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Can you explain cents and ratios notation?

Jan 14th, 2002 20:10
Robert Walker, http://members.tripod.com/~tuning_archive/on_site_tree/robertwalker/cents_and_ratios.html

100 cents = 1 semitone. Ratio = ratio between the frequencies of two 
notes in herz.
The thing that can confuse is that you add cents, and multiply ratios.
So, a fifth on the piano is seven semitones, made up of a major third 
of four semitiones and a minor third of three. As cents, it's 700 
cents, made up of a major third of 400 cents, and a minor third of 300.
So you just add the cents as 300 + 400 = 700.
A bit like using percentages - it's easier to say 30 percent of a 
semitone rather than 0.3 semitones.
To go up an octave you add 1200 cents, i.e. 12 semitones.
Now, to go up an octave from any frequency in ratio notation, you 
multiply by 2. 
To go down, divide by 2.
E.g. when you go up from a at 440 hz to a' at 880 hz, you multiply by 
2. a'' is at 1760 hz, so one keeps on multiplying by 2 for each new 
octave, rather than adding.
So, 13/1, 13/2, 13/4 and 13/8 are all the same note, in different 
octaves. One can get used to looking at the powers of 2 in a ratio and 
thinking of them as octaves.
The overtone series from middle c 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 ones in brackets are in the cracks between the keys of a 
The a''' flat is the 13th harmonic 13/1.
Dropping it down to an a flat, and you need to go down three octaves,
= divide by 8, 13/8.
The e'' is the fifth overtone 5/1, which drops down to e = 5/4.
The third overtone g' drops down to g = 3/2.
So, 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. 
Now to find the minor third, one looks at the interval from the e'' to 
the g''. That is between the 5th and the 6th overtones. The ratio 
between these is 6/5, which is how one does it with ratios - instead of 
subtracting, you divide, just as you multiply instead of adding. 
So, to go up a minor third from any frequency, you multiply it by 6/5.
E.g. to go up a minor third from 440 hz, it's 440*6/5 = 528 hz.
So, in fact if one is working with herz, then ratios notation is 
actually easier to use than semitones or cents - it's much harder to 
work out what note is exactly three semitones above 440 hz.
So to go up by a major third followed by a minor third you multiply 
first by 5/4, then by 6/5, and (5/4)*(6/5) = 3/2
so you end up with a fifth, as one expects.
Notes from the overtone series sound especially good in harmonic 
When one goes to inharmonic timbres - bells, various types of 
percussion, specially constructed timbres, or whatever, all the rules 
change completely. You can make almost any notes sound good together 
using a suitable timbre. E.g. 11 equally spaced notes to an octave, as 
in a clip Bill Sethares posted recently to the MakeMicroMusic group. 
Also some timbres just work well 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, one might want to have some beating of notes etc for whatever 
reason, can sound great too. That seems to work in 12 tone equal 
temperament - we get some beating, e.g. of major thirds especially, but 
they sound okay in the music written for the idiom.