[Harp-L] an explanation of limit in JI

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.

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