#fourier approximation

[bobacupcake]

Can you do one about the Sea of Thieves water?

OK

so . there was a biiig long talk about this at siggraph one year!! you can watch that here if you'd like . in the time between me getting this ask and me fully recreating the water, acerola also released a great video about it . the biiig underlying thing they do and the reason why it looks so good is they are making a Really Detailed Ocean Mesh in realtime using something called an FFT (fast fourier transform) to simulate hundreds of thousands of waves, based on a paper by TESSENDORF

WHAT IS AN FFT - we'll get to that. first we have to talk about the DFT - the discreet fourier transform. let's say you have a SOUND. it is a c chord - a C, an E, and a G, being played at the same time. all sounds are waves!!! so when you play multiple sounds at the same time, those waves combine!!! like here: the top is all 3 notes playing together, so they form the waveform at the bottom!!

now if someone handed you the bottom wave, could you figure out each individual note that was being played? how about if someone handed you a wave of One Hundred Notes. you would think it would be very hard. and well, it would be, if not for the Discreet Fourier Transform.

essentially, there is a way to take a bunch of points on a waveform comprised of a bunch of different waves, add them all together, do some messed up stuff with imaginary numbers, that will spit back out at you what individual waves are present. i made a little test program at the start of all this: the left are the waves i am putting into my Big Waveform, the top right is what that ends up looking like, and all the little rainbow points on it are being sampled to spit out the graph at the bottom right: it shows which frequency bands the DFT is finding (here it is animated)

this has enormous use cases in anything that deals with audio and image processing, and also,

THE OCEAN

tessendorf is basically like, hey, People Who Are Good At The Ocean say that a buuuunch of sine waves do a pretty good job of approximating what it looks like. and by a bunch they mean like, hundreds of thousands to millions. oh no.... if only there was a way we could easily deal with millions of sine waves..........

well GREAT news. not only can you do the DFT in one direction, but you can also do it in REVERSE. if you were to be given the frequency graph of a noise for example, you could use an INVERSE DFT to calculate what the combined wave graph looks like at any given time. so if you were to have say, the frequency graph of an oceaaaan, for example, you could calculate what the Ocean wave looks like at any given time. and lucky for us, it works in two dimensions. and thats the foundation of the simulation !!!!!

BUT WAIT

as incredible as the DFT is, it doesn't scale very well. the more times you have to do it, the slower it gets, exponentially, and we are working with potentially millions of sine waves here

THE FAST FOURIER TRANSFORM here we are . the fast fourier transform is a way of doing the discreet fourier transform, except, well, fast. i am Not going to explain the intricacies of it because its very complex, but if you want to learn more there are a ton of good 30 minute long videos on youtube about it . but essentially, due to the nature of sine waves repeating, you can borrow values as you go, and make the calculation Much faster (from exponential growth to logarithmic growth which is much much slower, and scales very well at higher numbers). it's, complicated, but the important part is it's so much faster and the diagram kind of looks like the shadow the hedgehog story plot

so if we use the inverse FFT on a graph of a rough estimate of what frequency of waves in the ocean (called a spectrum, basically tells us things like how many small waves, how many big waves, how different waves follow the wind direction. sea of thieves uses one called the phillips spectrum but there are better ones out there!!) now we have our waves !!!!!!! we can also use another inverse FFT to get the normals of the waves, and horizontal displacement of the waves (sharpening peaks and broadening valleys) through some derivatives . yayy calculus

OK MATH IS OVER. WE HAVE OUR WAVES!!! they are solid pink and look like pepto bismol. WHAT NOW

i cheated a bit here they look better than not being shaded because i am using the normals to reflect a CUBEMAP to make it look shiny. i think sea of thieves does this too but they didnt mention it in their talk. they did mention a FEW THINGS THEY DID THOUGH

FIRST OFF - SUBSURFACE SCATTERING. this is where the sun pokes through since water is translucent. SSS IS REALLY EXPENSIVE !!!!!! so they just faked it. do you remember the wave sharpening displacement i mentioned earlier? they just take the value where the waves are being sharpened and this will pretty naturally show off the areas that should have subsurface scattering (the sides of waves). they make it shine through any time you are looking towards the sun. they also add a bit of specular ! sss here is that nice blue color, and specular is the shiny bits coming off the sun. the rest of the lighting is the cubemap i mentioned earlier, i dont know if thats what they use but it looks nice !!!!!

then the other big thing that they do is the FOAM !! sorry i lied. there's more math. last one. you remember the wave sharpening displacement i Just mentioned. well they used that to find something called the JACOBIAN and well im not even going to begin to try and explain what it means but functionally what it does, is when the jacobian is NEGATIVE it means waves are clipping into eachother. and that means we should draw some foam!!! we can also blur and fade out the foam texture over time and continuously write to it to give it some movement, and bias this value a bit to make more or less foam. they do both of these!!!

YAYYYYY !! OK !! THAT'S SEA OF THIEVES WATER!!!!! THANKS FOR WAITING ALL THIS TIME. you can see my journey here if you would like to i have tagged it all oceanquest2023

thank you everyone for joining me :) i had fun

[yaldabaothadeez]

How fast does a wave move?

According to my poll 9/11 people (never forget) didn't already know this, so here's my attempt to explain the difference between group and phase velocities of waves with minimal (but not no) background.

Honestly, you're better off reading the wikipedia page "group velocity" from which I am stealing all my pictures, but I'm doing this anyway as an exercise in pop-science writing.

Summary: a small ripple on the surface of water actually looks something like this

i.e. not a fixed pattern of peaks and troughs moving together as a block, but a bunch of wiggles moving more slowly than the peaks of the waves.

The rest of this post is an attempt to explain why this happens, and that in the particular case of water surface waves the ripple is traveling at half the apparent speed of the waves that make it up.

Waves and dispersion

First, let's think about waves in general.

An idealized uniform plane wave (in one dimension) looks like this

the y-axis is amplitude (e.g. the height of the surface for a water wave[1], pressure for a sound wave, field strength for radio waves etc.)

The x-axis is, well, it could be either space or time, right? The wave varies left-to-right, and the amplitude of any fixed point wobbles up and down, something like:

y = h(x,t) = sin(k x - w t)

for some constants k and w. w is the frequency of the wave, 2π/ (the time between successive peaks) , k is the wavenumber, 2π/ (the distance between successive peaks) [2]

The physics of the situation, whatever they happen to be, are going to have solutions with certain frequencies at certain wavenumbers. So we can treat w as a (currently undetermined) function of k

w = w(k)

this is the "dispersion relation", and is all we'll need to work out the rest of this.

Phase velocity

How fast do the peaks of the wave move?

The peaks occur when sin(k x - w t) = 1, so we need (k x - w t) to be a constant. That is, the peaks move at a speed

v_p = w/k

This is called the "phase velocity" of the wave, and is given by the ratio of frequency to wavenumber.

In general, this will vary with k, unless the dispersion relation is linear.

Ripples

Now, we want to think about something more realistic - not a uniform sine wave extending out to infinity, but a small localized disturbance to an otherwise flat medium (caused by a duck landing in a pond or something).

We'll assume that at a time t=0, we have

h(x,0) = f(x)

for some localized function f(x). We want to know how this little bump propagates over time.

Now f(x) is obviously not a plane wave, but if it is well behaved we can approximate it by a sum of plane waves of different amplitudes, phases and wavenumbers (we'd like to do this, as we understand how the plane waves move). In the continuous limit, we get the fourier transform

h(x,0) = ʃ H(k) sin( k x + b(k) ) dk

for some function H(k) [3].

If this perturbation is small enough that the system can be treated as linear, then solutions just add together and

h(x,t) = ʃ H(k) sin( k x - w(k) t+ b(k) ) dk

this is true but not very helpful - how does our ripple actually move?

Group velocity

From the stuff with the phase velocity, we can guess that this is going to depend on k.

So let's think about wavenumbers close to some fixed k0 and ignore the rest (this is fine because everything's linear). For convenience, I'll assume k0 is a local maximum of the function H(k), so I can ignore its derivative.

We'll write k = k0 + δk and expand the argument of the sine above to first order in the small value δk

k x - w(k) t+ b(k) = k0 x -w(k0) t + b(k0) + δk(x - w' t + b') +...

where w'(k0) = dw/dk is the first derivative (i.e gradient) of w(k) at k0

substituting that back into our expression for h(x,t), dropping any higher powers of δk and doing some basic trigonometry, we get

h(x,t) ~ cos[k0 x -w(k0) t + b(k0)] ʃ H(k) sin[δk(x - w' t + b')] dk +sin[k0 x -w(k0) t + b(k0)] ʃ H(k) cos[δk(x - w' t + b')] dk

The sine and cosine terms outside the integral that I've colored purple are just a plane wave at k0. The terms inside the integral in red show us how the pattern propagates: it moves at a velocity

v_g = dw/dk

This is the "group velocity" of the wave, and is given by the gradient of the frequency with wavenumber.

Of course, this is also generally a function of k, and so the ripple will gradually separate out into its constituent frequencies (hence the name "dispersion relation") [4]

Sound and light

For most types of wave you have probably encountered before (e.g. sound, electromagnetic waves in a vacuum), the dispersion relation is

w = c k

for some constant c. These types of waves are called non-dispersive, as v_p = c = v_g, for all values of k. Sound really does travel as fast as the pressure wave crests.

Waves on the surface of water

Okay, finally getting to the point. We're going to think about waves on the surface of a fluid (e.g. water) under gravity.

Assumptions: the waves are very small, the fluid is very deep (and of a uniform depth), surface tension and viscosity can be ignored.

Now, the "right" way to do this is to solve an appropriate limit of the Navier-Stokes equations, but that's too much like hard work:

k has units 1/length, w has units 1/time, we've assumed away everything in the problem except gravity so up to a constant

w = √(gk)

Where g (~9.81 m/s^2) is the acceleration due to gravity.

So, immediately, we can say

v_p = √(g/k)

these waves are dispersive, narrower ones go slower.

v_g =0.5 * √(g/k) = 0.5 * v_p

ripples travel at half the speed of waves that make them up.

footnotes

[1] called a "gravity wave" by fluids people, as that's what the restoring force is. Which is why LIGO and such are always careful to say they are measuring gravitational waves.

[2] my angles are in radians

[3] The function b is there because these waves might be out of phase with each other, if I were using complex numbers I could absorb that into H, but I'm trying to not use more types of math than I have to

[4] Clearly v_p and v_g can be different, but in more than one dimension, they might not even be pointing the same way.

[canmom]

Music theory notes (for science bitches) - part 2: pentatonics and friends

or, the West ain't all that.

Hello again everyone! I'm grateful for the warm reception to the first music theory notes post (aka 'what is music? from first principles'). If you haven't read it, take a look~

In that stab at a first step towards 'what is music', I tried to distinguish between what's a relatively universal mathematical structure (nearly all musical systems have the octave) and what's an arbitrary convention. But in the end I did consciously limit myself, and make a beeline for the widely used 12TET tuning system and the diatonic scales used in Western music. I wanted to avoid overwhelming myself, and... 👻 it's all around us...👻

But! But but but. This is a series on music theory. Not just one music theory. The whole damn thing. I think I'm doing a huge disservice to everyone, not least me, if that's where we stop.

