19-tone equal temperament is a magic temperament. However, magic temperaments can get a good deal more accurate than 19-equal. So, musicians currently working with 19-tone music might be interested in magic temperament as a bridge between equal temperament and just intonation. On this page, I'll explain how to do that.

The Pengcheng Scale

Magic temperaments include a 19 tone scale that's usually fairly evenly spaced. I call this scale "pengcheng". Because it's consistent with 19-equal it can be uniquely notated in conventional notation. With the mapping of note names I used in the Tripod Notation article article, it looks like this on a 5-limit lattice:

.        A♯----E♯----B♯----Fx
        / \   / \   / \   /
       /   \ /   \ /   \ /
      F♯----C♯----G♯----D♯
       \   / \   / \   /
        \ /   \ /   \ /
         A-----E-----B
        / \   / \   / \
       /   \ /   \ /   \
      F-----C-----G-----D
     / \   / \   / \   /
    /   \ /   \ /   \ /
   D♭----A♭----E♭----B♭
  / \   / \   / \   /
 /   \ /   \ /   \ /
B♭♭---F♭----C♭----G♭

Pitches on the top and bottom rows of that diagram are duplicated. So, E♯ is the same as F♭, B♯ is the same as C♭, and Fx is the same as G♭. These really are the same in any magic temperament, and you can see that all 19 pitches are connected by a single chain of major thirds.

The names are chosen so that you have a full C major key. If you move by fifths, you'll eventually fall of the edge, which means that some chords will end up more out of tune than in equal temperament. However, chords you play within the lattice will be more in tune.

7-Limit Harmony

4 steps of 19 could be an approximation of either 8:7 or 7:6. In equal temperament, neither of these approximations is very good because the interval between them (49:48) has to be tempered out.

On the pengcheng scale, each interval of 4 scale steps is a good approximation of either 8:7 or 7:6. There are 12 approximations of 7:6 and 7 approximations of 8:7 in the scale. The approximations of 8:7 are: B♭–C♯, B–D♭, D–F♭ (or D–E♯), D♯–F, E♭–F♯, G♭–A (or Fx–A), and G–A♯. The higher note of each 8:7 approximation is the root of a full 7-limit tetrad.

9-Limit Harmony

In fact, wherever you have a 7-limit tetrad, you can extend it to a 9-limit pentad. That means roots of A, A♯, C♯, D♭, E♯/F♭, F, and F♯. These are all on a chain of major thirds from F♯ to E♯. Magic temperament will naturally emphasize major thirds.

3 steps of 19 could be an approximation of either 10:9 or 9:8. Like 8:7 and 7:6, these intervals can be approximated better in magic than 19 tone equal temperament, but each 3 steps of the pengcheng scale can only approximate one of these intervals. You get about as many of each, and you can recognize the 9:8 approximations because they're two horizontal steps on the lattice.

Because a 3 step whole tone is always a 9-limit interval, so is a 16 step minor seventh. So, you can take any major triad and add either a 9:5 or 16:9 to it, but you never get to choose which as long as you keep to the pengcheng scale. Whenever you have a 7:4 subminor seventh (or augmented sixth), you also have a 9:5 minor seventh on the same root. As a result, you never get a choice of a 16:9 minor seventh or a 7:4 subminor seventh.

Tuning Tables

If you're used to 19-equal, you'll probably want to know how the pengcheng scale deviates from it. The standard units for measuring small pitch differences are cents, based on 1200 divisions of the octave. That explains the mess of units in this table. It also shows each scale degree in terms of the larger "changan" scale with 41 notes, which happens to be a good magic tuning when equally tempered.

The last column shows a ratio that the magic interval (relative to C) approximates. These ratios are not tuned exactly. In some cases (like 6:5), they are tuned worse than in equal temperament.

There are many ways of tuning magic temperaments. These numbers follow the pure octaves Tenney optimal (POTE) method.

Scala file for the optimal pengcheng scale. Use it with your favorite 19 key octave.

! pengcheng_te.scl
19 notes of TE magic
19
58.997
117.994
203.704
262.701
321.698
380.695
439.692
498.689
557.686
643.396
702.393
761.390
820.387
879.384
938.381
1024.091
1083.088
1142.085
1201.082

Adjusted for pure octaves for cases where that's easier.

! pengcheng_pote.scl
19 notes of pure octaves TE magic
19
58.944
117.888
203.520
262.464
321.408
380.352
439.296
498.240
557.184
642.816
701.760
760.704
819.648
878.592
937.536
1023.168
1082.112
1141.056
1200.000

Haizhou Extensions

If you run out of notes, you can add more pitches to the chain of thirds. The next moments of symmetry (MOS) scale is the 22 note haizhou. As a 19-tone musician, you probably won't bother with this. After all, why use a bigger scale than you were happy with, all in the name of accuracy? That's why I left this part until the end. You can skip it if you like, and come back if you ever decide you need a few more notes.

Because all spellings map to the pengcheng scale, you need some form of extended notation to name the new pitches. I'll use tricycle notation here. The idea is that you split the octave into three wheels and each pitch sits on one of the wheels. The haizhou scale then looks like this:

.         A♯----E♯----B♯----Fx
         / \   / \   / \   /
        /   \ /   \ /   \ /
      F♯2----C♯----G♯----D♯
      / \   / \   / \   / \
     /   \ /   \ /   \ /   \
    D1----A-----E-----B----F♯3
   / \   / \   / \   / \   /
  /   \ /   \ /   \ /   \ /
B♭3----F-----C-----G-----D2
  \   / \   / \   / \   /
   \ /   \ /   \ /   \ /
    D♭----A♭----E♭---B♭1
     \   / \   / \   /
      \ /   \ /   \ /
       F♭----C♭----G♭

In 5-limit just intonation, D2 is a syntonic comma (81:80 or 21½ cents) sharper than D1. In magic temperaments, it is can be a bit larger: around 29 cents for 41 note equal temperament. F♯3 is sharp of F♯2 by the same amount, and B♭1 is sharp of B♭3 because the wheels revolve so that 1 is above 3.

There's no need to specify which wheels the other pitches sit on because there's no ambiguity. However, you can work them out easily enough by knowing that the three notes of a major or minor triad sit on three different wheels.

Haizhou has only three more notes than pengcheng, so it will sound mostly the same. It means you have a few places to handle pitches that would be the same in 19-equal, but different in magic temperament. This includes the white-note diatonic with two tunings of D that you need in just intonation.

Here's how to tune it:

Magic temperament page

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