____twoism

zeoevil wrote:How in the world did they think of that let alone implement it? Cheers, OP.

Thank you, sir or madam! And yes, it is truly incredible to imagine how someone could build something like that.

apriorion wrote:I'm sorry to be the only naysayer in here, but I can't see where you're coming from. I mean, I like Jaquard Causeway, and I'll be prepared to admit that there's interesting mathematical shit going on in there, but some things about your picture don't make any sense to me. For example, why is your picture going around in a circle? I get that BOC is all about looking at song structures in different ways (I remember an interview where one of them say something about branches spiraling off of a stem, or something like that), but given that we are finite temporal beings who are locked in a progression from past-to-present-to-future, songs are typically represented according to the flow of time, right? So I don't follow that aspect of your picture. I mean, I get why you picked hexagons as the monads in this diagram, since that's BOC's thing and all, but how it makes up a larger hexagon seems like a bit of a projection on your part more than anything inherent to the track. Also, you drew some colored lines in various patterns across the hexagons and proclaimed that the sums for each are 69 and 70, respectively. I can't see where you're coming from with that at all. If you could explain the steps in between to show how this analysis relates to the track I'm hearing, that would help me a bit. Because when I hear this track, I hear some very emotional melodies running through a delay/echo thingy, and a percussion that runs in a slightly weird time signature. I cant see how a cyclical hexagon pattern can be made from that. I would like to, though.

1. "Why is your picture going around in a circle?"

I think the most basic answer to your question is that loops are an integral part of music. But to be more specific, I'm using collections of monads, as you said, or patterns, to represent those musical loops. "Jacquard Causeway" has a constant 3/4 tempo and time-signature during the body of the song.

On the smallest level, a tiny hexagon of say "orange" or "blue" color, as you see it in the image, represents a single bar of the 3/4 time signature in "Jacquard Causeway". The rhythm of this song is 3/4. If you don't know what that means, I can't help you. But any music that occurs in 3/4 can be subdivided to contain 6 notes during a single bar. Because each count can have an "and". This is why I represent each bar as having 6 "nodes" as a regular hexagon: because that faithfully represents the [KICK; HAT; SNARE] rhythm when viewed on a doubly more intricate level of scale. (There's also a 6/4 delay I hear on the synth, but that's another story.)

2. "given that we are finite temporal beings who are locked in a progression from past-to-present-to-future, songs are typically represented according to the flow of time, right? So I don't follow that aspect of your picture.

Since when did forward time have to mean "new stuff" all the time? A hexagonal latticework makes sense to consider as a structural framework for this song because of the name of the song ("Jacquard Causeway"), in conjunction with the 3/4 time signature.

BoC make electronic music, often with patterns repeated in loops. And any shape that completes itself (such as a hexagon, square, n-gon) could be considered analogous to a certain sound loop. For example, you could say for a square, each side of the square represents 0.5 seconds, so it'd take 2.0 seconds to start the loop again.

So while a song might move horizontally with respect to absolute time; parts of the song can also move backward to an earlier part of memory or tape and play again. Not only was this possible with tape (probably what BoC use a lot of the time), but it's too easy with modern DAW programs.

3. I get why you picked hexagons as the monads in this diagram, since that's BOC's thing and all, but how it makes up a larger hexagon seems like a bit of a projection on your part more than anything inherent to the track."

I think your proposed problem is vague.

Is the problem that the atomic elements of my diagram (i.e., all the tiny orange hexagons ) have been cast as hexagons? These shapes could be replaced with any shape and/or color you like (pink floyd... ). But regarding the hexagonal latticework that embeds those simple-colored atoms, I did not arbitrarily pick that hexagonal latticework in order to fit the BoC motif. It's natural to subdivide the space into 6 parts with a 3/4 rhythm, for reasons already stated.

I will say, the song structure could be represented in different ways. But according to my observations, I hear 39 eighth notes in the intro and 39 groups of 6 bars of 3/4. The reason I posted this was because I found an interesting isomorphism between those two.

4. Also, you drew some colored lines in various patterns across the hexagons and proclaimed that the sums for each are 69 and 70, respectively. I can't see where you're coming from with that at all.

No, I said the sum of image 1 was 66; and that you can add 4 more to 66 to get 70 total.

