#cyclic permutation

[math1089]

Mathematical Beauties of the Number 142587

There are some mysteries that the human mind will never penetrate. To convince ourselves we have only to cast a glance at tables of primes and we should perceive that there reigns neither order nor rule.Leonhard Euler Welcome to the blog Math1089 – Mathematics for All. I’m glad you came by. I wanted to let you know I appreciate your spending time here on the blog very much. I do appreciate your…

[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!)

[reductionisms]

if the gintama “human” = gintoki = perpetually becoming human. then why not the permutation group composed of permutations of gintoki (including himself inside it as most composed identity) and the whole group itself is what it is to be human. cyclically connecting others inside of you and making you in the process, eternally…

[as-if-and-only-if]

so, coding challenge! what’s the quickest way (in a real, normal computer—ok, a very quick way) to generate all and only all length-N sequences of bits such that:

among all bit sequences equivalent to each other under cyclic permutation, only the lexicographically minimal one is generated.

For example:

if N = 4, we only want to wind up with

0000

0001 (equivalent to: 0010, 0100, 1000)

0011 (equivalent to: 0110, 1100, 1001)

0101 (equivalent to: 1010)

0111 (equivalent to: 1110, 1101, 1011)

1111

They don’t need to be generated in lexicographic order, though. Output order is irrelevant.

Also, I’m willing to sacrifice lexicographic minimality for simply generating a single element of each equivalence class if need be—the core of the problem is generating these representatives without redundancy!

If somehow it happens to be convenient, we can even exclude ones that are periodic (like 0000 and 0101, i.e. with nontrivial stabilizer under the cyclic permutation action) if we like. (Doing so is technically better for my application, but only very very slightly, so it’s not necessary in practice.)

Bonus points for an approach that is lazy and parallelizable, e.g. one that bijectively maps integers up to some known quantity to sequences directly; it would be good to sample randomly from things here. Even more bonus points if you can specify N, k, i, and get out the i’th bit sequence of length N (not necessarily in any particular order) that has exactly k 1’s in it! But speed is more important than both of these.

N should be able to be, like, pretty big. At least 25.

Does this problem have a name? Has someone figured this out already?

Note: this is equivalent to generating all lists of integers that sum to N and, similarly, choosing a representative from each orbit-under-cyclic-permutations which is lexicographically maximal among those in the orbit: just look at all runs of the form 0…01 for nonzero bit sequences after rotating the sequence at hand so that it ends with a 1 (e.g. 100011010 ↦ 010001101 = (01)(0001)(1)(01) ↦ [2,4,1,2]; we would not generate this one, and instead generate [4,1,2,2]). (fun exercise: prove this makes sense!)

[disorderofsevenstars]

Gender and Divinity

At one end of the gender spectrum lies the Feminine, associated with Black and darkness, the Earth and mundane matters, collectivity, interconnection, receptivity, cyclicality, synthesis, and heterogeneity. At the other end lies the Masculine, associated with White and illumination, the Heavens and spiritual matters, individuality, separation, exclusivity, linearity, analysis, and homogeneity. These two extremes, Yin and Yang, interact to create a multitude of permutations which lie somewhere in between the purely Feminine and the purely Masculine. Everything that exists as part of what we usually call "reality" is located somewhere in this intermediate space. The extremes themselves are conceptual, and do not exist in material reality, as they are characterized by a purity which cannot be found in Nature.

Sleeping, dreaming, and Death-as-a-state (in contrast with Death-as-a-process) fall within the domain of the Feminine, as do gravity and magnetism, Zero and the even numbers, the elements of Water and Earth, the Moon, and the Cosmic Void or World Egg. Femininity denotes a state of rest, stability, and equilibrium. It promotes slowness and longevity, regeneration and even immortality, characteristic of oceanic or cold environments and the organisms that live there, shielded from the mutagenic properties of solar radiation. The influence of the Feminine on human societies manifests through tribalism, collectivism, and egalitarianism, giving rise to social structures such as animist and polytheistic religions, cults, clans, matrilineal descent systems, polygamy and polyamory, democracy, communism, hedonism, mysticism, subcultures and countercultures. It is driven by the Dionysian impulse to shed the Ego, losing the Self within something larger, like a drop of rain falling into the ocean. The Moon is a mirror; it does not generate its own light.

Femininity is the original and final state of all things. It is prominent in small children and the elderly, more so in non-human animals (especially large and slow-moving animals, those animals which are more primordial or lower on the food chain, as well as small animals with collectivistic tendencies such as ants), even more so in plants, and still more so in rocks, soil, and water. In animals, it is highly oriented toward the senses of touch, taste, and smell. During the course of our evolution, we developed eyes, learned to hunt, learned to walk on land, learned to stand upright, became taller, acquired language and logic, mastered the use of tools and fire, and lost our fur; some of us also lost our melanin. The development of an embryo into an infant, into a child, into an adult, involves a loss of plasticity; bones harden and fuse, the percentage of water in the body decreases, and neural pathways become more rigid. All of these traits indicate a process of Masculinization.

The Masculine is the domain of awareness, alertness, problem-solving, and conflict. It governs changes, transitions, and boundaries. Masculinity is a property of electricity, odd numbers, prime numbers, the elements of Fire and Air, the Sun, and the Axis Mundi. It is dynamic and always in motion, striving to reach the Feminine rest-state, like an arrow flying towards a target, or a key inserted into a lock. The influence of the Masculine on human societies manifests through individualism and inequality, giving rise to colonialism, capitalism, competition, war, monotheism, monogamy, patriarchy, the nation-state, asceticism, and scientific thought. Driven by the Apollonian impulse towards separation and clarity, it prioritizes facts over feelings.

Unable to coexist and seeing plurality as a threat, Masculinity seeks to dominate the Other and propagate the Self, often through violent means; Western culture, Christianity, and Islam are examples of this. Monotheism and patriarchy have a mutual affinity, and in many cases, one promotes the other. The god of a monotheistic religion is usually male. The Sun is the central axis around which all bodies in the Solar System revolve. This tendency of the Masculine to see itself as superior lies behind the association of Light with Good, and Darkness with Evil. Humans, considered as a whole, lean towards Masculinity, and serve as a Masculine counterpart to the natural environment of Earth.

The supreme divinity is like a polygon with an infinite number of sides: simultaneously circular and linear. Fate arises from the interconnection of individual wills, a product of emergent complexity, many individual entities inadvertently working together in a larger system. "GOD" can be described as genderless, or a perfect balance of all possible genders. It has the Feminine quality of Being, and the Masculine quality of Unity. It is an all-pervasive energy field of pure universal consciousness, which can be channeled into various manifestations that possess genders and other specific attributes, like white light being split by a prism into its spectral components. These facets of divinity are conceptualized in every pantheon of deities, in the 12 signs of the Zodiac, in the 22 Major Arcana or 22 letters of the Hebrew Alphabet, in Kabbalah as the 10 Sefirot, in Christianity as the Holy Trinity, in the Tzolk'in as the 20 Naguales, and in Chinese cosmology as the Ba Gua (Eight Trigrams) and 64 Hexagrams of the I Ching.

[3/16/2024]

[animebw]

Short Reflection: Attack on Titan: The Final Chapters

The impossible problem of reviewing the ending of Attack on Titan is that there's way too goddamn much to talk about.

