Hz-cents

Hz/cents

This is about the relation between Hz and cents, for most of you straightforward, but for students sometimes maybe useful.
Here I am trying to explain this relation and come up with a simple rule.
Let me give the simple rule first.

If we want to calculate the next semitone in Hz in equal temperament we get it reasonably precise by multiplying the frequency by 1.06.

So from 415 we get the next by using 415*1.06=439.9 . This, for any musical practice is sufficiently precise.
Of course, if we were to go from one side of the piano to the other this may not be precise enough.

But the essence is that there are two ways of thinking about tones. The one is by multiplying, the other by counting. Multiplying by for instance 1.06 or counting 415, 416, 417 etcetera.
The difference is in using Hz or cents. Let us try to understand it a bit better.

Pythagoras already discovered that it was the ratio of the frequency of the notes that makes us feel good or bad. So 415-830 is an octave. That is a factor 2 on the frequency in Hz. That is thinking in a ratio of course. The octave factor 2 is the basis for everything. The distance in Hz is 415.

An octave is 12 semitones.
415 to 440, a semitone and 440-415 is 25Hz. But this is not 1/12 of the number 415, the whole octave distance. Instead, 12*25=300. How is that possible. We will need to think in ratios!!

So the issue is counting 440,441,442 etcetera or having a look at the ratio in Hz between two different notes. That gives a different way of thinking.

Let us have a look at twelve equal ratios. In other words at equal temperament.
So the issue is the ratio between the Hz of different notes. If the ratio for an octave is 2 and there are 12 equal ratios for twelve different half tones then in Hz: semitone-2/semitone-1 must be the same as semitone-3/semitone-2 etcetera.
Let us call this ratio semitone-2/semitone-1 “r”. So semitone2/semitone1=r . Then: r * r * r * r * r ……., ,12 multiplications must be 2. So more mathematically r to the power 12 must be 2. Or r**12=2.
Therefore r is 2 to the power 1/12. Writing it differently r=2**(1/12)= 1.0596…….and infinitely more decimals.
This means that if we want to calculate the next semitone in Hz in equal temperament we get it approximately by multiplying the frequency by 1.06.
For instance 415*1.06=439.9 and 392*1.06= 415,52 .

So for equal temperament given the Hz of a tone we can calculate the next semitone by multiplying by 1.06 and the next and the next. That is sufficient precision for a whole octave because 1.06**12=2.012.

We have a further agreement that we subdivide a half tone in 100 equal steps. This again is thinking in ratios. So 1 cent is 2 to the power 1/1200. Therefore, if one plays 10 cents above 415 this is not 415.10 Hz or something like that. The Hz is found from 415*2**(1/120) = 417.404 So for seeing how much higher 10 cents will get us we have to multiply by (rounded) 1.0058.

However, life is not so easy because basically we may not want equal temperament. For that there is more information on https://www.earlyflute.com/pages/traversotuning.html

Simon Polak: Early Flutes.

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