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something here. But I think this is like so deeply, intimately connected with so many areas of life and the universe and everything that I think it deserves the name of a philosophical toy. But okay, what else is a philosophical toy? The more general version of these is called the harmonograph, of which there are two kinds. There are like circular and what are they called lateral, rotary and lateral harmonographs, where basically you have two pendulums, feeding their oscillations into another pendulum. And depending on, you know, the specific like relative frequency between them and, you know, like the speed and things like that, and the phase, you will get completely different shapes and the shapes tend to be really pretty and really cool. And I mean, this is like actually super profound and I'll see in a second, you know, when the frequencies are exactly the same, you get this thing that is a unison. Basically, yeah, it's kind of just a circle. And really just depending on the relative phase of the two pendulum, you will get like either a circle or a line. So that's a unison. Okay, so if you're seeing kind of like oscillations that look like that, yeah, you can infer that there are two pendulums that are feeding into it and maybe they're out of phase or not, depending on the particular shape that happens. Now, what happens when they are like in near unison? Well, then you get this family of alternative behaviors. Now, here is where this starts to connect with, yeah, like actually very deep philosophy and qualia research institute paradigms. That in a sense, when you're close to unison, when you're close to the same frequency as somebody else, but not quite, then actually that gives rise to dissonance and some kind of stress. And we can, in a sense, like, yeah, there's a lot of research that we're doing in that space. So basically, like misalignments actually causing, you know, here, like with increasing degree of difference in the frequencies, it becomes more and more erratic until it gets like this crazy mesh, which is, yeah, not particularly pleasantly looking. You know, it looks kind of in disarray, essentially. Now, essentially what this also gives you is musical scales, right? Because you can, for example, put two pendulums that are at a ratio frequency of one to two. And then you will get these, yeah, basically an octave, basically double frequency. And you will get this, yeah, beautiful kind of like infinite shape. And depending on the phase, you will get shapes of this sort. Again, if you see any of these shapes, which are called Lissajous shapes, you can know it's two pendulums and they're in an octave of relationship. And depending on the specific shape, you can infer what the relative phase is between these two oscillators. But it gets even crazier. I mean, like, you know, rotary, like if they're rotating, you get

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text:
like these other beautiful patterns. And it gets like more complex. Let me show you the kind of like the cheat sheet, I think, is that what I'm calling it? I mean, essentially, yeah, ordered by the level of dissonance, in a sense, which actually has to do with, as we'll see in a second, this is actually like just the temperament notion of dissonance, which is like, okay, how many of them are actually in an integer relationship ratio, such that the denominator is small. So basically, these are like the possible ratios of the frequencies. And you can see that the largest number in the ratios will be the one that determines how dissonant a particular shape is. And essentially, these are ordered by a degree of dissonance, this one being lower dissonance, this one being higher dissonance. And yeah, I mean, essentially, yeah, the higher the denominator, this would be nine, because it's nine against eight, and that fraction doesn't reduce further, you get this kind of like very detailed and intricate, but still, they're like integer ratios, right? Like, they're not close to unison, where it would actually become kind of this crazy mess. So even though, you know, this is not as consonant as the one on top, they're still fairly good. You know, they're still fairly good. Okay, so that is the harmonograph and the kalé de faune. Okay, so what about the numbers I was talking about? Well, okay, so it turns out those numbers in particular, you will see them, you will find them in ancient, you know, carvings and artwork about the Pythagoreans. So this is art about the Pythagoreans. You can see, for example, here, the sizes of the bells are in that relationship, like four, six, eight, nine, 12, 16. And here are the lengths of the flutes or the size of the of the rocks, maybe the weight of the rocks that are making these variously tense, and so on. So why do these numbers show up in here? Like what's going on? And they seem to be across various mediums, right? So here's the answer. That particular set of numbers has a outstanding level of global consonants, meaning that the pairwise relationships between these numbers have reduced fractions that give rise to denominators. That are very small. So for example, four and eight, there is an octave of difference. They're in a one to two ratio, and then eight and 16. They're also in a one to two ratio. And now let me kind of like write it in a way that kind of like shows all of those relationships. Again, this is the sort of thing that, yeah, in a different context, you probably would judge me as being completely crazy. But yeah, I mean, here, here's what I'm talking about. So and I'll explain in a second also like the way in which I'm including information in this diagram. Doopie doopie doo. Okay, so. Okay,

