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compare: Web-Projects:master
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Web-Projects:music-theory Web-Projects:master Web-Projects:i-love-programming-languages Web-Projects:donations Web-Projects:search Web-Projects:localization Web-Projects:colors Web-Projects:table-of-contents Web-Projects:margin-rework Web-Projects:sidenotes

10 Commits
music-theo ... master

Author SHA1 Message Date
Danila Fedorin
bdd2be48bd Update theme 
Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
 2026-03-22 01:31:20 -05:00
Danila Fedorin
a350cd14f8 Publich 'On Spiders' 
Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
 2026-03-22 01:08:11 -05:00
Danila Fedorin
49ab3982f1 Extract writing CSS into a separate file from the void. 
Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
 2026-03-22 01:04:58 -05:00
Danila Fedorin
4d35ca04fe Slightly tweak wording 
Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
 2026-01-19 10:23:02 -08:00
Danila Fedorin
5e117e3f48 Ignore warnings encoded as syntax highlighting 
Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
 2026-01-04 16:33:43 -08:00
Danila Fedorin
a6f3cd3f9a Update theme. 
Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
 2026-01-03 18:08:51 -08:00
Danila Fedorin
ccc8d6f0eb Re-enable hiding LaTeX 
Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
 2026-01-03 16:34:57 -08:00
Danila Fedorin
7a088d6739 Edit, finish, and publish post on PL 
Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
 2025-12-31 21:27:55 -08:00
Danila Fedorin
626baefd76 Do some aggressive trimming and editing. 
Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
 2025-12-30 00:06:17 -08:00
Danila Fedorin
4602d02720 [WIP] 2/3 draft 
Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
 2025-12-29 23:07:55 -08:00
20 changed files with 1287 additions and 581 deletions
Expand all files Collapse all files

3
.gitignore vendored
View File

@@ -1,4 +1 @@
**/build/*
/.quarto/
**/*.quarto_ipynb

3
_quarto.yml
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@@ -1,3 +0,0 @@
format:
gfm:
variant: +yaml_metadata_block

12
agda.rb
View File

@@ -46,7 +46,17 @@ class AgdaContext
# assumes that links can't span multiple pages, and that links
# aren't nested, so ensure that the parent of the textual node
# is the preformatted block itself.
raise "unsupported Agda HTML output" if at.parent.name != "pre"
if at.parent.name != "pre"
# Costmetic highlight warnings are sometimes applied to newlines.
# If they don't have content, treat them as normal newlines at the
# top level.
#
# This is an <a class="CosmeticProblem">\n</a> node.
unless at.parent.name == "a" and at.parent['class'] == "CosmeticProblem" and at.content.strip.empty?
raise "unsupported Agda HTML output in file #{file} at line #{line} (content #{at.content.inspect})"
end
end
# Increase the line and track the final offset. Written as a loop
# in case we eventually want to add some handling for the pieces

19
assets/scss/onspiders.scss Normal file
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@@ -0,0 +1,19 @@
@import "variables.scss";
@import "mixins.scss";
.spiderweb {
position: fixed;
top: 0;
right: 0;
opacity: 0.25;
max-width: 900px;
@include below-container-width {
width: $container-width-threshold;
}
}
code {
background: none;
border: none;
}

48
assets/scss/thevoid.scss
View File

@@ -1,51 +1,3 @@
@import "variables.scss";
body {
background-color: #1c1e26;
--text-color: white;
font-family: $font-code;
}
h1, h2, h3, h4, h5, h6 {
text-align: left;
font-family: $font-code;
}
h1::after {
content: "(writing)";
font-size: 1rem;
margin-left: 0.5em;
position: relative;
bottom: -0.5em;
color: $primary-color;
}
nav .container {
justify-content: flex-start;
a {
padding-left: 0;
margin-right: 1em;
}
}
.header-divider {
visibility: hidden;
}
hr {
height: auto;
border: none;
&::after {
content: "* * *";
color: $primary-color;
font-size: 2rem;
display: block;
text-align: center;
}
}
/* Code for the CSS glitch effect. Originally from: https://codepen.io/mattgrosswork/pen/VwprebG */
.glitch {

47
assets/scss/writing.scss Normal file
View File

@@ -0,0 +1,47 @@
@import "variables.scss";
body {
background-color: #1c1e26;
--text-color: white;
font-family: $font-code;
}
h1, h2, h3, h4, h5, h6 {
text-align: left;
font-family: $font-code;
}
h1::after {
content: "(writing)";
font-size: 1rem;
margin-left: 0.5em;
position: relative;
bottom: -0.5em;
color: $primary-color;
}
nav .container {
justify-content: flex-start;
a {
padding-left: 0;
margin-right: 1em;
}
}
.header-divider {
visibility: hidden;
}
hr {
height: auto;
border: none;
&::after {
content: "* * *";
color: $primary-color;
font-size: 2rem;
display: block;
text-align: center;
}
}

