My First Cents


The tempo and pitch of a sound are closely related and here I want to deepen the mathematical aspects of the relationship time/picht.

There are various ways to “keep time” of rhythmic loops, now all sequencers and VST provide different functions of time-stretching.

In software samplers each midi note can start an audio loop.

In the section Mod / Tune of the sampler Steinberg Halion, which we will use for our evaluations, for pitch checking a resolution up to “cent” of “coarse” (hundredths of a semitone) is expected.

If I lower the pitch of a drum loop (with the option “NoTraspose” selected) this sounds in a slower tempo and vice versa.

If I want to use this feature to “keep time”audio loops and use them as a rhythmic base for a song I must set a rule to match the tempo of the loop with that my song.

For a first evaluation I took a 100bpm loop and executed pitch corrections in semitoni (coarse).

The results “by ear”, and therefore rounded up, are the following: ‑12coarse=50bpm, ‑11=52, ‑10=55, ‑9=58,5, ‑8=62, ‑7=65, ‑6=70, ‑5=74, ‑4=78, ‑3=83,5, ‑2=90, ‑1=96,5, +1=106, +2=113, +3=120, +4=126, +5=134, +6=142, +7=150, +8=159, +9=168,  +10=178, +11=189, +12=200

The time difference decreases by lowering the tone (just 2 bpm going from -11 to -12 semitones) and increases by raising the pitch (11 bpm going from +11 to +12)

The constant factor is a ratio of about 1.06 between a time and the following one.

The frequency of the middle LA is 440Hz, of the upper octave is 880Hz, exactly doubled.

This coincides with a doubling time of the loop (from 100 bpm to 200bpm) by increasing the pitch by one octave and halvening (from 100bpm 50 bpm) decreasing the pitch by one octave.

Being divided in cents (100cents = 1 half-step) from an octave to another there are 1200 cents.

The number that raised to the power of 1200 (the cents in an octave) gives me 2 (the ratio between the frequency of a sound and the one of a superior octave) is:

If I want to synchronize a drum loop with original time 80bpm with a song on 84bpm time I must find the exponent n of 1.0005777895 I resulting in the ratio between the highest and the lowest value: 84/80 = 5.1

As an example I attach a table with the rounded-up results corresponding to the 1200 cent of pitch correction.

In the table the closest value to the ratio 1.05 is 84 cents.

From the example above it is clear that the values shown in the table (corresponding to 1200 cents) will not coincide with the relationship between the two times (original loop time and song time)

original loop time: 80 bpm
song time: 84 bpm
relationship between the two times (84/80): 1.05
nearest table value 84 cents: 1.04971668362255
Time loop + 84 cents (80 x 1,04971668362255): 83.977 bpm

In case, as for the previous example, the ratio is approximately in between two values I can possibly listen to both decide which one to adopt.

Higher table value (85 cent): 1.0503231989072
Time loop + 85 cents (80 x 1,0503231989072): 84.026 bpm

If the rhythm loop is a few bars the deviation does not affect the loop as this “imperceptible” advance or delay is to be part of the loop “groove”

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