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text:
so one way that I was playing with earlier today that I found how to lay down these numbers in a very significant way is like this. So eight, like, let's say the vertical direction is essentially octave relationships. So four, eight, 16, six and 12 are also in an octave relationship. So those would be the most consonant relationships. So in this group, you have four octaves, four ways of generating an octave of consonants. Now, these double lines or double bonds are where the relationship is a three, where the denominator is a three, which could be two thirds or one third, either way. So you get kind of these like squiggly, like eight goes to 12, which goes to four, which goes to six, which goes to nine. And then like the more squiggly ones are relationships where the denominator reduces to four. And that's about it. I mean, like, there's also really like the black sheep in the set is basically nine, because, you know, nine actually only reduces to essentially 16 when it's connected to 16. And also it reduces to nine when you're connected with eight. So anyway, nine here would be like the one that causes more dissonance, but also that's a little bit of variety. Now, that said, yes, I mean, basically, this kind of set is almost optimal at, in a sense, like using small integers that are like four or higher, where you have like a non-trivial configuration that minimizes dissonance or maximizes the consonant relationships between them. And in that sense, it is a harmonious group. Okay. So that was cool. Now, what I did today, I guess, as part of my birthday celebration, a little side project, essentially, yeah, well, I started doing numerology. So I wrote a program where basically I was like searching through the space of possible, yeah, you know, sets of six numbers to find like others that also, in a sense, minimize some notion of consonants, of dissonance. And there's like various ways of doing that. I mean, like the objective function here, you know, it could be like adding up, you know, and getting the mean for the denominators of all of the pairs. Another one is maybe taking like the top, you know, 50% of the dominators and just looking at those. Another one is like taking the square root. Another one is like even, you know, squaring them, but basically there's like different priorities and actually which priority you have will depend if you live in a geometric space. Actually, it maybe doesn't matter. You know that there's like a few of those that are like very decent with one another, because maybe you can put them further away. Right. So they don't interfere very much. In a high dimensional space, everything will be interacting with everything. And in that sense, yeah, maybe actually what you want is like to reduce the average, you know, then again, dissonance doesn't increase

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text:
linearly as a function of the denominator, which actually justifies, you know, taking the square root. Anyway, in most of the experiments, all of these measures were actually very highly correlated, meaning that, yeah, I mean, basically the whole consonants reinforces itself because, you know, it's every pair in a sense. So, yeah, yeah, basically if a subset of the pairs, they're correlated. It's a subset of the pairs are consonant that increases the probability that the remaining ones are also consonant. So, yeah, in a sense, all the measures are correlated. But the one that I ended up settling in was the average square root of the largest denominator. So basically, very simple program, you know, actually just taking these like sets of numbers, randomly sample, you know, without replacement and then looking at what is their reduced fraction and then looking at the denominator. Okay, and then, okay, a little bit of intelligence into how to carve the combinatorial space so that you don't, you know, you're not wasting your time basically just sampling primes because a bunch of primes put together will not be very consonant for sure, at least in this notion of consonants and harmony. So, well, here's a cool thing. So let me write it down. So now, yeah, and I found a few other numbers, you know, sets that were pretty good, actually. Excuse me for a second. Some ginger juice to carry me through this pretty, pretty elaborate discussion. I also recommend, yeah, if you have a, yeah, having something to drink to keep your mind sharp as we get into these very crazy territory together. Okay. So now imagine, right, like, so these, you know, these Pythagorean relationships, maybe they're like kind of optimal for a brain like ours in a normal state of consciousness, where in a sense, we can't really tell apart like denominators are like larger because they kind of just like blend the resolution of our experience is not like that high. But what if you were, let's say, kind of like an angel or like a divine being or something like that, and you had kind of like much higher resolution, you had like a much better ear and you could, in a sense, like, rather than, let's say, like, you know, playing a symphony with like 60 instruments, you were playing it with like 600 instruments because you can detect all the complex relationships between them and, and or something of that sort. Okay. So I was thinking like, okay, if you're some, like an entity like that, maybe rather than finding like six, you know, like numbers that are like very consonant with each other, why do we find 10? Okay. So I was exploring like 10 numbers and I found out that the following one was actually really good. Like it, it was like scoring the best in a bunch of different metrics. So here you have folks, you have essentially, well, it's