718
assets/svg/spiderweb.svg Normal file
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@@ -0,0 +1,718 @@
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inkscape:window-width="1728"
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---
title: "Reasons to Love the Field of Programming Languages"
date: 2025-12-31
tags: ["Programming Languages", "Compilers", "Type Systems"]
---
I work at HPE on the
[Chapel Programming Language](https://chapel-lang.org). Recently, another HPE
person asked me:
> So, you work on the programming language. What's next for you?
This caught me off-guard because I hadn't even conceived of moving on.
I don't want to move on, because __I love the field of programming languages__.
In addition, I have come to think there is something in PL for everyone, from
theorists to developers to laypeople.
So, in that spirit, I am writing this list as a non-exhaustive survey that holds
the dual purpose of explaining my personal infatuation with PL, and providing
others with ways to engage with PL that align with their existing interests.
I try to provide rationale for each claim, but you can just read the reasons
themselves and skip the rest.
My general thesis goes something like this: programming languages are a unique
mix of the __inherently human and social__ and the __deeply mathematical__,
a mix that often remains deeply grounded in the practical, __low-level realities of
our hardware__.
Personally, I find all of these properties equally important, but we have to
start somewhere. Let's begin with the human aspect of programming languages.
### Human Aspects of PL
> Programs must be written for people to read, and only incidentally for machines
> to execute.
>
> --- Abelson & Sussman, _Structure and Interpretation of Computer Programs_.
As we learn more about the other creatures that inhabit our world, we discover
that they are similar to us in ways that we didn't expect. However, our
language is unique to us. It gives us the ability to go far beyond
the simple sharing of information: we communicate abstract concepts,
social dynamics, stories. In my view, storytelling is our birthright more
so than anything else.
I think this has always been reflected in the broader discipline of programming.
_Code should always tell a story_, I've heard throughout my education and career.
_It should explain itself_. In paradigms such as
[literate programming](https://en.wikipedia.org/wiki/Literate_programming),
we explicitly mix prose and code. Notebook technologies
like [Jupyter](https://jupyter.org/) intersperse computation with explanations
thereof.
* __Reason 1__: programming languages provide the foundation of expressing
human thought and stories through code.
From flowery prose to clinical report, human expression takes a wide variety
of forms. The need to vary our descriptions is also well-served by the diversity
of PL paradigms. From stateful transformations in languages like Python and C++,
through pure and immutable functions in Haskell and Lean, to fully declarative
statements-of-fact in Nix, various languages have evolved to
support the many ways in which we wish to describe our world and our needs.
* __Reason 2__: diverse programming languages enable different perspectives
and ways of storytelling, allowing us choice in how to express our thoughts
and solve our problems.
Those human thoughts of ours are not fundamentally grounded in logic,
mathematics, or anything else. They are a product of millennia of evolution
through natural selection, of adaptation to ever-changing conditions.
Our cognition is limited, rife with blind spots, and partial to the subject
matter at hand. We lean on objects, actors, contracts, and more as helpful,
mammal-compatible analogies. I find this to be beautiful; here is something
we can really call ours.
* __Reason 3__: programming languages imbue the universe's fundamental rules of
computation with humanity's identity and idiosyncrasies. They carve out
a home for us within impersonal reality.
Storytelling (and, more generally, writing) is not just about communicating
with others. Writing helps clarify one's own thoughts, and to think deeper.
In his 1979 Turing Award lecture,
[Notation as a Tool of Thought](https://www.eecg.utoronto.ca/~jzhu/csc326/readings/iverson.pdf),
Kenneth Iverson, the creator of [APL](https://tryapl.org/), highlighted ways
in which programming languages, with their notation, can help express patterns
and facilitate thinking.
Throughout computing history, programming languages built abstractions that ---
together with advances in hardware --- made it possible to create ever more
complex software. Dijkstra's
[structured programming](https://en.wikipedia.org/wiki/Structured_programming)
crystallized the familiar patterns of `if`/`else` and `while` out of
a sea of control flow. Structures and objects partitioned data and state
into bundles that could be reasoned about, or put out of mind when irrelevant.
Recently, I dare say that notions of ownership and lifetimes popularized
by Rust have clarified how we think about memory.
* __Reason 4__: programming languages combat complexity, and give us tools to
think and reason about unwieldy and difficult problems.
The fight against complexity occurs on more battlegrounds than PL design alone.
Besides its syntax and semantics, a programming language is comprised of its
surrounding tooling: its interpreter or compiler, perhaps its package manager
or even its editor. Language designers and developers take great care to
[improve the quality of error messages](https://elm-lang.org/news/compiler-errors-for-humans),
to provide [convenient editor tooling](https://chapel-lang.org/blog/posts/chapel-lsp/),
and build powerful package managers
like [Yarn](https://yarnpkg.com/). Thus, in each language project, there is
room for folks who, even if they are not particularly interested in grammars or
semantics, care about the user experience.
* __Reason 5__: programming languages provide numerous opportunities for