The left image of my post that you're referring to demonstrates the four lowest-level patterns that the broadest-level pattern contains. Look at the left image. Notice how the most-inner hexagon is blue. Imagine that the blue-hexagon-line-loop is a barbeque skewer, holding skewered slices of juicy peppers. As you follow the blue hexagon loop around, you see the blue hexagon-line has 6 pepper slices on it, and you could infinitely tessellate its shape within the hexagonal latticework.

Now consider the orange-hexagon-line-loop. It has 12 pepper slices on it, and you could infinitely tessellate it. If you follow the Green and Red loops, you'll see that each of those paths are also infinitely tessellatable, and they contain 18 and 30 pepper slices, respectively.


6 + 12 + 18 + 30 = 36 + 30 = 66

The two shapes on the right image of my post refer to the final 3 & 1 that can be added to 66 to make a total of 70. There are 3 broader shapes which connect to make an infinitely-tesselatable hexagon: the three purple shapes. They could have been (probably should have been) looped-around by a purple line in my diagram. And the red line is the entire shape, a factor of itself. In other words, it is the broadest shape that could be infintely tesselatable, if given the correct boundary.


-=-=-=


I agree that the song has some very emotional qualities. I'm just pointing out a structural property. I have other observations about this song but it's not right to mention them at a post already this long. It's truly a great song and I hope my explanations made it clearer why I'm going on about it.

Thank you for taking the time to reply to my post. As I said, I don't want to be a naysayer, and I respect your effort. I still have some issues with the representation, which I will explain below.

JESUSANDHOES wrote:

1. "Why is your picture going around in a circle?"

I think the most basic answer to your question is that loops are an integral part of music. But to be more specific, I'm using collections of monads, as you said, or patterns, to represent those musical loops. "Jacquard Causeway" has a constant 3/4 tempo and time-signature during the body of the song.

This is where I'm having trouble still. I mean, I listen to a lot of music with repetitive patterns and loops. It seems that by this reasoning, then, I could represent a lot of my favorite tracks as a big hexagonal loop of little hexagon atoms. In particular, I'd imagine many of Aphex Twin's tracks from "I care because you do" (the album that changed my life back in 1995) will be representable in this fashion.

On the smallest level, a tiny hexagon of say "orange" or "blue" color, as you see it in the image, represents a single bar of the 3/4 time signature in "Jacquard Causeway". The rhythm of this song is 3/4. If you don't know what that means, I can't help you. But any music that occurs in 3/4 can be subdivided to contain 6 notes during a single bar. Because each count can have an "and". This is why I represent each bar as having 6 "nodes" as a regular hexagon: because that faithfully represents the [KICK; HAT; SNARE] rhythm when viewed on a doubly more intricate level of scale. (There's also a 6/4 delay I hear on the synth, but that's another story.)

I'd be very interested to hear your thoughts on that. I suspected that something about the delay was what caused my disorientation when listening to the track. Please, expand on this.

2. "given that we are finite temporal beings who are locked in a progression from past-to-present-to-future, songs are typically represented according to the flow of time, right? So I don't follow that aspect of your picture.

Since when did forward time have to mean "new stuff" all the time? A hexagonal latticework makes sense to consider as a structural framework for this song because of the name of the song ("Jacquard Causeway"), in conjunction with the 3/4 time signature.

BoC make electronic music, often with patterns repeated in loops. And any shape that completes itself (such as a hexagon, square, n-gon) could be considered analogous to a certain sound loop. For example, you could say for a square, each side of the square represents 0.5 seconds, so it'd take 2.0 seconds to start the loop again.

Right. Again, this is where I worry that your diagram doesn't quite get at what's unique about the track. Actually, it would help me quite a bit if you could contrast this BOC track with another track that has repetitive loops and overall sound structure. I'd imagine that some older Aphex tracks (like the ones on "I care...") or Autechre tracks (from "Tri Repetae", e.g.) might work well for this task.

So while a song might move horizontally with respect to absolute time; parts of the song can also move backward to an earlier part of memory or tape and play again. Not only was this possible with tape (probably what BoC use a lot of the time), but it's too easy with modern DAW programs.

3. I get why you picked hexagons as the monads in this diagram, since that's BOC's thing and all, but how it makes up a larger hexagon seems like a bit of a projection on your part more than anything inherent to the track."