Today, then! For our second installation: 'Music theory notes (for science bitches)' will take a quick look through some examples that diverge from the diatonic scale: the erhu, Japanese pentatonic scales, gamelan, klezmer, and blues.

Also since the first part was quite abstract, we'll also having a go at using the tools we've built so far on a specific piece, the Edo-period folk song Sakura, Sakura.

Sound fun? Let's fucking gooooo

The story so far

To recap: in the first post we started by saying we're gonna be looking at tonal music, which isn't the only type of music. We introduced the idea of notes and frequencies by invoking the magic name of Fourier.

We said music can be approximated (for now) as an idealised pressure wave, which we can divide into brief windows called 'notes', and these notes are usually made of a strong sine wave at the 'fundamental frequency', plus a stack of further sine waves at integer multiples of that frequency called 'overtones'.

Then, we started constructing a culturally specific but extremely widespread system of creating structure between notes, known as '12 Tone Equal Temperament' or 12TET. The main character of this story is the interval, which is the ratio between the fundamental frequencies of two notes; we talked about how small-integer ratios of frequencies tend to be especially 'consonant' or nice-sounding.

We introduced the idea of the 'octave', which is when two notes have a frequency ratio of 2. We established the convention treating notes an octave apart as deeply related, to the point that we give them the same name. We also brought in the 'fifth', the ratio of 1.5, and talked about the idea of constructing a scale using small-integer ratios.

But we argued that if you try and build everything with those small-integer ratios you can dig yourself into a hole where moving around the musical space is rife with complications.

As a solution to this, I pulled out 'equal temperament' as an approximation with a lot of mathematical simplicity. Using a special irrational ratio called the "semitone" as a building block, we could construct the Western system of scales and modes and chords and such, where

a 'scale' is kind of like a palette for a piece of music, defined by a set of frequency ratios relative to a 'root' or 'tonic' note. this can be abstract, as in 'the major scale', or concrete, as in 'C major'.

a 'mode' is a cyclic permutation of an abstract scale. although it may contain the same notes, moving them around can change the feeling a lot!

a 'chord' is playing multiple notes at the same time. 'Triad' chords can be constructed from scales. There are other types which add or remove stuff from the triads. We'll come back to this.

I also summarised how sheet music works and the rather arbitrary choices in its construction, and at the end, I very briefly talked about chord notation.

There's a lot of ways to do this...

I recently watched a video by jazz musician and music theory youtuber Adam Neely, in which he and Philip Ewell discuss how much "music theory" is treated as synonymous with a very specific music theory which Neely glosses as "the harmonic style of 18th-century European composers". He argues, pretty convincingly imo, that 'music theory' pedagogy is seriously weakened by not taking non-white/Western models, such as Indian classical music theories, as a foil - citing Anuja Kamat's channel on Indian classical music as a great example of how to do things differently. Here's her introductory playlist on Indian classical music concepts, which I will hopefully be able to lean on in future posts:

There's two big pitfalls I wanna avoid as I teach myself music theory. I like maths a lot, and if I can fit something into a mathematical structure it's much easier for me to remember it - but I gotta be really careful not to mathwash some very arbitrary conventions and present them as more universal than they are. Music involves a lot of mathematics, but you can't reduce it to maths. It's a language for expressing emotion, not a predicate to prove.

One of the big goals of this series is to get straight in my head what has a good answer to 'why this way?', and what is just 'idk it's the convention we use'. And if something is an arbitrary convention, we gotta ask, what other conventions exist? Humans are inventive little buggers after all.

I also don't want to limit my analytical toolbox to a single 'hammer' of Western music theory, and try and force everything else into that frame. The reasons I'm learning music theory are... 1. to make my playing and singing better, and be more comfortable improvising; 2. to learn to compose stuff, which is currently a great mystery. How do they do it? I do like Western classical music, but honestly? Most of the music I enjoy is actually not Western. I want to be able to approach that music on its own terms.

For example, the erhu... for erhuample???

The instrument family I'm learning, erhu/zhonghu, is remarkably versatile - there are no frets (or even a soundboard!) to guide you, which is both a challenge and a huge freedom. You can absolutely play 12TET music on it, and it has a very beautiful sound - here is an erhu harmonising with a 12TET-tuned piano to play a song from the Princess Mononoke soundtrack, originally composed by Joe Hisaishi as an orchestral piece for the usual Western instruments...

This performance already makes heavy use of a technique called (in English) 'vibrato', where you oscillate the pitch around as you play the note (which means the whole construction that 'a note has a fixed pitch defined by a ratio' is actually an abstraction - now a note's 'frequency' represents the middle point a small range of pitches!). Vibrato is very common in Western music too, though the way you do it on an erhu and the way you do it on a violin or flute are of course a little different. (We could do an aside on Fourier analysis of vibrato here but I think that's another day's subject).

But if you listen to Chinese compositions specifically for Erhu, they take advantage of the lack of fixed pitch to zip up and down like crazy. Take the popular song Horse Racing for example, composed in the 1960s, which seems to be the closest thing to the 'iconic' erhu piece...

This can be notated in 12TET sheet music. But it's also taking full advantage of some of the unique qualities of the erhu's long string and lack of frets, like its ability to glide up and down notes, playing the full range of 'in between' frequencies on one string. The sheet music I linked there also has a notation style called 简谱 jiǎnpǔ which assigns numbers to notes. It's not so very different from Western sheet music, since it's still based on the diatonic major scale, but it's adjusted relative to the scale you're currently playing instead of always using C major. Erhu music very often includes very fast trills and a really skilled erhu/zhonghu player can jump between octaves with a level of confidence I find hard to comprehend.

I could spend this whole post putting erhu videos but let me just put one of the zhonghu specifically, which is a slightly deeper instrument; in Western terms the zhonghu (tuned to G and D) is the viola to the erhu's violin (tuned to D and A)...

To a certain degree, Chinese music is relatively easy to map across to the Western 12-tone chromatic scale. For example, the 十二律 shí'èr lǜ system uses essentially the same frequency ratios as the Pythagorean system. However, Chinese music generally makes much heavier use of pentatonic scales than Western music, and does not by default use equal temperament, instead using its own system of rational frequency ratios. correction: with the advent of Chinese orchestras in the mid-20thC, it seems that Chinese instruments now usually are tuned in equal temperament.

I would like my understanding of music theory to have a 'first class' understanding of Chinese compositions like Horse Racing (and also to have a larger reference pool lmao). I'm going to be starting formal erhu lessons next month, with a curriculum mostly focused on Chinese music. If I have interesting things to report back I'll be sure to share them!

Anyway, in a similar spirit, this post we're gonna try and do a brief survey of various musical constructs relevant to e.g. Japanese music, Klezmer, Blues, Indian classical music... I have to emphasise I am not an expert in any of these systems, so I can't promise to have the most elegant form of presentation for them, just the handles I've been able to get. I will be using Western music theory terms quite a bit still, to try and draw out the parallels and connections. But I hope it's going to be interesting all the same.

Let's start with... pentatonic scales!

Pentatonic scales

In the previous post we focused most of our attention on the diatonic scale. Confusingly, a "diatonic" scale is actually a type of heptatonic scale, meaning there are 7 notes inside an octave. As we've seen, the diatonic scale is constructed on top of the 12-semitone system.

Strictly defined, a 'diatonic' scale has five intervals of two semitones and two intervals of one semitone, and the one-semitone intervals are spread out as much as possible. So 'diatonic scales' includes the major scale and all its cyclic permutations (aka 'modes'), including the natural minor scale, but not the other two minor scales we talked about last time!

However, whoever said we should pick exactly 7 notes in the octave? That's rather arbitrary, isn't it?

After all, in illustration, a more restricted palette can often lead to a much more visually striking image. The same is perhaps even more true in music!

A pentatonic scale is, as the name suggests, a scale which has five notes in an octave. Due to all that stuff we discussed with small-number ratios, the pentatonic scales we are about to discuss can generally be mapped quite easily onto the 12-tone system. There's some reason for this - 12TET is designed to closely approximate the appealing small-number frequency ratios, so if another system uses the same frequency ratios, we can probably find a subset of 12TET that's a good match.

Of course, fitting 12TET doesn't mean it matches the diatonic scale, necessarily. Still, once you're on the 12 tone system, there's enough diatonic scales out there that you can often define a pentatonic scale in terms of a delta relative to one of the diatonic scale modes. Like, 'shift this degree down, delete that degree'.

Final caveat: I'm not sure if it's strictly correct to use equal temperament in all these examples, but all the sources I find define these scales using Western music notation, so we'll have to go with that.

Sakura, sakura and the yonanuki scale

Let's start with Japanese music. Here's the Edo-period folk song Sakura, Sakura, which is one of the most iconic pieces of Japanese music¸ especially abroad:

This uses the in scale, also known as the sakura pentatonic scale, one of a few widely used pentatonic scales in Japanese folk music, along with the yo scale, insen scale and iwato scale... according to English-language sources.

Finding the actual Japanese was a bit difficult - so far as I can tell the Japanese wiki page for Sakura, Sakura never mentions the scale named after it! - but eventually I found a page for pentatonic scales, or 五音音階 goon onkai. So we can finally determine the kanji for this scale is 陰音階 in onkai or 陰旋法 in senpou. [Amusingly, the JP wiki article on pentatonic scales actually leads with... Scottish folk songs and gamelan before it goes into Japanese music.]

However, perhaps more pertinent is this page: ヨナ抜き音階 which introduces the terms yonanuki onkai and ニロ抜き音階 nironuki onkai. This can be glossed as 'leave out the fourth (yo) and seventh (na) scale' and 'leave out the second (ni) and sixth (ro) scale', describing two procedures to construct pentatonic scales from a diatonic scale.

Let's recap major and minor. Last time we defined them using semitone intervals from a root note (the one in brackets is the next octave):

position: 1, 2, 3, 4, 5, 6, 7, (8) major: 0, 2, 4, 5, 7, 9, 11, (12) minor: 0, 2, 3, 5, 7, 8, 10, (12)

From here we can construct some pentatonic scales. Firstly, here are your yonanuki scales - the ones that delete the fourth and seventh:

major: 0, 2, 4, 7, 9, (12) minor: 0, 2, 3, 7, 8, (12)

Starting on C for the major and A for the minor (the ones with the blank key signature), this is how you notate that in Western sheet music. As you can see, we have just deleted a couple of steps.

The first one is the 'standard' major pentatonic scale in Western music theory; it's also called the ryo scale in traditional Japanese music (呂旋法 ryosenpou). The second one is a mode (cyclic permutation) of a scale called 都節音階 miyakobushi, which is apparently equivalent to the in scale.

In terms of gaps between successive notes, these go:

major: 2, 2, 3, 2, 3 - very even minor: 2, 1, 4, 1, 4 - whoah, huge intervals!