thoughtful forays into the realms of User Experience and Human-Computer
Interaction.
I hope you agree, by this point, that programming languages are fundamentally
tethered to the human. Like any human endeavor, then, they don't exist in
isolation. To speak a language, one usually wants a partner who understands
and speaks that same language. Likely, one wants a whole community, topics
to talk about, or even a set of shared beliefs or mythologies. This desire
maps onto the realm of programming languages. When using a particular PL,
you want to talk to others about your code, implement established design patterns,
use existing libraries.
I mentioned mythologies earlier. In some ways, language
communities do more than share know-how about writing code. In many
cases, I think language communities rally around ideals embodied by their
language. The most obvious example seems to be Rust. From what I've seen,
the Rust community believes in language design that protects its users
from the pitfalls of low-level programming. The Go community
believes in radical simplicity. Julia actively incorporates contributions from
diverse research projects into an interoperable set of scientific packages.
* __Reason 6__: programming languages are complex collaborative social projects
that have the power to champion innovative ideas within the field of
computer science.
So far, I've presented interpretations of the field of PL as tools for expression and thought,
human harbor to the universe's ocean, and collaborative social projects.
These interpretations coexist and superimpose, but they are only a fraction of
the whole. What has kept me enamored with PL is that it blends these human
aspects with a mathematical ground truth, through fundamental connections to
computation and mathematics.
### The Mathematics of PL
> Like buses: you wait two thousand years for a definition of “effectively
> calculable”, and then three come along at once.
>
> --- Philip Wadler, _Propositions as Types_
There are two foundations,
[lambda calculus](https://en.wikipedia.org/wiki/Lambda_calculus) and
[Turing machines](https://en.wikipedia.org/wiki/Turing_machine), that underpin
most modern PLs. The abstract notion of Turing machines
is closely related to, and most similar among the "famous" computational models,
to the
[von Neumann Architecture](https://en.wikipedia.org/wiki/Von_Neumann_architecture).
Through bottom-up organization of "control unit instructions" into
"structured programs" into the imperative high-level languages today, we can
trace the influence of Turing machines in C++, Python, Java, and many others.
At the same time, and running on the same hardware functional programming
languages like Haskell represent a chain of succession from the lambda calculus,
embellished today with types and numerous other niceties. These two lineages
are inseparably linked: they have been mathematically proven to be equivalent.
They are two worlds coexisting.
The two foundations have a crucial property in common: they are descriptions
of what can be computed. Both were developed initially as mathematical formalisms.
They are rooted not only in pragmatic concerns of "what can I do with
these transistors?", but in the deeper questions of "what can be done
with a computer?".
* __Reason 7__: general-purpose programming languages are built on foundations of computation,
and wield the power to compute anything we consider "effectively computable at all".
Because of these mathematical beginnings, we have long had precise and powerful
ways to talk about what code written in a particular language _means_.
This is the domain of _semantics_. Instead of reference implementations
of languages (CPython for Python, `rustc` for Rust), and instead of textual
specifications, we can explicitly map constructs in languages either to
mathematical objects ([denotational semantics](https://en.wikipedia.org/wiki/Denotational_semantics))
or to (abstractly) execute them ([operational semantics](https://en.wikipedia.org/wiki/Operational_semantics)).
To be honest, the precise and mathematical nature of these tools is, for me,
justification enough to love them. However, precise semantics for languages
have real advantages. For one, they allow us to compare programs' real
behavior with what we _expect_, giving us a "ground truth" when trying to
fix bugs or evolve the language. For another, they allow us to confidently
make optimizations: if you can _prove_ that a transformation won't affect
a program's behavior, but make it faster, you can safely use it. Finally,
the discipline of formalizing programming language semantics usually entails
boiling them down to their most essential components. Stripping the
[syntax sugar](https://en.wikipedia.org/wiki/Syntactic_sugar) helps clarify
how complex combinations of features should behave together.
Some of these techniques bear a noticeable resemblance to the study of
semantics in linguistics. Given our preceding discussion on the humanity
of programming languages, perhaps that's not too surprising.
* __Reason 8__: programming languages can be precisely formalized, giving
exact, mathematical descriptions of how they should work.
In talking about how programs behave, we run into an important limitation
of reasoning about Turing machines and lambda calculus, stated precisely in
[Rice's theorem](https://en.wikipedia.org/wiki/Rice%27s_theorem):
all non-trivial semantic properties of programs (termination, throwing errors)
are undecidable. There will always be programs that elude not only human analysis,
but algorithmic understanding.
It is in the context of this constraint that I like to think about type systems.
The beauty of type systems, to me, is in how they tame the impossible.
Depending on the design of a type system, a well-typed program may well be
guaranteed not to produce any errors, or produce only the "expected" sort of
errors. By constructing reasonable _approximations_ of program
behavior, type systems allow us to verify that programs are well-behaved in
