Interesting Stuff

I’ve been reading about the history of musical tuning (which is way more interesting than it sounds; no pun intended, but please bear with me), and it recently occurred to me that these days, you can actually get all your notes in tune (which has not truly been possible at any point in history more than about 50 years ago). I can’t tell if this will make a significant difference in most music, but I think it would be interesting to try. Before I explain my idea in more detail, let me summarize the various tuning systems over the years, to give you the background of where this came from. Wikipedia, while not necessarily wrong here, is too incomprehensible to really read; I got a lot of information from this old temperament overview.

The trick to getting a chord to sound good is to have the ratios of frequencies of the notes be easily expressible in nice, small numbers. For instance, an octave is a ratio of 2:1 (which is to say, if the base note has frequency x, the note one octave above it has frequency 2x). Other commonly used intervals are equally simple: a perfect fifth (like C-G) has a ratio of 3:2, a perfect fourth is 4:3, a major third is 5:4, a minor third is 6:5, and that’s usually enough to get you by. The higher the numbers in the ratio, the more dissonant the notes sound. For instance, major chords should ideally have a 4:5:6 ratio of the frequencies, which is very consonant because the numbers are so small. Minor chords are 10:12:15 (pairs of them are 3:2, 6:5, and 5:4), which is still nice-sounding, but introduces a bit more dissonance than the major. A seventh is even more dissonant: it’s typically a ratio of 15:8. It’s not nearly as bad as having a note out of tune, but there’s way more dissonance there than the major chord.

If you tune your instrument (for the sake of argument, let’s call it a piano) using these perfect ratios, you’re using what’s called a “just intonation” system, and there have been a bunch of them over the years. The problem with just intonation is that things don’t quite add up right when you move to any other key (and since most songs these days have a chord progression where you move through several keys, that’s important).

Take, for instance, the Pythagorean tuning system. In Pythagorean tuning, you start with one note that you designate “in tune,” and you tune all its octaves. You then tune the perfect fifth and perfect fourth of it, call those “in tune,” and repeat. So, if you start with a note called C (for the sake of easy math, we’ll say it has frequency 1), you can tune higher C’s (frequencies 2, 4, 8, etc), lower C’s (1/2, 1/4, 1/8, etc). You then tune G (3/2) and F (4/3). This lets you tune all G’s (…, 3/8, 3/4, 3/2, 3, 6, 12, …) and all F’s (…, 1/3, 2/3, 4/3, 8/3, 16/3, …). Tuning the G’s lets you tune a perfect fifth up from that to D (note that tuning the perfect fourth is C again, but it won’t change because so far everything is consistent: its “new” frequency would be 2, which is a perfect octave from the old one and it therefore needs no modification). So now you’ve got D’s (…, 9/16, 9/8, 9/4, 9/2, 9, …). and from the F you’ve already tuned, you can grab a perfect fourth and get the Bb’s in tune, &c.

Where’s the problem, you ask? There are two of them here. The first is that intervals besides fifths, fourths, and octaves are off. For instance, the C-E major third actually has a ratio of 81:64, while an ideal major third has a ratio of 5:4 (or 80:64). So, things are a little off there. As a corollary, you’ve really tuned to your starting note, C: if you picked a different starting note and did the same Pythagorean tuning, all the notes would be slightly different. This is why really old music (like Gregorian chants) doesn’t have a lot of different chords in it: all the other chords they didn’t tune to are somewhat off because all the other scales are somewhat wrong. However, a bigger problem is getting that last note in tune. In the above example, the last note you tune will be F#. There are two ways of tuning it: you can get there by tuning C-G-D-A-E-B-F# (giving it a frequency of 729/512), or you can go C-F-Bb-D#-Ab-C#-F# (so it has frequency 1024/729). The problem is that these frequencies are pretty far apart (1.4238 versus 1.4045; nearly 2% different), so whichever you pick, the other will sound awful. This is what’s known as a “wolf,” and music played in that key sounds very out of tune.

