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Bruce
What is the relationship between Hz and cents? I have always wondered which is the most accurate.
Does anyone know? Cheers
John Norris
Dear Bruce,
in instrument tuning, the terms "Hertz" and "Cent" mean two totally different things. If you think of a roadmap, Hertz describes the location of the towns (notes), and Cents is the unit of measurement used to measure the distance (intervals) between those towns. It is similar to map co-ordinates and miles (or kilometers).
As a note increases in pitch, its location changes and also its distance from other static notes but the Hz value and the cent value are not measured one-to-one, since often one value increases while the other decreases.

For example:
In Equal temperament, given a concert A4 of 440.000Hz
A in the second octave will be 110.000 Hz
If we move that A one cent away in pitch, its location is now 110.064 Hz

A in the 8th octave is located at 7,040.000 Hz
If we move that A one cent away in pitch, its location is now 7,044.068 Hz

So at one end of the scale, one cent = 0.064 Hz, whereas at the other end of the scale, one cent = 4.068 Hz

Our tuners work at a resolution of 0.1 cent and are also adjustable in 0.1 cent increments. They are outfitted with multiple temperaments and in some cases even multiple roots, so very fine measurements can be made.

John N.
Bruce
How did you work that out? Is it a feature of Peterson tuners?
Marc Savage
This site includes a cents to hz calculator. I find the chart available at the following URL more useful: http://www.pianosupply.com/cents-hz/

Both cents and hertz are valid ways to measure pitch or intervals. As John N. has pointed out, hertz refer to an absolute pitch, whereas cents are used to measure the size of intervals. However, there is also overlap in their usage because we can talk about the number of hz between intervals (though this isn't always that useful), and we can also identify a specific pitch by the number of cents it varies from a reference pitch.

Looking at an interval such as the octave using both systems clarifies the differences:

Hertz: We begin, for example, with a given pitch A = 110 Hz. Then if we wish to identify the pitch exactly one octave higher, we must always double the Hz. So one octave above A = 110 is A = 220. The ratio the the frequencies of the two pitches, one octave apart, is 2/1 ("two to one"). In this case, we have increased the pitch by 110 Hz (from 110 to 220) to increase the pitch by one octave. If we want to go up another octave we must double the Hz again, this time from 220 to 440. So this time we've had to increase the number of Hz by 220 to go up an octave -- twice as much as the previous octave. Each time we go up an octave we have to double the Hz, so each higher octave has twice as many Hz to traverse to cover one octave of pitch change (defined as doubling the frequency, or a ratio of 2/1).

Cents: There are always 1200 cents per octave no matter which pitch the octave begins with. From A = 110 to A = 220 is 1200 cents. Similarly, from A = 220 to A - 440 is also 1200 cents. We can think of cents as very small equally-tempered intervals. There are always 12 equally-tempered half steps per octave. Just the same, there are always 1200 equally-tempered cents in an octave.

So asking which is "more accurate", cents or hertz, isn't exactly the right question. Each unit of measurement has it appropriate place and is accurate in that context. When you want to identify a given pitch, you might specify it most accurately by stating its frequency in hertz. When you want to identify the size of a musical interval, you would be able to do so most accurately by stating its ratio, or its size in cents. There is actually much more of a correspondence between RATIOS (of frequencies) and CENTS than there is between hertz and cents. This is because both ratios and cents describe intervals, not single pitches.
Bruce
Thanks, but that chart is only accurate to one cent and is fixed at 440 hertz, is there something better?
John Norris
Hi Bruce,
yes, there is, you can get to it on this site by going to the petersontuners.com page and going to "Support", then click on "Cents to Frequency Chart" or just click here

The upper box allows you to set the Concert A frequency to anywhere from 350Hz to 550Hz, choose an octave and then a chart will be calculated and displayed accordingly.
Marc's link was to our older chart used by a customer on his site which was fixed at A=440Hz.
In the lower box, you will be able to convert any Hz frequency (in 0.001 Hz increments) to its cent equivalent (in 0.1 cent increments).

Have fun!

John N.
Marc Savage
I just gave the Peterson calculator, which is on this site, a bit more of a checkout. It's cool! I didn't realize it would generate charts like that. I thought it just did single frequency conversions. Of course it does that too. Very useful and flexible tool! Thanks!
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