I have just recently written a Sibelius plugin that allows playback in 19-tone equal temperament, Offtonic 19-TET. The 19-tone scale has also been available on the Offtonic Microtonal Synthesizer since its creation. So what is it? What is it about? What is the deal with 19-tone equal temperament, or 19-TET?
To begin with, there are many ways to generate an EDO, an Equal Division of the Octave. The most obvious way is to simply divide the octave into n equal parts, and voilà, n-TET. You can think of this as starting from note 0 and adding 1 each time, and when you get to n, that’s really the same as note 0. For example, in 12-TET, I can start at C:
B#/C (0), C#/Db (1), D (2), D#/Eb (3), E/Fb (4), E#/F (5), F#/Gb (6), G (7), G#/Ab (8), A (9), A#/Bb (10), B/Cb (11), B#/C (0 again), …
Note that I went out of my way to indicate notes with multiple names. These are called enharmonic, because in tonal music, they serve different functions. In C major, an A# resolves up to B, but a Bb resolves down to A. You can’t tell the difference just from hearing it, but you can definitely tell in context. Another way to generate the same 12 tones is to go up by 7/12 of the octave each time. This works because 7 is relatively prime to 12; if you went up 6/12 of the octave each time, you’d end up with C, F#, C, F#, C, F#, … and never get anywhere. Starting at C, let’s see how this one works:
C (0), G (7), D (2), A (9), E (4), B/Cb (11), F#/Gb (6), C#/Db (1), Ab (8), Eb (3), Bb (10), F (5), C (0 again), …
This is known as the Circle of Fifths, usually in a circle, since you go up by a perfect fifth each time (7 half steps). If you go backwards, it’s the Circle of Fourths, since you go up by a perfect fourth each time (5 half steps). You’re always adding mod 12, so it makes sense that forwards it’s fifths and backwards it’s fourths, since 7 = -5 (mod 12).
Now, this perfect fifth is actually an approximation to the perfect fifth, the real deal. When you divide the octave into 12 equal parts, the frequency ratio corresponding to a single half step is 2^(1/12), and for 7 half steps, it’s 2^(7/12) ≈ 1.4983… This is mighty close to the ratio 3/2, at the bottom of the overtone series (check out the Harmonic (C1) preset on the synth to hear how a really in-tune fifth sounds). Believe it or not, this perfect fifth, and its cousin the 5/4 major third, have been the source of our notes for centuries, and this equal-tempered notion is fairly new, from the 18th century! The thing is that they sound so good that our entire system of harmony is based on the idea of the fifth. C and C# aren’t very closely related, but C and G are. The issue is that if you go up twelve perfect fifths, you don’t really end up all the way back at C. The ratio is 531441/524288 ≈ 1.0136…, the Pythagorean comma. If you make a scale with those notes, it won’t be equal-tempered — check out the Pythagorean preset on the synth. The scale, going up by fifths, is this:
Eb, Bb, F, C, G, D, A, E, B, F#, C#, G#
Note that I didn’t mark enharmonics. That’s because they aren’t there. If you go down a fifth from Eb, you get Ab, but that’s not the same note as G#. That’s not even a circle of fifths, because they don’t connect at the ends! The interval between the G# and Eb is called a wolf fifth, since it’s the fifth that doesn’t sound good while all the others do. It makes it hard to play in Ab, huh? That’s what you had to deal with until equal temperament came around. With the advent of 12-tone equal temperament, this line of fifths was turned into the circle above, where G# is the same as Ab, and all can live happily ever after.
The idea of 19-TET is that, surprise, there are 19 notes in the octave instead of 12. If you go up 19 perfect fifths, you get 1162261467/1073741824 ≈ 1.0824…, which is actually fairly large; a half step is 1.0595… So you use narrower fifths. Instead of 1.5, or 1.4983… like 12-TET, you use 2^(11/19) ≈ 1.4938…, and that’s very close! It adds up over 19 intervals, though. You get the following scale, with 19 tones now:
C (0), G (11), D (3), A (14), E (6), B (17), F# (9), C# (1), G# (12), D# (4), A# (15), E#/Fb (7), B#/Cb (18), Gb (10), Db (2), Ab (13), Eb (5), Bb (16), F (8), C (0 again), …
Note where I put the enharmonics. C# is not equivalent to Db, but B# is equivalent to Cb. I got the note names simply from going in diatonic fifths, and the numbers are just adding 11 (mod 19) each time starting at C, which is 0. I can put this in order:
C, C#, Db, D, D#, Eb, E, E#/Fb, F, F#, Gb, G, G#, Ab, A, A#, Bb, B, B#/Cb, C
This nice thing here is that each note is very close to its cousin of the same name in 12-TET. Compared to 12-TET, anything involving enharmonic equivalents breaks — for example, an augmented 6th chord is no longer the same notes as a dominant 7th chord — but because tonal music is so heavily based on the fifth, nothing else breaks. Cool, huh? You do get leading tones that are too low, since the diatonic minor second is wider than the half step, but otherwise, you’re good. Oh, and the third! The perfectly in-tune major third has ratio 5/4 = 1.25. In 12-TET, this is 4 half steps, or 2^(4/12) ≈ 1.2599… In 19-TET, it’s 6 steps, or 2^(6/19) ≈ 1.2447… It’s lower, but it’s much closer to the in-tune value! As a consequence, major and minor chords sound really, really nice in 19-TET, even while dominant 7ths don’t so much because the leading tone is too low and the seventh is too high.
Oh, yeah. I’ve got recordings.
Can’t hear the difference so well? Fear not; since every measure of this Prelude is divided into two equal parts (except the last three bars, but we’ll ignore those), I made the first half 12-TET and the second in 19-TET:
So that you can get a feel for the sound of the intervals, I took away the rhythm and made just the chords, in 19-TET:
And because a Prelude is not much without a Fugue, here are some more:
These were all made in Sibelius, the 19-TET recordings using the Offtonic 19-TET plugin. The irony of using Bach’s Well-Tempered Clavier to demonstrate a different equal temperament should not be lost.
19-TET can be very similar to 12-TET, and this is very interesting, because it’s not true of many other temperaments. Many aspects of 12-TET do not exist in 19-TET, however: since enharmonics are out, many interesting situations caused by ambiguities between enharmonic equivalents are no longer possible. On the other hand, there are certain to be new and different possibilities in 19-TET to be explored, and that exploration is much easier now that there’s a plugin. Since 19 is prime, the symmetric “scales” possible in 12-TET — the whole tone scale, the diminished chord, the augmented chord — are no longer symmetrical, meaning that a sequence of major thirds or minor thirds — or any other interval — will go on and on until every single note is hit.
I don’t yet know what can be done with 19-TET. If you know, tell me in the comments. But I do intend to find out!