How could there not be? After ten years and close to a hundred episodes, after almost single-handedly jump-starting anime's mainstream takeover back when the first episode dropped in 2013, after so many twists and reversals that completely flip your perspective on what kind of show you're ever watching, Attack on Titan has become, fittingly enough, one of the most colossal media properties on the planet. It's a story that bursts at the seams with ideas and messages and Things Worth Talking About, and far from putting a nice neat bow on things, the ending somehow only explodes further outward with discussion-worthy topics. I can pick out at least four separate moments just from the final 30 minutes that could support an entire thesis paper all on their own. I could talk about its portrayal of fascism and genocide, its ultimate statement on the nature of war, its portrayal of trauma bonds, its observations on the cyclical nature of generational violence, whether or not it's ultimately hopeful or cynical, how the characters' final choices echo all the way back to their first appearances... and I still wouldn't cover everything meaningful there is to talk about in this two-part two-and-a-half-hour showstopper finale. I could spend all week writing analyses of Attack on Titan's ending and there would still be more to cover. Even trying to figure out how to tackle this monstrosity feels as overwhelming as staring up at the Colossal Titan itself.

But perhaps that's only fitting. Over the years, Attack on Titan has remained one of the boldest, bravest, most staggeringly ambitious properties to ever come out of anime, swinging for the fences as relentlessly as the Beast Titan's rocky barrages of death, never taking the easy way out of an impossible question or desperate situation. It's survived a sea-chance in the landscape of the medium that it itself helped usher in, gone through so many permutations of genre and form and production, yet never once lost its luster. From the first sight of Eren's home being destroyed to the final, haunting shots of this finale, Attack on Titan has been one of the most important anime of all time. So of course its finale would demand just as much of us as the rest of the show. Of course it wouldn't be satisfied if it didn't go out with one last jaw-dropper outing that blew everything you thought you knew out of the water. Of course it would close out its domination of the anime sphere just as inescapable and undeniable as when it first stomped on our screens 10 years ago. After the incredible journey it's taken us on, I wouldn't expect anything less.

Still, I've got to try and say something. This is the end of one of anime's defining properties, the epic conclusion to a story that stands head and shoulders above all its contemporaries. I can't let that occasion pass without commemorating the way it chose to close things out. Just know that this review in no way captures the full extent of my thoughts on the finale, in no way portrays all the countless whirling ideas it inspires in me. There will be time in the future, perhaps, to explore all its nuances in more depth, to give proper space and weight to the thousand component parts making up this massive whole.

For now, though?

For now, I simply want to celebrate the end of one of my favorite anime series of all time.

For brevity's sake, I won't spend too much time on the first two hours of breathless action that make up the majority of these two super-long episodes. There are far more interesting things to talk about in the final act, and I simply don't have time to dig into anything more. All I'll say is that this final battle was Attack on Titan operating at the absolute peak of its powers, and it was truly something to behold. For all the chaos that's historically plagued this series' production, Wit and Mappa alike, this was an absolute masterpiece of a climactic showdown. Stunning action fit to stand with all the series' most iconic moments, heartwrenching grief and tragedy in the face of armageddon, characters barely escaping death and facing down the apocalypse with all their strength, refusing to yield against impossible odds no matter how many walls stand in their way. I can't count the number of times I cheered in delirious glee or broke down in tears from the sheer majesty of it all. As spectacle, as mission statement, as payoff for ten years of escalating action and human drama, it's an unimpeachable triumph.

And then we reach the denoument. The battle ends, the day is saved, the war is over, the bloodshed has stopped, the centuries of pain and suffering the world has endured finally takes a long, haggard breath of fresh air. The Rumbling has been stopped. Eren has been defeated. Peace, at long last, has prevailed.

And then the show throws one more curveball.

Ever since the manga ended, I've been hearing how much manga readers hated it. I've seen every insult under the sun slung its way, seen it called the worst ending of all time, an insult to the series and everyone who loved it. But I've always had my doubts. Considering how Loudly Wrong so many of these people were about many other things, how many of their criticisms seemed to boil down to being annoyed at Eren not being venerated as a gigachad and treated like the deranged psychopath he actually was, I had a sneaking suspicion the hate was overblown nonsense. And while I've heard the adaptation changes a few lines here and there to maybe make some of the final scenes less clunky, if the finale I watched in the show was anything like the finale in the manga, then I can only conclude my skepticism was right. Far from destroying the story or insulting the fanbase, this ending is one of the single most audacious and ambitious ways I've ever seen a mainstream anime conclude. It's an ending that could only come from a creative spirit dedicated not just to writing a cool story, but creative capital-a Art(tm). And long after the Twitter salt of the moment has faded, it will linger in my mind as the punctuation mark that seals this series' legacy until the end of time.

It's maddeningly hard to discuss why without spoilers, but I'll do my best. In short, Eren's plan actually kind of ends up succeeding... in a way that only reveals how pointless it was to begin with. The world is left irreparably scarred by the Rumbling in ways that we are told and shown, in no uncertain terms, will only ensure the cycle of violence will continue long after the concept of Titans themselves have faded from memory. The best way I can describe it is a dark subversion of Lelouch's famous gambit from the finale of Code Geass. In that show, Lelouch took all the world's evil upon himself to give the world a common enemy, sacrificing his own life to unite the warring nations in a desire to defeat him. Here, Eren does much the same thing... but it only leaves the world more broken and insecure than when he started. All the madness he inflicted, all the carnage he wrought, and in the end, all he accomplished was guaranteeing an uncertain, unstable peace that we have no way of knowing how long it lasted before things once again fell to ruin.

On the surface, it's a bitter, bleak ending that spits in the face of all our heroes' hard work. And yet, it's also the most strangely honest portrayal of humanity I've seen in a long time. What this finale gets right about Eren- what all those screaming fanboys were so pants-pissingly angry about- is that he's not some maniacal genius or machiavellan mastermind. In his own words, he's just a garden-variety idiot, an ordinary person with ordinary fallibility and biases and prejudices who was given far more power than he could handle. Right up to the end, he was a scared, lonely boy wearing the hollow shell of mature, stoic rage as a shield from the pain he could never process. He was weak. He was foolish. He was immature and scared and completely unequipped to handle the incredible responsibility of holding the world's fate in the palm of his hand.

Now you tell me: what end result could such a person achieve, if not this ugly, imperfect half-measure?

The truth is, violent individuals in our real world aren't grand schemers like Lelouch who always have the answers and make everyone dance in the palm of their hand. They're people like Eren; flawed, broken, consumed by hatred, making irrational decisions off imperfect assumptions and forcing everyone else to deal with the consequences. Eren doesn't fail because of character assassination, or because the mystical future sight be picked up from the Founding Titan told him so. He fails because no matter how many times he runs through the future in his head, his hatred leaves him incapable of seeing any path forward that doesn't involve annihilation. His fate was not set in stone by intangible destiny, but by his own inability to accept a peaceful path when his heart screamed endlessly for blood. This messy, transient, tenuous peace is the best that someone like him could hope to accomplish. It's a miracle he ended up with a result that good, and that's only because his friends were brave enough and strong enough to stop his hatred in its tracks before it could completely swallow the world whole.

But that messy, transient, tenuous result is what we're left with. All the chaos of the past ten years, and we're left not with closure, but uncertainty. How long will this hold out before the cycle of violence begins anew? How much peace did we earn through such miserable ends? What, if anything, can be done to ensure the next turn of the cycle doesn't end up like this? What better ways were there to keep this from happening in the first place?

What do you do when the answer you've been looking for only leaves you with more questions?