I think your proposed problem is vague.

Is the problem that the atomic elements of my diagram (i.e., all the tiny orange hexagons ) have been cast as hexagons? These shapes could be replaced with any shape and/or color you like (pink floyd... ). But regarding the hexagonal latticework that embeds those simple-colored atoms, I did not arbitrarily pick that hexagonal latticework in order to fit the BoC motif. It's natural to subdivide the space into 6 parts with a 3/4 rhythm, for reasons already stated.

I will say, the song structure could be represented in different ways. But according to my observations, I hear 39 eighth notes in the intro and 39 groups of 6 bars of 3/4. The reason I posted this was because I found an interesting isomorphism between those two.

4. Also, you drew some colored lines in various patterns across the hexagons and proclaimed that the sums for each are 69 and 70, respectively. I can't see where you're coming from with that at all.

No, I said the sum of image 1 was 66; and that you can add 4 more to 66 to get 70 total.

Sorry, that was a typo on my part.

The left image of my post that you're referring to demonstrates the four lowest-level patterns that the broadest-level pattern contains. Look at the left image. Notice how the most-inner hexagon is blue. Imagine that the blue-hexagon-line-loop is a barbeque skewer, holding skewered slices of juicy peppers. As you follow the blue hexagon loop around, you see the blue hexagon-line has 6 pepper slices on it, and you could infinitely tessellate its shape within the hexagonal latticework.

Sorry, I'm a bit confused on this still, and I would like to understand why you ended up with this picture. Looking at the diagram again closely, you skewered some blue hexagons and not others. There are a lot of blue hexagon monads left out in the cold. Why did you chose to skewer the ones you did, and not the others?

Now consider the orange-hexagon-line-loop. It has 12 pepper slices on it, and you could infinitely tessellate it. If you follow the Green and Red loops, you'll see that each of those paths are also infinitely tessellatable, and they contain 18 and 30 pepper slices, respectively.


6 + 12 + 18 + 30 = 36 + 30 = 66

The two shapes on the right image of my post refer to the final 3 & 1 that can be added to 66 to make a total of 70. There are 3 broader shapes which connect to make an infinitely-tesselatable hexagon: the three purple shapes. They could have been (probably should have been) looped-around by a purple line in my diagram. And the red line is the entire shape, a factor of itself. In other words, it is the broadest shape that could be infintely tesselatable, if given the correct boundary.


-=-=-=


I agree that the song has some very emotional qualities. I'm just pointing out a structural property. I have other observations about this song but it's not right to mention them at a post already this long. It's truly a great song and I hope my explanations made it clearer why I'm going on about it.

I would like to hear about your other observations. Again, I'm just a skeptical guy by nature, and if something confuses me, I'm inclined to ask about it (even at the risk of revealing my ignorance).

great stuff, and yes we'd love to hear more..

looking at it one way, it's a beautiful expression of arithmetic overflow. the embedded theme really speaks to the whole album. does this song sound like the 'flashback' or break in the linear progression of the first three tracks to tell the story of how the system grew? that would make a nice progression into the concepts of the following tracks too.

apriorion wrote:

I: why the shape
-=-=-=-=-=-=-=


This shape does not necessarily represent the song's structure in any direct way. I'm just visualizing the numbers that BoC seem to be referring to!

I'm saying I have observed the occurrence of 39 in two different parts of the song's structure. Then I began thinking about 39 as 36 + 3 because of the prominence of sixes and thirty-sixes, not only on this album, but also at least on MHTRTC. In "Aquarius" the woman's voice starts speaking random numbers after counting upwards to 36. 36 also is connected to roulette which might also be a connection to the random numbers the woman speaks, but I digress!

Anyway once I noticed the double-occurrence of 39, I began to consider how the number might correspond to a visual pattern. I think it's clear that the song has a "programmatic" feel to it. The sounds all come in perfectly right on time, but the noodly synth just effortlessly meanders around into some really unexpected strange places. There are a lot of tiny details in the continual one-off synth lines that ebb in and then echo off into the distance. I've counted the length the song probably 4 or 5 times, and it's not easy to focus for that long. Playing these synth parts 'live' would require an unbelievable amount of focus and practice. I don't want to say it couldn't be done, but I imagine it would be quite difficult. It's much easier to collate those types of perfectly loose sounds from multiple takes, rather than playing 50 unique phrases perfectly in real time. But who knows. They are good musicians..