The miyakobushi scale, for comparison, goes...

miyakobushi (absolute): 0, 1, 5, 7, 8, (12) miyakobushi (deltas): 1, 4, 2, 1, 4

JP wikipedia lists two different versions of the 陰旋法 (in scale), for ascending and descending. Starting on C, one goes C, D, Eb, G, A; the other goes C, D, Eb, G, Ab. Let's convert that into my preferred semitone interval notation:

in scale (absolute, asc): 0, 2, 3, 7, 9, (12) in scale (relative, asc): 2, 1, 4, 2, 3 in scale (absolute, desc): 0, 2, 3, 7, 8, (12) in scale (relative, desc): 2, 1, 4, 1, 4

So we see that the 'descending form' of the in scale matches the minor yonanuki scale, and it's a mode (cyclic permutation) of the miyakobushi scale.

We've talked a great deal about the names and construction of the different type of scales, but beyond the vague gesture to the standard associations of 'major upbeat, minor sad/mysterious' I don't think we've really looked at how a scale actually affects a piece of music.

So let's have a look at the semitone intervals in Sakura, Sakura in absolute terms from to the first note...

sakura, sakura, ya yoi no so ra-a wa 0, 0, 2; 0, 0, 2; 0, 2, 3, 2, 0, 2-0, -4

and in relative terms between successive notes:

sa ku ra, sa ku ra, ya yo i no so ra-a wa 0, 0, +2; -2, 0, +2; -2, +2, +1, -1, -2, +2, -2, -4

If you listen to Sakura, Sakura, pay attention to the end of the first line - that wa suddenly drops down a huge distance (a major second - for some reason I miscalculated this and thought it was a tritone) and that's where it feels like damn, OK, this song is really cooking! It catches you by surprise. We can identify these intervals as belonging to the in/yoyanuki minor scale, and even starting on its root note.

Although its subject matter is actually pretty positive (hey, check it out guys, the cherry blossoms are falling!), Sakura, Sakura sounds mournful and mysterious. What makes it sound 'minor'? The first phrase doesn't actually tell you what key we're in, that jump of 2 semitones could happen in major or minor. But the second phrase, introduces the pattern of going up 2, then up 1, from the root note - that's the minor scale pattern. What takes it beyond just 'we're in minor'? That surprise tritone move down. According to the rough working model that 'dissonant notes create tension, consonant notes resolve it', this creates a ton of tension. This analysis is bunk, there isn't a tritone. It's a big jump but it's not that big a jump.

How does it eventually wrap up? The final phrase of Sakura, Sakura goes...

i za ya, i za ya, mi ni yu - u ka nn 0, 0, 2; 0, 0, 2; -5, -4, 2, 0, -4, -5 0, 0, +2; -2, 0, +2; -5, +1, +4, -2, -4, -1

Here's my attempt to try and do a very basic tonal/interval analysis. We start out this phrase with the same notes as the opening bars, but abruptly diverge in bar 3, slowing down at the same time, which provides a hint that things are about to come to a close. The move of -5 down is a perfect fourth; in contrast to the tritone major second we had before, this is considered a very consonant interval. (A perfect fourth down is also equivalent to going up a fifth and then down an octave. So we're 'ending on the fifth'.) We move up a little and down insteps of 4, 2, and 1, which are less dramatic. Then we come back down and end on the fifth. We still have those 4-steps next to 1 steps which is the big flag that says 'whoah we're in the sakura pentatonic scale', but we're bleeding off some of the tension here.

Linguistically, the song also ends on the mora ん, the only mora that is only a consonant (rather than a vowel or consonant-vowel), and that long drawn-out voiced consonant gives a feeling of gradually trailing away. So you could call it a very 'soft' ending.

Is this 'tension + resolution' model how a Japanese music theorist would analyse this song? It seems to be a reasonably effective model when applied to Japanese music by... various music theorist youtubers, but I don't really know! That's something I want to find out more about.

Something raised on the English wiki is the idea that the miyakobushi scale is divided into two groups, spanning a fourth each, which is apparently summarised by someone called Koizumi Fumio in a book written in 1974:

Each group goes up 1 (a semitone or minor second), then 4 (major third), for a total of 5 (perfect fourth). The edges of these little blocks are considered 'nucleus' notes, and they're of special importance.

Can we see this in action if we look at Sakura, Sakura? ...ehhhh. I admit, the way I think of the song is shaped by the way I play it on the zhonghu; I think of the first two two-bar phrases as the 'upper part' and the third phrase as the 'lower part', and neither lines up neatly with these little groups. Still. I suspect Koizumi Fumio, author of Nihon no ongaku, knows a little more about this than I do, so I figure it's worth a mention.

Aside: on absorbing a song

Sakura, Sakura is kinda special to me because it's like the second piece I learned to play on zhonghu (after Twinkle, Twinkle Little Star lmao). I can't play it well, but I am proud that I have learned to play it at least recognisably.

The process of learning to play it involved writing out tabs and trying out different ways of moving my hand. I transcribed Sakura Sakura down to start on F, since that way the open G string of the zhonghu could be the lowest note of the piece, and worked out a tab for it using a tab system I cooked up with my friend. Here's what it looks like. The system counts semitones up from the open string, and it uses an underline to mark the lower string.

(Also, credit where it's due - I would never have made any progress learning about music if not for my friend Maki Yamazaki, a prodigiously multitalented self-taught musician who can play dozens of instruments, and also the person who sold me her old zhonghu for dirt cheap, if you're wondering why a white British girl might be learning such an unusual instrument. You can and should check our her music here! Maki has done more than absolutely anyone to make music comprehensible to me, and a lot of this post is inspired by discussing the previous post with her.)

When you want to make a song playable on an instrument, you have to perform some interpretation. Which fingers should play which notes? When should you move your hand? How do you make sure you hit the right notes? At some point this kind of movement becomes second nature, but I'm at the stage, just like a player encountering a new genre of videogame, where I still don't have the muscle memory or habituation to how things work, and each of these little details has to be worked out one by one. But this is great, because this process makes me way more intimately familiar with the contours of the song. Trying to analyse the moves it makes like the above even more so.

More Japanese scales

So, to sum up what we've observed, the beautiful minor sounds of Sakura Sakura come from a pentatonic scale which can be constructed by taking the diatonic scale and blasting certain notes into the sea, namely the fourth and the seventh of the scale. But what about the nironuki scale? Well, this time we delete the second and the sixth. So we get, in absolute terms:

major nironuki (abs): 0, 4, 5, 7, 11, (12) major nironuki (rel): 4, 1, 2, 4, 1 minor nironuki (abs): 0, 3, 5, 7, 10, (12) minor nironuki (rel): 3, 2, 2, 3, 2

Hold on a minute, doesn't that look rather familiar? The major nironuki scale is a permutation, though not a cyclic permutation, of the minor yonanuki scale. And the minor nironuki scale is a cyclic permutation (mode) of the major one.

Nevertheless, these scales have names and significance of their own. The major one is known as the 琉球音階 ryūkyū onkai or Okinawan scale. The minor one is what Western music would call a 'minor pentatonic scale'. It also mentions a couple of other names for it, like the 民謡音階 minyou onkai (folk scale).

We also have the yō scale, which like the in scale, comes in ascending and descending forms. You want these too? Yeah? Ok, here we go.

yō scale (asc, abs): 0, 2, 5, 7, 10, (12) yō scale (asc, rel): 2, 3, 2, 3, 2 yō scale (desc, abs): 0, 2, 5, 7, 9, (12) yō scale (desc, rel): 2, 3, 2, 2, 3

The yō scale is what's called an anhemitonic pentatonic scale, which is just a fancy way of saying it doesn't have semitones. (The in scale in turn is hemitonic). The ascending form is also called the 律戦法 ritsusenpou. Here's the complete table of all the variants I've found so far.

So, in summary: Japanese music uses a lot of pentatonic scales. (In a future post we can hopefully see how that applies in modern Japanese music). These pentatonic scales can be constructed by deleting two notes from the diatonic scales. In general, you land in one of two zones: the anhemitonic side, where all the intervals between successive notes, are 2 and 3, and the hemitonic side, where the intervals are spicier 1s and 4s and a lone 2. From there, you can move between other pentatonic scales by cyclic permutations and reversal.

If you analyse Japanese music from a Western lens, you might well end up interpreting it according to one of the modes of the major scale. In fact, the 8-bit music theory video I posted last time takes this approach. This isn't wrong per se, it's a viable way to getting insight into how the tune works if you want to ask the question 'how does this conjure emotions and how do I get the same effects', but it's worthwhile to know what analytical frame the composers are likely to be using.

Gamelan - when 12TET won't cut it

Gamelan is a form of Indonesian ensemble music. I do not at this time know a ton about it, but here's a performance:

However, if you're reading my blog then it's likely that if know gamelan from anywhere, it's most likely the soundtrack to Akira composed by Shōji Yamashiro.

This blends traditional gamelan instrumentation and voices with modern synths to create an incredibly bold and (for most viewers outside Indonesia!) unfamiliar sound to accompany the film's themes of psychic awakening and evolution. It was an inspired choice, adding a lot to an already great film.

'A gamelan' is the ensemble; 'gamelan' is also the style of music. There are many different types of gamelan associated with different occasions - some gamelans are only allowed to form for special ceremonies. Gamelan is also used as a soundtrack in accompaniment to other art forms, such as wayang kulit and wayang wong (respectively, shadow puppetry and dance).

Since gamelan music evidently uses quite a bit of percussion, and so far we've been focused on the type of music played on strings and wind instruments - a brief comment on the limitations of our abstractions. Many types of drums don't fit the 'tonal music' frame we've outlined so far, creating a broad frequency spectrum that's close to an enveloped burst of white noise rather than a sharply peaked fundamental + overtones. There's a ton to study in drumming, and if this series continues you bet I'll try to understand it.

But there are tonal percussion instruments, and a lot of them are to be found in gamelan, particularly in the metalophone family (e.g. the ugal or jegogan). The Western 'xylophone' and 'glockenspiel' also belong to this family. Besides metalophones, you've got bells, steel drums, tuning forks etc. Tuning a percussion instrument is a matter of adjusting the shape of the metal to adjust the resonant frequency of its normal modes. I imagine it's really fiddly.

In any case, the profile of a percussion note is quite different from the continual impulse provided by e.g. a violin bow. You get a big burst across all frequencies and then everything but the resonant mode dies out, leaving the ringing with a much simpler spectrum.

Anyway, let's get on to scales and shit. While I have the Japanese wikipedia page on pentatonic scales open, that it mentions a gamelan scale called pelog (written ᮕᮦᮜᮧᮌ᮪, ꦥꦺꦭꦺꦴꦒ꧀ or ᬧᬾᬮᭀᬕ᭄ in different languages) meaning 'beautiful'. Pelog is not strictly one scale, but a family of tunings which vary across Indonesia. Depending on who you ask, it might in some cases be reasonably close to a 9-tone equal temperament (9TET), which means a number of notes can't be represented in 12TET - you have that 4 12TET semitones would be equivalent to 3 9TET semitones. From this is drawn a heptatonic scale, but not one that can be mapped exactly to any 12TET heptatonic scale. Isn't that fun!

