music in the shape of a houndstooth, using the square root of 2 extensively
I wrote this piece by translating the houndstooth pattern into music in a couple different ways.
 It's close to 7:5, though, if that helps you feel it at all.
This is analagous to steps of the traditional Western pitch system.
Used in a single unified rhythm, this √2 beat may simply add a bit of shuffle to your step. However, the Houndstoothtopia Theme boasts two different underlying rhythms, so its voices quickly sort themselves into two separate rhythmic groups. In this type of situation, playing loosely will not cut it; the accuracy of the interval must be respected or the piece would degenerate into sludge.
Because √2 is irrational, the only way for voices to realign after such dis-alignment would be to eventually have moved by the same count of √2's. Fortunately, the Houndstoothtopia Theme's two different underlying rhythms are designed to have the same count of √2's, so they do realign after each segment of the piece. And in the middle of each segment we get to enjoy some wacky staggering, creating new meta-intervals such as √2 - 1.
And now for a discussion of the two different underlying rhythms of the piece.
Here is the supertile rhythm:
short, long,
short, short, long,
short, short, long, long,
short, long, long,
where each short note has a duration of 1 and each long note has a duration of √2.
In order to answer the question: "Why is it called the Supertile Rhythm?" we must first understand what the supertile is.
The houndstooth pattern can be thought of as a repetition of four basic square tiles: a solid black tile, a solid white tile, and two different tiles which are both diagonally striped black and white but one of them begins with a black stripe and the other begins with a white stripe.
(diagram showing these)
A supertile is a group of connected tiles which function together like a tile. In the case of the houndstooth pattern, the simplest supertile is a 2x2 square containing one each of the four aforementioned basic tiles.
(supertile diagram, which is just the highlighted from the previous blown up)
And now, how to get from this supertile to the above rhythm?
Think of each tile as a series of diagonal stripes, even if it is solid. In the case of the white tile, for example, you could think of it as four white stripes - you can't tell that they're stripes because they're all the same color. (For these purposes, we ignore the fact that the first and last stripes, being adjacent to the corners of the tile, are clipped so that one side is reduced to a point causing them to be triangles, lacking parallel lines and thus not quite feeling like stripes.)
For each tile of the supertile, find the minimum number of stripe colors necessary to describe the tile in terms of the repeating pattern starting from the top left. For solid tiles this would be 1. For striped tiles it is 2.
(image)
Take the maximum of those minima, and use that as your standard. In other words, you are now describing your supertile like this:
[ [ white, white ], [ white, black ] ], [ [ black, white ], [ black, black ] ]
Now create a cycle by choosing elements counter-clockwise around that nested array, like this:
[ [ white, white ], [ black, white ], [ black, black ], [ white, black ], ]
Now substitute in 1 for white, 2 for black.
[ [ 1, 1 ], [ 2, 1 ], [ 2, 2 ], [ 1, 2 ], ]
Now, map [ short, long ] over the sequence, meaning that if short matches up with a 1, use 2 short beats:
[ [ [ short ], [ long ] ], [ [ short, short ], [ long ] ], [ [ short, short ], [ long, long ] ], [ [ short ], [ long, long ] ], ]
And that's it! Once we flatten completely / clean up all the nested array structure, we have our supertile rhythm:
short, long, short, short, long, short, short, long, long, short, long, long,
The end result I hear more readily as beats in clusters of 2, 3, 3, 1, 2, and 1. But under the hood this is how it works.
Here is the perimeter rhythm:
short, long, short, short, long, short, double-long,
short, long, short, short, long, short, double-long,
where a short beat has duration 1, a long has duration √2, and a double-long has duration 2√2.
(Oh yeah, I stretched the truth a bit earlier; there are three durations in the Houndstoothtopia Theme: 1,√2, and 2√2).
This is arrived at by starting on one vertex of the houndstooth shape, tracing its perimeter edge by edge, and recording their lengths.
Interestingly, the perimeter rhythm begins similarly to the supertile rhythm, despite being generated in a very different way.
(show them overlaid)
The perimeter rhythm also has harmonic content, however, and this is where it gets a bit complicated. For each note in the perimeter rhythm, attribute to it a pitch scaled by the height of the vertex.
You can rotate the houndstooth perimeter, and while the edge lengths won't change and thus the rhythm won't change, the heights of the vertices will. The Houndstoothtopia them uses four different rotations of the houndsooth.
(show them)
The perimeter rhythm is longer than the supertile rhythm, in a ratio of 4:3.
Okay, so again, I lied a bit earlier. The supertile rhythm and the perimeter rhythm do not have the exact same number of √2 beats in them. They just have a simple ratio of them to each other. This way, after repeating 3 perimeter rhythms and 4 supertile rhythms, they've realigned.
Weirdly, though, the perimeter rhythm does not hypermetrically organize in groups of 3. It goes by groups of 4. This way, the supertile rhythm keeps sliding into different alignments with the hypermetrical structure of the perimeter rhythm.
What is that hypermetrical structure? First, recognize four perimeter rhythm voices. Every one plays the perimeter rhythm, but each one plays a different rotation of it, thus contributing different pitches to the chords they make together.
Repeat the perimeter rhythm four times for each of the four rotations of the perimeter rhythm: - first repetition: no voice sounds. - second repetition: just the voice for the current rotation sounds. - third repetition: all the voices sound. - fourth repetition: all the voices except the one for the current rotation sound.
A parallelism exists between this hypermetrical structure for the perimeter rhythm voices and the supertile rhythm. Imagine if starting from this point:
[ [ white, white ], [ black, white ], [ black, black ], [ white, black ], ]
You instead substituted in "sound" and "no sound":
[ [ no sound, no sound ], [ sound, no sound ], [ sound, sound ], [ no sound, sound ], ]
And then instead of [ short, long ] you mapped over that [ current, others ]:
[ [ current - no sound, others - no sound ], [ current - sound, others - no sound ], [ current - sound, others - sound ], [ current - no sound, others - sound ], ]


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light - light handle
inner_angle# - inner angle of cone outer_angle# - outer angle of cone |
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Sets the 'cone' angle for a 'spot' light.
The default light cone angles setting is 0,90. See also: CreateLight, LightRange, LightColor. |
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Graphics3D 640,480
camera = CreateCamera() MoveEntity camera,0,0,-40 flat = CreatePlane(10) TurnEntity flat,-90,0,0 lite = CreateLight(3) ; try different lights 1 to 3 MoveEntity lite,0,0,-15 While Not KeyDown(1) RenderWorld:Flip If KeyHit(57) Then ; press SPACEBAR to randomly change the 'cone' of light LightConeAngles lite, Rand(120),Rand(120) EndIf Wend End |


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