spite of Rice's theorem. Much of the time, too, we can do so in a way that is
straightforward for humans to understand and machines to execute.
* __Reason 9__: in the face of the fundamentally impossible, type systems
pragmatically grant us confidence in our programs for surprisingly little
conceptual cost.
At first, type systems look like engineering formalisms. That
may well be the original intention, but in our invention of type systems,
we have actually completed a quadrant of a deeper connection: the
[Curry-Howard isomorphism](https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence).
[Propositions](https://en.wikipedia.org/wiki/Proposition), in the logical sense,
correspond one-to-one with types of programs, and proofs of these propositions
correspond to programs that have the matching type.
This is an incredibly deep connection. In adding parametric polymorphism
to a type system (think Java generics, or C++ templates without specialization),
we augment the corresponding logic with the "for all x" (\(\forall x\)) quantifier.
Restrict the copying of values in a way similar to Rust, and you get an
[affine logic](https://en.wikipedia.org/wiki/Affine_logic), capable of reasoning about resources and their use.
In languages like Agda with [dependent types](https://en.wikipedia.org/wiki/Dependent_type),
you get a system powerful enough [to serve as a foundation for mathematics](https://en.wikipedia.org/wiki/Intuitionistic_type_theory).
Suddenly, you can write code and mathematically prove properties about that
code in the same language. I've done this in my work with
[formally-verified static program analysis]({{< relref "series/static-program-analysis-in-agda" >}}).
This connection proves appealing even from the perspective of "regular"
mathematics. We have developed established engineering practices
for writing code: review, deployment, documentation. What if we could use
the same techniques for doing mathematics? What if, through the deep
connection of programming languages to logic, we could turn mathematics
into a computer-verified, collaborative endeavor?
I therefore present:
* __Reason 10__: type systems for programming languages deeply correspond
to logic, allowing us to mathematically prove properties about code,
using code, and to advance mathematics through the practices of software engineering.
{{< details summary="Bonus meta-reason to love the mathy side of PL!" >}}
In addition to the theoretical depth, I also find great enjoyment in the way that PL is practiced.
Here more than elsewhere, creativity and artfulness come into
play. In PL, [inference rules](https://en.wikipedia.org/wiki/Rule_of_inference) are a
lingua franca through which the formalisms I've mentioned above are expressed
and shared. They are such a central tool in the field that I've
developed [a system for exploring them interactively]({{< relref "blog/bergamot" >}})
on this blog.
In me personally, inference rules spark joy. They are a concise and elegant
way to do much of the formal heavy-lifting I described in this section;
we use them for operational semantics, type systems, and sometimes more.
When navigating the variety and complexity of the many languages and type
systems out there, we can count on inference rules to take us directly to
what we need to know. This same variety naturally demands flexibility in
how rules are constructed, and what notation is used. Though this can sometimes
be troublesome (one [paper](https://labs.oracle.com/pls/apex/f?p=LABS%3A0%3A%3AAPPLICATION_PROCESS%3DGETDOC_INLINE%3A%3A%3ADOC_ID%3A959")
I've seen describes __27__ different ways of writing the simple operation of substitution in literature!),
it also creates opportunities for novel and elegant ways of formalizing
PL.
* __Bonus Reason__: the field of programming languages has a standard technique
for expressing its formalisms, which precisely highlights core concepts
and leaves room for creative expression and elegance.
{{< /details >}}
I know that mathematics is a polarizing subject. Often, I find myself
torn between wanting precision and eschewing overzealous formalism. The
cusp between the two is probably determined by my own tolerance for abstraction.
Regardless of how much abstraction you are interested in learning about,
PL has another dimension, close to the ground: more often than not, our languages
need to execute on real hardware.
### Pragmatics of PL
Your perfectly-designed language can be completely useless if there is no
way to
{{< sidenote "right" "execute-note" "execute it" >}}
Technically, there are language that don't care if you execute them at all.
Many programs in theorem-proving languages like Agda and Rocq exist only
to be type-checked. So, you could nitpick this claim; or, you could take
it more generally: your language can be useless if there's no
way to make it efficiently do what it's been made to do.
{{< /sidenote >}} efficiently. Thus, the field of PL subsumes not only
the theoretical foundations of languages and their human-centric design; it
includes also their realization as software.
The overall point of this section is that there is much depth to the techniques
involved in bringing a programming language to life. If you are a tinkerer
or engineer at heart, you will never run out of avenues of exploration.
The reasons are all framed from this perspective.
One fascinating aspect to programming languages is the "direction" from
which they have grown. On one side, you have languages that came
together from the need to control and describe hardware. I'd say that
this is the case for C and C++, Fortran, and others. More often than not,
these languages are compiled to machine code. Still subject to human
constraints, these languages often evolve more user-facing features as time
goes on. On the other side, you have languages developed to enable
people to write software, later faced constraints of actually working
efficiently. These are languages like Python, Ruby, and JavaScript. These
languages are often interpreted (executed by a dedicated program), with