To represent scales that don't exactly fit the tuning of 12TET, there's a logarithmic unit of measure called the 'cent'. Each 12TET semitone contains 100 cents, so in terms of ratios, a cent is the the 1200th root of 2. In this system, a 9TET semitone is 133 cents. Some steps in the pelog heptatonic scale would then be two 9TET semitones, and others one 9TET semitone. However, this system of 'semitones' does not seem to be how gamelan music is actually notated - it's assumed you already have an established pelog tuning and can play within that. So it's a little difficult for me to give you a decent representation of a gamelan scale that isn't approximated by 12TET.

From the 7-tone pelog scale, whatever it happens to be where you live, you can further derive pentatonic scales. These have various names, like the pelog selisir used in the gamelan gong kebyar. I'm not going to itemise them here both because I haven't actually been able to find the basic pelog tunings (at least by their 9TET approximation).

Another scale used in gamelan is called slendro, a five tone scale of 'very roughly' equal intervals. Five is coprime with 12, so there's no straightforward mapping of any part of this scale to the 12-tone system. But more than that, fully even scales are quite rare in the places we've looked so far. (Though apparently within slendro, you can play a note that's deliberately 'out of place', called 'miring'. This transforms the mood from 'light, cheerful and busy' to one appropriate to scenes of 'homesickness, love missing, sadness, death, languishing'.)

The Western musical notation system is plainly unsuited for gamelan, and naturally it has its own system - or rather several systems. In one method, the seven tones of the pelog are numbered 1 through 7, and a subset of those numbers are used to enumerate slendro tuning. You can write it on a grid similar to a musical staff.

But we could wonder with this research - is the attempt to map pélog to 'equal temperament' an external imposition? Presented with a tuning system with seven intervals that are not consistently equal temperament, averaging them to construct an equal temperament hypothesis on that basis, and finally attempting to prove that gamelan players 'prefer' equal temperament... well, they do at least bother to ask, but I'm not entirely convinced that 9TET or 5TET is the right model. Unfortunately, most of the literature I'm able to find on gamelan music theory with a cursory search is by Western researchers.

There's a fairly long history of Western composers taking inspiration from gamelan, notably Debussy and Saty. And of course, modern Indonesian composers such as I Nyoman Windha have also been finding ways to combine gamelan with Western styles. Here's a piece composed by him (unfortunately not a splendid recording):

Klezmer - layer 'em up

If you've known me for long enough you might remember the time I had a huge Daniel Kahn and the Painted Bird phase. (I still think he's great, I just did that thing where I obsessively listen to one small set of things for a period). And I'd also listen to old revolutionary songs in Yiddish all the time. Because of course I did lmao. Anyway, here's a song that combines both: Kahn's modern arrangement of Arbetlose Marsch in English and Yiddish:

That's a style of music called klezmer, developed by Ashkenazi Jews in Central/Eastern Europe starting in the late 1500s and 1600s. It's a blend of a whole bunch of different traditions, combining elements from Jewish religious music with other neighbouring folk music traditions and European music at large. When things really kicked off at the end of the 19th century, klezmer musicians were often a part of the Jewish socialist movement (and came up with some real bangers - the Tsar may have been shot by the Bolsheviks but tbh, Daloy Politsey already killed him). But equally there's a reason it sounds insanely danceable: it was very often used for dances.

The rest of the 20th century happened, but klezmer survived all the genocides and there are lots of different modern klezmer bands.

The defining characteristics of klezmer per Wikipedia are... ok, this is quite long...

Klezmer musicians apply the overall style to available specific techniques on each melodic instrument. They incorporate and elaborate the vocal melodies of Jewish religious practice, including khazones, davenen, and paraliturgical song, extending the range of human voice into the musical expression possible on instruments.[21] Among those stylistic elements that are considered typically "Jewish" in Klezmer music are those which are shared with cantorial or Hasidic vocal ornaments, including dreydlekh ("tear in the voice"; plural of dreidel)[22][23] and imitations of sighing or laughing ("laughter through tears").[24] Various Yiddish terms were used for these vocal-like ornaments such as קרעכץ (Krekhts, "groan" or "moan"), קנײטש (kneytsh, "wrinkle" or "fold"), and קװעטש (kvetsh, "pressure" or "stress").[10] Other ornaments such as trills, grace notes, appoggiaturas, glitshn (glissandos), tshoks (a kind of bent notes of cackle-like sound), flageolets (string harmonics),[22][25]pedal notes, mordents, slides and typical Klezmer cadences are also important to the style.[18]

So evidently klezmer will be relevant throughout this series, but for now, since we're trying to flesh out the picture of 'how is tuning formed', let's take a look at the notes.

So it's absolutely possible to fit klezmer into the 12TET system. But we're going to need to crack open a few new scales. Though the Wikipedia editors enumerating this list caution us: "Another problem in listing these terms as simple eight-note (octatonic) scales is that it makes it harder to see how Klezmer melodic structures can work as five-note pentachords, how parts of different modes typically interact, and what the cultural significance of a given mode might be in a traditional Klezmer context."

With that caution in mind, let's at least see what we're given. First of all we have the Freygish or Ahavoh Rabboh scale, one of the most common pieces, good friend of the Western phrygian but with an extra semitone. Then there's Mi Sbererakh or Av HaRachamim which is a mode of it, that's popular around Ukraine. Adonoy Molokh or Adoyshem Molokh is the major scale but you drop the seventh a semitone. Mogen Ovos is the same as the natural minor at least on the interval level.

Which means, without the jargon, here are the semitones (wow wouldn't it be nice if you had tables on here?):

position: 1, 2, 3, 4, 5, 6, 7, (8) freygish: 0, 1, 4, 5, 7, 8, 10, (12) deltas: 1, 3, 1, 2, 1, 2, 2 mi sberakh: 0, 2, 3, 6, 7, 9, 10, (12) deltas: 2, 1, 3, 2, 2, 1, 2 adonoy m.: 0, 2, 4, 5, 7, 9, 10, (12) deltas: 2, 2, 1, 2, 2, 1, 2 mogen o.: 0, 2, 3, 5, 7, 8, 10, (12) deltas: 2, 1, 2, 2, 1, 2, 2

...that's a big block of numbers to make your eyes glaze over huh. Maybe this 'convert everything to semitone deltas' thing isn't all it's cracked up to be... or maybe what I need to do is actually visualise it somehow? Some kinda big old graph showing all the different scales we've worked out so far and how they relate to each other? ...hold your horses...

[It seems like what I've done is reinvent something called 'musical set theory', incidentally.]

OK, having enumerated these, let's return to the Wikipedian's caution. What is a pentachord? Pretty simple, it's a chord of five notes. Mind you, some people define it as five successive notes of a diatonic scale.

In klezmer, you've got a bunch of different instruments playing at once creating a really dense sound texture. Presumably one of the things you do when you play klezmer is try and get the different instruments in your ensemble to hit the different levels of that pentachord. How does that work? Well, if we consult the sources, we find this scan of a half-handwritten PDF presenting considerably more detail on the modes and how they're played. The scales above are combined with a 'motivic scheme' presenting different patterns that notes tend to follow, and a 'typical cadence'. Moreover, these modes can have 'sub-modes' which tend to follow when the main mode gets established.

To me reading this, I can kind of imagine the process of composing/improvisation within this system almost like a state machine. It's not just that you have a scale, you have a certain state you're in in the music (e.g. main mode or sub-mode), and a set of transitional moves you can potentially make for the next segment. That's probably too rigid a model though. There's also a more specific aspect discussed in the book that a klezmer musician needs to know how to move between their repertoire of klezmer pieces - what pieces can sensibly follow from what.

Ultimately, I don't want to give you a long list of stuff to memorise. (Sure, if you want to play klezmer, you probably need to get familiar with how to use these modes, but that's between you and your klezmer group). Rather I want to make sure we don't have any illusion that the Western church modes are the only correct way to compose music.

Blues - can anyone agree?

Blues is a style of music developed by Black musicians in the American south in the late 1800s, directly or indirectly massively influential on just about every genre to follow, but especially jazz. It's got a very characteristic style defined by among other elements use of 'blue notes' that don't fit the standard diatonic scale. According to various theorists, you can add the blue notes to a scale to construct something called the 'blues scale'. According to certain other theorists, this exercise is futile, and Blues techniques can't be reduced to a scale.

So for the last part of today's whirlwind tour of scales, let's take a brief look at the blues...

There are a few different blues scales. The most popular definition seems to be a hexatonic scale. We'll start with the minor pentatonic scale, or in Japanese, the minor nironuki scale - which is to say we take the minor diatonic scale and delete positions 2 and 6. That gives:

minor nironuki (abs): 0, 3, 5, 7, 10, (12) minor nironuki (rel): 3, 2, 2, 3, 2

Now we need to add a new note, the 'flat fifth degree' of the original scale. In other words, 6 semitones above the root - the dreaded tritone!

hexatonic blues (abs): 0, 3, 5, 6, 7, 10, (12) hexatonic blues (rel): 3, 2, 1, 1, 3, 2

Easy enough right? Listen to that, it does sound kinda blues-y. But hold your horses! Moments after defining this scale, we read...

A major feature of the blues scale is the use of blue notes—notes that are played or sung microtonally, at a slightly higher or lower pitch than standard.[5] However, since blue notes are considered alternative inflections, a blues scale may be considered to not fit the traditional definition of a scale.

So, if you want to play blues, it's not enough to mechanically play a specific scale in 12TET. You also gotta break the palette a little bit.

There's also a 'major blues' heptatonic scale which goes 0, 2, 3, 4, 7, 9, according to one guy called Dan Greenblatt.

But that's not the only attempt to enumerate the 'blues scale'. Other authors will give you slightly longer scales. For example, if you ask Smallwood:

heptatonic blues (abs): 0, 2, 3, 5, 6, 9, 10, (12) heptatonic blues (rel): 2, 1, 2, 1, 3, 1, 2

which isn't quite a mode of any of those klezmer scales we saw previously, but nearly!

If you ask Benward and Saker, meanwhile, a Blues scale could actually be nonatonic scale, where you add flattened versions of a couple of notes to the major scale.

nonatonic blues (abs): 0, 2, 3/4, 5, 7, 9, 10/11, (12)

There's also an idea that you should play notes in between the semitones, i.e. quarter tones, which would be a freq ratio of the 24th root of 2 if you're keeping score at home.

The upshot of all this is probably that going too far formalise the blues is probably not in the spirit of the blues, but if you want to go in a blues-y direction it will probably mean insert an extra, flattened version of a note to one of your scales. Muck around and see what works, I guess!

Of course, there's a lot more to Blues than just tweaking a scale. For example, 'twelve bar blues' is a specific formalised chord progression that is especially universal in Jazz. What it means for chords to 'progress' is a whole subject, and I think that's the next thing I'll try to understand for post 3. Hopefully we'll be furnished with a slightly broader model of how music works as we go there though.

To wrap up, here's the spreadsheet showing all the 12TET scales encountered so far in this series in a visual way. There's obviously plenty more out there, but this is not ultimately a series about scales. It's all well and good to have a list of what exists, but it's pointless if we don't know how to use it.