This ambiguity, in all its raw messiness, is the ultimate message of Attack on Titan. This is not a story where the heroes save the day and defeat hatred and save the world for good. This is a story where the heroes do the best they can and accomplish as much as they're able... and then leave us to reckon with the failures they left behind. It's not even really a nihilistic ending; as bleak as it seems on the surface, there are countless moments of hope and light that speak to humanity's desire for love over hatred. Like all the great tragedies of the world, it doesn't claim that humanity is irredeemable and we can never escape our flaws. Instead, it forces us to consider how easy it can be to fall short of our ideals, and how much pain and misery they leave in their wake. It shows us the consequences of giving in to hatred not as an inescapable fact of humanity, but as a warning of what we sacrifice when we lack the courage to prevent our darker instincts from directing our actions. It takes these characters and this world you've fallen in love with for a full decade and, rather than bidding them happily after after, asks you to carry the weight of their failures into your own life, a reminder for every single time you're tempted to let blind rage guide your own actions from now on. "This is what their lives have amounted to," the show says. "Don't let your own life go the same way."

I've watched a lot of anime over the years. Since I started getting into it back in 2017, I've tackled countless series and movies with all different kinds of endings. But I don't think I've ever seen one that so perfectly nailed this kind of hopeful, agonizing ambiguity. I don't think I've ever seen a show commit to an ending this uncompromising or pull it off so astoundingly, let alone in a series this long. And in an anime industry that more and more seeks to only pursue the safest, least threatening options, flattening this medium's remarkable creative potential with art that doesn't say anything or mean anything, the fact that this, of all ways, is how the most popular anime on the face of the planet chose to end its tale is nothing short of remarkable. Attack on Titan is a runaway success story of epic proportions, a single anime that arguably paved the way for the medium's increasing mainstream popularity. And it married that mainstream success with a story that dared to ask hard questions of its audience and send them home with a plea to see their own world a little deeper rather than half-ass a happy ending for easy closure. It's the rare mega-popular franchise that didn't have to sacrifice an inch of depth or complexity to appeal to the broadest common demoninator. And it accomplished all that while still being a never-ending thrill ride of sheer entertainment value the likes of which we may never see again.

It's easy to be cynical about anime once you've watched too much of it. Honestly, I find myself struggling a lot these days to keep the bad parts of this medium from swallowing my enjoyment of it. But in an anime landscape that too often settles for taking the path of least resistance, Attack on Titan is a soaring reminder of what it looks like to be brave. It's a series that never compromised on itself, never gave anything less than 100%, and never once lost its magic through all its ups and downs. It represents the overwhelming power that anime is capable of when it trusts its audience to follow it off the beaten path and face more challenging questions than whether Ultra-Instinct Goku can beat Gear 5 Luffy. It's an achievement as staggering and colossal as any of the titans within Isayama's pages. And this ending, in all its powerful uncertainty, only cements its legacy as one of the single greatest works of art ever produced by this medium, and one my my new personal top 10 anime of all time.

I wish I had time and space to keep talking. I wish I could go in-depth about the show's willingness to let Eren be truly pathetic in hilarious fashion. I wish I could write an essay about Mikasa and Ymir's parallels as women trapped by their irrational love for monstrous men and why that makes Mikasa's final choice so fucking impactful. I wish I could go into every last detail of that ending montage and all the implications it raises about the future of this world. But there will be time later to dig into the guts of this ending and truly pick it apart. There will be time aplenty to appreciate all the moments that make this show such a titan in its own right. For now, all I can say is this:

Thank you.

To Isayama, to all the voice actors, to all the staff at Wit and Mappa who were forced to work unreasonable schedules to bring this story to life, to every hand that touched this series and ensured its continued majesty all the way to the end... thank you. I dedicate my heart to you.

Attack on Titan is over.

It's the end of the world as we know it.

And I feel fine.

10/10

[shimyereh]

Spring, spring! How many variations On fleeting youth can poets write? I sift through years of permutations — Their words are passionate but trite, An iconography of yearning. In well-worn phrases, brightly burning With uninventive ardor, they Reiterate each old cliché Of meadows, nightingales, and roses. And yet — they struggle to define: Is time a circle, or a line? Spring’s cyclical return exposes A loophole, letting some conclude That even youth may be renewed.

[excludedmiddle]

Back on the horse (a little)

Let's write up a codeforces problem!

A couple important nuances - this problem revolves around fixed points - namely, array elements where the index of the element matches the value of the element (indexing from 1, which codeforces inexplicably loves to do).

My first instinct is that we can conceptualize this as a directed graph. Each permutation of the array is a different node on the graph, and each fixed point transformation is a directed edge from the previous state to the shifted state.

In this model, what we're looking for is a path through the graph with k edges. This could be a cycle or it could be a non-cyclic path, but there are only n nodes in the graph (they are shifts, not true permutations) so either way our answer is bounded.

This alone can get you the solution - construct the graph, find the connections, run DFS - but that solution is slow and annoying to implement.

The missing insight that gets you to a nice solution is that after such a transformation, the fixed point that you used to transform the array must be the last element, simply because of how fixed points work. Then all we have to do is check that element.

Egor has a very nice implementation here:

[iit-jam]

IIT JAM Syllabus 2025: A Comprehensive Guide

The IIT JAM (Joint Admission Test for Masters) is one of the most competitive exams for students aspiring to pursue postgraduate studies in esteemed institutions like IITs and IISc. Mathematics, being a core subject, attracts candidates with strong analytical and problem-solving skills. To excel in this exam, a thorough understanding of the IIT JAM Mathematics Syllabus 2025 is essential. This blog outlines the syllabus in detail and provides tips to help candidates prepare effectively.

Overview of IIT JAM Mathematics Syllabus 2025

The IIT JAM Mathematics Syllabus 2025 is crafted to test the candidates' knowledge of fundamental mathematical concepts covered at the undergraduate level. The syllabus is broad, covering topics such as calculus, linear algebra, differential equations, and numerical analysis. Each section focuses on key areas that are crucial for advanced studies and professional applications.

Key Topics in the Syllabus

1. Sequences and Series

This section includes the convergence of sequences and series, tests for convergence (such as comparison, ratio, and root tests), and the study of power series and their radius of convergence.

2. Differential Calculus

Candidates must understand single-variable calculus concepts like limits, continuity, and differentiability. Topics also include Taylor series, mean value theorem, and indeterminate forms. For multivariable calculus, partial derivatives, maxima, minima, saddle points, and the method of Lagrange multipliers are essential.

3. Integral Calculus

This section covers definite and indefinite integrals, improper integrals, and special functions like beta and gamma functions. The application of double and triple integrals is also emphasized.

4. Linear Algebra

A critical area of the syllabus, it focuses on vector spaces, subspaces, linear transformations, rank, nullity, eigenvalues, eigenvectors, and matrix diagonalization. Understanding the solution of systems of linear equations is vital.

5. Real Analysis

This section involves the properties of real numbers, limits, continuity, differentiability, and Riemann integration. Candidates must also be familiar with sequences, Cauchy sequences, and uniform continuity.

6. Ordinary Differential Equations (ODEs)

This includes first-order ODEs, linear differential equations with constant coefficients, systems of linear ODEs, and Laplace transform techniques for solutions.

7. Vector Calculus

Important topics include gradient, divergence, curl, line integrals, surface integrals, and volume integrals, along with Green’s, Stokes’, and Gauss divergence theorems.

8. Group Theory

The basics of groups, subgroups, cyclic groups, Lagrange’s theorem, permutation groups, and homomorphisms are covered.

9. Numerical Analysis

This section focuses on numerical solutions for non-linear equations, numerical integration and differentiation, interpolation methods, and error analysis.

Tips for Preparing the Syllabus

Understand the Weightage: Review past papers to prioritize high-scoring topics like Linear Algebra, Real Analysis, and Differential Calculus.

Strategize Your Study Plan: Divide the syllabus into manageable sections, set achievable goals, and stick to a consistent schedule.

Practice Regularly: Solve previous years’ papers and mock tests to familiarize yourself with the question patterns and improve speed.

