Tuning is the final frontier of my musical understanding. I start reading about it, and then I hit a big table of fractions or logarithms and my eyes immediately glaze over. However, tuning is important and interesting! So I continue to struggle on. Fortunately, as with so many music theory concepts, the right computer software can open up lots of new learning avenues. I have been having a great time with MTS-ESP by Oddsound. It was designed to help you hear and play different tuning systems, but it also visualizes them in an attractive circular way. If you read this blog, you know how much I love a good circular music visualization scheme.
So here is the basic problem with tuning. An ideal system (for Western people) would be based on the natural harmonic series, because we love how harmonics sound. This kind of tuning system is called just intonation. It sounds lovely! Unfortunately, just intonation makes it impossible to change keys or tune your guitar. Let’s use MTS-ESP to figure out why that is.
First, we need a starting note. Let’s use middle C, and let’s say that it has a frequency of 1 Hz. (It doesn’t! Middle C’s actual frequency is 261.626 Hz. Multiply all the numbers in this post by 261.626 to get actual frequencies of all the notes.) If you double the frequency of middle C, you get another C an octave higher. MTS-ESP represents the octave between these two notes as a circle, with both notes occupying the twelve o’clock spot.
Now let’s take a look at the natural harmonics of C. These are integer multiples of C’s frequency, divided by two enough times to bring them down into the same octave.
MTS-ESP doesn’t show the even-numbered harmonics because they are just higher-octave copies of the ones you see here. The fainter, unlabeled harmonics are the eleventh, thirteenth and seventeenth, which are not very audible and not relevant to this discussion.
The first thing we’re going to do is take the lowest harmonics of C and turn them into notes unto themselves. The third harmonic of C is G at 3/2 Hz. (We’re multiplying C’s frequency by 3 and then dividing it by 2 to bring it down an octave.) The fifth harmonic of C is E at 5/4 Hz. (We’re multiplying C’s frequency by 5 and then dividing it by 2 a couple of times to bring it down two octaves.)
So now we have C at 1 Hz, E at 5/4 Hz, and G at 3/2 Hz. We can make a C major triad, but not much else. We’re going to need some more notes. To do that, let’s take a look at the harmonics of G.
The third harmonic of G is D at 9/8 Hz. (We’re multiplying 3/2 by 3 and then octave normalizing the result.) The fifth harmonic of G is B at 15/8 Hz. (We’re multiplying 3/2 by 5 and then octave normalizing.)
So far, we have C, D, E, G, and B. To fill out the C major scale, we’re going to look at the “subharmonics” of C, that is, notes whose own harmonic series include C. In the image below, the subharmonics are red .
The third subharmonic of C is F at 4/3 Hz.
Now we can make some more notes by looking at the harmonics of F.
The third harmonic of F is C at 1 Hz (no surprise there, that’s the definition of the third subharmonic of C). The fifth harmonic of F is A at 5/3 Hz.
Now we have the entire C major scale. Next, we can start finding some black key notes. Let’s look at the harmonics of D at 9/8 Hz.
The fifth harmonic of D is F-sharp at 45/32 Hz. The third harmonic of D is A at 27/16 Hz. Uh-oh. That does not line up with our existing A at 5/3 Hz. In other words, while D and A are both in tune compared to C, they are out of tune with each other! We could solve the problem by just declaring A to be 27/16 Hz, but then it wouldn’t be in tune with F.
So maybe harmonics are not the way to go. Maybe we should add black key notes using subharmonics instead. Let’s take a look at the subharmonics of F.
The third subharmonic of F is B-flat at 16/9 Hz.
Maybe we can safely make some more notes using the harmonics of B-flat?
The third harmonic of B-flat is F at 4/3 Hz, by definition. The fifth harmonic of B-flat is D at… oy gevalt… 10/9 Hz. That is going to conflict with our existing D at 9/8 Hz. B-flat and D are both in tune with C, but not with each other. If we declare D to be 10/9 Hz, it won’t be in tune with G.
Okay. Maybe the problem is mixing harmonics and subharmonics of C. What if we only used harmonics? Everything would be in tune that way… but then we wouldn’t be able to make F. The best we could do would be E-sharp, which is the third harmonic of the third harmonic of the third harmonic of the third harmonic of the third harmonic of the third harmonic of the third harmonic of the third harmonic of the third harmonic of the third harmonic of the third harmonic of the third harmonic of C. As you might expect, E-sharp would not sound good. We also wouldn’t be able to make B-flat; we’d have to use A-sharp, and that would sound pretty grim too.
Meanwhile, if we only used subharmonics, we’d have a similarly miserable time trying to make the notes from the sharp side of the circle of fifths. Instead of B, we would have to use C-flat. Instead of E, we would have to use F-flat. Instead of A, we would have B-flat-flat. And it would only get worse from there.
Ultimately, your tuning choices are these:
- Tune with harmonics and stick close to the key of C. Handle your black-key notes with extreme caution. This is what medieval Europeans did.
- Modify (“temper”) your harmonics-based intervals to spread the out-of-tune-ness around so you can use keys other than C. None of them will be perfectly in tune, but more of them will sound good than they would in just intonation. If you use a very sophisticated temperament system, you can get all of the keys to sound at least okay. The canonical Western European composers used a wide variety of temperaments.
- Modify your harmonics-based intervals by spreading the out-of-tune-ness across every possible interval (except octaves). None of them will be perfect (except octaves), but none of them will be terrible either. This approach is called twelve-tone equal temperament (12-TET), and it has been the uniform tuning standard throughout the Western world for the past hundred-ish years.
- Use higher harmonics to make a wider variety of intervals. You still wouldn’t be able to easily change keys, but at least you would have more good-sounding notes to choose from. Unfortunately, if you do this, you will have a hard time finding other people to play with, at least in the US or Europe.
None of these choices are ideal. Most of us use 12-TET by default, but we unconsciously feel its deficiencies. Skilled musicians instinctively aim for just intonation intervals when they sing or play instruments with fine pitch control. Some of the canonical European composers resisted 12-TET because they thought it sounded bland or out of tune. A guy named Ross Duffin thinks that 12-TET ruined harmony. Maybe you don’t feel that strongly about it, but you should at least know that there is more to harmony than the way that instruments usually get tuned.