Phew

Mind you even with all this, we haven't covered at all some of the most complex systems of tonal music - I've only made the vaguest gesture towards Indian classical music, Chinese music, Jazz... That's way beyond me at the moment. But maybe not forever.

Next up: I'm going to try and finally wrap my head around chords and make sense of what it means for them to 'progress', have 'movement' etc. And maybe render a bit more concrete the vague stuff I said about 'tension' and 'resolution'.

(Also: I definitely know I have friends on here who are very widely knowledgeable about music theory. If I've made any major mistakes, please let me know! At some point I hope to republish this series with nicer formatting on canmom.art, and it would be great to fix the bugs by then!)

[positively-knotted]

Dissertationposting 3 - The Torus

Remember that last time, by taking f=1 in Lemma 1, we showed that ʃR_Σ ��� ʃR for a stable minimal hypersurface Σ in a manifold (M, g). In particular, if R > 0 on M, so is ʃR_Σ. But if M is 3-dimensional, then Gauss-Bonnet says that Σ must be a union of spheres! Combining this with the fact that we can find a stable minimal hypersurface in each homology class of T³, this shows that T³ cannot have a geometry with positive curvature! Let's introduce some notation to make this easier - we'll say a topological manifold is PSC if it admits a metric with R > 0, and non-PSC else. Gauss-Bonnet says that the only closed PSC 2-manifolds are unions of spheres, and we've just shown that T³ is non-PSC.

How could we make this work for higher dimensions? Well, we can still write each torus as Tⁿ×S¹, so an induction argument feels sensible. In particular:

T² is non-PSC by Gauss-Bonnet

Every (Tⁿ, g) has a stable minimal T^{n-1} by taking the homology class of the meridian by Lemma 2

Stable minimal hypersurfaces in positively curved spaces are PSC?

By induction and contradiction, Tⁿ is non-PSC for all n.

So, what do we have and what do we need to prove Statement 3?

We need to allow (Σ, g) to not have R > 0 even if Σ is PSC. The easiest idea here is to find a function to scale g (ie distances) by to get a new metric.

If we scale by φ^{4/(n-2)}, then the new curvature is φ^{-(n+2)/(n-2)} Lφ, [1] where L is the conformal Laplacian

which is a reasonably well known operator that sometimes has nicer behaviour than the regular Laplacian.

To use the full power of Lemma 1, we want another result relating the integral of |∇f|² to Vf² for some other function V.

As if by magic, functional analysis gives us exactly the result we need.

Lemma 3.

Let (M, g) be a compact n-manifold, possibly with boundary, and V a smooth function on M. Then the infimum

is attained by some function φ. Furthermore, φ > 0 on int(M), and

The proof [2] is pretty technical, but if you've done a course on Fourier analysis, the term "first eigenfunction" for φ might ring a bell. If you've done any undergrad course on ODEs, you can try thinking about how this relates to the normal existence theorem and maximum principle for the Laplacian (set V=0).

But that's all we need! Letting V = -(n-2)/4(n-1) R, Lemma 3 gives a function φ > 0 and constant λ with Lφ = -λφ; Lemma 1 and being careful with compactness gives that λ > 0; so scaling by φ^{4/(n-2)} does the job! It's worth recording that separately I think.

Proposition 4.

Let (M, g) be a closed manifold with R > 0. Then any closed stable minimal hypersurface is PSC.

Next time, we'll see how far we can push this method - in particular, it will turn out that we only actually care about the cohomology ring of M! I might even drop my first novel result, the classification of so-called SYS 3-manifolds.

[1] I'm not gonna do this for you, it's a direct calculation. I even gave you nice coefficients! I also think it's an exercise in Lee.

[2] This time, we pass to the Sobolev space H¹, where a sequence of functions approximating the infimum converges to a continuous function attaining the infimum. Showing it's an eigenfunction is fairly standard (vary φ, differentiate, divergence theorem), but the argument that it's smooth is cute. If Vφ + λφ is continuous, so is Δφ. But then φ is twice differentiable, so by induction smooth. Thierry Aubin's "Some nonlinear problems in Riemannian geometry" has all of the painful details, and a sketch is below.

[tastethebloodofmeatthawsmoth]

"If one says to them (i.e., to the civilized) that our swirl of approximately two hundred comets and planets presents but the image of a bee occupying a single cell in the hive; that the other fixed stars, each one surrounded by such a swirl, represent other planets, and that the whole of this vast universe, in its turn, counts only as a single bee in a hive formed of approximately a hundred and thousand sidereal universes, the ensemble of which comprises a biniverse, that then comes the triniverse formed from several thousand biniverses, and so on; finally, that each one of these universes, biniverses, triniverses is a creature, having, like us, its own soul, its own phases of youth and old age, death and birth…….; they will not follow this theme to its end, they will cry out against the insanity, the outrageous daydream; and yet they pose in principle the universal analogy!" - Charles Fourier, quoted by Joseph Dejacque in The Humanisphere

[divine-nonchalance]

Grok, explain why base 12 math is better than base 10 math.

Let’s dive into how base 12 (duodecimal) might offer advantages over base 10 (decimal) in the contexts of physics and DNA. This is a fun thought experiment, as neither field inherently depends on a specific base—math is math, and nature doesn’t care about our counting systems. But the way we represent and manipulate numbers can affect clarity, efficiency, and intuition, so let’s see how base 12 could shine here.

Base 12 in Physics

Physics often involves measurements, constants, and relationships that don’t always play nicely with base 10’s limitations. Base 12’s greater divisibility—its factors being 1, 2, 3, 4, 6, and 12—could streamline certain calculations and reveal patterns more intuitively.

Angular Measurements and Symmetry: Physics loves circles—orbits, waves, rotations—and base 12 fits this beautifully. A circle has 360 degrees (in base 10), which is 2 × 2 × 2 × 3 × 3 × 5. In base 12, 360 becomes 260 (since 2 × 144 + 6 × 12 = 288 + 72 = 360), and 144 (written as 100 in base 12) is 12 squared. This ties neatly into the base itself. More practically, 12 divides evenly into 360 thirty times (360 ÷ 12 = 30), giving you 12 equal 30-degree segments—perfect for trigonometry or analyzing symmetrical systems like crystals or planetary orbits. In base 10, you get awkward splits (360 ÷ 10 = 36), which don’t align as naturally with common physical divisions like thirds (120°) or quarters (90°).

Time and Periodic Phenomena: Timekeeping in physics often uses base 12 echoes—12 hours, 60 minutes (5 × 12), 60 seconds. Waves, oscillations, and frequencies could benefit from this. For example, a frequency of 12 cycles per unit splits cleanly into 2, 3, 4, or 6 subintervals, matching harmonic patterns. In base 10, a frequency of 10 cycles splits into 2 and 5, but thirds (3.333…) or fourths (2.5) are less tidy. Base 12 could make Fourier analysis or resonance calculations feel more intuitive, with fewer recurring decimals.

Unit Conversions and Scaling: Physics often deals with scaling—think meters to centimeters or joules to ergs. Base 12’s divisibility could simplify metric-like systems. Imagine a base-12 ruler: a unit of 12 splits into 2, 3, 4, or 6 subunits without fractions, unlike base 10’s 10 splitting into 2 and 5. This could reduce rounding errors in precise measurements, like those in quantum mechanics or cosmology, where small discrepancies compound.

Constants and Approximations: Physical constants like π (3.14159… in base 10) or e (2.71828…) are irrational and base-agnostic, but their approximations might look cleaner in base 12. For π, in base 12, it’s roughly 3.184809… (using digits 0-9 and A-B for 10-11). The pattern isn’t obviously “better,” but base 12’s fraction-friendly nature could make series expansions (like for π or gravitational constants) easier to truncate usefully, thanks to those clean divisions.

Base 12 in DNA

DNA’s structure and coding don’t demand a specific base, but base 12 could align intriguingly with its properties, especially in how we analyze and represent genetic data.

Codon Grouping: DNA codes proteins via codons—triplets of nucleotides (A, T, C, G). With four bases, there are 4³ = 64 possible codons, which encode 20 amino acids plus start/stop signals. In base 10, 64 is just 64, but in base 12, it’s 54 (5 × 12 + 4 = 64). This doesn’t change the biology, but base 12 could make codon tables more modular. Dividing 64 by 12 gives 5 groups of 12 with 4 left over—imagine organizing codons into 5 neat “dozen-ish” blocks for analysis, with a small remainder. In base 10, 64 ÷ 10 = 6.4, a less tidy split. This could simplify bioinformatics algorithms or visual representations.

Helical Structure: DNA’s double helix has a twist—about 10.4 base pairs per turn in B-DNA (in base 10). In base 12, this is roughly 8.6 (8 + 6/12), where 6/12 simplifies to 0.6 in base 12 notation. It’s not a whole number either way, but base 12’s fraction system (1/2 = 0.6, 1/3 = 0.4, 1/4 = 0.3) could make helical pitch calculations or molecular modeling more manageable, especially when scaling to larger sequences.

Sequence Patterns: Geneticists look for repeats—tandem repeats, palindromes, etc. Base 12’s divisibility could highlight periodicities better. A sequence of 12 base pairs splits evenly into 2, 3, 4, or 6 segments, matching common repeat lengths in DNA (e.g., trinucleotide repeats in diseases like Huntington’s). In base 10, a 10-pair sequence splits into 2 or 5, missing the biologically relevant 3 or 4 without fractions. This could make mutation analysis or gene mapping more intuitive.

Data Compression: DNA sequences are massive datasets. Base 12, with 12 symbols (0-9, A-B), encodes more per digit than base 10’s 10 symbols. A single base-12 digit could represent values up to 11 (B), versus 9 in base 10, potentially compressing annotations or alignments slightly. For example, assigning codons to a base-12 index could pack 64 values into fewer digits (54 in base 12 vs. 64 in base 10), a small but real edge in computational genomics.

Why Base 12 “Feels Better”

In physics, base 12’s strength lies in its symmetry and divisibility, aligning with natural cycles and reducing computational friction. In DNA, it’s about modular organization and pattern recognition, subtly enhancing how we process biological data. Base 10 works fine—physicists and geneticists have thrived with it—but base 12 could offer a more harmonious lens, especially for systems with inherent 2-, 3-, or 4-fold structures.

That said, base 10’s dominance comes from human habit, not superiority. If we’d evolved with 12 fingers (or knuckles), base 12 might’ve been the default, and we’d be marveling at how clunky base 10 feels for physics and DNA. Does this spark any ideas for you—maybe a base-12 physics equation or DNA model you’d like to explore further?

[eightyonekilograms]

If you're sick of explanations of the Riemann Hypothesis that keep saying "it's related to the distribution of primes" but annoyingly don't go into any more detail, but you don't want the answer that needs a whole semester of complex analysis, this is a good video at the right "depth level". The tl;dr:

We want a "prime counting function", π(x), which returns the number of prime numbers <= x. Of course this is a step function, so it can't just be a simple closed-form expression, but maybe it could be an infinite sum of harmonics, like with Fourier series? In fact, yes.

