[Harp-L] an explanation of limit in JI

[bad_hat@xxxxxxxx]

This from Pat Missin's site, www.patmissin.com/tunings/tun1.html

You really can't go wrong by heading to Pat's page any time you have a 
harmonica theory question.  I'm excerpting the part on the tunings/tun1 
page that talks about Limit and what it means.  It's a great primer on 
Just Intonation.  So the content below is Pat's and is from his site.




I did have a couple of offlist questions regarding the term "limit" in 
JI theory. For the most part, it's not something you need to worry about 
too much, but here is a quick cursory coverage.

All intervals in JI are determined by ratios, with a preference for 
simpler ones. Basically, the simpler the relationship between two notes, 
the easier it is for our brain to process them, so the sweeter they 
sound to us.

If you take two notes of identical pitch, they are said to be in the 
ratio of 1:1, ie a single helping of one note is the same size as a 
single helping of the other note. The highest prime number (in fact, the 
only number) in this ratio is 1, therefore this could be said to have a 
prime limit of 1. A 1-limit tuning system would not be very exciting, as 
you would only have one note!

You probably all know that if you take a note and double its frequency, 
you will raise its pitch by an octave. Therefore, two notes an octave 
apart are said to be in the ratio 1:2. The highest prime number here is 
2. If you were to have a tuning system with a prime limit of 2, then you 
could have only one note, but you could have it in any octave you 
wanted. Obviously not a very practical tuning system for most music.

If you were to take two numbers in the ratio of 2:3, the second number 
would be a perfect fifth higher than the first. So, if you were to take 
a note and raise its pitch by 3/2 (ie multiply it by one and a half) you 
would raise it by a fifth. For example, if you were to start with 440Hz 
and multiply by 3/2, you would get 660Hz. In our standard system, 440Hz 
is an A, so 660Hz would be a perfect 5th above it. By taking any note as 
a starting point and multiplying it by 3/2, then multiplying the result 
by 3/2, over and over, you can generate a series of perfect fifths. 
Unlike in our typical tempered tuning system, there is no circle of 
perfect fifths, it is an infinite spiral. Starting with C you would have 
C, G, D, A, E, B, F#, C#, G#, D#, A#, E#, B#, F##, C##, etc., etc. The 
B# in this system would be a little sharper than C, the C## would be a 
little sharper than D, etc. In fact, this is how most Western music was 
tuned for several centuries until someone decided to make B# the same as 
C and limit the scale to 12 notes. Tuning this way by perfect fifths is 
often called Pythagorean tuning, although Pythagoras was by no means the 
first or the only person to discover it. As the scale is generated by 
the ratio 2:3 and the highest prime number in this ratio is 3, then this 
is termed a 3-limit system. It makes a fine scale for melodic purposes, 
but the "major thirds" produced by the 3-limit are a little harsh when 
used harmonically. For this reason, early Western art music treated the 
major third as a dissonant interval.

OK. So we've had 1:2 and 2:3, so the next obvious pair of numbers would 
be 3:4. If you take any frequency and raise it by 4/3 (or multiply by 1 
and a third), you raise the pitch of the note by a perfect fourth. 
Actually, the highest prime in the ratio 3:4 is also 3, so this is still 
a 3-limit system, because a perfect fourth is simply a perfect fifth 
going off towards infinity in the opposite direction: C, F, Bb, Eb, Ab, 
Db, Gb, Cb, Fb, Bbb, etc.

So, after 1:2, 2:3, and 3:4, we would logically arrive at 4:5 and if you 
were to raise a frequency by 5/4 (ie multiply it by one and a quarter) 
you would get a note a major third higher. So if we take our 440Hz and 
multiply it by 5/4, we get 550Hz. This would be a justly intonated C#. 
If we were to throw in our E of 660Hz, we would have a chord of 440Hz, 
550Hz and 660Hz. This would be a pure major chord in the ratios 4:5:6. 
The highest prime number in this ratio is 5, so this is part of a 
5-limit system. As I said, this is the theoretical basis of much Western 
music. Both 3-limit and 5 limit JI give acceptable diatonic and 
chromatic scales, but 5-limit gives nice smooth harmonies in thirds and 
sixths.

The next ratio after 1:2, 2:3, 3:4, and 4:5 would be 5:6. Actually, we 
already encountered that in our chord of 440Hz 550Hz and 660Hz. The C 
and E in this A major chord are in the ratio of 5:6, so multiplying any 
frequency by 6/5 raises it by a minor third. This is still part of the 
5-limit.

So our next basic ratio would be 6:7. This is where the "problems" come 
in. If we take our major chord in the ratio of 4:5:6 and add a note to 
make it 4:5:6:7, we get a pure seventh chord, the same as you would find 
in holes 2, 3, 4 and 5 draw on a traditionally just intonated harp. So, 
added to our A of 440Hz, our C# of 550Hz and our E of 660Hz, we would 
have a slightly flat G of 770Hz. The highest prime in the ratio 6:7 is 
7, so we are now dealing with a 7-limit system.

Now pause for second and remember that to raise a note by a perfect 
fourth you multiply it by 4/3. So, let's take our A, raise it by a 
fourth to get a D, then raise it by another fourth to get a G.

440Hz x 4/3 = 586.666Hz
586.666Hz x 4/3 = 782.222Hz

Hm. So the 7-limit G (the upper note of our pure A7 chord) is actually 
more than 12Hz flatter than the 3-limit G (two perfect fourths above the 
A). In fact, although both the 3-limit and 5-limit systems give us stuff 
that sounds like our familiar diatonic scales, the 7-limit introduces 
notes that are often about 1/3 of a semitone away from our familiar 
scales. This can make for nice sweet barbershop harmony, or for piquant 
blues notes, but you really don't want to be playing a 7-limit interval 
when your ears (or your audience's ears) want to hear a 5-limit or 
3-limit interval.

The next prime numbers after 7 are 11 and 13. The 11-limit and 13-limit 
intervals involve even greater deviation from familiar territory, often 
involving intervals of around a quartertone (half a tempered semitone), 
giving such exotic things as neutral thirds and sevenths (ie neither 
major nor minor, but somewhere in between). Not particularly useful for 
conventional harmonic purposes, but you do often encounter such melodic 
intervals in blues and some Middle Eastern music.

The next couple of primes are 17 and 19. These start to take us back to 
more familiar sounds, as 17-limit and 19-limit intervals often come 
quite close to approximating our common 12TET notes. Beyond the 
19-limit, things tend not to have much of an identifiable quality of 
their own.