Yer: Difference between revisions

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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.

[[File:Yer comma pump families.png|none|thumb|Yer comma pump families]]

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.

Each of these families has a mirror image. You can achieve it by either reversing the order you move by those harmonics, or flipping the harmonic directions you move (from harmonic to subharmonic, or vice versa). I call a reversal like this a “counter”, as in “counterclockwise”, (because in this case we are reversing a cycle, so counter is appropriate, and there is another place soon where I need the word “reverse” and I didn’t want to use “retrograde”) and I call a flipping like this an inversion, as in inverting a fraction’s numerator and denominator, which it is.

That is you could counter:

up 19, down 17, down 13, down 11

to

down 11, down 13, down 17, up 19

Or you could invert:

up 19, down 17, down 13, down 11

to

down 19, up 17, up 13, up 11.

The difference between a counter and an invert is only in which direction you flow through the comma; the shape is the same between them, i.e. the mirror image of the original. So if you do *both*, you stay in the same position/shape, but the flow through your shape is in the opposite direction. That’s “reverse”.

[[File:Yer comma pump operations - invert, reverse.png|none|thumb|

Yer comma pump operations

]]

There’s another operation which does not change the shape, and that’s “cycling” which is changing which of the four notes you start on.

[[File:Yer comma pump operations - cycle.png|none|thumb|

Yer comma pump operations - cycle

]]

Finally there are several possible rotations of each comma within Yer, a rotation being a changing of the position within the octave circle. There are five each, to be exact: one for where each of its four corners touches the Blumeyer comma interval, plus one which uses the Blume commas.

[[File:Yer comma pump operations - rotate.png|none|thumb|

Yer comma pump operations - rotate

]]

So with three comma pump families, each with four total forms (base, inverse, converse, and reverse), each of those with four possible starting positions or cyclings, and each of those with five possible rotations, Yer boasts 3 * 4 * 4 * 5 = 240 total possible comma pumps.

''I haven’t much explored the Blume comma’s pumps, but those would be much simpler. Basically for each 17 interval, make a long skinny triangle across the circle out of its two 11s.''

So you could write music with these comma pumps. The first piece I’ve written with Yer, in fact, uses three variations on one comma pump family, interrupting each other to create a sort of “meta-pump”. That’s a horrible term, though, don’t use that. Because they’re not really full pumps anymore, they’re partial ones. I’ll think of something later.

== Another Lens ==

There’s another lens I’d like to look at Yer through. This will take a few steps. As I mentioned before, you can think of our EFG as two EFGs of the next smallest order, with each node in one connected to the analogous node of the other. EFGs build on each other like that. And in fact, of this order 4 EFG, you could pick any other of our four harmonics, not just 19 works this way, split the lattice on it, and you’ll end up with a pair of cubes. Here I’ve shown all of these, just without the bonds of the harmonics dividing them.

[[File:Yer - paired cube faces of tesseract.png|none|thumb|

Yer - paired cube faces of tesseract

]]

Considered as a whole, these are the 8 cubic faces of the tesseract, or hypercube, of our order 4 EFG.

But 19 still is the more interesting one. First, look at the lattice view on the left, to get a sense of the blue cube, the base EFG, and the red cube, which represents the EFG with every node multiplied by 19. Next, look at the cycle view, and see how that coloring transfers there.

[[File:Yer - two cubes off by 19.png|none|thumb|

Yer - two cubes off by 19

]]

Now suppose we take the red cube and the blue cube, and for each of them instead color code for their constituent harmonic intervals. The blue becomes this on the left, the red becomes that on the right. Now it becomes clear how rotating the blue cube around the cycle by a 19th brings those three pairs of pitches together.

[[File:Yer - two cubes separated by 19, relationship.png|none|thumb|

Yer - two cubes separated by 19, relationship

]]

Here’s an extreme subtlety, but worth mentioning I think in case it confuses anyone else as bad as it did me. It’s really easy to look at these two diagrams and think “mirror image” (across the vertical axis), but that’s deceptive. The unison and the aota are *not* connected by a diameter here. They are not on opposite sides of the octave circle. That is because a movement by all intervals is not 600 cents, but only 594. Remember that the 19th harmonic is 298 cents, and the other three harmonics in the system together are almost exactly the same. So if you wanted to orient Yer so that its line of symmetry is perfectly vertical (which I submit makes a good amount of sense) Then you must also stay aware that the unsion and aota are not at the 180 degree and 0 degree positions, but more like 179.5 and 0.5 degree positions.

[[File:Yer - EFG nodes.png|none|thumb|

Yer - EFG nodes

]]

We can also delve a little deeper into Yer in terms of its constituent CPS’s. In the lattice view on the left, you can see how the unison and aota are opposite extreme corners, and their immediately adjacent nodes are all tetrany members — choose 1 wrapping unison, choose 3 wrapping the aota — and all nodes two moves away from them are in the hexany. In other words, no node of a given color touches any other nodes of its own color. Only green touches yellow, only blue touches brown, and red buffers between green and blue. I could have drawn these diagrams with the black lines replaced/augmented with dotted lines or arrows of color indicating the two types they each bridge.