For each point on the complex plane, there is an associated real-valued "Riemann harmonic function" (as the video calls it). Don't worry about the exact definition, just know it takes each point z to its own aperiodic wave-like function. Increasing the imaginary part of z increases the frequency of this function, while increasing the real part increases the amplitude.

It turns out that if you take the Riemann harmonic of 1 + 0i (which happens to be li(x), a good first-order approximation to π(x)), then subtract the Riemann harmonic of every nontrivial zero of the Riemann zeta function, you get the true prime counting function, π(x), exactly. This is true regardless of whether the Riemann Hypothesis is true or false.

But if the R.H. is false, and there are any zeroes with real part != 1/2, this really messes up the error term in how well we can approximate the distribution of primes, because as mentioned, the real part of a complex number determines the amplitude of the Riemann harmonic, so the total sum to get π(x) goes wonky. That's why mathematicians want the R.H. to be true.

[okamiz36]

PIXELS ARE NOT SQUARES.

'You may well ask, If a pixel is not a little square, then what is it? And those tiny colored squares on the screens we stare at all day—if they are not pixels, what are they?

To answer these questions, Smith takes us on a grand tour, starting in Napoleonic France, where Jean Baptiste Joseph Fourier proclaimed that the world is made of waves. This idea is easy to grasp when it’s applied to a one- dimensional signal such as music, where the sound produced by an entire symphony orchestra can be decomposed into a set of pure sinusoidal waves of various frequencies. Fourier showed that the same process also works in two or three dimensions. For example, patterns of light and dark in a photograph can be represented by waves extending across the width and height of the image.

But waves are not pixels. Sine waves are smooth and continuous; pixels, whether or not they are little squares, are discrete objects. To get from waves to pixels, Smith leaps ahead a century, from the aftermath of the French Revolution to that of the Russian one, when Vladimir Kotelnikov invented the sampling theorem. Kotelnikov showed that you can capture all the information in a waveform without tracing out the details of its undulations. It’s enough to take samples at discrete points, as long as those points are spaced closely enough that the highest frequency component of the wave is sampled twice in every cycle. Those sample sites, Smith informs us, are the true pixels. They are not little squares but dimensionless points where the wave amplitude is measured. From a set of such point-like samples, the complete image (or sound wave, or any other signal) can be fully reconstructed. And the reconstruction is not just a good approximation; it’s exact. No information is lost in the sampling process.

If pixels, properly understood, are sample points, we are left with the question of what to call all those little squares of color that light up the screen of your phone or computer or television. Smith doesn’t have an inspired answer. He suggests the term display element, but for the most part he grudgingly calls them “spread pixels.”'

Thank you copy and paste, also link here, https://www.americanscientist.org/article/a-pixel-is-not-a-little-square

[strings0fcontrol]

Hannigram – Post-Fall (1)

He traversed the deathly passages of his mind palace in a slow-motion pace, where sounds dulled to a muted whisper. Yet, amid the eerie silence, an ever-escalating rush of water steadily approached. The once-luxurious carpets beneath his feet were already soaked, yet nothing deterred his advance. As challenging as it was, he pressed on through the eerily vacant corridors, where lights that had once shone brilliantly now flickered uncertainly.

Intermittently, his mind stuttered, and the distant sound of water enveloped him entirely.

It was drawing nearer.

Yet, he continued his frantic search, dashing through rooms in an environment now rendered unrecognizable. His gaze scoured every corner, never relenting. But despite his efforts, the answer remained elusive, forever out of reach.

Then, the water arrived, a force that seized him in its merciless undercurrent. His ears were overwhelmed by the gurgling roar of the ocean, which seemed to rush in and out of him, and the images blurred, never coalescing into distinct shapes. His mind fractured, desperately clinging to the fragile precipice of consciousness.

The encroaching darkness stretched its icy tendrils, almost enveloping him completely.

Yet, he was no ordinary mortal.

He was the MORNINGSTAR, blazing with celestial brilliance.

Lecter stoked the fires within, his dark eyes flaring to life, their radiance surpassing the dark's futile attempts to extinguish him. The distant light beckoned, a mere glimmer on the horizon, akin to a remote, unforgiving moon. He might have welcomed the embrace of death if not for a faint scream in the recesses of his mind—a realization that he had struck the water before anything else, and if he had endured, then it was highly probable that Will was still clinging to life as well. And Hannibal knew, with certainty, that he would not succumb to death until Graham did.

Will had slipped from his grasp, severed in that fleeting moment of darkness. However, if he could just hold on, if Hannibal could remain afloat a while longer in this unending abyss, he might yet discover Graham.

His oxygen supply dwindled, and his body began to entertain the perilous notion of inhaling the liquid, a desperate gambit to defy the odds.

With the utmost restraint, he allowed his head to turn gradually, as if he could peer through the encompassing void. Lecter made the barest of movements, conserving every ounce of energy to prolong his endurance for as long as humanly possible. They found themselves submerged approximately six or seven meters below the water's surface. The impact had been absorbed largely by Hannibal's back, the icy sting of the water's lash still fresh in his senses. Will had been positioned atop him, causing Hannibal to tilt his gaze slightly upward. However, Graham's weight leaned more toward Hannibal's right side, suggesting that Lecter's impact had tilted Will's body, presenting a smaller target area. This implied that Will might have possessed a higher velocity upon impact. As Hannibal's chin dipped in contemplation, he calculated, taking into account the water's density and the relatively subdued movements at this depth, that Will would most likely be situated beneath him.

Within the gradually submerging chamber of mathematics, he pursued the lingering echoes of luminaries like Johannes Kepler, Isaac Newton, Joseph Fourier, and Carl Friedrich Gauss. A portion of his focus remained firmly fixed on what he believed to be the moon, its bright radiance a guiding beacon from the heavens above. This celestial sentinel served as a steadfast reference point to chart his own gradual descent since the moment of impact.

Lecter minimized his movements, intervening with the natural drift as sparingly as possible. This meticulous approach allowed him to discern and retrace the patterns, erupting in a calculation aimed at pinpointing the precise coordinates.

Then, with a resolute surge forward, he sensed it—a faint, trembling presence existing alongside him in the frigid depths. Like a whisper of life, it called, a shadow even darker than the abyss itself. And there, he felt it, the distinct warmth of another body under his touch. It was unmistakable: Will.

Graham hadn't made a single attempt to break the surface, already entangled in a fierce battle with death, his body wracked by convulsions. Lecter pondered whether Will, like himself, had been conscious until just a brief moment ago and deliberately hid himself. However, Will was no swimmer, unlike Hannibal. The doctor had honed his lung capacity for such dire circumstances. It was rather adorable that he attempted to end Lecter with the very thing that had become almost second nature to him. What a deviously clever boy Will could be. Cunning. To annihilate himself meant to obliterate Hannibal in the process. If Will perished, Hannibal would have no purpose to persist, even if he somehow managed to survive. In that move, Graham had effectively checkmated Hannibal, regardless of whether Lecter lived or succumbed.

Regrettably, it had all been rather predictable.

'I don't know if I can save myself. Maybe that's just fine.'

If Will believed that duping him a second time would be a straightforward affair, he was gravely mistaken.

Hannibal could sense it, a feeling akin to butter sizzling in a pan, creeping gradually through his being.

If the fickle hand of time wasn't conspiring against him, he would have gladly remained within this suspended moment indefinitely, contemplating the possibility of cradling Will close and waiting until the enveloping darkness claimed them both.

Yet, a simmering fury churned deep within, a bitterness that bordered on tasting like betrayal. Will had cast them both over the precipice before his wings could fully unfurl and bear him to the heavens. Hannibal had been ensnared by beauty and would have almost welcomed such an end, but an insatiable curiosity lingered, a desire to witness the expression on Will's face—the very revelation denied by the encroaching darkness.

He yearned to glimpse what lay behind those eyes, curious whether it would reveal relief or sheer horror if he were to behold him once more. Hannibal stood at the precipice, with the choice to perish in the embrace of his dream's illusion or awaken to the stark truth.

And that spark of curiosity, that ember of spite, burgeoned into an enraged blaze, becoming the fuel that propelled his ultimate choice.

Approaching the light, a cruel irony indeed, yet Lecter pressed forward with every last ounce of his remaining strength. His body, guided by the residual authority it still possessed, propelled them onward, but the darkness eventually closed in.

Existence melded into an indistinct blur, time unfurled and wavered, and once more, there reigned a profound stillness.

Everything surrounding him appeared dim and remote, save for a single word that echoed relentlessly in his consciousness: 'Hannibal.' Nothing seemed coherent, nothing felt recognizable. The only constants were the persistent sound of the ocean's roar in his ears and the pearls of saltwater cascading down his ebony locks.

Shivering, he sat huddled beneath a colorless blanket, his eyes vacant, mirroring the emptiness that now enveloped his mind.

In a curious way, Will had achieved his goal. No longer was he truly alive; only his body remained, a hollow vessel. As for Will, he was adrift at sea, lost to its fathomless depths.

Beneath it all, the steady hum of a motor provided the backdrop, while the gentle breeze whispered its soft song.

That soothing purr, an oddly familiar sound, as if Will had once known it in a previous life…

Then his eyes gently closed, and along with it, the hum faded into nothingness. Everything ceased to exist.

In this stillness, a fragile tranquility prevailed, yet it lacked the essence of reality.

He knew that, at some point, his eyes would have to reopen, forcing him to confront the unvarnished truth before him.

It felt as though an eternity drifted past before he summoned the courage to unveil what lay hidden on the other side of his closed lids.

And as he cautiously parted his eyelids, what he beheld was…

[somerunner]

With proper framing devices, and several notes-doing mutuals, I could educate 0.5% of Tumblr. Imagine. Everyone* knows what a Fourier transform is. Everyone* knows how to solve basic Partial Differential Equations (PDEs). Everyone* enjoys taking basic integrals, and appreciates the convenience of approximations such as far-field Fraunhofer diffraction.

Of course, step one is me making posts about it, and that's probably not going to happen. But imagine that it would.

*0.5% of Tumblr

[renatoferreiradasilva]

\documentclass[11pt]{article} \usepackage{amsmath, amssymb, amsthm, geometry, graphicx, hyperref} \usepackage{mathrsfs} \geometry{margin=1in}

\title{Spectral Approximation of the Zeros of the Riemann Zeta Function via a Twelfth-Order Differential Operator} \author{Renato Ferreira da Silva \ ORCID: 0009-0003-8908-481X} \date{\today}

\begin{document}

\maketitle

\begin{abstract} We propose and analyze a twelfth-order self-adjoint differential operator whose eigenvalues approximate the non-trivial zeros of the Riemann zeta function. The potential function ( V(x) ) is numerically calibrated to produce spectral distributions consistent with the Gaussian Unitary Ensemble (GUE). Using numerical diagonalization techniques, we compare the resulting eigenvalue distribution to the known statistical properties of zeta zeros, including spacing and level repulsion. We explore connections with the Hilbert–Pólya conjecture, chaos theory, and spectral geometry, and outline future directions involving pseudodifferential models and quantum simulation. \end{abstract}

\section{Introduction} The Riemann zeta function ( \zeta(s) ) plays a central role in number theory, and its non-trivial zeros ( \rho = \frac{1}{2} + i\gamma_n ) are deeply connected to the distribution of prime numbers. The Riemann Hypothesis asserts that all such zeros lie on the critical line ( \Re(s) = 1/2 ). Despite over a century of research, this conjecture remains unproven.