techniques such as [just-in-time compilation](https://en.wikipedia.org/wiki/Just-in-time_compilation).
There is no one-size-fits-all way to execute a language, and as a result,
* __Reason 11__: the techniques of executing programming languages are varied
and rich. From compilation, to JIT, to interpretation, the field
has many sub-disciplines, each with its own know-hows and tricks.
At the same time, someone whose goal is to actually develop a compiler
likely doesn't want to develop everything from scratch. To do so would
be a daunting task, especially if you want the compiler to run beyond
the confines of a personal machine. CPU [architectures](https://en.wikipedia.org/wiki/Instruction_set_architecture)
and operating system differences are hard for any individual to keep up with.
Fortunately, we have a gargantuan ongoing effort in the field:
the [LLVM Project](https://llvm.org/). LLVM spans numerous architectures
and targets, and has become a common back-end for languages like C++
(via [Clang](https://clang.llvm.org/get_started.html)), Swift, and Rust.
LLVM helps share and distribute the load of keeping up with the ongoing
march of architectures and OSes. It also provides a shared playground upon
which to experiment with language implementations, optimizations, and more.
* __Reason 12__: large projects like LLVM enable language designers to
lean on decades of precedent to develop a compiler for their language.
Though LLVM is powerful, it does not automatically grant languages implemented
with it good performance. In fact, no other tool does. To make a language
run fast requires a deep understanding of the language itself, the hardware
upon which it runs, and the tools used to execute it. That is a big ask!
Modern computers are extraordinarily complex. Techniques such as
[out-of-order execution](https://en.wikipedia.org/wiki/Out-of-order_execution),
[caching](https://en.wikipedia.org/wiki/Cache_(computing)#HARDWARE),
and [speculative execution](https://en.wikipedia.org/wiki/Speculative_execution)
are constantly at play. This means that any program is subject to hard-to-predict
and often unintuitive effects. On top of that, depending on your language's
capabilities, performance work can often entail working with additional
hardware, such as GPUs and NICs, which have their own distinct performance
characteristics. This applies both to compiled and interpreted languages.
Therefore, I give you:
* __Reason 13__: improving the performance of a programming language is rife
with opportunities to engage with low-level details of the hardware
and operating system.
In the [mathematics section](#the-mathematics-of-pl), we talked about how constructing correct
optimizations requires an understanding of the language's semantics. It
was one of the practical uses for having a mathematical definition of a language.
Reason 13 is where that comes in, but the synthesis is not automatic. In fact,
a discipline sits in-between defining how a language behaves and
optimizing programs: program analysis. Algorithms that analyze
properties of programs such as [reaching definitions](https://en.wikipedia.org/wiki/Reaching_definition)
enable optimizations such as [loop-invariant code motion](https://en.wikipedia.org/wiki/Loop-invariant_code_motion),
which can have very significant performance impact. At the same time, for an
analysis to be correct, it must be grounded in the program's mathematical
semantics. There are many fascinating techniques in this discipline,
including [ones that use lattice theory](https://cs.au.dk/~amoeller/spa/spa.pdf).
* __Reason 14__: the sub-discipline of program analysis serves as a grounded
application of PL theory to PL practice, enabling numerous optimizations
and transformations.
The programs your compiler generates are software, and, as we just saw,
may need to be tweaked for performance. But the compiler and/or interpreter
is itself a piece of software, and its own performance. Today's language
implementations are subject to demands that hadn't been there historically.
For instance, languages are used to provide [language servers](https://microsoft.github.io/language-server-protocol/)
to enable editors to give users deeper insights into their code. Today,
a language implementation may be called upon every keystroke to provide
a typing user live updates. This has led to the introduction of
techniques like the [query architecture](https://ollef.github.io/blog/posts/query-based-compilers.html)
(see also [salsa](https://github.com/salsa-rs/salsa)) to avoid
redundant work and re-used intermediate results. New language implementations
like that of [Carbon](https://github.com/carbon-language/carbon-lang)
are exploring alternative representations of programs in memory. In
short,
* __Reason 15__: language implementations are themselves pieces of software,
subject to unique constraints and requiring careful and innovative
engineering.
### Conclusion
I've now given a tour of ways in which I found the PL field compelling,
organized across three broad categories. There is just one more reason
I'd like to share.
I was 16 years old when I got involved with the world of programming
languages and compilers. Though I made efforts to learn about it through
literature (the _Dragon Book_, and _Modern Compiler Design_), I simply
didn't have the background to find these resources accessible. However, all
was not lost. The PL community online has been, and still is, a vibrant and
enthusiastic place. I have found it to be welcoming of folks with backgrounds
spanning complete beginners and experts alike. Back then, it gave me
accessible introductions to anything I wanted. Now, every week I see new
articles go by that challenge my intuitions, teach me new things, or take PL
ideas to absurd and humorous extremes. So, my final reason:
* __Reason 16__: the programming languages community is full of brilliant,
kind, welcoming and enthusiastic people, who dedicate much of their
time to spreading the joy of the field.
I ❤️ the programming languages community.