Anyway I guess overall, I began to consider what 'plan' there was behind the music, particularly the 39s. Other people also wondered about the song length... I've noticed some people saying they think it goes on too long (an opinion I disagree with). Mainly I just felt there might be some purpose or plan behind it, so I decided to start drawing stuff out.

The 39 that relates to the song's body (39 groups of 6 bars of 3/4) could be visualized like this, which would more accurately represent the durations of each respective broad section of the tune:


-=-=-=-=-=-=-=-=-=-=-=-=-=
X // intro(1)
X X X X X X //-=-=-=-=
X X X X X X //
X X X X X X // body(35)
X X X X X X //
X X X X X //-=-=-=-=
X X X // outro(3)
-=-=-=-=-=-=-=-=-=-=-=-=-=
1 + 35 + 3 = 39
-=-=-=-=-=-=-=-=-=-=-=-=-=


But considering the song's name is Jacquard Causeway, I decided to also consider other ways that the 39 could be visualized. As I mentioned, 36 + 3 made sense because of the history of the number 36. We also know that BoC have stated they like fractals and sometimes write songs with mathematical patterns in them.

So, the three levels in my image. First the "monads", of all six colors. These could be any shape and color. Each monad represents a bar of 3/4 in the song, ie, a single occurrence of the [KICK; HAT; SNARE] loop. There are 234 monads in my drawing, and the song's length is almost exactly 234 bars of 3/4.

To recap: have 39 occurring in the intro. We have 234 occurring in the song's length. 234 / 39 = 6. We can find sixes also as the delay on certain synth phrases. I'd have to explain too much other stuff to explain how I definitely hear those 6/4 delays, but suffice it to say that those synth phrases repeat on a longer delay every 6 counts, or every 2 bars of drums. Maybe other people can hear this too. But the point is basically from what I've observed, the song's rhythmic and structural properties seem to be factors of each other.

The next level in my image is the first level of actual hexagons. Since the song's title refers to a hexagonal latticework, and since we know the band likes fractals, I decided to represent the 39 as a hexagon, made out of hexagons, made out of monads.. with three additional outer-adornments, making the broadest-level shape total 39.

So, I hope that helped to explain what this shape represents, and why I decided to make it.


II: why the 70
-=-=-=-=-=-=


As I mentioned in I, this shape does not directly represent the song structure of Jacquard Causeway. It demonstrates how proportional and self-contained the song's rhythmic properties are.

But the shape's really cool because the sum of all unique broadest-level sub-patterns around the center of the shape that are hexagon-like, ie, that have 3-axes of symmetry, in the shape is 70. Just to be clear, to be a true 'factor'-shape of the broadest-level pattern, each unique sub-pattern around the center of the shape must have all identical monads, and have 3 axes of symmetry. The number of monads that each sub-pattern contains is added to the total, which is 70.

Look at this image http://i.imgur.com/lT4VCTN.png and start on the inner-blue hexagon line, which is on the left side of the image. Pick a ROYGBIV hexagon and follow the blue line around the shape. Once you arrive back to your home base, you will have passed over 6 ROYGBIV hexagons. The same activity can be followed for all the other colors on the left side of the image. The important concept is that each of those 4 colors on the left creates a unique sub-pattern of a hexagon within the image. Each of those shapes could be infinitely tessellated. And each unique shape has a different number of 'monads' comprising it. Like in a fractal, the hexagon reappears on multiple levels of scale. And of course, hexagons have 3 axes of symmetry.

Now for the image on the right. There are three purple shapes that can combine to create another hexagon sub-pattern. You'd actually have to tesselate this pattern out of a bigger latticework than I've 'framed' the image with, but it's geometrically regular and symmetrical on 3 axes just like a regular hexagon. These purple shapes are the 'monads' of this particular hexagon-like sub-pattern. And of course, the dotted red line represents the fact that the most outer shape, as a monad of itself, would also be infinitely tessellatable, but just like with the purple shape, you'd have to embed the entire shape into a larger latticework in order to tessellate it.