Strengthen Fundamentals: Focus on core concepts by revisiting undergraduate textbooks and seeking clarity on challenging topics.

Leverage Online Resources: Utilize tutorials, study materials, and practice tests available online to supplement your preparation.

Conclusion

The IIT JAM Mathematics Syllabus 2025 is extensive yet well-structured, providing a clear framework for aspirants to plan their preparation. By mastering the syllabus and practicing diligently, candidates can confidently tackle the exam and achieve their dream of joining top postgraduate programs. Dedicate time, stay consistent, and focus on strengthening your mathematical foundations to excel in IIT JAM Mathematics 2025.

[geopolicraticus]

Friday 11 October 2024

Grand Strategy Newsletter

The View from Oregon – 310

Permutation of Cyclicality and Linearity

…in which I discuss linear and cyclical history, reductivist formulations of history, Nietzsche’s eternal return, inter-definability of permutations, temporal iterations, uniqueness in time, repetition, plate tectonics, and the supercontinent cycle…

Substack: https://geopolicraticus.substack.com/p/permutations-of-cyclicality-and-linearity

Medium: https://jnnielsen.medium.com/permutations-of-cyclicality-and-linearity-1dead7293ef4

Reddit:

[russian--wind]

My last three days in New orleans.

If I could visualize time, I often see the threads of a chaotic attractor, zipping through an invisible ether like a craft in an asteroid belt. Trajectories. People think that the immaterial and intangible have no limitations or structure, which is where the concept of the "butterfly effect" comes from, as well as the common pitfall of FOMO and feeling like you have no control over the events around you. But time like all good things has a tight order to it. Most paths circle back around. Sometimes you run into a dead end. It's like in a game, where what you're really playing with is the higher order but the effects are only visible in the tangible. So you have to abstract a level out, and mastery comes when you don't get too rigid about it, and kind of just flow with it. Often eventspace is like water. For instance, all the houses I visited to potentially move in have all been in the same neighborhood. And they're all dead ends now. I would've ended up here no matter which set of paths I took, and it would have ended in some permutation of what happened. For an intuitive life experience fills in your calibration of what navigating through time, making decisions, and manipulating (in the same way you manipulate a ball in a game) events around you can look like.

When I was 20, I was fascinated by mathematics. I threw it away like the suicidal protagonist of Cormac McCarthy's swan song, Stella Maris. Alicia, like me at 20, is a brilliant math prodigy, who, afflicted with hallucinations and the belief that her brother whom she's in love with is dead, commits suicide at 20. I lived, but I threw away the math until the underlying emotion beneath it resurfaced in New Orleans this year. Did I tell you that Alicia's brother, who was in love with her, ends up drifting in New Orleans a decade after her death? As a salvage diver? And that the other day, the mailman walked into my workplace and handed me an anonymous postcard with a quote from Cormac's The Road? And that I'm currently in a round of interviews with the institute where he holed up in the last two decades of his life and wrote those books, the same institute that studies the complex systems phenomena I was entranced with as a 20 year old?

I could feel in the patterns of dynamical systems and wave flows, the shapes of fractal geometry, and scaling theory, the shape of the universe around us expressed in some gestalt way -- that's what drew me to those topics.

Synchronicities point to some trajectories that are so dense in our personal life path that they are inescapable. Not in a doom-y way, more in a way that engenders faith. Your destiny won't leave you. You can't miss your destiny because you went down the wrong aisle at the grocery store. Gravity doesn't work like that and even the intangible is ruled by physical laws.

So! Free from FOMO, what will you do? How will you meet the universe halfway?

Time is also a wave, and I think about it less like this. There are cycles that repeat themselves with different periodic frequencies. Astrology is really a study of time. I wonder if people struggle with mixing the music of the spheres (the zodiac) with the cyclical nature of time. How do you know when exactly it stops being Aries and starts being Gemini? I don't know, the same way you know it stops being a fourth and starts being a fifth? Or that there is a fourth and a fifth? We don't expect order to arise out of chaos, much less a geometric, harmonious, implicitly beautiful order.

My last three days in New Orleans. I understand this time's coming to an end. I'm taking it all in. Old decision points are returning to me, magnifying. The job interview. Camp. Timelines I didn't take in 2018 are being offered to me once more. Like a promised reunion, or just the second verse. Neutral Milk Hotel's In the Aeroplane Over the Sea spinning through my head, through the galaxies, the half decade separating me from when I dreamed of this city and didn't know it was a real place. I sat in Jackson Square and played the song again and I could feel myself on the couch reaching for a place, reaching for the song, reaching for myself. I connected the two times and felt them click. Magic. Ever have a vision and then you're in the vision?

[rnomics]

Transfer #RNA halves are found as nicked tRNAs in cells: evidence that nicked tRNAs regulate expression of an #RNA repair operon [Report]

Transfer RNA fragments are proposed to regulate numerous processes in eukaryotes, including translation inhibition, epigenetic inheritance and #cancer. In the bacterium Salmonella enterica serovar Typhimurium, 5' tRNA halves ending in 2', 3' cyclic phosphate are proposed to bind the RtcR transcriptional activator, resulting in transcription of an RNA repair operon. However, since 5' and 3' tRNA halves can remain base paired after cleavage, the 5' tRNA halves could potentially bind RtcR as nicked tRNAs. Here we report that nicked tRNAs are ligands for RtcR. By isolating RNA from bacteria under conditions that preserve base pairing, we show that many tRNA halves are in the form of nicked tRNAs. Using a circularly permuted tRNA that mimics a nicked tRNA, we show that nicked tRNA ending in 2', 3' cyclic phosphate is a better ligand for RtcR than the corresponding 5' tRNA half. In human cells, we show that some tRNA halves similarly remain base paired as nicked tRNAs following cleavage by anticodon nucleases. Our work supports a role for the RNA repair operon in repairing nicked tRNAs and has implications for the functions proposed for tRNA fragments in eukaryotes. http://rnajournal.cshlp.org/cgi/content/short/rna.079575.122v1?rss=1&utm_source=dlvr.it&utm_medium=tumblr

[canmom]

aaah tysm! you're one of the most knowledgeable people I know when it comes to music theory, so I was hoping you'd read this and like it ^^'

That's fascinating about the historical context to the church modes! I fully admit I skimmed right over all the parts of the wiki article on medieval church modes to understand the version of the idea used in modern day composition, but evidently that's very reductive ^^'

I've had someone else talk to me a bit about modes and mode progressions, so for the next part I definitely want to build a much stronger understanding of the idea than just 'oh it's cyclic permutations'.

Music Theory Notes (for science bitches) 1: chords & such

This is one of these series where I use my blog as a kind of study blog type thing. If you're knowledgeable about music theory, it will be very basic. But that's kind of the problem, I've really struggled to absorb those basics!

When I was a teenager I learned to play violin and played in orchestras. I could read music, and play decently enough, but I didn't really understand music. I just read what was on the page, and played the scales I had to play for exams.

Lately I've been trying to learn music again. This time my instruments are zhonghu, voice, and DAWs. At some point I might get my violin back too. But really, I'm a total beginner again, and this time I want to do it properly.

For a long time when I tried to learn about music I would get overwhelmed with terminology and jargon and conventions. I might watch videos on composition and they'd be interesting but a lot of it would just fly over my head, I'd just have to nod along because I had no idea what all the different types of chord and such were. I tried to learn from sites like musictheory.net, but I found it hard to figure out the logical structure to fit it all into.

I feel like I'm finally making a bit of headway, so it's time to take some notes. The idea here is not just to answer the what, but also to give some sense of why, a motivation. So in a sense this is a first attempt at writing the introduction to music theory I wish I'd had. This is going to assume you know a little bit about physics, but basically nothing about music.

