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.

Enter 19-TET.

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.

Prelude 1 in C, Well-Tempered Clavier Book I, J. S. Bach, in 12-TET

Prelude 1 in C, Well-Tempered Clavier Book I, J. S. Bach, in 19-TET

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:

Prelude 1 in C, Well-Tempered Clavier Book I, J. S. Bach, half in 12-TET and half 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:

Prelude 1 in C, Well-Tempered Clavier Book I, J. S. Bach, in 19-TET, just the chords

And because a Prelude is not much without a Fugue, here are some more:

Fugue 1 in C, Well-Tempered Clavier Book I, J. S. Bach, in 12-TET

Fugue 1 in C, Well-Tempered Clavier Book I, J. S. Bach, in 19-TET

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!

## 2 Comments

Dear Mauro,

Thanks for you page featuring Bach in one 19-ET tuning (one of 12 plausible close-matches to 12-ET, some of them better matches than others; see below…)

Music written in 12 equally tempered pitches can be translated to a system of 19 equally tempered tones in 12 different ways, analogous to our 7 modes.

In other words: a musical piece written in 7 equally tempered tones could be translated to our 12-tone world in 7 different ways, corresponding to the 7 modes of our traditional music – each mode starting on a different white key of the piano. Similarly, a musical piece originally in 5 equally tempered tones could be rendered in 5 different ways on our 12-tone keyboard, each starting on a different black key.

So again, there are 12 plausible ways to match a 12-ET piece (one from Bach for instance) to a 19-ET system, each way starting on one of the twelve white keys on a 19-tone keyboard. The remaining seven black keys are A, B, C, D, E, F, G naturals. As you point out, the only enharmonic equivalents among the white keys turn out to be exactly those that are different pitches in 12-ET: B# = Cb and E# = Fb.

So have you experimented with the Bach WTC opening piece in any of these different modes?

Do you know of any programs that make this possible?

Thanks,

Damon Minvielle

Jefferson Parish, Louisiana

(would “five” work for the “one + 4 =” challenge?)

Hm, I’m not sure I’m following you. What’s the difference between starting on C versus starting on any other note? C is really arbitrary, but the relationships stay the same. The idea here is that if you translate the fifth of 12-ET, 2^(7/12), into a fifth in 19-ET, 2^(11/19), you can get a circle of fifths that maps easily to 12-ET provided you don’t mess around with enharmonics. Starting on C is really just a convenience; if we started on, say, F, we’d get exactly the same notes except that the whole thing would be shifted by some very small amount. It would be like picking A440 instead of A443 or something like that. The seven modes we have in our diatonic scale are different because the scale itself is not symmetric. We pick seven notes to be our major scale, right? The intervals between them aren’t always the same — D to E is different from E to F, for instance. So if you take the C major scale and start on C, you’ll get different intervals from if you start on F, so we call those different modes. In this case, we’re really translating a symmetric 12-tone scale to a symmetric 19-tone scale, so there’s no real difference.

On the other hand, if we were to take a 47-tone scale, there actually are two different “modes”, because there are two different fifths we can use. I’m not sure how that would sound, though — maybe I’ll investigate that at some point!

By the way, the “challenge” thing is a captcha I found on WordPress, so I don’t actually know how it works. It doesn’t seem to keep all the spam out, though…