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like four, five, six, eight, 10, 12, 16, you know, 20, 24, 32, you know, whatever. Okay. I think I'm missing 12 in here. Sorry. So, well, that one is really good. And because, you know, for the, how large it gets like 32 and the fact that it's 10, it actually is like really good in terms of like how consonant it is, you know, and it's, I think it's like the best out of a hundred thousand that I tried after even like carving the space. Now I'm sure there's like even betters and maybe this is not the optimal metric, but you know, you're doing what you can. Okay. So, and then I tried to arrange it in kind of like a scale musical or configuration, you know, where like you could see more clearly the correspondences between these numbers. So, and here's what I have. So essentially, and the philosophy here is like, we've got to, you know, like lay them out in a way that is actually insightful in a way that tells us that you look at it. And in a sense, like it gives you information, like the spatial layout gives you ordered information. So this is more or less how I am laying it out. So essentially the vertical axis, all of a sudden becomes octaves. So six goes to 12, goes to 24. Again, I know this looks completely crazy. Numerological correspondences. Okay. So getting back to it, five connects to 10, which connects to 20, four to eight, eight to 16, 16 to 32. Now let's say kind of like a slightly different color. Hopefully this works for, for example, relationships where the denominator reduces to three. So that would be like this one. We also have, you know, four goes to 12. We also have like eight goes to 12, eight goes to 24 and 16, which is pretty cool. It's kind of like this squiggly thing. And the awesome thing about this layout is that you have kind of this mirror symmetry where in the right, what is like where the denominator reduces to three, in the left, it reduces to five. And I'm going to represent that as a kind of like, Oh, this one, hopefully this one, like a squiggly, squiggly line. Okay. So, and it's essentially a mirror here. So basically four, 10 reduces to also like a denominator of five. Um, four, like eight, 10, eight, 20, 20, 16. It also does that. And, uh, and another really cool relationship is horizontal. So horizontal, you actually have where the denominator reduces to six. So essentially you, you do this for all of the relationships that have high consonants and, uh, and it's a pretty cool. So let me just like quickly show you like what, what happens. So you get these, you get these, these, these, uh, apologies. I'm not really used to, you know, draw live as it were. Uh, but, uh, but

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text:
yeah, here we are. So I'm, I'm, uh, almost done, almost done, uh, getting there, getting there. So there's this one too, this one dooby doo doo doo doo. And I think, no, no, no, uh, there's so yeah, well, this is roughly it. Um, Oh no, it's this one too. And you get, uh, this shape. So I don't know. I mean, it looks kind of cool if, um, I mean, I'm sure it's like just random noise or something, but like, if any of you find anything related to this shape, please let me know. It's just cool. It's just like what emerged, you know, from the thought experiment of imagine there's like kind of like divine beings that can understand, you know, scales with more numbers and they play kind of divine music. And, uh, what is kind of the shape of, you know, the scale that emerges something like this anyway, I'm sure completely random, you know, but just, just in case if any of you, you know, notice this any relationship with anything else that exists already, please, please let me know in the comments. Okay. Uh, so that was pretty cool. Something I was doing today, um, which is, yeah, in a sense, kind of steel manning all of these, um, yeah, numerological correlations and correspondences, but from the point of view of actually valence, you know, combinations that feel really good, really, really, really, really good together. Okay. So, um, what next? Well, okay. Where do we get actually that, um, you know, an octave feels good when, when you have like a string and then a string that is twice the size when you plug them together, why does that sound good? Okay. So that is, uh, basically something that Helmholtz, uh, figured out. Uh, some people say probably rightly that, uh, Helmholtz was the last true polymath, the last person who like truly was actually the most knowledgeable of every scientific subject of his time. Um, at simultaneously and, uh, yeah, so basically he wrote books about everything, uh, extremely brilliant guy and, uh, and he figured out where decent and consonants come from in musical instruments. And the way he did it is amazing. It's basically, you know, when people have this aesthetic picture of science is like, no, you need kind of like a bio lab and you need like supercomputers and, and like, you know, researchers with, uh, uh, yeah, like, uh, basically like looking like a scientist with glasses and like a professor, like stuff like that. Well, no Helmholtz revolutionized completely our understanding of music by doing kitchen experiments. Essentially he, he would make these glass tubes and glass bubbles. And he literally has instructions in these pages about like how to use wax in order to put it into your ear and seal it properly. Like it's all, all sounds like super amateurish, like, you know, not serious work. Right. But, but then it turns