What is music? From first principles.

This is impossible to answer in full generality, especially since as certain people would be quick to remind me, there's a whole corner of avant-garde composers who will cook up counterexamples to whatever claim you make. So let's narrow our focus: I'm talking about the 'most common' type of music in the society I inhabit, which is called 'tonal music'. (However some observations may be relevant to other types of music such as noise or purely rhythmic music.)

Music is generally an art form involving arranging sound waves in time into patterns (in the sense that illustration is about creating patterns on a 2D surface with light, animation is arranging illustrations in time, etc.).

Physically, sound is a pressure wave propagating through a medium, primarily air. As sound waves propagate, they will reflect off surfaces and go into superposition, and depending on the materials around, certain frequencies might be attenuated or amplified. So the way sound waves propagate in a space is very complicated!

But in general we've found we can pretty decently approximate the experience of listening to something using one or two 'audio tracks', which are played back at just one or two points. So for the sake of making headway, we will make an approximation: rather than worry about the entire sound field, we're going to talk about a one-dimensional function of time, namely the pressure at the idealised audio source. This is what gets displayed inside an audio editor. For example, here's me playing the zhonghu, recorded on a mic, as seen inside Audacity.

A wrinkle that is not relevant for this discussion: The idealised 'pressure wave' is a continuous real function of the reals (time to pressure). By contrast, computer audio is quantised in both the pressure level and time, and this is used to reconstruct a continuous pressure wave by convolution at playback time. (Just like a pixel is not a little square, an audio sample is not a constant pressure!) But I'm going to talk about real numbers until quantisation becomes relevant.

When the human eye receives light, the cone cells in the eye respond to the frequencies of EM radiation, creating just three different neural signals, but with incredibly high sensitivity to direction. By contrast, when the human ear receives sound, it is directed into an organ called the cochlea which is kind of like a cone rolled up into a spiral...

Inside this organ, the sound wave moves around the spiral, which has a fascinatingly complex structure that means different frequencies of wave will excite tiny hairs at different points along the tube. In effect, the cochlea performs a short-time Fourier transform of the incoming sound wave. Information about the direction of the incoming wave is given by the way it reflects off the shape of the ear, the difference between ears, and the movement of our head.

So! In contrast to light, where the brain receives a huge amount of information about directions of incoming light but only limited information of the frequency spectrum, with sound we receive a huge amount of information about the frequency spectrum but only quite limited information about its direction.

Music thus generally involves creating patterns with vibration frequencies in the sound wave. More than this, it's also generally about creating repeating patterns on a longer timescale, which is known as rhythm. This has something to do with the way neurons respond to signals but that's something I'm not well-versed in, and in any case it is heavily culturally mediated.

All right, so, this is the medium we have to play with. When we analyse an audio signal that represents music, we chop it up into small windows, and use a Fourier transform to find out the 'frequencies that are present in the signal'.

Most musical instruments are designed to make sounds that are combinations of certain frequencies at integer ratios. For example here is a plot of the [discrete] Fourier transform of a note played on the zhonghu:

The intensity of the signal is written in decibels, so it's actually a logarithmic scale despite looking linear. The frequency of the wave is written in Hertz, and plotted logarithmically as well. A pure sine wave would look like a thin vertical line; a slightly wider spike means it's a combination of a bunch of sine waves of very close frequencies.

The signal consists of one strong peak at 397Hz and nearby frequencies, and a series of peaks at (roughly) integer multiples of this frequency. In this case the second and third peaks are measured at 786Hz, and 1176Hz. Exact integer ratios would give us 794Hz and 1191Hz, but because the first peak is quite wide we'd expect there to be some error.

Some terminology: The first peak is called the fundamental, and the remaining peaks are known as overtones. The frequency of the fundamental is what defines this signal as a particular musical note, and the intensities of the overtone and widths of the peaks define the quality of the note - the thing that makes a flute and a violin playing the same fundamental frequency sound different when we listen to them. If you played two different notes at the same time, you'd get the spectrums of both notes added together - each note has its own fundamental and overtones.

OK, so far that's just basic audio analysis, nothing is specific to music. To go further we need to start imposing some kind of logical structure on the sound, defining relationships between the different notes.

The twelve-tone music system

There are many ways to do this, but in the West, one specific system has evolved as a kind of 'common language' that the vast majority of music is written in. As a language, it gives names to the notes, and defines a space of emotional connotations. We unconsciously learn this language as we go through the process of socialisation, just as we learn to interpret pictures, watch films, etc.

The system I'm about to outline is known as 12-tone equal temperament or "12TET". It was first cooked up in the 16th century almost simultaneously in China and Europe, but it truly became the standard tuning in the West around the 18th century, distilled from a hodgepodge of musical systems in use previously. In the 20th century, classical composers became rather bored of it and started experimenting with other systems of tonality. Nevertheless, it's the system used for the vast majority of popular music, film and game soundtracks, etc.

Other systems exist, just as complex. Western music tends to create scales of seven notes in an octave, but there are variants that use other amounts, like 6. And for example classical Indian music uses its own variant of a seven-note scale; there are also nuances within Western music such as 'just intonation' which we'll discuss in a bit; really, everything in music is really fucking complicated!

I'll be primarily discussing 12TET because 1. it's hard enough to understand just one system and this one is the most accessible; 2. this has a very nice mathematical structure which tickles my autismbrain. However, along the way we'll visit some variants, such as 'Pythagorean intervals'.

The goal is to try and not just say 'this is what the notation means' but explain why we might construct music this way. Since a lot of musical stuff is kept around for historical reasons, that will require some detours into history.

Octaves

So, what's the big idea here? Well, let's start with the idea of an octave. If you have two notes, let's call then M and N, and the frequency of N is twice the frequency of M... well, to the human ear, they sound very very closely related. In fact N is the first overtone of M - if you play M on almost any instrument, you're also hearing N.

Harmony, which we'll talk about in a minute, is the idea that two notes sound especially pleasant together - but this goes even further. So in many many music systems around the world, these two notes with frequency ratio of 2 are actually identified - they are in some sense 'the same note', and they're given the same name. This also means that further powers of 2, of e.g. 4, 8, 16, and so on, are also 'the same note'. We call the relationship between M and N an octave - we say if two notes are 'an octave apart', one has twice the frequency of the other.

For example, a note whose fundamental frequency is 261.626Hz is known as 'C' in the convention of 'concert pitch'. This implies an infinite series of other Cs, but since the human ear has a limited range of frequencies, in practice you have Cs from 8.176Hz up through 16744.036. These are given a series of numbers by convention, so 261.626Hz is called C4, often 'middle C'. 523.251Hz is C5, 1046.502Hz is C6, and so on. However, a lot of the time it doesn't matter which C you're talking about, so you just say 'C'.

But the identification of "C" with 261.626Hz * 2^N is just a convention (known as 'concert pitch'). Nothing is stopping you tuning to any other frequency: to build up the rest of the structure you just need some note to start with, and the rest unfolds using ratios.

Harmony and intervals

Music is less about individual notes, and more about the relationship between notes - either notes played at the same time, or in succession.

Between any two notes we have something called an interval determined by the ratio of their fundamental frequencies. We've already seen one interval: the octave, which has ratio 2.

The next interval to bring up is the 'fifth'. There are a few different variants of this idea, but generally speaking if two notes have a ratio of 1.5, they sound really really nice together. Why is this called a 'fifth'? Historical reasons, there is no way to shake this terminology, we're stuck with it. Just bear with me here, it will become semi-clear in a minute.