Fast forward a few centuries, and Europe started using “meantone” temperaments to replace the Pythagorean one. The idea behind these is that all major thirds are perfectly in tune (C-E-G#, for instance, have frequencies 1, 5/4, 25/16; the high C has frequency 2 although there’s a case to be made that it should be 125/64, which is roughly 1.95). There are a bunch of variations on meantone tuning, but a common one (“quarter-comma meantone”) makes whole tones simply the arithmetic mean of the major third (so D would have frequency \sqrt{5}/2, for example), and then does this screwy thing with two different versions of halftones in order to try to make the fourths and fifths somewhat close to what they should be.

This has the same problems that Pythagorean tuning had (intervals besides thirds and octaves are slightly wrong; there will be a wolf key due to the crap you’ve got to pull with different halftones). However, we have lessened the problems with moving between different keys, and you can start having a few accidentals in your music and a couple key changes (think Renaissance music).

Eventually, people like Bach got tired of this crap, and started playing around with “well tempered” tunings. Part of the problem with these is that the whole idea is somewhat poorly defined, and you have lots of different tuning systems that all claim to be well-tempered (such as the Werckmeister system and the Kirnberger system). I don’t really understand how these work, except that they spread out the badness throughout all the scales so there is no wolf key. However, they don’t spread things out evenly, which gives different keys different “colors.” This is why “sad” and “happy” pieces are traditionally played in different keys: if you use a well-tempered system, certain keys sound sadder or happier than others. This was great in that it gave the composer extra stuff to work with (you could really mean something in your key switches) and it got rid of that wolf that had been hounding musicians for centuries. However, at this point we’d gotten so far away from just temperament that most intervals were no longer the ideal frequency ratio.

A couple centuries later, great improvements were made to tuning fork technology (which isn’t really a career you can have any more; it’s hard to get a job in tuning fork research these days), and by the early 1900’s the centuries-old dream of having equal temperament could be realized (yes, it had been used in guitars and clarinets and stuff since they were invented, but now you could have it in pianos and organs and glockenspiels and things, too; now you could have a guitar/piano duet and have them be in tune with each other). This is the system we have today: the frequency ratio of any two consecutive notes is 2^{1/12}. This preserves perfect octaves (which are 12 halftones apart) and spreads everything else out evenly. This eliminates the colors of well tempered systems (so you can transpose any song into any key and have it sound the same), and it keeps the wolf banished. However, now every (non-octave) interval is guaranteed not to be its ideal frequency ratio, so while everything sounds alright, nothing sounds perfect.

…right. So, that was a quick summary of the history of tuning systems in Europe. When you use a just intonation system, some intervals are perfect but you’re stuck with music in only a few keys and you need to watch out for nasty accidentals and wolf keys. When you use a more evenly tempered system, you can get rid of the wolf and use more accidentals, but you lose the perfect frequency ratios in your chords. No matter how you tune your instrument, you can’t fix both problems at the same time.

So here’s my idea: we now have synthesizers that can let you play tones at any frequency. Why not switch your tuning every time you switch chords? Typically when you move from one chord to another, you keep one or two notes the same and move just a couple others around. You know what the chord is supposed to sound like (major chord, or minor, or diminished or whatever), and I think that’s enough to construct the new perfect chord. Which is to say, when you go from chord X to chord Y, don’t change the notes they have in common, and retune the others to have perfect frequency ratios (if the notes they have in common must change, keep the most important note the same and move the other one). If you want to add in a color, go for it; you can use any color in any key because you can adjust each note individually. Sure, I suspect that every time you play through the entire chord progression your “base” frequency will move by a few hertz, but if you keep your music to a reasonable length, you probably won’t drift too high or low. and while you play, every single chord will be perfectly in tune, every time.

Yes, it would be a pain in the butt to do this correctly, and yes, you can’t do this with traditional instruments (you’d be limited to synthesizers and computers), but it seems like a way to have your cake and eat it too. What do you think? Do you know if anyone has tried this out already? Do you think people would notice a difference in the tuning? The electric keyboard I play supports historical tunings, and although I can totally hear the wolfs and how out of tune they are, I have trouble picking up the colors of different keys, so it’s possible that people wouldn’t even notice when you tune your chords perfectly. I dunno, but it was interesting to think about this.