484
content/blog/music_theory/index.qmd
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@@ -1,484 +0,0 @@
---
title: "Some Music Theory From (Computational) First Principles"
date: 2025-09-20T18:36:28-07:00
draft: true
filters: ["./to-parens.lua"]
custom_js: ["playsound.js"]
---
Sound is a perturbation in air pressure that our ear recognizes and interprets.
A note, which is a fundamental building block of music, is a perturbation
that can be described by a sine wave. All sine waves have a specific and
unique frequency. The frequency of a note determines how it sounds (its
_pitch_). Pitch is a matter of our perception; however, it happens to
correlate with frequency, such that notes with higher frequencies
are perceived as higher pitches.
Let's encode a frequency as a class in Python.
```{python}
#| echo: false
import math
import colorsys
```
```{python}
class Frequency:
def __init__(self, hz):
self.hz = hz
```
```{python}
#| echo: false
def points_to_polyline(points, color):
return """<polyline fill="none" stroke="{color}"
stroke-width="4"
points="{points_str}" />""".format(
color=color,
points_str=" ".join(f"{x},{y}" for x, y in points)
)
def wrap_svg(inner_svg, width, height):
return f"""<svg xmlns="http://www.w3.org/2000/svg"
width="{width}" height="{height}"
viewBox="0 0 {width} {height}">
{inner_svg}
</svg>"""
OVERLAY_HUE = 0
class Superimpose:
def __init__(self, *args):
global OVERLAY_HUE
self.args = args
self.color = hex_from_hsv(OVERLAY_HUE, 1, 1)
OVERLAY_HUE = (OVERLAY_HUE + 1.618033988) % 1
def points(self, width, height):
points = []
if not self.args:
return [0 for _ in range(width)]
first_points = [(x, y - height/2) for (x, y) in self.args[0].points(width, height)]
for thing in self.args[1:]:
other_points = thing.points(width, height)
for (i, ((x1, y1), (x2, y2))) in enumerate(zip(first_points, other_points)):
assert(x1 == x2)
first_points[i] = (x1, y1 + (y2 - height/2))
# normalize
max_y = max(abs(y) for x, y in first_points)
if max_y > height / 2:
first_points = [(x, y * (0.9 * height / 2) / max_y) for x, y in first_points]
return [(x, height/2 + y) for x, y in first_points]
def get_color(self):
return self.color
def _repr_svg_(self):
width = 720
height = 100
points = self.points(width, height)
return wrap_svg(
points_to_polyline(points, self.get_color()),
width, height
)
def hugo_shortcode(body):
return "{{" + "< " + body + " >" + "}}"
class PlayNotes:
def __init__(self, *hzs):
self.hzs = hzs
def _repr_html_(self):
toplay = ",".join([str(hz) for hz in self.hzs])
return hugo_shortcode(f"playsound \"{toplay}\"")
class VerticalStack:
def __init__(self, *args):
self.args = args
def _repr_svg_(self):
width = 720
height = 100
buffer = 10
polylines = []
for (i, arg) in enumerate(self.args):
offset = i * (height + buffer)
points = [(x, y+offset) for (x,y) in arg.points(width, height)]
polylines.append(points_to_polyline(points, arg.get_color()))
return wrap_svg(
"".join(polylines),
width, len(self.args) * (height + buffer)
)
def hex_from_hsv(h, s, v):
r, g, b = colorsys.hsv_to_rgb(h, s, v)
return f"#{int(r*255):02x}{int(g*255):02x}{int(b*255):02x}"
def color_for_frequency(hz):
while hz < 261.63:
hz *= 2
while hz > 523.25:
hz /= 2
hue = (math.log2(hz / 261.63) * 360)
return hex_from_hsv(hue / 360, 1, 1)
def Frequency_points(self, width=720, height=100):
# let 261.63 Hz be 5 periods in the width
points = []
period = width / 5
for x in range(width):
y = 0.9 * height / 2 * math.sin(x/period * self.hz / 261.63 * 2 * math.pi)
points.append((x, height/2 - y))
return points
def Frequency_get_color(self):
return color_for_frequency(self.hz)
def Frequency_repr_svg_(self):
# the container on the blog is 720 pixels wide. Use that.
width = 720
height = 100
points = self.points(width, height)
points_str = " ".join(f"{x},{y}" for x, y in points)
return wrap_svg(
points_to_polyline(points, self.get_color()),
width, height
)
Frequency.points = Frequency_points
Frequency.get_color = Frequency_get_color
Frequency._repr_svg_ = Frequency_repr_svg_
```
Let's take a look at a particular frequency. For reason that are historical
and not particularly interesting to me, this frequency is called "middle C".
```{python}
middleC = Frequency(261.63)
middleC
```
```{python}
#| echo: false
PlayNotes(middleC.hz)
```
Great! Now, if you're a composer, you can play this note and make music out
of it. Except, music made with just one note is a bit boring, just like saying
the same word over and over again won't make for an interesting story.
No big deal -- we can construct a whole variety of notes by picking any
other frequency.
```{python}
g4 = Frequency(392.445)
g4
```
```{python}
#| echo: false
PlayNotes(g4.hz)
```
```{python}
fSharp4 = Frequency(370.000694) # we write this F#
fSharp4
```
```{python}
#| echo: false
PlayNotes(fSharp4.hz)
```
This is pretty cool. You can start making melodies with these notes, and sing
some jingles. However, if your friend sings along with you, and happens to
sing F# while you're singing the middle C, it's going to sound pretty awful.
So awful does it sound that somewhere around the 18th century, people started
calling it _diabolus in musica_ (the devil in music).
Why does it sound so bad? Let's take a look at the
{{< sidenote "right" "superposition-note" "superposition" >}}
When waves combine, they follow the principle of superposition. One way
to explain this is that their graphs are added to each other. In practice,
what this means is that two peaks in the same spot combine to a larger
peak, as do two troughs; on the other hand, a peak and a trough "cancel out"
and produce a "flatter" line.
{{< /sidenote >}} of these two
notes, which is what happens when they are played at the same time. For reason I'm
going to explain later, I will multiply each frequency by 4. These frequencies
still sound bad together, but playing them higher lets me "zoom out" and
show you the bigger picture.
```{python}
Superimpose(Frequency(middleC.hz*4), Frequency(fSharp4.hz*4))
```
```{python}
#| echo: false
PlayNotes(middleC.hz, fSharp4.hz)
```
Looking at this picture, we can see that it's far more disordered than the
pure sine waves we've been looking at so far. There's not much of a pattern
to the peaks. This is interpreted by our brain as unpleasant.
{{< dialog >}}
{{< message "question" "reader" >}}
So there's no fundamental reason why these notes sound bad together?
{{< /message >}}
{{< message "answer" "Daniel" >}}
That's right. We might objectively characterize the combination of these
two notes as having a less clear periodicity, but that doesn't mean
that fundamentally it should sound bad. Them sounding good is a purely human
judgement.
{{< /message >}}
{{< /dialog >}}
If picking two notes whose frequencies don't combine into a nice pattern
makes for a bad sound, then to make a good sound we ought to pick two notes
whose frequencies *do* combine into a nice pattern.
