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When you have scales in temperaments and tunings with maybe more than twelve notes per octave, or less, or some other type of tuning, then you need special notations to describe them. The twelve tone notation of note names and numbers of semitones is no longer adequate. Special note name systems have been designed for many of these- either based on the twelve tone system with notations for fine shades of accidentals - or sometimes a scale will have a special notation system all of its own. However, it is also useful to have a system that is universal for all the types of scales and tunings and can be used to compare them.
The most common notations used for this are cents and ratios. So here is an introduction for those who may be familiar with semitones, herz, and so forth, but not know much about cents or ratios.
Cents are just percentages of an equal tempered semitione. So 100 cents = 1 semitone.
Ratios are the ratios between the frequencies of two notes in herz.
Here is a script to convert ratios to cents:
The thing that can confuse newbies to this subject is that you add cents, and multiply ratios.
A fifth on the piano such as C to G is seven semitones, made up of a major third of four semitiones C to E and a minor third of three semitones from E to G. 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.
To go up an octave you add 1200 cents, i.e. 12 semitones.
Now, in ratio notation, you double the frequency to go up by an octave..
E.g. when you go up from a at 440 hz to a' at 880 hz, you multiply by 2. Then the note in the next octave is a'' at 1760 hz, so one keeps on multiplying by 2 for each new octave, instead of adding a number.
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 keyboard.
The e'' is the fifth overtone 5/1. When you divide it by two a couple of times to get into the same octave as the unison 1/1, it drops down to e = 5/4, which is the major third c to e.
The third overtone g' becomes g = 3/2 which is the fifth c to g.
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. Most musical instruments we use have harmonic timbres - but bells and a few others have inharmonic timbres which mean that some of their constituent frequencies don't fall into the overtone series. Many others have overtones, but they may sometimes be rather weak - for instance the ocarina may have very few overtones at all, which means you can play it in almost any kind of a tuning and won't notice much in the way of roughness of chords or beats.
Now to find the minor third, one looks at the interval from the e'' to the g'' in this overtone series on c. Here it is again:
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)
We see that the interval we want is between the 5th and the 6th overtones. The ratio between these is 6/5, which is how one works out the intervals for ratios - instead of subtracting, you divide, just as you multiply instead of adding.
So the minor third is 6/5. This means that to go up a minor third from any frequency, you multiply it by 6/5.
Here is an example: to go up a minor third from 440 hz, multiply by 6/5 to get 440*6/5 = 528 hz.
We see that 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. You will probably need to use logs - or use my javascript applet (or Fractal Tune Smithy).
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 timbres.
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.