look like:
(tap) 0.2 s / 0.2 s = 1 beat(s)
(tap) 0.2 s / 0.2 s = 1 beat(s)
(tap) 0.4 s / 0.2 s = 2 beat(s)
(tap) 0.6 s / 0.2 s = 3 beat(s)
(tap) 0.4 s / 0.2 s = 2 beat(s)
(tap) 0.2 s / 0.2 s = 1 beat(s)
(tap) 0.4 s / 0.2 s = 2 beat(s)
(tap) 0.4 s / 0.2 s = 2 beat(s)
(tap) 0.4 s / 0.2 s = 2 beat(s)
or
(tap) 0.2 s / 0.2 s = 1 beat(s)
(tap) 0.2 s / 0.2 s = 1 beat(s)
(tap) 0.4 s / 0.2 s = 2 beat(s)
(tap) 0.6 s / 0.2 s = 3 beat(s)
(tap) 0.4 s / 0.2 s = 2 beat(s)
(tap) 0.2 s / 0.2 s = 1 beat(s)
(tap) 0.4 s / 0.2 s = 2 beat(s)
(tap) 0.2 s / 0.2 s = 1 beat(s)
(tap) 0.2 s / 0.2 s = 1 beat(s)
(tap) 0.4 s / 0.2 s = 2 beat(s)
where the added portion has been bolded.
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