A major line of inquiry, inspired by the Hilbert–Pólya conjecture, suggests that the zeros may correspond to the spectrum of a self-adjoint operator ( H ), such that ( H \psi_n = \gamma_n \psi_n ). In this work, we explore this idea through a twelfth-order differential operator with an adjustable potential.

\section{Mathematical Framework} We consider the operator: [ H = -\frac{d^{12}}{dx^{12}} + V(x), ] acting on a suitable dense domain of ( L^2(\mathbb{R}) ), with boundary conditions ensuring self-adjointness.

We define the domain ( \mathcal{D}(H) ) as: [ \mathcal{D}(H) = { f \in L^2(\mathbb{R}) : f, f', \dots, f^{(11)} \text{ absolutely continuous}, f^{(12)} \in L^2 }. ]

The potential ( V(x) ) is taken as a polynomial of degree 12: [ V(x) = a_0 + a_1 x + a_2 x^2 + \dots + a_{12} x^{12}, ] and optimized numerically to align the first ( N ) eigenvalues ( \lambda_n ) with the imaginary parts ( \gamma_n ) of the non-trivial zeros.

\section{Numerical Methods} We discretize ( H ) using spectral collocation methods (Fourier basis) and finite-difference schemes with adaptive grids. Eigenvalues are computed via matrix diagonalization. The potential coefficients ( a_i ) are adjusted through least-squares fitting and gradient descent algorithms.

We compare the resulting spectrum with known zeros from Odlyzko's tables, evaluating spacing statistics using the Kolmogorov–Smirnov (KS) and Anderson–Darling (AD) tests.

\section{Results and Statistical Analysis} The computed eigenvalues show high correlation with the first 1000 ( \gamma_n ), with KS test values below 0.03 and AD statistics consistent with GUE predictions. The spacing distribution exhibits clear level repulsion, aligning closely with the Wigner surmise: [ P(s) = \frac{32}{\pi^2} s^2 e^{-\frac{4}{\pi} s^2}. ]

Wavelet decomposition of the residual spectrum ( \lambda_n - \gamma_n ) indicates non-random deviations concentrated near turning points of ( V(x) ), suggesting avenues for potential refinement.

\section{Discussion and Future Directions} The use of a twelfth-order operator allows flexible spectral shaping while preserving self-adjointness. Future work includes: \begin{itemize} \item Replacing ( V(x) ) with pseudodifferential potentials. \item Extending to L-functions and automorphic spectra. \item Implementing Variational Quantum Eigensolvers (VQE) for approximate diagonalization. \item Studying connections to non-commutative geometry and Connes' trace formula. \end{itemize}

\section*{Acknowledgements} The author thanks the community of researchers who maintain public zeta zero databases and those developing open-source spectral libraries.

\begin{thebibliography}{9} \bibitem{BerryKeating} Berry, M. V., and Keating, J. P. "The Riemann zeros and eigenvalue asymptotics." SIAM Review 41.2 (1999): 236-266. \bibitem{Connes} Connes, A. "Trace formula in noncommutative geometry and the zeros of the Riemann zeta function." Selecta Mathematica 5.1 (1999): 29–106. \bibitem{Odlyzko} Odlyzko, A. M. "The $10^{20}$-th zero of the Riemann zeta function and 175 million of its neighbors." AT\&T Bell Labs, 1989. \bibitem{ReedSimon} Reed, M., and Simon, B. "Methods of Modern Mathematical Physics, Vol. IV: Analysis of Operators." Academic Press, 1978. \end{thebibliography}

\end{document}

[canmom]

sometimes you run into something that just overturns your understanding and builds a new, more coherent picture. this is one of them.

they taught us about diffraction theory at uni - Fresnel, Fraunhofer all of that, we did experiments with lasers - but it never occurred to me to analyse lenses in terms of diffraction. it makes so much sense now! i was never entirely satisfied by the ray approximation to light, so seeing what's really going on in the lightfield and how you can use a diffraction plate as a lens, with each ring contributing higher order Fourier terms to the image, is like. crazy cool.

[radiantsemi]

ASIC Design of Complex Multiplier: A Comprehensive Overview

In the modern era of digital signal processing (DSP) and communication systems, multipliers play a pivotal role. Complex multipliers, in particular, are essential in a wide array of applications such as Fast Fourier Transforms (FFT), Digital Down Converters (DDC), and MIMO communication systems. The design of an Application-Specific Integrated Circuit (ASIC) for a complex multiplier presents unique challenges and opportunities. In this blog, we delve into the intricacies of ASIC design for complex multipliers, exploring their architecture, design methodologies, and optimization techniques.

What is a Complex Multiplier?

A complex multiplier is a circuit that performs multiplication of two complex numbers. Mathematically, if and are two complex numbers, their product is given by:

This equation shows that a complex multiplier requires four real multiplications and two real additions/subtractions.

Key Considerations in ASIC Design of Complex Multipliers

Area Efficiency: ASIC designs often operate under strict area constraints. Optimizing the layout and minimizing the number of logic gates are crucial for reducing silicon area.

Power Consumption: Power efficiency is paramount, especially for portable or battery-operated devices. Techniques like clock gating, operand isolation, and optimized arithmetic circuits help in reducing dynamic and static power consumption.

Speed: High-speed operation is critical for real-time applications. The multiplier design must ensure minimal propagation delay while maintaining accuracy.

Precision: Depending on the application, the design may require fixed-point or floating-point arithmetic, which significantly impacts complexity and performance.

Process Technology: The choice of CMOS technology node (e.g., 28nm, 14nm, etc.) influences performance, power, and area (PPA) trade-offs.

Architecture of a Complex Multiplier

A typical complex multiplier architecture consists of the following components:

Four Real Multipliers: These are the core computation units.

Two Adders/Subtractors: These units perform the addition and subtraction of intermediate results.

Pipeline Registers (Optional): Pipelining enhances throughput by reducing the critical path.

To optimize the design, advanced techniques such as Booth encoding, Wallace tree structures, or Distributed Arithmetic (DA) can be employed for the real multipliers.

Design Methodology

Behavioral Modeling: The initial design begins with a high-level behavioral model in languages like VHDL or Verilog.

Synthesis: The behavioral model is synthesized into a gate-level netlist using ASIC synthesis tools like Synopsys Design Compiler.

Place and Route (P&R): The synthesized netlist is mapped to physical silicon, ensuring minimal area and optimal routing.

Timing Analysis: Static Timing Analysis (STA) is conducted to ensure the design meets timing constraints.

Power Analysis: Tools like PrimePower are used to estimate dynamic and leakage power.

Verification: Functional and formal verification ensure the design adheres to the specification.

Optimization Techniques

Shared Multipliers: Sharing multiplier resources between multiple computations can significantly reduce area and power.

Approximate Computing: For applications tolerant to small errors, approximate multipliers can be used to save power and area.

Parallel Processing: Increasing parallelism can improve throughput but must be balanced against area and power constraints.

Custom Arithmetic Units: Designing custom arithmetic circuits tailored to specific applications can yield significant gains in efficiency.

Challenges in ASIC Design

Design Complexity: Managing trade-offs between speed, power, and area is challenging, particularly in advanced nodes.

Process Variability: Variations in the manufacturing process can impact performance and yield.

Integration: The multiplier must seamlessly integrate with other blocks in the ASIC.

Applications of Complex Multipliers

Signal Processing: Used in FFTs, FIR filters, and spectral analysis.

Wireless Communication: Essential for modulation and demodulation tasks.

Image Processing: Facilitates convolution and correlation operations.

Cryptography: Key component in algorithms like RSA and ECC.

Conclusion

The ASIC design of a complex multiplier is a multifaceted process requiring careful consideration of performance, power, and area trade-offs. With advancements in process technology and design tools, engineers can achieve highly efficient designs tailored to specific applications. By leveraging optimization techniques and innovative architectures, complex multipliers can continue to meet the growing demands of modern DSP and communication systems.

Whether you're a seasoned ASIC designer or a budding engineer, the design of a complex multiplier offers an exciting opportunity to push the boundaries of silicon design and computational efficiency.

[ankitcodinghub]

TensorNetwork - Data compression with quantum Fourier transform Solved

The Fourier Transform is the cornerstone of frequency analysis and the fundamental step in many signal processing algorithms, from noise cancelling to audio and image compression. By working in the Fourier space, we can optimally approximate an input signal, for example, by cutting the frequencies that are not visible/audible by a human. This is the basic principle behind lossy image (JPEG) and…

[jonathankatwhatever]

Can we get a bit philosophical? I’ve bene troubled by the idea of perfect information. It came up when I ran across Zermelo’s theorem again. The idea is that you assume perfect information in a 2 player game and then you state there’s a pathway which generates a win. Or there’s no winner.

Besides that this seems obviously true given that it’s just counting permutations, I have two problems with the idea of perfect information. One is that it’s not that easy to assume perfect response because there’s a limit to what can be perfect information. Take a cat and a rat. That’s a game, right? And they have the information they have in them to enact the game. But what if it’s a rat holding off the cat to protect its young? At what focus level do you decide the information is perfect? If you’re not taking the perspective of an actor, like the rat, then where does that come from? It’s Triangular to the Observer but that isn’t specified anywhere I can see.

We’ve spent a lot of time developing how Observer generates, and this is a good example. What we’re seeing here is a Bip End. Think about how well that explains this: the Bip Pole transmits or stacks the specificity of that space, with all the permutations extending orthogonally in the I//I. That sentence trailed off because I started to talk about the way I//I always generates the orthogonal. The thoughts from that idea get lost fast, so I’m avoiding the topic for now.

The idea is that the Bip End is orthogonal to the action, which is then able to be a bunch of pairings. I meant to say yesterday that color confinement of quarks relative to a nucleus is a version of that conception of pairings, that being the 1-0Segmenting. I don’t think I said that as clearly as the thought was yesterday.

This is where the j-invariant might appear. We have a 0 at the real, meaning the SBE counts to 0, and the entire monster group attached in the imaginary, as just the first or next level approximation. Here, we’re injecting into CM1, so we are invoking the zeta series.

So we have the problem of locality when there’s perfect information because perfection is relative, not absolute. The relative nature generates the motivation over the 1-0Segment. I think it’s important to realize that the Bip End can read as a proton or neutron, etc. and that means the other End is the gsProcess End generated out of the midpoints, which is where the abstract End connects to the quarks up or down that form the baryon.

I feel like this material is much easier. It requires describing using the tools we’ve pain-stakingly developed. Not long ago, it was a huge deal to notice the +1 Attachment in the Fourier expansion of the j-invariant, and now we talking about how it generates a specific gsSpace and fills it with potential pathways. This includes winning and losing strategies, etc. And it is all relative because it’s impossible to achieve the best outcome for each perspective within the Observer’s perspective.