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text:
out that doing these filters, the sound, basically it works as a, uh, narrow band filter where you get rid of all of the high frequencies and the low well frequencies above a certain level frequencies below a certain level. And you get just a tiny fraction passing through. And by doing that, what he found was that, uh, he was able to isolate for any given note, for example, in the piano, he was able to isolate, um, all the ways in which, uh, that note actually was made out of a fundamental frequency and the integer multiples of that fundamental frequency. Okay. So, um, and once he did that, then he was able to, in a sense, kind of like dissect very carefully, um, any sound that is produced by musical instruments and, and in a sense, figure out in what way there's kind of interference between these harmonics, these integer multiples. And, uh, yeah, just like speaking very simply, uh, if you think of a, um, basically a spectrum, uh, when you play a piano key, let's say this is like 200 Hertz, you will have some amplitude, some intensity in the 200 Hertz range. Then you will have also in the 400, then in the 600, 800, 1000, and so on. Uh, it kind of like decays slowly as you increase the frequency. And if you play an octave, you play twice the frequency. That is like playing something that starts here, jumps one harmonic, then moves another, like has amplitude over here, jumps another harmonic. But what you will notice is that the harmonics align perfectly. They actually overlap. And when you have something that is in a relationship where there's like, you know, two to three, or like, you know, a fraction that reduces to a denominator of three, you will also have that relationship, except that kind of like it will align only partially. It will sometimes align, maybe I think like a half of them or two thirds will align and the, and the other ones just will fall through. Um, and, and, and so on, as you increase the denominator, you will have less and less alignment, but whenever you have, yeah, basically something that is in a whole integer relationship with each other, you will have some alignment. And with that some alignment, you will be preventing harmonics being close to each other, which as we saw in the harmonograph is exactly what generates these problems, which in the case of sound, it's called beat patterns. Uh, for, for a lot of you, this is probably going to be, yeah, kind of like just a, just review of something you already know, but you know, not, not everybody knows this about the history of, um, of music and our understanding of it. Well, it turns out that whenever you have like two frequencies that are close together and you play them simultaneously, let's say these two frequencies, you will get

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text:
beat patterns essentially and trigonometrically, you know, when you have like two signs added together with that, you know, slightly different frequency that is trigonometrically equivalent to a cosine multiplied by a sine. So that's kind of like, you have like this way larger wave multiplied by a tiny one. So you get kind of this amplitude modulation. So you get like this, like, whoa, whoa, whoa, whoa, whoa, which is basically, yeah, beat patterns. And it turns out that the stronger the beat patterns are, the rougher the experience sounds, the rougher the sound becomes. So, um, and the relationship ends up being, I mean, something you can empirically test and it has been done lots of times, uh, link in the description, but basically you have a relationship that looks like, uh, apologies, looks more or less like this. It's kind of like this sharp skewed, kind of like long tail. So if the X axis corresponds to the frequency difference and the Y axis corresponds to the amount of dissonance, well, for most frequencies, and this varies depending on the base frequency a little bit, but basically you will have something that like, yeah, I mean, around like 10% difference, you will have like maximum dissonance. Um, again, it varies like for higher frequencies, this is smaller for lower frequencies, this is larger, but the point is that there is kind of this, what's called critical band that like, if you have the frequencies too close together like that, the beat patterns are actually just noticeable and it becomes rough and you can't quite separate the two sounds from each other and they get confused in your perception. So if you look at the spectrum of a complex symphony, uh, that is not trying to maximize dissonance and very few symphonies, even those that are very dissonant for the most part, either well done, they will essentially, yeah, it's kind of like this jigsaw puzzle of how do you arrange the harmonics of all of these different instruments and all of these different amplitudes with their, you know, characteristic timbres in such a way that there is a spacing, enough spacing between them. And so, yeah, I mean, I recommend like taking your favorite symphony and put it on, you know, uh, Adobe Audition or something like that, or, or, you know, just on Python and getting the spectrum and you will see that, yeah, it's a work of art. Basically the harmonics are kind of like laid out exactly in such a way that they don't interfere very much with one another. Uh, which, yeah, it gives rise to tolerable, interesting textures of sound that are very rich and packed across the spectrum. That is, I think like, in a sense, musically speaking, key to understand for, for making sense of a musical, musical valence. Now, okay. So there's a, uh, additional information here to, to give you, again, this is kind of a review for