In the same vein, other ratios of small integers tend to sound 'harmonious'. They're satisfying to hear together. Ratios of larger integers, by contrast, feel unsatisfying. But this creates an idea of 'tension' and 'resolution'. If you play two notes together that don't harmonise as nicely, you create a feeling of expectation and tension; when you you play some notes that harmonise really well, that 'resolves' the tension and creates a sense of relief.

Building a scale - just intonation

The exact 3:2 integer ratio used in two tuning systems called 'Pythagorean tuning' and 'just intonation'. Using these kinds of integer ratios, you can unfold out a whole series of other notes, and that's how the Europeans generally did things before 12TET came along. For example, in 'just intonation', you might start with some frequency, and then procede in the ratios 9/8, 5/4, 4/3, 3/2, 5/3, 15/8, and at last 2 (the octave). These would be given a series of letters, creating a 'scale'.

What is a scale? A scale is something like the 'colour palette' for a piece of music. It's a set of notes you use. You might use notes from outside the scale but only very occasionally. Different scales are associated with different feelings - for example, the 'major scale' generally feels happy and triumphant, while a 'minor scale' tends to feel sad and forlorn. We'll talk a lot more about scales soon.

In the European musical tradition, a 'scale' consists of seven notes in each octave, so the notes are named by the first seven notes of the alphabet, i.e. A B C D E F G. A scale has a 'base note', and then you'd unfold the other frequencies using the ratios. An instrument such as a piano would be tuned to play a particular scale. The ratios above are one definition of a 'major scale', and starting with C as the base note, the resulting set of notes is called 'C Major'.

All these nice small-number ratios tend to sound really good together. But it becomes rather tricky if you want to play multiple scales on the same instrument. For example, say your piano is tuned in just intonation to C Major. This means, assuming you have a starting frequency we'll call C, you have the following notes available in a given octave:

C, D=(9/8)C, E=(5/4)C, F=(4/3)C, G=(3/2)C [the fifth!], A=(5/3)C, B=(15/8)C, and 2C [the start of the next octave].

Note: the interval we named the 'fifth' is the fifth note in this scale. It's actually the fifth note in the various minor scales too.

But now suppose you want to play with some different notes - let's say a scale we'll call 'A major', which has the same frequency ratios starting on the note we previously called A. Does our piano have the right keys to play this scale?

Well, the next note up from A would be (9/8)A, which would be (9/8)(5/3)C=(15/8)C - that's our B key, so far so good. Then (5/4)A=(5/4)(5/3)C=(25/12)C and... uh oh! We don't have a (25/12)C key, we have 2C, so if we start at A and go up two keys, we have a note that is slightly lower frequency than the one we're looking for.

What this means is that, depending on your tuning, you could only approximate the pretty integer ratios for any scale besides C major. (25/12) is pretty close to 2, so that might not seem so bad, but sometimes we'd land right in between two notes. We can approximate these notes by adding some more 'in between' piano keys. How should we work out what 'extra' keys to include? Well, there were multiple conventions, but we'll see there is some logic to it...

[You might ask, why are you spending so long on this historical system that is now considered obsolete? Well, intervals and their harmonious qualities are still really important in modern music, and it makes most sense to introduce them with the idea of 'small-integer ratios'.]

The semitone

We've seen if we build the 'major scale' using a bunch of 'nice' ratios, we have trouble playing other scales. The gap above may look rather haphazard and arbitrary, but hold on, we're working in exponential space here - shouldn't we be using a logarithmic scale? If I switch to a logarithmic x-axis, we suddenly get a rather appealing pattern...

All the gaps between successive notes are about the same size, except for the gap between E and F, and B and C, which are about half that size. If you try to work that out exactly, you run into the problems we saw above, where C to D is 9/8 or 1.125, but D to E is 10/9 or 1.11111... Even so, you can imagine how people who were playing around with sounds might notice, damn, these are nice even steps we have here. Though you might also notice places where, in this scheme, it's not completely even - for example G to A (ratio 10/9) is noticeably smaller than A to B (ratio 9/8).

We've obliquely approached the idea of dividing the octave up into 12 steps, where each step is about the size of the gap between E and F or B and C. We call each of these steps a 'semitone'. Two semitones make a 'whole tone'. We might fill in all the missing semitones in our scale here using whole-number ratios, which gives you the black keys on the piano. There are multiple schemes for doing this, and the ratios tend to get a bit uglier. In the system we've outlined so far, a 'semitone' is not a fixed ratio, even though it's always somewhere around 1.06.

The set of 12 semitones is called the 'chromatic scale'. It is something like the 'colour space' for Western music. When you compose a piece, you select some subset of the 12 semitones as your 'palette' - the 'scale of' a piece of music.

But we still have a problem here, which is the unevenness of the gaps we discussed above. This could be considered a feature, not a bug, since each scale would have its own 'character' - it's defined by a slightly different set of ratios. But it does add a lot of complication when moving between scales.

So let's say we take all this irregularity as a bug, and try to fix it. The solution is 'equal temperament', which is the idea that the semitone should always be the exact same ratio, allowing the instrument to play any scale you please without difficulty.

Posed like this, it's easy to work out what that ratio should be: if you want 12 equal steps to be an octave, each step must be the 12th root of 2. Which is an irrational number that is about 1.05946...

At this point you say, wait, Bryn, didn't you just start this all off by saying that the human ear likes to hear nice simple integer ratios of frequencies? And now you're telling me that we should actually use an irrational number, which can't be represented by any integer ratio? What gives? But it turns out the human ear isn't quite that picky. If you have a ratio of 7 semitones, or a ratio of 2^(7/12)=1.4983..., that's close enough to 1.5 to feel almost as good. And this brings a lot of huge advantages: you can easily move ('transpose') between different scales of the same type, and trust that all the relevant ratios will be the same.

Equal temperament was the eventual standard, but there was a gradual process of approaching it called stuff like 'well-tempered' or 'good temperament'. One of the major steps along the way was Bach's collection 'the well-tempered klavier', showing how a keyboard instrument with a suitable tuning could play music in every single established scale. Here's one of those pieces:

Although we're using these irrational numbers, inside the scale are certain intervals that are considered to have certain meanings - some that are 'consonant' and some that are 'dissonant'. We've already mentioned the 'fifth', which is the 'most consonant' ratio. The fifth consists of 7 semitones and it's roughly a 1.5 ratio in equal temperament. Its close cousin is the 'fourth', which consists of 5 semitones. Because it's so nice, the fifth is kind of 'neutral' - it's just there but it doesn't mean a lot on its own.

For the other important intervals we've got to introduce different types of scale.

The scale zoo

So, up above we introduced the 'major' scale. In semitones, the major scale is intervals of 2, 2, 1, 2, 2, 2, 1. This is also called a 'mode', specifically the 'Ionian mode'. There are seven different 'modes', representing different permutations of these intervals, which all have funky Greek names.

The major scale generally connotes "upbeat, happy, triumphant". There are 12 different major scales, taking the 12 different notes of the chromatic scale as the starting point for each one.

Next is the minor scale, which tends to feel more sad or mysterious. Actually there are a few different minor scales. The 'natural minor' goes 2, 1, 2, 2, 1, 2, 2. You might notice this is a cyclic permutation of the major scale! So in fact a natural minor scale is the same set of notes as a major scale. What makes it different?

Well, remember when we talked about tension and resolution? It's about how the notes are organised. Our starting note is the 'root' note of the scale, usually established early on in the piece of music - quite often the very first note of the piece. The way you move around that root note determines whether the piece 'feels' major or minor. So every major scale has a companion natural minor scale, and vice versa. The set of notes in a piece is enough to narrow it down to one minor and one major, but you have to look closer to figure out which one is most relevant.

