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My question comes from MUS 204: At what BPM (Beats per Minute; quarter note = beat) will we hear the top note as F4?

Now, I understand this polyrhythm would generate a perfect fourth when sped up to an audio frequency. However, I do not know how to find the BPM of a note if given this information. How would you answer the question correctly?

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    Do you know the standard frequency for F4? If you don't you can find it easily. If you do, you know that you'll listen to that many beats per seconds, and each beat is one of the note above. Since a bar is 4 of those notes, you then know the frequency of that bar. Then you can compute the frequency of the beats below. Nov 22 at 1:27
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    There's something I'm not understanding here. How are sound-wave cycles supposed to correspond to rhythmic beats? Aren't the two concepts unrelated (or at least, independent)? Is there information missing from the question, like the starting pitch and tempo of one or both notes?
    – Aaron
    Nov 22 at 1:49
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    @Aaron I think in this question, it is considered that the notes are played extremely fast, with the F4 note frequency. Nov 22 at 1:51
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    @Aaron Like something Adam Neely demonstrates here: youtu.be/-tRAkWaeepg Nov 22 at 1:56
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    @ToddWilcox the question title informs us that the quarter note gets the beat.
    – phoog
    Nov 22 at 3:14
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The target frequency (F4) is approximately 349.23Hz.

You have to find a common divisor between those two frequencies, which coincidentally is the bar.

So, assuming the frequency above, you can find that each bar happens at frequency/4. Then, you can find the frequency for the beat by multiplying it by 3 (since the meter is 3/4 - aka, 3 beats per bar), and finally get the actual BPM by multiplying by 60:

F4 = 349.23
barsPerSecond = F4 / 4
>>> 87.3075
beatsPerSecond = barsPerSecond * 3
>>> 261.9225
beatsPerMinute = beatsPerSecond * 60
>>> 15715.35

Consider that the answer from Aaron is perfectly right, but it is based on the ratio of an interval that uses the frequency of the second note as reference. Since you might not know the precision of that reference, it's up to you to decide if you should use the known ratio of the interval (not knowing the tuning it's based on), or the rhythm ratio shown in the example.

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    A calculation doesn't yield a greater precision than the input; you should round that final number to five significant figures Nov 22 at 2:49
  • @ElementsinSpace you're technically right (especially considering the frequency in use), but I think we still have to consider that not all platforms/languages use the same kind of approximation/precision, and, most importantly, precision might be much more important for lower frequencies. While we might not hear any difference between 15715 and 15715.35, it's not a difference that difficult to consider, and we should probably consider it anyway, at least for consistency, in case we need to do further computations. For instance, Aaron's answer already shows an interesting difference. Nov 22 at 3:14
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    You've claimed that F4 is an approximation, so your conclusion can only be an approximation of the same precision. Your calculation is fine [now], and you should leave the calculated number as is, but you should add another line that concludes BPM = 15715. The '.35' is meaningless/inaccurate for a conclusion. For consistency if you want a 7 sig. fig. conclusion you need to define/assume a 7 figure value for F4. It is more accurate [and easier] to compare [your] 15715 to [Aaron's] 15698. Nov 22 at 4:10
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    I appreciate the shout-out, but can you clarify the part about interval vs. rhythm ratios. OP says the interval is a perfect fourth based on the polyrhythm, so both are 4:3. There's no difference in the reference point except the order of operations of the math. But since multiplication is commutative, that doesn't influence the result.
    – Aaron
    Nov 22 at 6:26
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The top and bottom notes are in a speed ratio of 4:3. This means that if the upper pitch is F4, the lower pitch will be C4.

Since BPM = quarter-notes per minute, and one quarter-note = one frequency cycle, then to convert beats to Hertz, we divide BPM by 60. (That is, convert quarter-notes per minute to quarter-notes per second.)

C4 is approximately 261.63 Hz (SOURCE), so X BPM / 60 = 261.63. Therefore X = 261.63 x 60 = 15,697.8

Thus, to produce an upper pitch of F4, the BPM must be approximately 15,697.8.

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  • Uhm, but the OP is asking about F4, and the BPM has to be related to that. So, assuming that F4 = 349.23 Hz ("beat per second"), we get the common division (the bar) at bar = F4 / 4 and the ratio based on the actual beat at beatPerSec = bar * 3, which results in BPM = beatPerSec * 60: 15715.35. Nov 22 at 2:25
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    @musicamante It's just a rounding error. Freq(F4) = Freq(C4) * 4/3. So given C4 = 261.63 Hz, that would put F4 at 348.84 Hz. The table gives F4 = 349.23, which would result in C4 = 261.92. 261.92 * 60 = 15715.2.
    – Aaron
    Nov 22 at 2:34
  • oh, right, now I get it, sorry... Nov 22 at 2:48
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    F4 is given; C4 is not. Therefore, the pitch that is out of tune with equal temperament is the lower pitch, not the upper one. The lower pitch, in other words, is not 261.63 Hz.
    – phoog
    Nov 22 at 3:20
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    @phoog I don't understand your comment. The exercise is clearly based on the idea that perfect fourths are in an exact 4:3 ratio. At that level, it makes no difference whether one uses C4 or F4 as the "fixed" frequency. Any difference based on that is just rounding error, as musicamante and I discussed in the previous comments. The exercise is designed to focus on the mathematical process, which is effectively the same either way.
    – Aaron
    Nov 22 at 3:24

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