So, each unique 3-axis-symmetric sub-pattern contains a different number of monads, skewered onto their respective line(s). The sum of this set of unique 3-axis-symmetric sub-patterns is 70:

6 + 12 + 18 + 30 + 3 + 1 = 70


III: what about other songs?
-=-=-=-=-=-=-=-=-=-=-=-=


Other songs or songs' properties can be visualized if you wanted to count them out, figure out their time signature, etc. This song works especially well because its factors seem to fit together, its tempo is consistent, its rhythm loops the entire song, the song's title and time signature seem to point towards the number 6 to use geometrically, etc. Other artists certainly have used diagrams and other schemas to make art, so you can definitely reverse-engineer stuff to some extent. But I don't really have the time to get into it any more than that! I don't think it's a problem for what I'm trying to show Twoism, because I'm just illustrating some interesting properties of the song.


EDIT: I think I mis-defined a concept; lol; think I did better the second time.
EDIT: I a word added added
EDIT: K now I'm done

I'm sorry but I think you are stretching it by a huge margin. And it's a little misleading to make others think the track somehow represents hexagons when it's only the visual representation you get personally. Good analysis on the signature though, and nice graphic regardless.

Radiobuzz wrote:I'm sorry but I think you are stretching it by a huge margin. And it's a little misleading to make others think the track somehow represents hexagons when it's only the visual representation you get personally. Good analysis on the signature though, and nice graphic regardless.

I don't want to mislead people, but I really think this pattern exists. And you haven't said anything about how my ideas are misleading. Let me know why you think my representation is misleading or too much of a purely personal representation. I don't know if you read my post right above you, but I think I explained it a little better there, so I would recommend reading it if you haven't.

This image isn't a direct representation of the song's broad structure. It's a representation of how the time-dividing properties of the song seem to subdivide evenly.

The broadest-level shape has 3 lines of symmetry; the song has 3 beats per bar. Each unique hexagon-like sub-pattern within the broadest-level pattern whose monads are all identical adds up to 70. I found 39 twice in this song, so the design has 39 pieces in it. There are 234 pieces in this design, and I counted 234 bars of 3/4 including the intro, body, and outro.

The imagery of a shape being made out of shapes relates to a Jacquard loom. It's a mechanized programmable loom that creates intricate meta-shapes out of simpler shapes of woven materials. The engineer presses a "piano" to cut the design into cards, 8 presses at a time. The cards are looped together and the machine produces loops of textile designs.

So anyway, I appreciate your comment.. But I do think this design corresponds well to the song's structure and possible meanings in a handful ways simultaneously, which is at very least rare and interesting, and possibly even meaningful. But I'm open to other people's opinions. That's why I posted it!

JESUSANDHOES wrote:So anyway, I appreciate your comment.. But I do think this design corresponds well to the song's structure and possible meanings in a handful ways simultaneously, which is at very least rare and interesting, and possibly even meaningful.

That is your own, personal, subjective, biased view. Passing it off as some mathematical inherent fact of the song is, if anything, misleading. And considering that, due to the band's mysterious and cryptic nature, many fans are likely to latch onto any hypothesis that makes any slight semblance of sense, yeah, your post is exactly that: misleading.

But aside from that, I wonder how this song is related, mathematically and geometrically, to Queen's song '39. The possibilities!

ok, wow. maybe not the safest place to try to engage in intelligent discourse. i guess it was worth a fuck and a t-shirt, that's something..
/splits.

Wow people need to get over themselves.
Seriously, take the pole out your collective arses once in a while.
It's a fun theory, nothing more. The guy didn't say it was gospel.

Cheers for sharing Jesusandhoes. For one, I thought it was a cool read, albeit in the vein of the shipping forecast. ie. fascinating, but mainly indecipherable to a maths halfwit like myself.

JESUSANDHOES was it you indeed...

Mexicola wrote:It's a fun theory, nothing more. The guy didn't say it was gospel.

Well, it was written more in a Leviticus sort of way.

frank chipboard wrote:I want this on a t shirt.

I will get it tatooed

apriorion wrote: I mean, I get why you picked hexagons as the monads in this diagram, since that's BOC's thing and all, but how it makes up a larger hexagon seems like a bit of a projection on your part more than anything inherent to the track.

sorry man, but I agree with this guy^

seems like a real stretch to me.

it's strange so many people have jumped on board with this one as everyone is usually so quick to put down the really "out-there" theories.