The 'harmonic minor' is almost the same, but it raises the second-last note (the 7th) a semitone. So its semitone intervals are 2, 1, 2, 2, 1, 3, 1.

The 'melodic minor' raises both the 6th and 7th by one semitone. So its semitone intervals are 2, 1, 2, 2, 2, 2, 1.

If you talk about a 'minor scale' unqualified, you mean the natural minor. It's also the 'Aeolian mode' in that system of funky Greek names I mentioned earlier.

So that leads to a set of 24 scales, a major and minor scale for every semitone. These are the most common scale types that almost all Western tonal music is written in.

But we ain't done. Because remember I said there were all those other "modes"? These are actually just cyclic permutations of the major scale. There's a really nerdy Youtube channel called '8-bit music theory' that has a bunch of videos analysing them in the context of videogame music which I'm going to watch at some point now I finally have enough background to understand wtf he's talking about.

And of top of that you have all sorts of other variants that come from shifting a note up or down a semitone.

The cast of intervals

OK, so we've established the idea of scales. Now let's talk intervals. As you might guess from the 'fifth', the intervals are named after their position in the scale.

Let me repeat the two most common scale modes, in terms of number of semitones relative to the root note:

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

So you can see the fourth and fifth are the same in both. But there's a difference in three places: the third, the sixth, and the seventh. In each case, the minor is down a semitone from the major.

The interval names are... not quite as simple as 'place in the scale', but that's mostly how it works. e.g. the 'major third' is four semitones and the 'minor third' is three.

The fourth and fifth, which are dual to each other (meaning going up a fifth takes you to the same note as going down a fourth, and vice versa) are called 'perfect'. The note right in between them, an interval of 6 semitones, is called the 'tritone'.

(You can also refer to these intervals as 'augmented' or 'diminished' versions of adjacent intervals. Just in case there wasn't enough terminology in the air. See the table for the names of every interval.)

So, with these names, what's the significance of each one? The thirds, sixths and sevenths are important, because they tell us whether we're in minor or major land when we're building chords. (More on that soon.)

The fifth and the octave are super consonant, as we've said. But the notes that are close to them, like the seventh, the second and even more so the tritone, are quite dissonant - they're near to a nice thing and ironically that leads to awkward ratios which feel uncomfy to our ears. So generally speaking, you use them to build tension and anticipation and set up for a resolution later. (Or don't, and deliberately leave them hanging.)

Of course all of these positions in the scale also have funky Latin names that describe their function.

There's a lot more complicated nuances that make the meaning of a particular interval very contextual, and I certainly couldn't claim to really understand in much depth, but that's basically what I understand about intervals so far.

Our goofy-ass musical notation system

So if semitones are the building block of everything, naturally the musical notation system we use in the modern 12TET era spaces everything out neatly in terms of semitones, right?

Right...?

Lmao no. Actually sheet music is written so that each row of the stave (or staff, the five lines you write notes on) represents a note of the C major scale. All the notes that aren't on the C major scale are represented with special symbols, namely ♯ (read 'sharp') which means 'go up a semitone', and ♭ (read 'flat') which means 'go down a semitone'. That means the same note can be notated in two different ways: A♯ and B♭ are the same note.

The above image shows the chromatic scale, notated in two different ways. Every step is exactly one semitone.

Since a given scale might end up using one of these 'in between' notes that has to be marked sharp or flat, and you don't want to do that for every single time that note appears. Luckily, it turns out that each major/minor scale pair ends up defining a unique set of notes to be adjusted up or down a semitone, called the 'key signature'. So you can write the key signature at the beginning of the piece, and it lasts until you change key signature. For example, the key of 'A♭ major' ends up having four sharps:

There is a formula you can use to work out the set of sharps or flats to write for a given key. (That's about the point I checked out on musictheory.net.)

There is some advantage to this system, which is that it very clearly tells you when the composer intends to shift into a different scale, and it saves space since with the usual scales there are no wasted lines. But it's also annoyingly arbitrary. You just have to remember that B to C is only a semitone, and the same for E to F.

What are those weird squiggly symbols? Those are 'clefs'. Each one assigns notes to specific lines. The first one 𝄞 is the 'treble clef', the second one 𝄢 is the 'bass clef'. Well, actually these are the 'G-clef' and the 'F-clef', and where they go on the stave determines note assignment, but thankfully this has been standardised and you will only ever see them in one place. The treble clef declares the lines to be E G B D F and the bass clef G B D F A.

There is also a rarer 'C-clef' which looks like 𝄡. This is usually used as the 'Alto clef' which means F A C E G.

This notation system seems needlessly convoluted, but we're rather stuck with it, because most of the music has been written in it already. It's not uncommon for people to come up with alternative notations, though, such as 'tabs' for a stringed instrument which indicate which position should be played on each string. Nowadays on computers, a lot of DAWs will instead use a 'piano roll' presentation which is organised by semitone.

And then there's chords.

Chords! And arpeggios!

A chord is when you play 3 or more notes at the same time.

Simple enough right? But if you wanna talk about it, you gotta have a way to give them names. And that's where things get fucking nuts.

But the basic chord type is a 'triad', consisting of three notes, separated by certain intervals. There are two standard types, which you basically assemble by taking every other note of a scale. In terms of semitones, these are:

Major triad: 0 - 4 - 7 Minor triad: 0 - 3 - 7

Then there's a bunch of variations, for example:

Augmented: 0 - 4 - 8 Diminished: 0 - 3 - 6 Suspended: 0 - 2 - 7 (sus2) or 0 - 5 - 7 (sus4) Dominant seventh: 0 - 4 - 7 - 10 Power: 0 - 7

There is a notation scheme for chords in pop, jazz, rock, etc., which starts with a root note and then adds a bunch of superscripts to tell you about any special features of a chord. So 'C' means the C Major triad (namely C,E,G) and 'Cm' or 'c' means the C Minor triad (namely C,E♭,G).

In musical composition, you usually tend to surround the melody (single voice) with a 'chord progression' that both harmonises and creates a sense of 'movement' from one chord to another. Some instruments like guitar and piano are really good at playing chords. On instruments that can't play chords, they can still play 'arpeggios', which is what happens if you take a chord and unroll it into a sequence of notes. Or you play in an ensemble and harmonise with the other players to create a chord together. Awww.

Given a scale, you can construct a series of seven triad chords, starting from each note of the scale. These are generally given scale-specific Roman numerals corresponding to the position in the scale, and they're used to analyse the progression of chords in a song. I pretty much learned about this today while writing this post, so I can't tell you much more than that.

Right now, that's about as far as I've gotten with chords. On a violin, you can play just two strings at the same time after all - I never had much need to learn about them so it remains a huge hole in my understanding of music. I can't recognise chords by ear at all. So I gotta learn more about them.

As much as I wrote this for my own benefit... if you found this post interesting, let me know. I might write more if people find this style of presentation appealing. ^^'

[satyam-mathematics]

1.चक्रीय क्रमचय (Cyclic Permutations),पक्षान्तरण (Transposition)-

चक्रीय क्रमचय (Cyclic Permutations),पक्षान्तरण (Transposition) को समझने के लिए पहले इनको परिभाषा के द्वारा समझते हैं।

चक्रीय क्रमचय की परिभाषा (Definition of Cyclic Permutations)-माना कि S  एक परिमित समुच्चय है तो यदि f इस समुच्चय का ऐसा क्रमचय है कि

Read More.

[azurefishnets]

Yuletide Letter 2021

Dear Author,

I’m excited to have you join me for this Yuletide Exchange! I hope you find something that resonates with you here. Feel free to mix and match as necessary; in all cases, all listed characters do not have to be included. In most cases, I've bulleted the ones I'm extra-interested in, but in all cases I've requested any nommed character and would be more than thrilled to get any nommed character.