Then here’s what we get when we actually render the CPSs and their connections within themselves. You’ll notice that these lines are completely different from the black ones. The black ones connect pitches separated by single harmonic factors. The nature of CPSs, though, is that pitches are connected when they are separated by '''two''' harmonic factors. For example, in the choose-1 tetrany, the members are just the single harmonics 11, 13, 17, 19. So 11 and 17 are related not by a single harmonic factor, but by 11:17 or 17:11; two factors.

[[File:EFG lines on cycle & lattice.png|none|thumb|

EFG lines on cycle & lattice

]]

In the tetrany, every node connects to every other node, But in the hexany that is not the case. Remember the hexany is our choose-2 set, for consider for example the pair of nodes 11*13, and 17*19. They do not share a factor, so they are not connected. The pair of nodes 11*13 and 11*19, however, share an 11, so the 11’s cancel out, and again you have a relationship based on two factors, in this case 13:19. This view is terribly busy, though. Too much information. But if you pull away the black lines for single harmonic relationships, which we already talked about a lot with respect to commas and pumps thereof...

[[File:EFG as lattice & cycle.png|none|thumb|

EFG as lattice & cycle

]]

...you can actually begin to see stuff. In the lattice view on the left, notice how the tetranies take form as tetrahedrons and the hexany as an octahedron. In the cyclical view on the right, you can see how the choose-1 and choose-3 tetranies, the blue and green shapes, are reflections of each other across a vertical line drawn down this circle, as well as how the choose 2 hexany is symmetrical across the same line, and the remaining two notes - the unison and the aota - are located symmetrically to each other. You could write music with just these CPSs.

== Scala file ==

Line 228:

Line 300:

[http://chrisvaisvil.com/now-yer-talkin-ji-piano Chris Vaisvil - Now Yer Talkin']

== See also ==

[https://musical-patterns.douglasblumeyer.com Musical Patterns] - select Tsraxcfaubdj (GitHub page is here to explore the programmatic reasoning behind how this piece uses Yer: https://github.com/MusicalPatterns/pattern-tsraxcfaubdj)