Playing the same frequency twice at the same time certainly will do it,
because both waves will have the exact same peaks and troughs.
```{python}
Superimpose(middleC, middleC)
```
In fact, this is just like playing one note, but louder. The fact of the matter
is that *playing any other frequency will mean that not all extremes of the graph align*.
We'll get graphs that are at least a little bink wonky. Intuitively, let's say
that our wonky-ish graph has a nice pattern when they repeat quickly. That way,
there's less time for the graph to do other, unpredictable things.
What's the soonest we can get our combined graph to repeat? It can't
repeat any sooner than either one of the individual notes --- how could it?
We can work with that, though. If we make one note have exactly twice
the frequency of the other, then exactly at the moment the less frequent
note completes its first repetition, the more frequent note will complete
its second. That puts us right back where we started. Here's what this looks
like graphically:
```{python}
twiceMiddleC = Frequency(middleC.hz*2)
VerticalStack(
middleC,
twiceMiddleC,
Superimpose(middleC, twiceMiddleC)
)
```
```{python}
#| echo: false
PlayNotes(middleC.hz, twiceMiddleC.hz)
```
You can easily inspect the new graph to verify that it has a repeating pattern,
and that this pattern repeats exactly as frequently as the lower-frequency
note at the top. Indeed, these two notes sound quite good together. It turns
out, our brains consider them the same in some sense. If you have ever tried to
sing a song that was outside of your range (like me singing along to Taylor Swift),
chances are you sang notes that had half the frequency of the original.
We say that these notes are _in the same pitch class_. While only the first
of the two notes I showed above was the _middle_ C, we call both notes C.
To distinguish different-frequency notes of the same pitch class, we sometimes
number them. The ones in this example were C4 and C5.
We can keep applying this trick to get C6, C7, and so on.
```{python}
C = {}
note = middleC
for i in range(4, 10):
C[i] = note
note = Frequency(note.hz * 2)
C[4]
```
To get C3 from C4, we do the reverse, and halve the frequency.
```{python}
note = middleC
for i in range(4, 0, -1):
C[i] = note
note = Frequency(note.hz / 2)
C[1]
```
You might've already noticed, but I set up this page so that individual
sine waves in the same pitch class have the same color.
All of this puts us almost right back where we started. We might have
different pithes, but we've only got one pitch _class_. Let's try again.
Previously, we made it so the second repeat of one note lined up with the
first repeat of another. What if we pick another note that repeats _three_
times as often instead?
```{python}
thriceMiddleC = Frequency(middleC.hz*3)
VerticalStack(
middleC,
thriceMiddleC,
Superimpose(middleC, thriceMiddleC)
)
```
```{python}
#| echo: false
PlayNotes(middleC.hz, thriceMiddleC.hz)
```
That's not bad! These two sound good together as well, but they are not
in the same pitch class. There's only one problem: these notes are a bit
far apart in terms of pitch. That `triceMiddleC` note is really high!
Wait a minute --- weren't we just talking about singing notes that were too
high at half their original frequency? We can do that here. The result is a
note we've already seen:
```{python}
print(thriceMiddleC.hz/2)
print(g4.hz)
```
```{python}
#| echo: false
PlayNotes(middleC.hz, g4.hz)
```
In the end, we got G4 by multiplying our original frequency by $3/2$. What if
we keep applying this process to find more notes? Let's not even worry
about the specific frequencies (like `261.63`) for a moment. We'll start
with a frequency of $1$. This makes our next frequency $3/2$. Taking this
new frequency and again multiplying it by $3/2$, we get $9/4$. But that
again puts us a little bit high: $9/4 > 2$. We can apply our earlier trick
and divide the result, getting $9/16$.
```{python}
from fractions import Fraction
note = Fraction(1, 1)
seen = {note}
while len(seen) < 6:
new_note = note * 3 / 2
if new_note > 2:
new_note = new_note / 2
seen.add(new_note)
note = new_note
```
For an admittedly handwavy reason, let's also throw in one note that we
get from going _backwards_: dividing by $2/3$ instead of multiplying.
This division puts us below our original frequency, so let's double it.
```{python}
# Throw in one more by going *backwards*. More on that in a bit.
seen.add(Fraction(2/3) * 2)
fractions = sorted(list(seen))
fractions
```
```{python}
frequencies = [middleC.hz * float(frac) for frac in fractions]
frequencies
```
```{python}
steps = [frequencies[i+1]/frequencies[i] for i in range(len(frequencies)-1)]
minstep = min(steps)
print([math.log(step)/math.log(minstep) for step in steps])
```
Since peaks and troughs
in the final result arise when peaks and troughs in the individual waves align,
we want to pick two frequencies that align with a nice pattern.
But that begs the question: what determines
how quickly the pattern of two notes played together repeats?
We can look at things geometrically, by thinking about the distance
between two successive peaks in a single note. This is called the
*wavelength* of a wave. Take a wave with some wavelength $w$.
If we start at a peak, then travel a distance of $w$ from where we started,
there ought to be another peak. Arriving at a distance of $2w$ (still counting
from where we started), we'll see another peak. Continuing in the same pattern,
we'll see peaks at distances of $3w$, $4w$, and so on. For a second wave
with wavelength $v$, the same will be true: peaks at $v$, $2v$, $3v$, and so on.
If we travel a distance that happens to be a multiple of both $w$ and $v$,
then we'll have a place where both peaks are present. At that point, the
pattern starts again. Mathematically, we can
state this as follows:
$$
nw = mv,\ \text{for some integers}\ n, m
$$
As we decided above, we'll try to find a combination of wavelengths/frequencies
where the repetition happens early.
__TODO:__ Follow the logic from Digit Sum Patterns
The number of iterations of the smaller wave before we reach a cycle is:
$$
k = \frac{w}{\text{gcd}(w, v)}
$$
```{python}
Superimpose(Frequency(261.63), Frequency(261.63*2))
```
```{python}
Superimpose(Frequency(261.63*4), Frequency(392.445*4))
```
```{python}
VerticalStack(
Frequency(440*8),
Frequency(450*8),
Superimpose(Frequency(440*8), Frequency(450*8))
)
```
```{python}
from enum import Enum
class Note(Enum):
C = 0
Cs = 1
D = 2
Ds = 3
E = 4
F = 5
Fs = 6
G = 7
Gs = 8
A = 9
As = 10
B = 11
def __add__(self, other):
return Note((self.value + other.value) % 12)
def __sub__(self, other):
return Interval((self.value - other.value + 12) % 12)
```
```{python, echo=false}
class MyClass:
def _repr_svg_(self):
return """<svg xmlns="http://www.w3.org/2000/svg"
width="120" height="120" viewBox="0 0 120 120">
<circle cx="60" cy="60" r="50" fill="red"/>
</svg>"""
MyClass()
```

