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Just Intonation, Part 8: A Few Tunings to Try

The basic tuning given in the previous section is what's called by intonation theorists a "5-limit" tuning. This is because all 12 notes are tuned using only the ratios 3/2 and 5/4. It's a 5-limit tuning because the 5/4 ratio (the distance between the fourth and fifth harmonics, for those of you who read the section on acoustic theory) is as high as we had to go in the harmonic series to construct the scale.

We can construct many different 5-limit tunings. For instance, instead of tuning the A-flat so it's a major third below C, we could tune it so it's a major third above E, making it a G-sharp. Many of these tunings have distinctive intervals that can be used melodically and harmonically.

We don't have to use the 5/4 ratio at all if we don't want to. The Pythagorean tuning uses only a stack of perfect fifths, such as Eb, Bb, F, C, G, D, A, E, B, F#, C#, and G#. Try this tuning and you'll discover that nine of the major triads are fairly bad, though not much worse than in equal temperament, while three (in this case the B, F#, and C# triads) are much better than in equal temperament. If we want to play a tune in C major, we can shift the series of fifths around to make the F, C, and G triads sound pretty good while the others are bad. To do this, start the series of perfect fifths with A: A, E, B, F#, C#, G#, D#, A#, F, C, G, D. This puts the bad fifth back between D and A.

I don't find the Pythagorean tuning very interesting. A more adventurous possibility, for me, is to use the seventh harmonic for a couple of the intervals. This 7-limit tuning can get seriously exotic, depending on how you set it up. The basic idea is to tune the C-Bb interval to a 7:4 ratio. This interval gives a C dominant 7th chord a dark, bluesy sound quite unlike anything in the equal-tempered scale. I like to do the same thing with the G-F interval so I can play a G7 chord that uses it.

A diagram of such a tuning might look something like this:

Bb  F
|   |
C - G - D - A
|   |   |   |
E   B   F#  C#
|   |
G#  D#

The intervals between the top row and the second row are both 7:4 minor sevenths, while those between the second and third rows and those between the third and fourth rows are all 5:4 major thirds. This tuning is good for playing blues in G, by the way.

Adding two more 7:4 intervals to the top row makes the tuning even more interesting. Unfortunately, you can't do it with a 12-note tuning table. The 7:4 minor seventh above D is positioned between B and C, so you'd need an extra key on the keyboard. Either that, or sacrifice the B that's a 5:4 major third above G, which will deprive you of the in-tune G7 chord.

If you've followed along to this point, congratulations! If you compose any music using the tunings I've described here, I hope you'll let me know where I can download it (or buy the CD). Eventually I hope to add links to some recordings.