This is an edited version of a post written to harp-l in August 2003.

Ramblings on the general topic of Just Intonation, ratios and prime limits

This brings up a question about tunings.

I think that I know the answer, but maybe Pat Missin can give the definite answer.

I don't know about THE answer, but I'll try to give an answer...

Is it not true that just tuning produces slightly different pitches in every

key? If true, this means that a harp can have just tuning only in the

first-position key stamped on the cover? Doesn't this mean that when you

play it in any other position/key that you don't have just tuning any more?

Not really. There is a widespread belief that Just Intonation has to be based in a given key. This is not exactly the case, although there are many text books that talk about such things as "** the** Just Intonation Scale in the key of G", or something like that. This is misleading, as there is no single JI scale. JI can be used to generate an infinite variety of scales. Nor do they have to be in a specific key.

Just Intonation is more concerned with getting the optimal harmony for a set of pitches to be played simultaneously. The harmonica makes this a comparatively easy process, as there are only certain combinations of notes that can be played at any given time. For example, on a standard C diatonic, blowing produces the notes C, E and G. It is easy to tune these notes to produce a perfectly pure chord. Now, it doesn't matter whether you are using this C chord as the tonic chord in first position, the subdominant chord in 2nd, or the dominant chord in 12th, the subtonic in 3rd position, etc ; in each case it is a perfectly tuned C major chord.

If you draw on the lower end of the same harp, you get the notes G, B and D. Again, it is easy to tune these notes as a pure G major chord and again, it doesn't matter whether you are playing this as the tonic chord in 2nd position, the dominant chord in 1st position, the secondary dominant in 12th position, or whatever. In each case it is a perfectly tuned G major chord.

Now, one of the problems with Just Intonation is that in order to have every note perfectly in tune with every other note, you would have to have an infinite range of notes. Instruments such as the trombone, a choir of human voices, fretless stringed instruments, etc, have precisely this capability (although only for one note at a time), but many instruments have only a limited range of notes. One immediate problem is that you cannot make up a simple seven note diatonic scale in JI where each of the notes is in perfect tune with each of the others. For example, if you took our C chord above and tuned it purely, then built a G chord starting on the fifth of the C chord, then rearranged them to form a scale, you would have the notes C D E G B C. Now, the problems would start if you wanted to add an A to this scale. If you were to tune the A to be a perfect fifth above the D, it would be out of tune with the C. If you were to tune it to be a pure minor third below the C, it would now be out of tune with the D. Start adding more notes and you start adding more of these problems. This is how tempering becomes a vitally important factor in the history of fretted strings, harps (stringed variety) keyboard instruments, etc. and the music associated with them.

However, the harmonica doesn't have these problems, for the simple reason that there are only certain combinations of notes that can be played simultaneously. For example, on our C harp, you can play the A (6 draw) at the same time as the D (4 draw), but you cannot play it at the same time as a C (4 or 7 blow). Therefore you can tune the D and the A together, and not have to worry too much about whether it is perfectly in tune with the C, providing they are both tuned in such as way that they sound acceptable in a melodic context. As the difference in this case between the first A (the one that is a perfect 5th above the D) and the second A (the one that is a pure minor 3rd below the C, or a pure major 6th above the C - it's the same note either way) is only about 20 cents, either one will function quite well melodically.

In fact, for the most part, the most common JI intervals do not vary from their 12TET equivalent (12TET = 12 Tone Equal Temperament, the most common tuning system used in modern world) by any painfully huge amounts. Therefore it is mostly a simple case of getting the harp into tune with itself to produce the sweetest chords, then tweaking the whole tuning so that the average deviation from "standard concert pitch" is minimal. This is what good harmonica technicians do on a daily basis. Any small discrepancies that occur can be dealt with by the player's technique and most good players do this almost unconsciously. This is why I chuckle to myself when I read recommendations that single note players would sound better using an equal tempered harp. Actually, for the most part, if you are playing single notes, the precise intonation of your harp barely matters at all. Rather than 12TET sounding better for single note players, it is more a case of 12TET sounding worse for chordal players.

However...

... there is one aspect of the standard diatonic tuning that does pose a significant problem. When I said that "the most common JI intervals do not vary from their 12TET equivalent by any painfully huge amounts" I am referring to what is technically called 5-limit Just Intonation. This is the theoretical basis of most Western art music (although in practice things rarely achieve this ideal) and it is possible to tune certain harmonica layouts perfectly using this system - natural minor and Melody Maker, for example. However, the standard major diatonic tuning (and also the harmonic minor tuning) pose certain problems, in that they are impossible to tune purely by using 5-limit intervals. The main problem is that pesky 5 draw (and its octave partner 9 draw).

In JI, with the exception of octaves (and to a lesser extent fourths and fifths) two consecutive intervals of exactly the same size simply do not sound good - they produce harsh dissonance. So, if you were to tune 3 and 4 draw (B and D on our C harp) and 4 and 5 draw (D and F) so that there were both pure minor thirds, they would sound fine if you were to play either B and D, or D and F. However, play any combination that includes both B and F and it will sound like an angry hornet's nest. The way to solve this is to tune the D-F "minor third" interval so that is slightly smaller than the B-D minor third. This gives you a beautifully smooth and rich G7 chord, but the F note at the top of that G7 chord is much flatter than the F that is the fourth note of the C major scale. Whereas the typical 5-limit JI intervals are usually less than 20 cents away from their 12TET equivalents, this "flat" 5 draw (technically it is a 7-limit minor third, also called a subminor third) is more than 30 cents from the tempered version, which is a lot more noticeable. This means that if you tune your harp like this (which is how all diatonics were traditionally tuned), it will sound great for playing cross harp blues, but when you play first position stuff (especially if you use single notes rather than a chordal approach) it might not sound so good. It probably will also sound a little odd when you play third position, as you are technically playing a subminor minor mode rather than the typical dorian minor and it will probably sound pretty horrible if you play in 12 position, as your root note will be substantially "flat". There are various ways to deal with this, however.

If anyone is interested, the harmonic minor tuning presents even more of a problem. Not only do you have the minor third between 3 and 4 draw, you also have minor thirds between 4 and 5 draw AND 5 and 6 draw. The solution to this involves using higher level JI intervals drawn from the 17-limit, but I'll save that for a later article ...

Just tuning is the reason barbershop and string quartets sound so good,

because the instruments used (voice and strings) can be tuned by the ears of

the players who automatically use just regardless of the key.

This is often described as "adaptive JI". Rather than having a fixed palette of pitches, singers and players of certain instruments can continuously tweak each note for the optimum intonation at any point in the tune. Again, to some extent, a good harmonica player will be able to do micro-bends to keep their instrument in good relative tune with the accompaniment.

Tempered/equal (twelfth-root-of-two) tuning is a compromise that doesn't

sound quite as good in any key as just but does sound the same in all keys?

Thus equal tuning is the obvious choice for any instrument that uses fixed

pitches and is played in all keys.

It is a choice, but it is not the only one, nor is it always the best one, although it is the commonest. There are various other possibilities, depending on the context.

It would seem possible to have an electronic keyboard that could adjust the

pitches to give just tuning in any key. You would only have to push a

button to tell it which key to tune to.

There are such things, although you then run into the problems that I mention above, so in anything but the most harmonically simple piece of music, you would have to keep pushing that button all the way though the tune.

Hope this isn't too technical and of course, if anyone has any further questions, I'd be happy to try to answer them.

However, as I am here anyway...

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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