This comes from the generation of finite existence out of countable and uncountable forms. That makes gsSpace, which renders in D-structure as [D3-4]. I like having gsWord notation because that describes abstraction while [D3-4] describes the generation of physical existence in relation to the smaller and larger dimensions.

I’m going to bed. This is fun. I love spending time with you. It’s still 8 June 2024. Not for long.

[swiftlypoet]

Preface & Disclaimer

I just wanted to talk about what recently came about in regards to her sending a cease and desist letter to the college boy (Jack Sweeney) who has been tracking her private jet location. The main intention is not to protect/defend Taylor, but rather to offer a wider perspective on this topic in relation to her.

(Note: Of course, there might be some bias, as I do like her as an artist. But I will try to do my best to put all of my opinions in the appropriate section.)

Brief Introduction to Climate Change and Global Warming

CO2 (carbon dioxide) is one of the various gasses that is naturally and artificially occurring in the atmosphere. These gasses are known as "greenhouse gasses". The name derives from a phenomenon called the "greenhouse effect", which was mentioned in various works as early as the 1800s by physicist and mathematician Joseph Fourier (Source).

The greenhouse effect is a natural process that warms the surface of a planet, namely, the Earth. (Source)

In simple terms, when the Sun's rays hit the surface of the Earth, part of this energy is converted into heat and remains trapped in the atmosphere keeping an average of 33 degrees Celsius. Whereas, part of it is reflected back into space. The gases responsible for this effect, are the greenhouse gases, which predominantly include carbon dioxide and methane among other gasses.

Therefore, it is logical to create an association between greenhouse gasses emitted by humans and rapid climate change.

(Note: We make a distinction when talking about "global warming" and "climate change". They are both natural processes that are accelerated by human contributions. The former refers to the increase of the global average temperature compared to 1880. The latter refers to the changing weather patterns made more extreme and frequent by human pollution. These two phenomena have been shown to be positively correlated.)

I will not be talking about this in any further detail, as it is out of the scope of this post. If you want to learn more, there are plenty of resources online that you can easily find.

Carbon Emissions & Taylor Swift

A lot of discourse has arisen from this topic association. It is no secret that a lot of celebrities travel by means of private jets or yachts. This practice has been scrutinised a lot by the public, especially in regards to the carbon footprint they leave on the climate. As reported in The Guardian, in 2022, Taylor Swift's private jet has emitted a total of around 8,300 MtCO2, the highest amount among all of the other celebrities. People who are also huge emitters are people such as Floyd Mayweather, Drake, the Kardashians, Travis Scott, etc. (Source). The yearly global average CO2 emissions is at around 37 billion MtCO2, that is, approximately 101,000,000 MtCO2 per day. This makes Taylor's highest yearly emission in 2022 part of the 0.008% of the global average daily emissions.

The scrutiny The scrutiny towards celebrities is definitely a valid argument. They represent a group of people with wealth that is above the average person. Thus, they have the privilege to make better choices that the average person may not have or cannot afford to make. Therefore, to put them at a higher standard, even as role models for current and future generations, is a reasonable thing to do. However, much of this scrutiny has been victim to opportunism and plain hate towards certain people, especially Taylor Swift, who has a very polarised image in the public eye (especially after 2019/20). We can try to find causes and perks that define an exact model to the level of hate and love that she receives from people. However, it is undeniable that some of it comes from misogyny and anti-capitalist feelings. As of 2024, Taylor has been officially labelled as a billionaire (based on her net worth), and one of the only people who have achieved this status from selling her own music, concerts, and merch.

Action & Counteraction Taylor Swift has disclosed to the public that she tries to offset her "carbon footprint" by buying what is called "carbon credits". Carbon credits are a sort of investment in green and sustainable projects that aim at slowing down global warming rates or actively trying to reduce the greenhouse gasses in the atmosphere.

It is not really certain whether offsetting carbon emissions by buying carbon credits really effectively works. It is definitely helpful to invest in these technologies, researches, and projects, as they are the future of sustainability and they do need the funds. However, these investments don't actively help balance out the emissions from a private jet. At least, not in the immediate future. Therefore, it is a "buffered counteraction". That is, the rate at which these green and sustainable technologies come out and get used widespread cannot keep up with the emissions increase. This has been used as additional scrutiny towards Taylor Swift.

The "carbon footprint" brainwash Where does the term "carbon footprint" even come from? Well, this concept has always been on people's minds but took off after the second industrial revolution, as the number of manufacturers increased and consumerism became rampant across the globe. This growth became a problem since it generated a lot of trash and waste that polluted indigenous lands. In fact, in the 70s various commercials, notably the "Crying Indian", were used to point out these issues. In a way to deter these criticisms, in a way to avoid being stopped from profiting off of their current ways, the response from these industries was to push the blame of pollution onto the consumers, off of the shoulders of the manufacturers themselves.

In particular, the most controversial brainwashing campaign was started by BP in 2004, by establishing an awareness on the consumer's carbon footprint by their usage of natural gasses and oil, rather than taking accountability on their own practices in the first place. Alongside BP, ExxonMobil was already attempting at projecting this responsibility on the public in the 70s. All of this led to "carbon footprint" being part of our daily lexicon even now. Companies still talk about being "carbon neutral" or disclose their "carbon footprint". However, an issue with this concept is that being carbon neutral does not do anything pertaining to the climate. You are not making less emissions, in the sense that, the energy is still distributed somewhere else. Therefore, the power plants that generate that energy still created those emissions. The goal in terms of carbon footprint would be to be carbon negative/positive. That is, actively trying to contribute positively to the climate, perhaps by converting CO2 into something more sustainable.

In regards to carbon credits and offset, we have evidence that they actually aid the climate at a very low percentage. Therefore, it may not be a desirable long-term solution. As the NYT puts it, we need "government regulation, effective carbon taxes, national standards for renewable energy and electrification, the elimination of legacy subsidies for the fossil fuel industry, strict auto emission standards and new national building codes.". However, all of these initiatives, decrease the dependency on the fossil fuel industry which makes a ton of money to global governments. Therefore, the only effective way to actually tackle global warming and climate change is to force the hand of governments, to care about its citizens and to change their ways.

(Source)

Conclusions – A weaponised nuanced global problem "Solving" climate change is not a simple task. It's not like fixing your lifestyle, or going to sleep at a regular hour every night. It's a systemic issue, which similarly to other systemic issues (such as racism), it is very hard to actually do something about it as average people. Often times, this topic among others is weaponised to bring other people down. In the face of anti-capitalist sentiments, the general public does not have any issue in dehumanising people who are contributing more than them on average to polluting the Earth. However, as we have explored today, it is a much more nuanced topic, and these nuances cannot be overlooked. The "carbon footprint" issue is not going to solve anything, unless we all become carbon negative/positive. However, these solutions can only gain momentum if the culprits at the source take accountability and action about it. That is, mainly the fossil fuel and food industry, as well as the travel industry. Taylor Swift flying her private jet is contributing to the emissions actively. By not travelling, she still wouldn't be carbon neutral or positive like the rest of us. Those big industries are still generating that amount of emissions. These slight individual choices do not affect the industry enough to actually make any significant changes (Source). Those industries overproduce energy and supplies in excess, so whether you use them or not, they still find a way to be part of the problem. Of course, a long-term global scale boycott against these industries would effectively halt their gains in a significant way. But this is also not really a feasible solution. Especially given that we have become so accustomed to all of our needs being provided by them in the first place, and this global scale dependency on them is the reason why a lot of products can sell at such low prices. This also means that it's the only way people in the lower income classes can afford to survive, whether it is about food, clothes, or electricity. Therefore, in conclusion, the only way that we can actually make any changes on this matter is by holding our governments and these industries responsible. While holding the consumer responsible is not going to benefit anyone whatsoever. They will just find another greedy client and the cycle goes on. The thing is not to break the dependency on these providers and manufacturers, but to change the way they are able to provide us these things. The role of the government is to take care of its people, and this is part of their responsibility.

Opinion Thus, from a broader perspective, the scrutiny that Taylor Swift faces is non-sensical and clearly used as a way to find something to complain about, in an attempt to make her look bad and bring her down. This has been the case for several years, especially after the whole period of time where people just collectively decided that they could not tolerate Taylor Swift anymore. We have seen a rise in popularity recently, as she started the eras tour in 2023, she released a Grammy winning album "Midnights" in 2022 which broke several records, she made a lot of TV appearances following her relationship with Travis Kelce, and she has also been nominated by notable magazine publishers as a very powerful and remarkable figure in the music industry. All of this, culminated with her success reaching an extreme that has never been reached before, and with this the amount of hate also increased by the same factor. This "hate" is a separate topic that can be explored, which relates to internal online conflicts, personal beliefs, social norms, misogyny, and the dehumanisation of celebrities which is sadly encouraged by a lot of them in the first place (directly or indirectly).

To address, the situation with Jack Sweeney briefly: Taylor Swift has sent a cease and desist letter to this college student (who is known for tracking private jets, e.g. Elon Musk's private jet), who has been tracking her private jet location and emissions for a few years now. The reasoning provided by the student is that this is to provide more awareness on the "carbon emissions" celebrities produce. However, Taylor Swift's spokesperson argued that sharing her location in real time is dangerous for her, as it has been the case in the past that she had stalkers and malicious people intrude in her house (e.g. recently in NYC). All of this also contributes to creating a dangerous environment for herself and the people crowding at her destinations. She has not commented too much on the emissions, justifying it by saying that her plane is actually loaned out by other people and that she is not the only one riding it (Source). However, the main reason for the C&D is in regards to her safety and privacy.

(I also find it ironic that this college boy, who is obsessed with planes and allegedly rides them himself privately, uses the "emissions" argument to appear morally superior).

I have already shared my view on all of this above. I do feel like it's not necessarily bad to track the carbon emissions of these jets. As I mentioned, it is a good thing to still hold them accountable in a way for them to also take initiatives in contributing to green projects (financially at the least). However, I do believe that sharing her location is a very dangerous thing, and the very reason that she cannot travel on public transportation. A lot of people try to make examples out of other celebrities, but the thing is that compared to Taylor Swift, their celebrity and popularity status is just not on the same level. I do not mean this as an offence to those celebrities, but you cannot compare Keanu Reeves taking the subway to Taylor Swift taking it. Her celebrity status is beyond what other celebrities have reached (in positive and negative light). Therefore, there aren't a lot of people who you can actually compare her to. Given also the rise of hate towards her in recent years, it has become even more clear that she is unable to do this, and she has also been open about being anxious in public spaces due to this. As we have witnessed recently with the deepfakes, she has a valid reason to find it troublesome to be around other people in public.

Lastly, the most superficial and obvious reasoning for her emissions is that, especially with the start of the eras tour, she has to travel around the world. So those emissions are business-related, I guess. But yeah, don't suddenly forget all of the above after reading this paragraph.

Carbon tracker: https://carbontracker.myclimate.org