BPM to frame timing: how long to hold each GIF frame

How long one beat lasts at a given tempo, and how long to hold each frame so a whole number of frames fills it exactly.

Why this is not simply beat length divided by frames

It is, for a video. Not for a GIF. The format stores a delay per frame in hundredths of a second, so the only durations available are 10ms, 20ms, 30ms and so on. At 120 BPM over eight frames a beat wants 62.5ms and the nearest a GIF can hold is 60ms. That loses 20ms a beat, or 2.4 seconds a minute.

Short delays are worse: most browsers substitute a tenth of a second for one hundredth, so a loop written at 10ms a frame plays ten times too slow. This is why Gifsync writes an MP4, where timestamps go to the microsecond.

One beat, and one frame of it

Tempo	One beat	4 frames	8 frames	12 frames	16 frames
60 BPM	1000.0ms	250.0ms	125.0ms	83.3ms	62.5ms
70 BPM	857.1ms	214.3ms	107.1ms	71.4ms	53.6ms
80 BPM	750.0ms	187.5ms	93.8ms	62.5ms	46.9ms
90 BPM	666.7ms	166.7ms	83.3ms	55.6ms	41.7ms
100 BPM	600.0ms	150.0ms	75.0ms	50.0ms	37.5ms
110 BPM	545.5ms	136.4ms	68.2ms	45.5ms	34.1ms
120 BPM	500.0ms	125.0ms	62.5ms	41.7ms	31.2ms
128 BPM	468.8ms	117.2ms	58.6ms	39.1ms	29.3ms
130 BPM	461.5ms	115.4ms	57.7ms	38.5ms	28.8ms
140 BPM	428.6ms	107.1ms	53.6ms	35.7ms	26.8ms
150 BPM	400.0ms	100.0ms	50.0ms	33.3ms	25.0ms
160 BPM	375.0ms	93.8ms	46.9ms	31.2ms	23.4ms
170 BPM	352.9ms	88.2ms	44.1ms	29.4ms	22.1ms
174 BPM	344.8ms	86.2ms	43.1ms	28.7ms	21.6ms
180 BPM	333.3ms	83.3ms	41.7ms	27.8ms	20.8ms

Below about 16.7ms, one refresh of a 60Hz screen, a frame will not be seen however exactly it is timed. Nothing in the table is quite that short. 16 frames at 180 BPM is close.

Going the other way: beats from a frame count

Divide the frame count by the beats you want and check it comes out whole. Twenty frames over eight beats gives 2.5, so some beats hold three frames and others two, a 50% swing in speed you can see. Over ten beats it gives two, even all the way round.