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Intro , Basic facts , Convert a ratio to cents , Why are there two methods , Ratios , Overtone series , Stacking ratios , Landmarks , Composing considerations
If one wants to discuss nuances of historical tunings, or to explore scales that don't fall within the range of the twelve tone system, one needs a more exact way of representing pitch 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, which one will learn if one is going to be using that scale much.
So here is an introduction for those who may be familiar with semitones, hertz, and so forth, but not know much about cents or ratios.
100 cents = 1 semitone. Ratio = ratio between the frequencies of two notes in hertz.
To do it on a calculator: 1200*log( m/n )/log(2) where m/n is your ratio.
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, and are more useful when working with the herz values of pitches..
The thing that can confuse is that you add cents, and multiply ratios.
By way of example, a fifth on the twelve equal tuned piano is 700 cents, which is also got by stacking a major third of 400 cents, and a minor third of 300. Here you just add the cents as 300 + 400 = 700.
Cents work well for fractions of a semitone - it's easier to say 30 percent of a semitone rather than 0.3 semitones, and both are easier to understand at a glance ratios such as 40/39 etc.
To go up an octave you add 1200 cents, i.e. 12 semitones.
That basically is all one needs to know about cents, to get started. So lets now look at ratios.
To go up an octave from any frequency in ratio notation, you multiply by 2.
To go down, divide by 2. Generally to go up by an interval in ratio notation you mutliply and to go down you divide.
E.g. when you go up from a at 440 hz to a' at 880 hz, you multiply by 2. Then a'' is at 1760 hz, so one keeps on multiplying by 2 for each new octave, rather than adding.
So, 5/1, 5/2 and 5/4 are all the same note, in different octaves.
One of the main landmarks in ratio notation is the overtone series. One can start anywhere, but if one starts from middle c, 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 ones in brackets are in the cracks between the keys of a keyboard.
These are the notes you get by overblowing on a natural horn, or by touching the string in various places to bring out the harmonics on a string instrument.
So, for example, we see from the overtone series that the e'' is 5/1 .
To get it into the range of the octave c to c', you need to go down two octaves, i..e. divide by two twice, which drops it down to e = 5/4. So that gives us our major third.
Then g' is 3/1 which drops down to g = 3/2 which is our fifth. The 13/1 needs to drop down three octaves to 13/8 - that doesn't correspond exactly to any of the twelve equal notes, but is a pleasant interval for those who have the taste for it.
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.
To find the minor third, one looks at the interval in the overtone series from the e'' to the g''. That is between the 5th and the 6th overtones.
The ratio between these is 6/5 - that is how one does it with ratios - instead of subtracting, you divide in this case, you divide the 6 by the 5 to find the ratio from 5 to 6.
So, to go up a minor third from any frequency, you multiply it by 6/5. E.g. if you go up a minor third from 440 hz, you get to 440*6/5 = 528 hz.
We can now see that when one is working with hertz, then ratios notation is actually easier to use than twelve equal temperament semitones or cents - it's harder to work out the hertz value for an e flat exactly three semitones, or 300 cents above 440 hz than to find the herz value for the pure minor third above 440 Hz.
Now, let's 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) = (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, writing this as (3*2)/(2*2) another 2 cancels 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 fifth (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 if 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, we can see from our landmarks that the 11/8 is a bit over a quarter tone sharp - 33/32 is a little larger than 34/33.
Notes from the overtone series sound especially good in harmonic timbres such as strings, voice etc - a harmonic timbre is one that includes many frequencies from the overtone series. You only get beats because of these component frequences in a sound - if you choose a timbre with hardly a trace of higher frequencies from the overtone series such as the ocarina (say) then a pure ratio, an equal tempered interval, or some other interval of about the same size will all be non beating intervals..
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 one of Bill Sethares' pieces. 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.