Yer
Introduction
Yer is an octave-reduced Euler-Fokker genus of 11, 13, 17, and 19. As such it is a 2.11.13.17.19 limit just intonation tuning (ignoring the 3rd, 5th, and 7th harmonics) which is octave-repeating.
| name | frequency multiplier (definition) | frequency multiplier (decimal) | pitch (¢) |
|---|---|---|---|
| 1 | 1/1 | 1 | 0 |
| 13 * 17 * 19 | 4199/4096 | 1.025146484 | 42.9960874 |
| 17 | 17/16 | 1.0625 | 104.9554095 |
| 11 * 13 | 143/128 | 1.1171875 | 191.8456041 |
| 11 * 13 * 17 | 2431/2048 | 1.187011719 | 296.8010136 |
| 19 | 19/16 | 1.1875 | 297.5130161 |
| 17 * 19 | 323/256 | 1.26171875 | 402.4684256 |
| 11 * 13 * 19 | 2717/2048 | 1.326660156 | 489.3586203 |
| 11 | 11/8 | 1.375 | 551.3179424 |
| 11 * 13 * 17 * 19 | 46189/32768 | 1.409576416 | 594.3140298 |
| 11 * 17 | 187/128 | 1.4609375 | 656.2733519 |
| 13 | 13/8 | 1.625 | 840.5276618 |
| 11 * 19 | 209/128 | 1.6328125 | 848.8309585 |
| 13 * 17 | 221/128 | 1.7265625 | 945.4830713 |
| 11 * 17 * 19 | 3553/2048 | 1.734863281 | 953.786368 |
| 13 * 19 | 247/128 | 1.9296875 | 1138.040678 |
Throughout this post, the same four colors will be consistently associated with the four harmonics: green is 11, orange is 13, cyan is 17, and magenta is 19.
It has a minor third of 19:16 and a minor second 17:16.
EFG as combination of CPS's
Every pitch in the system is a combination of either zero, one, two, three, or four of 11, 13, 17, or 19, for a total of 1 + 4 + 6 + 4 + 1 = 16 pitches. In other words it is the powerset of {11, 13, 17, 19}. If familiar with Erv Wilson’s Combination Product Sets, or CPS's, it may help to think of an Euler-Fokker Genus as the intersection of the choose-1 tetrany, the choose-2 hexany, and the choose-3 tetrany, plus the unison, plus 11 * 13 * 17 * 19 all together which may be called the aota, an acronym for “all of the above”.
As an EFG, it has no subharmonic factors, that is, no division - only multiplication of them together, which leads to it being ambitonal, not otonal or utonal leaning. While it is nicely “balanced/symmetrical” as all CPSs are, pitches in the resultant system are not evenly distributed. There are two massive gaps of 185 cents, and two places where pitches land almost directly on top of each other.
Blumeyer comma
Those pitches right on top of each other are a feature, not a bug. The pitch 11 * 13 * 17 is 2431, only one off from the pitch 19, which when octave adjusted is 2432. So we end up with this superparticular ratio, 2432:2431, called the Blumeyer comma.
| name | value | cents | monzo |
|---|---|---|---|
| Blumeyer comma | 2432/2431 | 0.71200249782 | | 7 0 0 0 -1 -1 -1 1 > |
Lattice Play
Since the EFG system of Yer includes both 11 * 13 * 17 and 19 as pitches, that means the comma exists as an interval in the scale. Since it’s so tiny, though, the two notes related by it can hardly be treated as separate. This means that when you go to draw this tuning out as a JI lattice, you can do something you wouldn’t normally do, which is set a couple points right together.
You could think of this lattice as a pair of cubes. One is an Euler-Fokker genus of [11, 13, 17]. The other is that same Euler-Fokker genus, just with every node multiplied by 19. That’s why every point in the second cube has the same set of circles colored in as the analogous one in the other cube, just with the magenta 19 filled in as well.
So every point in the first cube is connected to the analogous node in the second cube. Normally the node for 19 would not have any direct connection with the node 11, 13, 17. It only directly connects with one node in the other cube, its analogous one, the unison. But here we see that not only is there another effect going on connecting these two nodes, that effect goes beyond connecting them, it straight up conflates them.