15
content/blog/music_theory/playsound.js
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@@ -1,15 +0,0 @@
window.addEventListener("load", (event) => {
for (const elt of document.getElementsByClassName("mt-sound-play-button")) {
elt.addEventListener("click", (event) => {
const audioCtx = new (window.AudioContext || window.webkitAudioContext)();
for (const freq of event.target.getAttribute("data-sound-info").split(",")) {
const oscillator = audioCtx.createOscillator();
oscillator.type = "sine"; // waveform: sine, square, sawtooth, triangle
oscillator.frequency.value = parseInt(freq); // Hz
oscillator.connect(audioCtx.destination);
oscillator.start();
oscillator.stop(audioCtx.currentTime + 2); // stop after 1 second
}
});
}
});

13
content/blog/music_theory/svg-inline.lua
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@@ -1,13 +0,0 @@
function Image(el)
local src = el.src
if src:match("%.svg$") then
local alt = el.caption and pandoc.utils.stringify(el.caption) or ""
local obj = string.format(
'{{< inlinesvg "%s" "%s" >}}',
src, alt
)
return pandoc.RawInline("html", obj)
else
return el
end
end

5
content/blog/music_theory/to-parens.lua
View File

@@ -1,5 +0,0 @@
function Math(el)
if el.mathtype == "InlineMath" then
return pandoc.RawInline("markdown", "\\(" .. el.text .. "\\)")
end
end

44
content/writing/onspiders.md Normal file
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@@ -0,0 +1,44 @@
---
title: "On Spiders"
date: 2026-03-22T01:03:00-05:00
type: onspiders
---
```
Whenever I stay still,
I feel the spiders weave their webs around me.
Their tiny legs entangle me in silk,
traverse my body as they seem to proudly
inspect their work and into darkness sink.
There're times I see the spiders' threads on others,
They shimmer gently in the purple dusk,
Or sway with zephyrs in the hair of lovers,
stuck.
To listen to the webs is to hear echoes
of social butterflies ensnared within
arachnid galleries of human ethos,
whose pieces come and go upon a whim.
Their curators are blind and want for stories,
delivered as vibrations through the links,
for righteous anger, anguish, joy and worries,
superimposing frequencies that mix,
reveberating through the sprawling complex
of radial lines and newly captured prey,
resisting first but ultimately hopeless,
these signals grow in urgency and they
form standing waves that make the victims sing
and even from myself bring out a resonant ring.
```
<!--
the signals cause a standing wave
intensify with every passing moment
arachnid galleries of human drama,
temples to the human mind
(traverse my body all while I sleep soundly,)
They shimmer gently in the setting sun, -->

11
convert.rb
View File

@@ -68,13 +68,12 @@ files.each do |file|
t.replace(rendered)
end
found_any ||= document.at('meta[name="needs-latex"]')
# If we didn't find any mathematical equations, no need to include KaTeX CSS.
# Disabled here because Bergamot technically doesn't require math blocks
# on the page but does need the CSS.
#
# unless found_any
# document.css('link[href$="katex.css"], link[href$="katex.min.css"]').each(&:remove)
# end
unless found_any
document.css('link[href$="katex.css"], link[href$="katex.min.css"]').each(&:remove)
end
File.write(file, document.to_html(encoding: 'UTF-8'))
end

25
layouts/onspiders/baseof.html Normal file
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@@ -0,0 +1,25 @@
{{- /* Note: changing the baseof template because the title, tags, etc. of a regular post are still valid. */ -}}
<!DOCTYPE html>
<html lang="{{ .Site.Language.Lang }}">
{{- partial "head.html" . -}}
<body>
{{ $writingcss := resources.Get "scss/writing.scss" | css.Sass | resources.Minify }}
<link rel="stylesheet" href="{{ $writingcss.Permalink }}">
{{ $spidercss := resources.Get "scss/onspiders.scss" | css.Sass | resources.Minify }}
<link rel="stylesheet" href="{{ $spidercss.Permalink }}">
<svg class="spiderweb" viewBox="0 0 197.21727 106.16592">
<use href="{{ (resources.Get "svg/spiderweb.svg").Permalink }}#mainlayer"></use>
</svg>
{{- partial "header.html" . -}}
<div class="container"><hr class="header-divider"></div>
<main class="container">
{{- if .Draft -}}
{{- partial "warning.html" (i18n "postDraft") -}}
{{- end -}}
{{- block "main" . }}{{- end }}
</main>
{{- block "after" . }}{{- end }}
</body>
</html>

1
layouts/shortcodes/inlinesvg.html
View File

@@ -1 +0,0 @@
<object type="image/svg+xml" data="{{ .Get 0 }}" aria-label="{{ .Get 1 }}"></object>

1
layouts/shortcodes/playsound.html
View File

@@ -1 +0,0 @@
<button class="mt-sound-play-button" data-sound-info="{{ .Get 0 }}">Play</button>

2
layouts/thevoid/baseof.html
View File

@@ -3,6 +3,8 @@
<html lang="{{ .Site.Language.Lang }}">
{{- partial "head.html" . -}}
<body>
{{ $writingcss := resources.Get "scss/writing.scss" | css.Sass | resources.Minify }}
<link rel="stylesheet" href="{{ $writingcss.Permalink }}">
{{ $voidcss := resources.Get "scss/thevoid.scss" | css.Sass | resources.Minify }}
<link rel="stylesheet" href="{{ $voidcss.Permalink }}">
{{- partial "header.html" . -}}

2
themes/vanilla

Submodule themes/vanilla updated: 952502e690...312d0d37c6