Cents and ratios
From Bounce Metronome
If one wants to discuss nuances of historical tunings, or to explore scales beyond the range of the twelve tone system, one needs a more exact pitch notation than the musical note names C, C# etc.. The most common notations used are cents and ratios. Some particular scales also have their own systems of note names, and accidentals as well, useful if you make frequent use of the scale
This page assumes you are familiar with musical intervals but not yet very familiar with the musical use of cents or ratios.
100 cents = 1 semitone. Ratio = ratio between the frequencies of two notes in hertz.
Convert a ratio to cents
To do it on a calculator: 1200*log( m/n )/log(2) where m/n is your ratio.
Why are there two methods, ratios and cents?
Cents make it very easy to compare intervals, and see which is the largest and by how much. Ratios make it easy to see which notes are at pure ratio intervals to each other.
One thing that confuses newbies is that you add cents, and multiply ratios.
A fifth on the twelve equal tuned piano is 700 cents. It is the same size as a major third of 400 cents, followed by a minor third of 300.
300 + 400 = 700.
To go up an octave in cents notation, you add 1200 cents - for 12 semitones at 100 cents each.
Cents are especially useful if you want to compare interval sizes.
The pure fifth is 3/2 which you can also get by playing a major third 5/4 followed by a minor sixth 6/5
5/4 * 6/5 = 3/2 (If you are rusty on multiplication of ratios, don't worry, we go through this example more slowly later on in the Stacking Ratios section).
To go up an octave in ratio notation you multiply by 2/1. So, 10/1, 5/1, 5/2 and 5/4 are all the same note, in different octaves.
That's because the ratio notation is a ratio of frequencies, and you double the frequency to go up an octave - 220 Hz, 440 Hz, 880 Hz and 1760 Hz are all the same note, A, in different octaves. When working with frequencies, the ratio notation is the natural one to use where possible.
That's enough to get started. So lets now look at ratios in more detail.
One of the main landmarks in ratio notation is the overtone series. One can start anywhere. Let's starts from middle c, then it goes
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 c, c', g', c'', e'', g'', (a'' flat), c''', d''', e''', (f'''), g''', (a''' flat)
where the pitches in brackets are in the cracks between the keys of a keyboard.
You can get these notes by overblowing on a natural horn, or by touching the string in various places to bring out the harmonics on a string instrument.
This overtone series contains many familiar musical intervals along with some that may be less familiar.
So, for example, we see from the overtone series that the e'' is 5/1 .
If you drop the fifth overtone down by two octaves (divide by 2 twice), it drops down to e = 5/4. That gives us our major third.
Then g' is 3/1 which drops down to g = 3/2 which is our pure fifth. The 13/1 needs to drop down three octaves to 13/8 - this is one of the ones likely to be unfamiliar as it doesn't correspond exactly to any of the twelve equal notes. But is a pleasant interval for those who have the taste for it.
To go up by a major third from any frequency, such as from c to e, you multiply by 5/4. This is pretty close to the 400 cents major third, a little flatter, and for those who get used to it, the interval has a particularly sweet feeling to it in harmonic timbres. A harmonic timbre is one such as voice, strings, etc, which has a 1 2 3 4 5,... type overtone series.
To find the minor third, one looks at the interval in the overtone series from the e'' to the g'' - the 5th and the 6th overtones.
The ratio between these is 6/5 - you divide the 6 by the 5 to find the ratio from 5 to 6.
To go up a minor third from any frequency, you multiply it by 6/5. E.g. why you go up a minor third from 440 hz, you get to 440*6/5 = 528 hz.
Now, let's try a major third followed by a minor third. You multiply first by 5/4, then by 6/5, and (5/4)*(6/5) = (5*6)/(4*5). For those who are a bit rusty in fraction manipulation - you can cancel factors in the top and bottom. Here the 5s cancel to give 6/4. Then, if you write 6/5 = (3*2)/(2*2) you can cancel by 2 as well, giving 3/2 as the answer. So you end up with a fifth, as one expects.
Similarly, if you see 9/5 then you know it will be a pure 3/2 above a 6/5 since (3*6)/(2*5) = (3*3*2)/(2*5) = 9/5 after cancelling the 2s.
A few landmarks may help:
34/33 ~= 50 cents - quarter tone 18/17 ~= 100 cents - semitone - more often one sees 16/15 = Pythagorean semitone 9/8 ~= 200 cents - Pythagorean whole tone 6/5 ~= 300 cents - minor third (6/5 is sharper than 300 cents) 5/4 ~= 400 cents - major third (flatter) 4/3 ~= 500 cents - perfect fourth (slightly flatter) 3/2 ~= 700 cents - perfect fwhyth (slightly sharper)
You can tell which is the larger of two ratios, and by how much with a short calculation
Ex. of the calculation, to see why 11/8 is larger than 4/3, you work out:
(11/8) / (4/3) = (11/8) *(3/4) = (11*3) / (4*8)= 33/32 which is greater than 1, so 11/8 is larger than 4/3.
In fact, 11/8 is a bit over a quarter tone sharp - as one might figure out from the value given for 34/33 - 33/32 is a little larger than 34/33.
Overtone series pitches sound especially harmonious and mellow when played using harmonic timbres such as strings, voice etc. A harmonic timbre is one rich in overtones - so it has many of the overtone pitches as component frequencies.
Timbres with very few overtones at all, such as ocarina etc, don't beat so noticebly, so the distinctions between pure ratios and equal tempered intervals etc are less strong.
If one uses inharmonic timbres - bells, various types of percussion, specially constructed timbres, or whatever, all the rules change. You can make almost any notes sound good together using a suitable timbre. E.g. 11 equally spaced notes to an octave, as in one of Bill Sethares' pieces. You can either construct timbres especially to work using his spectral mapping techniques - or find which timbres work experimentally. For instance, for some reason, I find that 13 equally spaced notes to an octave sounds great on the sitar voice of the SB Live!, even though that is a harmonic timbre, possibly something to do with it having lots of high overtones in it.
Also, there is no need to try to remove all beating intervals from ones music - one might want to have some beating of notes etc for whatever reason, can sound great too. In 12 tone equal temperament - we get a fair amount of beating, e.g. of major thirds especially, but they sound okay in the music written for the idiom. Just add a touch of brightness and excitement to the music perhaps.