But that's not all; recall that there were two other pairs of pitches that were almost the same, too. One of those pairs has such simple sounding members, it may surprise you: 13 and 11 * 19. In this view, we can see that they are nowhere near each other. So we’ll have to nudge our lattice around a little bit more; that's how we arrive at the final lattice shown in the introduction. There, the three pairs of intervals in the center connected by short solid black lines can be modulated between almost unnoticed.
Blume comma
The reason why 13 and 11 * 19 are so close is that it turns out there is a *another* comma existing in this world of 11’s, 13’s, 17’s, and 19’s — not nearly as exciting or colorful of one, but it’s there. If we move by an eleventh, a seventeenth, and another eleventh, we end up right back where you started, off by 7.591 cents. 13 and 11 * 19 are off from each other by an amount of one Blumeyer comma and one Blume comma. These two commas are just both so small that it doesn’t much matter. 7.591 + 0.712 is still just 8.303 cents.
| name | value | cents | monzo |
|---|---|---|---|
| Blume comma | 2057/2048 | 7.59129422992 | | -11 0 0 0 2 0 1 > |
Another consequence of the Blume comma is that a couple “extra” lattice connections appear. These are drawn in dotted green on the lattices in the introduction.
The idea is that if you were to try to go from 11 * 17 by an eleventh to 11 * 11 * 17, well, you’re not really allowed to do that because the EFG does not duplicate factors (you can’t have two 11’s), but since 11 * 11 * 17 is essentially 1, we’ll permit it.
Analogously, you can move from 11 * 13 * 17 * 19 by an eleventh to 11 * 11 * 13 * 17 * 19, or just 13 * 19. Though, for symmetry, we actually do 13 * 19 / 11 rather than 11 * 11 * 13 * 17 * 19.
And as long as we're changing the angle we look on the cube to bring the right pairs of pitches together, we have also taken care to balance these dotted lines with the real 11 lines, so that it’s the zig to the zag of the real 11, more strongly suggesting the dimension of the 11th harmonic’s relationship to that of the 17th (i.e that two 11's make a 17).
Now we could have drawn dotted lines connecting 11 * 13 * 17 to 13, but declined, considering that the ability to move between these two pitches is already achievable by modulating from from 11 * 13 * 17 to 19, then moving by that 11 to 11 * 19, then modulating to 13. The same goes for the connection between 11 * 17 * 19 and 19. This would be pretty obvious on the cycle view, because we’d just be drawing a dotted line right alongside an existing solid one.
Here’s something else interesting: we can move by four 11’s in a row. We can move from 17 * 19 to 11 * 17 * 19, which can be shifted to 13 * 17, then move to 11 * 13 * 17, which can be shifted to 19, then move to 11 * 19, which can be shifted to 13, then move to 11 * 13.
You can also move by 3 11’s in a row on the left and right edges; for the other factors, you can only ever move by two of them in a row (in sad 17’s case, only one spot where you can do that, which happens to be in parallel to the spot where you can 4x by 11).
Comma pumps
Yer is pure JI, but due to the five places where it boasts two pitches very close together but with very different harmonic compositions, it can achieve zero comma pumps by “comma shifting” at those key points, returning to exactly their original pitch.
In Yer, there are three classes of pumps of the Blumeyer comma. One way of defining them is by which harmonic is opposite the 19. So in class a, it’s the 11, in b it’s the 13, and in c it’s the 17.
This diagram shows real instances of these pumps in Yer and then rotates each of these three diagrams by a different amount so that the 19 would be horizontal across the top in each one, in order to better bring out their differences in shape.
Scala file
! blumeyer_ji.scl ! Blumeyer JI scale, combination of CPS's of 11, 13, 17, 19 16 ! 4199/4096 17/16 143/128 2431/2048 19/16 323/256 2717/2048 11/8 46189/32768 187/128 13/8 209/128 221/128 3553/2048 247/128 2/1
Video explanation
More details are presented in this video: Yer (pitch system)
Listening
Douglas Blumeyer - Tsraxcfaubdj