Just Intonation, Part 1 Music Technology Pages Return to MusicWords Home

Just Intonation, Part 7: How to Set Up Your Synthesizer

In this section I'll assume you have access to a synthesizer with a 12-note tuning table. We're going to create a scale in just intonation, proceeding along two parallel paths at the same time. We'll tune the synth and listen to the results, while simultaneously discussing the interval relationships we're creating.

Before we start, you'll need to find or create a suitable synth sound. What we want is a simple, pure sound, but not one that's too muted. It should have some overtones. If you don't have a usable sound already in your synth's memory, follow these steps:

1. Find a memory slot whose contents you don't mind overwriting with the sound we're about to create.
   
   
2. If your synth has an "initialize voice" command, use it. If not, it's okay to start with the sound you'll be overwriting.
   
   
3. Turn off all the effects processing in the sound. You may be able to do this by selecting "no effect," or by setting the effects' wet/dry parameters to completely dry.
   
   
4. Disable all but one oscillator. Multi-oscillator sounds tend to be rich-sounding, but the richness makes it hard to hear pure intervals. While you're at it, make sure this oscillator is set to zero detuning. If you're using an analog-type synth, make sure oscillator sync, ring modulation, and FM are switched off.
   
   
5. Make sure there's no vibrato affecting the oscillator's pitch, and no pitch envelope. If you're using a Roland synth, also check to make sure the "random tune" parameter is set to zero.
   
   
6. Set up an amplitude envelope with a fast attack, a high sustain level, and a reasonably short release time. We want to be able to hear sustained notes. While you're at it, get rid of most of the velocity response, so that the tone will be at a fairly uniform volume no matter how soft or hard you hit the keys.
   
   
7. Open the filter cutoff to a fairly high setting, so the sound isn't too muted. Get rid of any filter modulation, especially from LFOs and oscillators.
   
   
8. Choose a waveform that's bland-sounding but not dull. If you're using a sample playback synth, you might try an analog-type wave, such as a sawtooth wave. A wave that has a lot of internal animation or unusual overtones, such as a bagpipe or a bell, would be a poor choice.
   
   
9. Listen to the sound you've set up. When you hold down a single note, the tone should sustain, and should have absolutely no animation -- no chorusing (a swirling type of sound) in the tone, and no vibrato. The purpose of this exercise is not to create a sound to play music with. It's so you'll have a sound that makes it easy to hear tuning changes as you set up your scale. So name the patch "Tuning Test Tone" or something similar, and save it.

Tuning the Synth

Access your synth's tuning table (it's probably in the Global area) and make sure it's active. You may be able to switch it on or off in the Global area, or you may need to select it in the Tuning Test Tone patch and then re-save the patch. If any garbage data is in the table, get rid of it: We want to start with all 12 notes tuned to their normal equal-tempered pitches, so set all of the fine-tune parameters in the table to zero.

Ready? Let's get started. Play a sustaining C-G fifth near the middle of the keyboard, and listen to it. You should hear a slow beating between the two notes. This beating probably won't be obvious, but you'll hear a certain harshness or liveliness when the two tones are combined. If you listen closely to the upper overtones, you should be able to hear some beating - a faint "wah-wah-wah" type of sound.

The ability to hear this beating is central to this whole exercise. If you don't hear the beating, it's most likely because (a) you haven't set up the synth tone as described above, (b) the tuning table is not set to its default equal-tempered values, (c) the tuning table is not switched on, or (d) you just haven't learned what to listen for yet. To help you out, this mp3 file illustrates the sound you're looking for. I created it by playing an equal-tempered C-G fifth on a Korg 01/W.

Got it? Okay, now bring the fine-tune parameter for the G up by 2 or 3 increments, until the beating gets as close as possible to going away. Listen to the tone after each change in the parameter. (On some synths, you may need to re-strike the key in order to hear the new value of the tuning parameter. Others will adjust the tuning during the course of a sustaining note, which makes the process easier.) The two tones will almost certainly continue to beat slowly, but at a certain point the roughness of the interval will be at a minimum. It should sound more or less like this. This point is as close as your synth is going to get to just intonation.

What we've done by retuning the G is compensate for the out-of-tuneness of 12-tone equal temperament. We've put the G back where it belongs in the harmonic series. It's now at a perfect 3:2 ratio (or as close to that ratio as your synth allows) above the C.

The next step is to tune the E. The E will have to be tuned downward by somewhere between 9 and 14 increments. The exact value will depend on the resolution of your tuning table. My Korg 01/W, for example, which is used in the audio examples, has a tuning adjustment in which each key in the octave can be tuned from -50 to +50. This range (101 possible settings) covers a range of a half-step up or down. On the 01/W, the E produces the least beating with the C and G at a setting of -13. This file illustrates the sound of the C-E interval without retuning, and this file shows what it sounds like when I've retuned it to a close-to-perfect 5:4 interval.

Now you're ready to compare the full C major triad (C, E, and G together). Here is the sound of this triad in equal temperament; here is what it sounds like when retuned. The difference is striking. Just as striking, if you think about it: All your life you've been listening to the ugly sound of that equal-tempered triad and thinking it was pleasant, consonant, a point of resolution and repose.