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people who, who, who've studied music theory, but yeah, there's this difference between just intonation and equal temperament. Where in the just one is exactly kind of this Pythagorean ideal of like, oh, let's put the scale in such a way that they are perfect. Uh, you know, integer multiples of each other, or they, the reduced fraction has a large denominator, uh, or small denominator. So when you do that, um, essentially the harmonies are perfect. They are like completely perfect and in a sense that does maximize valence and cleanliness of the sound, but it can be a little bit boring. We also, there's also these like boredom part, uh, uh, boredom mechanism I've talked about in other videos. I recommend watching the symmetry theory valence presentation, uh, also LinkedIn in the description. And, uh, yeah, I mean, basically, uh, if you listen to very simple patterns for long enough, you just get bored and that's a dissonance of itself. Okay. Uh, that's an insight that, uh, Mikey Johnson actually came up with. Um, so, and, and that's one problem. The other problem is more serious, which is that they're just not very versatile in a sense. If you compose music for a particular scale in the just intonation, uh, it will not generalize to other scales. Actually, you need to produce music specifically for specific scales and the instruments will be very finely tuned specifically for those scales as well. Which is, yeah, kind of like a nightmare if you want to like modify music or use the same instrument to play many different kinds of song within the same night. You'll have to tune it again, every single transition. So instead you also have equal temperament, which is, uh, dividing an octave into 12 parts. And it just actually so happens that doing that division ends up being a fairly reasonable average consonance between the pair of relationships of the notes. Um, and, uh, and that's basically what most of Western music is based on basically equal temperament because it's just so convenient. So easy to, to, to in a sense, like tune and play different songs with the same instrument instrument in the same night. Um, now, uh, some people say that, yeah, this is kind of like, we are used to equal temperament, uh, so much that like you realize that actually it's almost kind of like has a, it's kind of like caffeine has a bit of a stimulating character because even like octave, well, even like major fifths or like, yeah, this kind of, uh, relationships that should be perfectly harmonious in a sense. They are not quite. And, uh, we get used to it. That's kind of like what we, we, we assume. So if you listen to like just intonation music, uh, you can actually almost kind of like fall asleep sometimes because he's like, oh, this, the, the, the, the background musical caffeine. I think that phrase comes from

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text:
a Quentin, Quentin Freerichs. Uh, uh, yeah, basically it shows up. Okay. So now what? Well, an additional twist here is that, um, this whole, you know, all of these music theory, you know, Pythagorean relationships and equal temperament and so on. All of it is based around instruments that are worth called, um, uh, harmonic, which is instruments whose notes, uh, have additional pure tones that are exactly integer multiples of each other. Now this makes physical sense when you have a string, yes, it's resonant modes will actually be integer, be integer multiples of each other because it's a string, a one dimensional string vibrating. Okay. Hopefully, hopefully that makes sense. It doesn't trivially generalize. It doesn't generalize to like two dimensions or three dimensions. Also, it doesn't generalize the music to like weird instruments that, uh, for example, are synthetic or like, you know, if you use a synthesizer or something like that or plain strange instruments, for example, tubular bells, the harmonic modes of tubular bells. Or sorry, the resonance of tubular bells are not in a harmonic relationship. There is an instrument that is called an inharmonic instrument. Um, so yeah, if you, if you put tubular bells in your piano, you know, in your, in your, uh, digital piano, uh, keep, I forget what it, what it's called. If you do that and, um, uh, and you try to, to play harmonies and, and, uh, chords, you will notice they're off and strange. So tubular bells are actually, it's very difficult to play with them, uh, and, and write music for it, especially symphonic music precisely because it's an inharmonic instrument. The other one is a xylophones that naturally because of the shape of it. Yeah. Basically the resonant modes are not in a harmonic relationship. So naturally already there are like some instruments, exotic instruments, admittedly that are inharmonic and for which, you know, these music theory doesn't work. And I do remember, okay. So like when I was, uh, uh, younger, uh, in my early youth, uh, as, uh, as some would say, um, I used to take a piano classes and I was very frustrated. Like learning these like musical scales, because I didn't understand why, why are they that way? You know, why do they skip the way they do? Like, it just seems arbitrary. And honestly, you know, my teachers, the teachers there and nobody could explain to me why they would say things like, oh yeah, music is very connected to math, but that's very, it seemed very superficial. But it's like, it's actually just like learning kind of arithmetic. It's like, you're not getting an insight. You don't understand it. There's also a lot of illusory understanding. A lot of people who studied music theory and they think they understand why it works that way. Actually, they just work. They just know how, you know, they know how to play with it. But not necessarily the deeper,