It’s probably worth noting that I am far, far more interested, in general, in gen fills and missing moments than I am in ship dynamics. Exceptions will be noted in the prompts themselves, but otherwise, I prefer erring on the side of friendly.

LIKES:

worldbuilding (I love prompts where it's like, pick a character and toss them in a location and see what happens)

sensory details (textures and smells especially)

unusual interactions (anything from slice of life to full adventure!)

puns and wordplay

cool symbolism & unexpected parallels

relationship level ups, whether that means platonic bonding or friends-to-more, etc

non-mundane AUs (no coffee shop, school, etc

fandom crossovers or fusions (but taken seriously, not cracky unless specifically requested)

missing moments

emotional support/intimacy

OKAY WITH:

any tense

any amount of plot or none

out-of-the-ordinary story structures

any amount of dialogue from none to screenplay

character explorations

characters appearing that aren’t requested as long as it doesn’t go against my DNWs

DNWS:

any kind of consent issues whatsoever

non-requested romantic relationships

angst without a happy ending or at least some kind of resolution

infidelity

underage relationships, non-canon children

disrespect of the holy & symbol

pandemic stuff (mentions of fantasy illness are fine but please, I don't want fantasy Covid).

GHOST TRICK: PHANTOM DETECTIVE

Worldbuilding

Alma

Cabanela

Jowd

As always I’m really really feeling my OT3 love, so any amount of goodness with them would be great please and thank you. You can split them, you can write just two of them or one of them, that’s OK, as long as no infidelity is implied.

That said, explorations of the times when some or all of them were dead are just fine? Or if you focus on one person…those five years when each of them are alone are of big interest to me in a bunch of ways; I just really really like the various plays on being haunted in this and anything that explores more of that will have my interest…

Or, if romance isn't your bag, any of the other tagged characters are up for grabs and I would LOVE some worldbuilding about Temsik or the Azul country or any kind of unusual interaction between any character. Emma goes on a book signing tour in a foreign country? Missile meets Pigeon Man at last?

Preferred pairings, if necessary:

Alma/Cabanela/Jowd (or any permutation thereof

Fond of Lynne/Memry

Fond of Emma/Justice Minister

LEGEND OF HEROES: TRAILS IN THE SKY

Worldbuilding

Cassius Bright

Celeste D. Auslese

These games are all about the found family and I love it but I also adore the absolute whiplash from adorable fluffiness to profound existential wtf-ness. I would be more than OK with something that explored that, maybe in conjunction with the multiple suggestions we’ve had of cyclical times, forced charisma, a civilization predating the Great Collapse… I don’t know, anything exploring the Mistwald dreams, Phantasma (carnivorous omnivorous bookcases!), the Tetracyclic Towers, or the time on the Ark would be super cool. Interweavings and references to the other two continuities are also extremely cool. Especially now, with Worldbuilding being a viable tag, I REALLY want any worldbuilding thing you care to write. Would love to see anything you've got about Celeste D. Auslese and her time in the Hermit's Garden with any of the other characters. Ooh, especially construct-to-construct conversation like Celeste & Cassius, sort of behind-the-scenes...

Preferred pairings:

Mueller/Olivier

Kevin/Ries

Estelle/Joshua is OK to mention but would rather it not be the focus

Please, no spoilers for Reverie or Kuro.

LEGEND OF HEROES: CROSSBELL ARC

Worldbuilding

Any

In Zero/Ao, explorations of KeA’s powers juxtaposed with her innocence are always cool… as far as locations go, I’m fascinated with the Temple of Moon, the Tower of Stargaze, and the Ancient Battlefield. How are they linked with themselves, and with the rest of Zemuria? What’s the connection to Erebonia? Any of the tag set is fair game. You can throw any character at any location and that would be cool, or I really enjoy Lloyd dadding people / Sergei tired-uncling people / Dudley being a SUPER JUSTICE NERD at people...

Preferred pairings: No ships please from the nominated tags. Fond of Ilya/Rixia if it happens to come up.

Please, no spoilers for Reverie or Kuro.

HEAVEN'S VAULT

Worldbuilding

Any other tag

While I'm fine with any nommed character or indeed any character at all, I'm really really interested in the implied life and culture of the Nebula both past and present, the Loop as either the Ancients understood it or as the current population of the Nebula understands it... also really interested in locations/their hopper links/the gods and how these intersect. Fascinated with robots' Ethical Cores and what they will do versus what they won't do, and how the concept of gardens seems inextricably woven into the Ancients' lives but not so much in the present, especially on Iox... (at least on the surface! Badum-tsh!)

Would love to see more interactions or just reactions from Huang to some of what Aliya is learning, his thoughts on robots, his thoughts post-ending. Or Oroi, prickle prickle prickle, particularly after meeting Enkei or, similarly, after the end of the game. I'm also very interested in Heaven's Vault and the disconnect between what the Ancient's knew, what Enkei knew, and what Aliya knows.

FAR: LONE SAILS

Worldbuilding

Lone Henriksson

The atmosphere of this game is just so fascinating to me. Unlike Heaven's Vault above, there is only ONE path, and yet the pathos is real! I loved the ways Lone made the okomotive a little more personal, a little more like both a home and a friend combined. What does she think about? What are her reactions to some of the places she sees? What are her thoughts on being the last Henriksson with, presumably, the last okomotive her family built? The angst potential is strong...

Or examine the past a bit more... delve into the lore. The resort towns left high n' dry by the sea... a missing moment either in a town or with some of the last wildlife or during the part with oppidumotive, especially in the offices! If Lone could have stopped and looked around, what would she have learned about herself or her family? Or what would she have liked to avoid learning...

CARTO

Worldbuilding

Granny Maldpo

Any other tag

I am really fascinated by the culture of the Navigators and how they can change the world to suit themselves. Explorations of the limits of that power would be cool--can you do it from the airship to the lands below? Or the story chalet... mischief and discoveries with how the chalet is constructed. Would be interesting to throw Carto or puppies or Granny Maldpo at the story chalet man... would LOVE more Granny Maldpo. Let her have cool adventures, let the story chalet man get cool stories about him... not in a shippy way but just like.... older people doing Cool Things way!

[king-of-men]

My solution below the cut:

Pick some integer N>2. For each set of N integers, first cyclically permute them, then swap the top integer with the lowest one in the next set. For example, with N=3:

123 456 789... becomes 314 627 9510...

Or in other words, still with N=3:

Move guest 1 to room 2

Otherwise move guests with n%3=1 to n-1, n%3=2 to n+3, and n%3=0 to n-2. Now, for any nonempty set of rooms, its smallest element is S. If S%3=2 then it is replaced by S-3, which by assumption is not in the set. If S%3=0 it is replaced by S+1, and if S%3=1 it's replaced by S+2, which are respectively 1 and 0 mod 3. Either those positions were in the set, or not. If not, the set satisfies the condition. If either one was in the set, we continue to replace by alternately adding 2 and 1 until we reach an element which isn't in the set and are done.

A little infinite puzzle for you.

We have an infinite hotel, that is a hotel, where the rooms are numbered by 1, 2, 3, ...

In each room there is a guest. Nothing interesting happened for a while, so the guests got very bored. They decide that each guest moves to a new room simultaneously. Furthermore, they want to move in such a way that for any nonempty finite set of rooms S there is a guest who did not live in the rooms of S but now moves into one of them. Can they achieve this?

For example, the guests in 1 and 2 are not allowed to switch places as there would be no new guest in S={1,2}.

Note that the condition implies that no room is left empty, as we could pick S to be that single room.