No need for ad hominem attacks. I respect the effort the jesusandhoes made, but I like to try to understand something, hence my questions. I don't think I had my head up my arse, and I don't mean any ill will against jesusandhoes. I look forward to hearing more of his thoughts, but I will always ask questions if (a) the subject is interesting to me, (b) I don't feel like I understand it, and (c) the person seems capable of answering me.

FernieCanto wrote:That is your own, personal, subjective, biased view. Passing it off as some mathematical inherent fact of the song is, if anything, misleading. And considering that, due to the band's mysterious and cryptic nature, many fans are likely to latch onto any hypothesis that makes any slight semblance of sense, yeah, your post is exactly that: misleading.

But aside from that, I wonder how this song is related, mathematically and geometrically, to Queen's song '39. The possibilities!

FernieCanto wrote:

Mexicola wrote: It's a fun theory, nothing more. The guy didn't say it was gospel.

Well, it was written more in a Leviticus sort of way.

FernieCanto, when I created this thread two days ago, I was still in the process of figuring the second part out. I presented it with words like 'seems' and 'observe' because I didn't want to present my theory as fact. And if someone disproves it, that's cool. But it is connected to observations and evidence, so I think that calling it 'misleading' without citing anything in particular that's wrong with it (aside from calling it 'subjective') is rather lazy.

If you could perhaps demonstrate why it's my personal, subjective, biased view that the song's body's tempo is 3/4; that there are 234 counts of 3/4 in the song from intro to outro; that there are 39 eighth note counts in the intro; that 234 / 39 is 6; that there are also delays that happen every 6/4 on the synth. These factors led to me thinking about 39, 36, hexagons, and shapes made out of shapes.

The design is just a shape that illustrates how nicely proportional the time-dividing properties of the song are, if you look at it the right way.

In the wake of interviews after Tomorrow's Harvest, BoC mentioned that there were more subliminal messages in the album than any other one. When I read that, I was shocked and excited because it meant that there were hidden aspects of the album. If you look at the threads on here for each song (01. Gemini, etc.) you'll see mostly comments about abstract surface-qualities of songs like mood, tone, influences, whether it's a good track or not -- and much less structural analysis of syntactic qualities like composition, time signatures, bars per measure, whether certain instruments are mixed mono (deducible via the AAAA - BBBB phase cancellation method), the imagery conjured by a specific series of musical notes, etc.

I think BoC have proven themselves capable of making interesting structural choices, so I'm analyzing the music with as much attention as I can. It's not my responsibility that people might mis-interpret what I'm saying. I'm trying to be as clear as I can about what I'm saying, which again, is connected to structural physical properties within the waves.

apriorion wrote:No need for ad hominem attacks. I respect the effort the jesusandhoes made, but I like to try to understand something, hence my questions. I don't think I had my head up my arse, and I don't mean any ill will against jesusandhoes. I look forward to hearing more of his thoughts, but I will always ask questions if (a) the subject is interesting to me, (b) I don't feel like I understand it, and (c) the person seems capable of answering me.

Thanks, I appreciate that. Hope my responses were useful.

ryanhewitt wrote:

apriorion wrote: I mean, I get why you picked hexagons as the monads in this diagram, since that's BOC's thing and all, but how it makes up a larger hexagon seems like a bit of a projection on your part more than anything inherent to the track.

sorry man, but I agree with this guy^

seems like a real stretch to me.

it's strange so many people have jumped on board with this one as everyone is usually so quick to put down the really "out-there" theories.

I think you've got it backwards... The monads, ie, the tiniest hexagons of 6 colors, could have been any shape. Triangle, circle, smiley face, whatever. It's the upper-level properties of this shape that correspond to the song's time-dividing properties.

-234 bars in the song = 234 tiny monads in the shape
-39 counts in the intro = 39 ROYGBIV hexagons
-3 ROYGBIV hexagons on the outside of the shape, giving it triangularity, would mean that a full rotation of this shape would produce a 3/4 rhythm.
-The factors of this shape add to 70. (It's a bit more complicated than that, but if you want to see an explanation go about 10 posts up.)

It's really not that "out-there." I'm measuring time!

Here's another great way to represent time:

music is math indeed

Plaudits to the OP