@@ Line 188: / Line 188: @@
 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.
+[[File:Yer comma pump families.png|none|thumb|Yer comma pump families]]
 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.
+Each of these families has a mirror image. You can achieve it by either reversing the order you move by those harmonics, or flipping the harmonic directions you move (from harmonic to subharmonic, or vice versa). I call a reversal like this a “counter”, as in “counterclockwise”, (because in this case we are reversing a cycle, so counter is appropriate, and there is another place soon where I need the word “reverse” and I didn’t want to use “retrograde”) and I call a flipping like this an inversion, as in inverting a fraction’s numerator and denominator, which it is.
+That is you could counter:
+up 19, down 17, down 13, down 11
+to
+down 11, down 13, down 17, up 19
+Or you could invert:
+up 19, down 17, down 13, down 11
+to
+down 19, up 17, up 13, up 11.
+The difference between a counter and an invert is only in which direction you flow through the comma; the shape is the same between them, i.e. the mirror image of the original. So if you do *both*, you stay in the same position/shape, but the flow through your shape is in the opposite direction. That’s “reverse”.
+[[File:Yer comma pump operations - invert, reverse.png|none|thumb|
+Yer comma pump operations
+]]
+There’s another operation which does not change the shape, and that’s “cycling” which is changing which of the four notes you start on.
+[[File:Yer comma pump operations - cycle.png|none|thumb|
+Yer comma pump operations - cycle
+]]
+Finally there are several possible rotations of each comma within Yer, a rotation being a changing of the position within the octave circle. There are five each, to be exact: one for where each of its four corners touches the Blumeyer comma interval, plus one which uses the Blume commas.
+[[File:Yer comma pump operations - rotate.png|none|thumb|
+Yer comma pump operations - rotate
+]]
+So with three comma pump families, each with four total forms (base, inverse, converse, and reverse), each of those with four possible starting positions or cyclings, and each of those with five possible rotations, Yer boasts 3 * 4 * 4 * 5 = 240 total possible comma pumps.
+''I haven’t much explored the Blume comma’s pumps, but those would be much simpler. Basically for each 17 interval, make a long skinny triangle across the circle out of its two 11s.''
+So you could write music with these comma pumps. The first piece I’ve written with Yer, in fact, uses three variations on one comma pump family, interrupting each other to create a sort of “meta-pump”. That’s a horrible term, though, don’t use that. Because they’re not really full pumps anymore, they’re partial ones. I’ll think of something later.
+== Another Lens ==
+There’s another lens I’d like to look at Yer through. This will take a few steps. As I mentioned before, you can think of our EFG as two EFGs of the next smallest order, with each node in one connected to the analogous node of the other. EFGs build on each other like that. And in fact, of this order 4 EFG, you could pick any other of our four harmonics, not just 19 works this way, split the lattice on it, and you’ll end up with a pair of cubes. Here I’ve shown all of these, just without the bonds of the harmonics dividing them.
+[[File:Yer - paired cube faces of tesseract.png|none|thumb|
+Yer - paired cube faces of tesseract
+]]
+Considered as a whole, these are the 8 cubic faces of the tesseract, or hypercube, of our order 4 EFG.
+But 19 still is the more interesting one. First, look at the lattice view on the left, to get a sense of the blue cube, the base EFG, and the red cube, which represents the EFG with every node multiplied by 19. Next, look at the cycle view, and see how that coloring transfers there.
+[[File:Yer - two cubes off by 19.png|none|thumb|
+Yer - two cubes off by 19
+]]
+Now suppose we take the red cube and the blue cube, and for each of them instead color code for their constituent harmonic intervals. The blue becomes this on the left, the red becomes that on the right. Now it becomes clear how rotating the blue cube around the cycle by a 19th brings those three pairs of pitches together.
+[[File:Yer - two cubes separated by 19, relationship.png|none|thumb|
+Yer - two cubes separated by 19, relationship
+]]
+Here’s an extreme subtlety, but worth mentioning I think in case it confuses anyone else as bad as it did me. It’s really easy to look at these two diagrams and think “mirror image” (across the vertical axis), but that’s deceptive. The unison and the aota are *not* connected by a diameter here. They are not on opposite sides of the octave circle. That is because a movement by all intervals is not 600 cents, but only 594. Remember that the 19th harmonic is 298 cents, and the other three harmonics in the system together are almost exactly the same. So if you wanted to orient Yer so that its line of symmetry is perfectly vertical (which I submit makes a good amount of sense) Then you must also stay aware that the unsion and aota are not at the 180 degree and 0 degree positions, but more like 179.5 and 0.5 degree positions.
+[[File:Yer - EFG nodes.png|none|thumb|
+Yer - EFG nodes
+]]
+We can also delve a little deeper into Yer in terms of its constituent CPS’s. In the lattice view on the left, you can see how the unison and aota are opposite extreme corners, and their immediately adjacent nodes are all tetrany members — choose 1 wrapping unison, choose 3 wrapping the aota — and all nodes two moves away from them are in the hexany. In other words, no node of a given color touches any other nodes of its own color. Only green touches yellow, only blue touches brown, and red buffers between green and blue. I could have drawn these diagrams with the black lines replaced/augmented with dotted lines or arrows of color indicating the two types they each bridge.
+Then here’s what we get when we actually render the CPSs and their connections within themselves. You’ll notice that these lines are completely different from the black ones. The black ones connect pitches separated by single harmonic factors. The nature of CPSs, though, is that pitches are connected when they are separated by '''two''' harmonic factors. For example, in the choose-1 tetrany, the members are just the single harmonics 11, 13, 17, 19. So 11 and 17 are related not by a single harmonic factor, but by 11:17 or 17:11; two factors.
+[[File:EFG lines on cycle & lattice.png|none|thumb|
+EFG lines on cycle & lattice
+]]
+In the tetrany, every node connects to every other node, But in the hexany that is not the case. Remember the hexany is our choose-2 set, for consider for example the pair of nodes 11*13, and 17*19. They do not share a factor, so they are not connected. The pair of nodes 11*13 and 11*19, however, share an 11, so the 11’s cancel out, and again you have a relationship based on two factors, in this case 13:19. This view is terribly busy, though. Too much information. But if you pull away the black lines for single harmonic relationships, which we already talked about a lot with respect to commas and pumps thereof...
+[[File:EFG as lattice & cycle.png|none|thumb|
+EFG as lattice & cycle
+]]
+...you can actually begin to see stuff. In the lattice view on the left, notice how the tetranies take form as tetrahedrons and the hexany as an octahedron. In the cyclical view on the right, you can see how the choose-1 and choose-3 tetranies, the blue and green shapes, are reflections of each other across a vertical line drawn down this circle, as well as how the choose 2 hexany is symmetrical across the same line, and the remaining two notes -  the unison and the aota - are located symmetrically to each other. You could write music with just these CPSs.
 == Scala file ==
@@ Line 228: / Line 300: @@
 [http://chrisvaisvil.com/now-yer-talkin-ji-piano Chris Vaisvil - Now Yer Talkin']
+== See also ==
+[https://musical-patterns.douglasblumeyer.com Musical Patterns] - select Tsraxcfaubdj (GitHub page is here to explore the programmatic reasoning behind how this piece uses Yer: https://github.com/MusicalPatterns/pattern-tsraxcfaubdj)