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text:
deeper why. But yeah, the deeper why is, well, musical scales. One way of understanding is, is carving the space of sounds, carving that state space such that the average pairwise or, or a triplet has a high degree of consonants. Or at least, you know, the top 50% or 20% of relationships are very consonant so that you can produce music that has that consonant quality. Of course, it is beneficial if in your scales, you also have the source of some dissonance. Some relationships are very dissonant such that you can create tension and then relieve it. And yeah, that's a very important part of music. But if you don't have the consonant core, kind of the, the consonant, kind of like large amount of the combinatorial space, in a sense like that actually will prevent you from producing like beautiful music. For the most part, you will be producing other, you know, more dissonance or noise based music. Okay. So what William Setharis did, who is a music theorist and mathematician, also link in the description, he generalized music theory to also inharmonic instruments. And this is something that honestly you could only have done after the advent of computers. Why? Because the way in which he did this was via a genetic algorithm. Where basically what he does is a little program computes the total dissonance of a particular sound constructed of pure tones. And he does it by looking at the pairwise relationships between them. And then what it does is it tries to find a way of spacing the notes, the frequencies, such that you again get a good scale and a good scale will be defined as, yeah, that the average pairwise dissonance within the notes that you have available is minimized. So, for example, tubular bells. Okay. Maybe a 12th, you know, tempered scale doesn't work, but maybe a 17th tempered scale does work really well because statistically speaking, the harmonics align better. Okay. Hopefully, hopefully this makes sense. But you see, this is a generalization of music theory. Then we kind of like need to ask the question of, does this also generalize to other dimensions, because as I was saying, all of these kind of music theory is based around one dimensional universe, like whether it's like cubes of air, columns of air or strings, you know, things like that. I guess percussions don't really count in that case, but yeah, basically you will have exactly this issue that the resonant modes will tend to be multiples of each other. But in other spaces, that may not be the case. So, and here is, yeah, basically a very important kind of like foretaste of what am I going to be discussing at a greater depth in this video and also later is, yeah, the wallpaper symmetry groups. I talk about it in relation to this effect in psychedelics that is called symmetrical pattern repetition, where you stare at a piece

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text:
of, for example, your wall, a textured wall, you know, statistical texture. And by staring at you, all of a sudden you will start to see kind of like copies of whatever is in your fovea and the detail will spread in one of these arrangements. And in a sense, the symmetry will actually reinforce itself via resonance, which is, yeah, I think something we can eventually prove is one of the important research paradigms that, yeah, basically when you have a wallpaper symmetry group appear in your visual field and psychedelics, that is a resonance effect. And you're actually getting the interlocking of resonant waves in several dimensions at once, such that their interference pattern gives rise to one of these symmetries. Okay, so all of a sudden, numerology again, right? Like the 17 wallpaper symmetry groups, because they're mathematically constrained. By the way, here's the other book by Conway, The Symmetry of Things, where, yeah, basically he explains and proves, well, this is a very well-known theorem. I think it's called the magic theorem, basically why there's only 17 wallpaper symmetry groups, two-dimensional ways of tessellating a two-dimensional space with symmetric elements. And more so, there's also a limited way when you include colors, which you can think of it as another degree of freedom or dimension here. And these are all of them. These are all the colorations. But yeah, I mean, that's in the real math and people who are interested in number theory and symmetry and so on. But for consciousness research, why does this matter? Well, they show up in psychedelics, but also they're extremely related with beauty and valence. And why? Right? Like this is so mysteriously beautiful. It's like an existential puzzle. Why? Why do these produce valence effects? And yeah, I mean, as we will go into further, basically, these are harmonious ways of putting together resonant modes in a way that their interference actually is consonant. So whenever you see something like this, actually, there's a signature of consonance in the resonant modes of your experiential field. And for that reason, oh, by the way, there's also kind of a crazy duality between spatial symmetry and temporal synchrony that arises in these states. Not only psychedelics, also in meditation, basically high energy states of consciousness. And so also, if you see one of these and it's very crisp and clear, it also means you have synchrony over time going on. It's not only that they are in a sense, yeah, it's not only spatial symmetry, it's also temporal synchrony, because otherwise they would be chaotic. You know, they would be, you might be kind of like one of these unison type of thing. But like, no, if this is a stable pattern, maybe, you know, it's rotating or something like that. But the point is that the symmetry elements are preserved. Then, yeah, you know, actually, it is a consonant relationship. And yes, also, again, you will have kind of

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