Before we tune the rest of the notes in the scale, I need to point out that in some synthesizers, the fine-tune parameters in the tuning table interact with the tone-generating algorithm in a slightly funky way. For example, if you've left C at 00 and found, by trial and error, that the G sounds closest to a perfect fifth with C when it's tuned to +3, it would be natural to assume that the D (being a perfect fifth above G) would be closest to a pure interval when tuned to +6. But you may find that the G-D interval sounds best when the D is tuned to +4 (as it is on the 01/W) or some other setting. Ultimately, the only way to tune a synthesizer is by ear, not by the numbers.

To tune all of the white keys, follow these steps:

1. Make C-G a perfect fifth by adjusting the pitch of the G.
   
   
2. Make G-D a perfect fifth by adjusting the D.
   
   
3. Make F-C a perfect fifth by adjusting the F.
   
   
4. Make C-E a perfect major third by adjusting the E. Test the tuning of the E by playing the C-E-G triad.
   
   
5. Tune the B the same way, by making G-B a perfect major third and then testing G-B-D.
   
   
6. Tune the A the same way with reference to F and C.

If you play a C major scale at this point, it will sound recognizable, but somewhat exotic. (Audio example coming soon.) In particular, the E, A, and B are somewhat lower than we're used to hearing, which -- to my ears, anyway -- makes the scale sound less aggressive. A more interesting discovery awaits you, however. Play the C-G fifth. Sounds good, doesn't it? So do all of the other perfect fifths among the white keys -- except for the D-A fifth. The D-A fifth sounds rather distinctly sour.

No, you didn't do anything wrong. That's the nature of just intonation. If you happen to want to play a piece that uses the D-A fifth a lot, you can easily tune your scale to make the problem go away. (At least, you can do it easily on a synthesizer.) But if you do, the problem will pop up somewhere else. It's like trying to deal with a hamster under a carpet by thwacking it with a tennis racquet. The hamster isn't going to flatten out, it's just going to crawl off to another spot. Tuning a 12-note scale so that all of the intervals have simple numerical relationships is just not possible.

Let's be clear about this. The D-A fifth is not out of tune; it's a mathematically perfect interval. It just isn't a 3:2 interval like the other fifths. In case you're curious, here's how to calculate the size of this interval:

First of all, the D is a perfect 3:2 fifth above G, and the G is a perfect 3:2 fifth above C. We start by defining C as our starting point: The C-to-C interval (like all unisons) is 1:1. So the D-C interval is defined by

3/2 x 3/2 = 9/4

Since 9/4 is greater than an octave (an octave being 2/1), we need to lower the D by an octave in order to look at the basic form of the interval. The major second between D and C is a ratio of 9:8, because

9/4 x 1/2 = 9/8

You'll recall that we tuned the A to be a perfect major third above F. The ratio between F and C is, again, 3:2, but the F is below the C, so we need to perform a little fraction magic. The 3/2 interval, when inverted, is 2/3, so if C is defined as 1/1, the F below it has a frequency ratio of 2/3. When we raise this F so it's in the octave above C, its relation to C is

2/3 x 2/1 = 4/3

A major third is an interval of 5/4. We add the C-F and F-A intervals by multiplying:

4/3 x 5/4 = 5/3

This A is 5/3 above C. Now we know that the D is tuned to 9/8 of the frequency of C. In order to find the ratio between this D and this A, we need to give them the same denominator:

9/8 = 27/24
5/3 = 40/24

So the ratio between D and A has the value 40/27. The fact that these are both large numbers might suggest that the interval won't sound pleasant. To finish our analysis, let's give 40/27 and 3/2 a common denominator:

40/27 = 80/54
3/2 = 81/54

This calculation shows us that if we were to tune two adjacent keys to two different A's, the first A being a major third above F, and the second a perfect fifth above D, the second A would be higher than the first by a ratio of 81/80. In some sense, equal temperament is a valiant but doomed attempt to spread that extra 81/80 around the keyboard so as to make it invisible.

We have several options for tuning the black keys. One simple method is to take our original row of perfect fifths -- F, C, G, D -- and tune four of the black keys (D-flat, A-flat, E-flat, and B-flat) so that each is a perfect major third below one of them. We've already tuned white keys to the pitches a perfect third above F, C, and G, so we can finish the scale by tuning the F-sharp so it's a perfect third above the D. This tuning can be represented by the following diagram:

A   E   B   F#
|   |   |   |
F - C - G - D
|   |   |   |
Db  Ab  Eb  Bb

(Note: For a full and lucid explanation of this diagram, I'd urge you to track down a copy of Harmonic Experience by W. A. Mathieu, published by Inner Traditions in Rochester, Vermont, in 1997.)

As you play this tuning, you'll make some discoveries. It contains pure major triads on six of the twelve roots: F, C, and G in the middle row of the diagram, and D-flat, A-flat, and E-flat in the bottom row. It also contains six pure minor triads, on F, C, G, A, E, and B. Nine of the twelve fifths in the scale are pure, but three are not. The Bb-F fifth has the same awkward sound as the D-A fifth, while the F#-Db fifth (technically, not a fifth at all but a diminished sixth) is really quite strained.