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Frequently, I want to mix two songs in the same key but different bpms. To transition one song into another, I need to match their bpms. For example, the currently playing song might be at 124 bpm, but the next song on deck might be 120 bpm. In this case, I'd increase the speed of the next song to 124 bpm. (This is known as beatmatching.)

Of course, the side effect of a tempo change is a pitch shift. A 120 bpm song played at 124 bpm sounds somewhere shy of a half step (100 cents) higher, roughly estimated by ear.

I'd like to know the approximate number of cents that pitch shifts from a 1 bpm increase or decrease, centered around 123 bpm, but I don't know how to calculate it.

I know there isn't an exact answer, since it's a logarithmic scale. But most of the songs I DJ are between 120 and 126 bpm, so I only need a practical measure, i.e. I want to be able to think, "4 bpm difference; that's about 4 times X cents higher," and turn the pitch knob accordingly to bring the next song closer to its original pitch.

Also, I'm curious about the math.

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The way I'd derive it is to Taylor expand around central frequency (BPM is a frequency):

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Using 123 BPM as the central frequency yields the same result as user37549: about 14 cents for a 1 BPM change.

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  • I'd like to point out that the just noticeable difference (JND) of a pitch is 5 cents. Any change more than 5 cents will be noticeable and if one source changes and the other doesn't, the result will sound very out of tune.
    – Dom ♦
    Mar 27, 2017 at 18:18
  • I never heard of JND and it's good to know its value, @Dom, thanks. Just to add a practical point of view though, this JND in a club setting may sound not only intentional, but sometimes even better than in-tune, because of detuned melodies giving the effect of a chorus, bigness, or retro rawness. Many techno and trance synths are synthesized with detuned waves, even. So, whether a DJ can pull off a 14- or 28-cent shift will depend on the songs and the DJ's proficiency. Regardless, I'm no longer inclined to attempt 42-cent shifts without pitch correction, and that's what I came here to know.
    – Andrew Cheong
    Mar 29, 2017 at 6:50
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Well, the speed shift ratio is 124/120 in that case, and the pitch will go up by 1200*log(124/120)/log(2) cents, namely about 57 cents, more than a quartertone but definitely less than a half step.

1 bpm of difference centered on 123bpm would be 1200*log(123.5/122.5)/log(2) cents, namely about 14 cents.

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  • Thanks! I guess my ears don't work so great under a semitone :^) I did read on some DJ forum that practically, you don't want to tempo shift more than 2 bpm if you plan to mix by key. Kind of makes sense now, as 2 bpm = 30 cents according to your figures. (I'm rounding to 15 cents / bpm, for easier math.) Maybe about 30 cents of relative off-tunedness is the limit before even a layman starts hearing that notes sound "wrong" / not good.
    – Andrew Cheong
    Mar 8, 2017 at 11:30
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    at 30 cents my ears would be screaming for the horrible noise to stop. Can't you use software that can change tempo without altering pitch?
    – Tetsujin
    Mar 8, 2017 at 12:21
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    Indeed. Changing the tempo with software designed for it can adjust to nearly any speed you wish without any perceivable change in tonality (the timbre of individual notes will start to suffer at slow speeds depending on the algorithm they used). Changing the tempo simply by speeding up or slowing down the waveform will always change the key of the original music, but it will still remain "in tune with itself". In this way, songs in one key may be shifted into a different key by accordingly increasing or decreasing their speed. Of course, there are also tools that do this at fixed tempo.
    – Darren Ringer
    Mar 8, 2017 at 13:55
  • @DarrenRinger - Thanks. I didn't know about timbre changes; interesting. As I just posted in a comment under the question, I'm not aware of live mixing software with auto-pitch shifting. I wonder if it's because, when it comes to bpm shifts, all it needs do is change the sample rate, i.e. drop samples or play slower, whereas pitch-shifting requires more complex transformations that require lag when administered live, e.g. user enables auto-pitch, auto-pitch doesn't actually start until 3.5 seconds later to create enough buffer for pitch-shifting ahead; still subject to CPU interruptions...
    – Andrew Cheong
    Mar 8, 2017 at 15:26
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    @AndrewCheong technically that's true, but not on the scale that you're thinking about. Rubberband (which is one of the best time-stretcher/pitch-shifters generally available) adds a delay of about 2-4 milliseconds when working with 44k or 48k input and stretching by a reasonable amount. Not enough to cause a perceptible difference between changing the control and hearing the result, unless some other part of the stack is causing the delay.
    – hobbs
    Mar 8, 2017 at 22:30
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The answer is... There is no standard (easy) Cent adjustment per bpm ...but it can be manually calculated. Here's a link to an article that unpacks why: http://www.musicmasterworks.com/WhereMathMeetsMusic.html

As a composer/remix artist, it's helpful to know this because, pitch-shifters don't always sound great on vocals or higher frequency instruments. If you're a producer who does soundtracks with any modern instruments, there are times when its better to adjust some things rather than others.

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