Western people like two things in harmony: intervals derived from the natural overtone series, and the ability to play in multiple keys. Unfortunately, it’s not possible to do both of these things within the same tuning system. If you want to use just intonation intervals derived from harmonics, then they will not work in every key. So we as a civilization have decided to use a tuning system that enables you to play in lots of different keys, even though it means that all of the keys are slightly out of tune. Fortunately, the computer makes it easy to explore alternative tuning systems. I have been experimenting with this cool tuning plugin called MTS-ESP.
I have struggled my whole life to understand how tuning works, so I made a track to demonstrate to myself how just intonation sounds when you use it in all twelve keys.
What you are hearing in my track is a tuning system that is “perfect” in C major, but not so perfect in other keys, and very not perfect in a few of them. Let’s figure out why!
An “ideal”, “pure” tuning system would be based on the natural harmonic series. The frequencies you find in the natural harmonics are related by ratios of integers: 2/1, 3/1, 4/1, 5/1, and so on. You can make a just intonation major scale by combining the ratios between the first five harmonics. For whatever frequency you define your “home base” pitch to be, you can have a major second that is 9/8 of that frequency, a major third that is 5/4 of that frequency, a perfect fourth that is 4/3 of that frequency, and so on. The tuning in my track is a version of just intonation called Ptolemy’s intense diatonic scale, as extended to a full chromatic scale by Kyle Gann. Here are the frequency ratios of each note.
- C: 1/1
- C#/Db: 16/15
- D: 9/8
- D#/Eb: 6/5
- E: 5/4
- F: 4/3
- F#/Gb: 45/32
- G: 3/2
- G#/Ab: 8/5
- A: 5/3
- A#/Bb: 9/5
- B: 15/8
- C: 2/1
If this tuning is so pure and perfect, why are we not all using it? The problem is that while these notes sound lovely compared to C, they don’t all sound good compared to each other. My track moves through all twelve dominant seventh chords around the circle of fifths. We can see which chords are out of tune with a little math.
For each dominant seventh chord, the major third should ideally be 5/4 times the root frequency, the fifth should ideally be 3/2 times the root frequency, and the flat seventh should ideally be… hmm. Sevenths are complicated. There are three possible choices:
- The harmonic seventh at 7/4. This is a cool sound, but it’s not something they historically used in Western Europe.
- The small just minor seventh or Pythagorean small minor seventh at 16/9, which is two perfect fourths stacked up.
- The large just minor seventh at 9/5, which is a perfect fifth plus a minor third.
Kyle Gann chose 9/5 for the B-flat in his scale, but I think 16/9 sounds better, so let’s go with that.
Now we can compare the ideal just intonation tuning of the notes in each dominant seventh chord to their actual values in the Gann scale. Let’s say we’re looking at D7. Its root note is the D at 9/8. The ideal major third above that root would be 5/4 times 9/8, or 45/32. The F-sharp in the Gann scale is indeed tuned at 45/32. Great! The ideal fifth above D would be 3/2 times 9/8, or 27/16. Unfortunately, the A in the Gann scale is 5/3, which is noticeably flatter than 27/16. This is why just intonation is a problem.
Here are comparisons for all the notes in all the dominant seventh chords.
- G7 (3/2, 15/8, 9/8, 4/3): Everything is perfect, by definition, because we are in C major and are using the overtones of C as the basis for our scale.
- C7 (1/1, 5/4, 3/2, 9/5): The seventh is too flat by a factor of 80/81.
- F7 (4/3, 5/3, 2/1, 6/5): The seventh is too sharp by a factor of 81/80.
- Bb7 (9/5, 9/8, 4/3, 8/5): The fifth is too flat by a factor of 80/81.
- Eb7 (6/5, 3/2, 9/5, 16/15): Everything in this chord is in tune, but there will be problems if we try to extend it. For example, if we want to have Eb9, the F will be out of tune.
- Ab7 (8/5, 1/1, 6/5, 45/32): The seventh is too flat, by a factor of 405/516.
- C#7 (16/15, 4/3, 8/5, 15/8): The seventh is too sharp by a factor of 135/128.
- F#7 (45/32, 9/5, 16/15, 5/4): The third is too sharp by a factor of 1152/1125, and the fifth is too flat by a factor of 2025/2048.
- B7 (15/8, 6/5, 45/32, 5/3): The third is too flat by a factor of 375/384.
- E7 (5/4, 8/5, 15/8, 9/8): The third is too sharp by a factor of 128/125, and the seventh is too sharp by a factor of 81/80.
- A7 (5/3, 16/15, 5/4, 3/2): The third is too sharp by a factor of 384/375, and the seventh is too sharp by a factor of 81/80.
- D7 (9/8, 45/32, 5/3, 2/1): The fifth is too flat by a factor of 80/81.
The bottom line is this: if you want to change keys and still keep those pure just intonation intervals, then you are going to have to retune all your instruments every time. There are many just-intonation-based tuning systems around the world, and it is no surprise that those musics don’t usually feature key changes. For example, in Hindustani classical tradition, you play everything over an unvarying drone. There’s reason to believe that the blues is based on seven-limit just intonation, and if that’s true, then it’s no coincidence that blues songs don’t change key either.
The way that we here in America tune our instruments is a temperament system, meaning that we temper (adjust) the intervals so that the out of tune ones are less obnoxious. There have been lots of temperament systems used throughout history. Some of them push all the out of tune intervals into certain keys (the ones with lots of sharps and flats in them), and then you just have to avoid those keys. Others try to distribute the out-of-tune-ness more evenly, so that all the keys are usable, even if all of them aren’t perfect. (When Bach wrote The Well-Tempered Clavier, this is what he meant.) The system that ultimately won out is 12-TET, which solves the problem by making every interval in every key a little bit out of tune. This makes all the keys sound equally good (or equally bad, depending on how much you care about these things.) In older systems, some keys were sweeter and others were more sour, but in 12-TET they are all interchangeable, like LEGOs that are different colors but all the same size.
There is one thing big caveat to everything in this post. The concepts of “in tune” and “out of tune” are highly subjective. Don’t be fooled by all the math; there is no intrinsic biological reason to prefer one tuning system over another. Remember that, technically, every interval in 12-TET except the octave is “out of tune”. We are just all so used to it that it feels “right” to us. After a lifetime of 12-TET, the “more pure” just intonation intervals can sound really weird at first. They certainly did to me! However, I have listened to my Ptolemaic scale track many times over the past few weeks, and with repeated exposure, even the most “out of tune” chords like E7 and A7 are starting to sound perfectly fine. They sound different from what I’m used to, but they no longer sound bad the way they did at first. You can get used to just about any tuning system with enough listening practice. Musicality is probably innate in humans, but music’s aesthetic qualities are a matter of enculturation.