The musical scale is arguably humanity’s oldest algorithm for beauty. Music theory can seem intimidating (for instance, I probably should have learned tuning theory back in 16 when I was sitting in orchestra). But it’s actually quite straightforward. Imagine you’re a piano designer: you want your instrument to sound nice, but you also don’t want it to have an infinite number of keys. Axiom 1 (Sounding nice) Two notes sound consonant when their frequencies form a ratio of small integers. The simpler the ratio, the more pleasant the sound. For example, a perfectly harmonic octave has a ratio of $2:1$, and a nice fifth has $3:2$. ...

AI can now make decent, impressive music. Check out Tunee and their example projects. Tunee’s interface features a conversational input, a project diagram, and options to preview AI-generated music with lyrics. However, there’s no way to export stems or sheet music yet. Out of curiosity, I impatiently splashed some random prompts on my keyboards into Tunee: the results were decent Bach, whimsical bouncy cool woodwinds, and a few suspiciously catchy ballads—of course, since AI is now quite good at generating lyrics: ...

After surviving a macro midterm where the True/False section felt like coin flipping, I rushed downtown for Dvořák’s Symphony No. 9 with the Chicago Symphony Orchestra under Sir Riccardo Muti (Concert Note digital version) Can you believe that Muti’s 84?! And he probably definitely counts bars better than I do. I sat in the center front row of the Terrace—right behind the stage. That seat feels almost illicit, as if you’ve joined the orchestra. Muti’s expressions are right there in front of you; his every gesture, every cue, goes directly under your eyes. He is astonishingly clear. Every motion carries intention—he shakes for trills, marks beats for the horns, molds phrases with his hands. His precision and humor are inseparable. It’s a masterclass in detail. Honestly, that seat should cost more when Muti’s conducting—as a orchestra flute player myself I’d say he’s the best conductor one could hope for. ...

Another note from info economics class. Recall Aumann’s common knowledge: fix an event $E$ and consider the two agents’ posteriors of event $E$ after agents each conduct their own deterministic partition experiment. (see this post for a recap) Theorem (Aumann 1976) If A’s posterior is $q_A$ and B’s posterior is $q_B$ is common knowledge for A and B, then $q_A = q_B$. But how come $q_A$ and $q_B$ become common knowledge in the first place? Geanakoplos and Polemarchakis’s paper We Can’t Disagree Forever (1982) gives a way that the two people can use (non-strategic) communication to arrive at a common understanding of each other’s posterior: ...

Who understands copyright best? IMSLP would definitely be shortlisted. They have an excellent article explaining general how copyright works: A work that is in the public domain is not protected by copyright and can be freely used for any purpose. Which works are in the public domain varies from country to country. For example, [in Canada] if a copyright lasts for 70 years after an author’s death, and the author who died on March 14, 1955, then the term lasts until December 31, 2025 and the work is in the public domain from 2026 onwards. ...

Consider the information game with one sender, one receiver, defined by $$ \lang \Omega, M, A, \mu_0, v, u\rang $$ where $\Omega$ is the state of the world $M$ is the space of messages $A$ is receiver’s action $\mu_0$ is the prior Sender’s payoff $v:A\times \Omega \to \R$ Receiver’s payoff: $u:A\times \Omega \to \R$. Sender’s strategy $\sigma:\Omega \to \Delta(M)$. Receiver’s strategy $\rho: M\to \Delta (A)$ A Cheap Talk Game proceeds as follows: ...

Beethoven has two adorable flute sonata. One is Anh.4, composed around his early 20. It was found amongst Beethoven’s papers after his death, remaining unpublished until 1906 (source wikipedia). Its Largo movement is absolutely gorgeous: The other is Op. 41, composed when he was around 30. Vivid, bouncy and cute: I was fascinated by these two pieces when first heard them in high school. While I later inevitably gradually become more found of other sophisticated works, they stayed as my loves — they’re so adorable. Happy Sunday!

Here’s a pretty cool and useless way to understand how to obtain the $\beta$ in linear regression. Let’s begin with (real valued) Hilbert space, $\mathcal H$ which comes with inner product $\lang \cdot, \cdot\rang$ that are symmetric, linear and positive definite. (Might as well think classical $x, y\in \R^m$ and $\lang x, y\rang = x^Ty = \sum_{i \in [m]}x_iy_i$ in standard linear algebra) Now suppose we are given $y\in \mathcal H$ and $X := [x_1, x_2, \ldots, x_N]\in \mathcal H^N$. The way to “project” $y$ onto $x$ is ...

Today The University of Chicago Symphony Orchestra performs our 2025 Halloween Concert. During the the William (Guillaume) Tell Overture by Gioachino Rossini, there’s a lovely slow passage in the middle of the piece: This pastorale section in G major and in an A-B-A-Coda form, signifying the calm after the storm, begins with a Ranz des vaches or “Call to the Cows”, featuring the cor anglais (English horn). The English horn then plays in alternating phrases with the flute, culminating in a duet with the triangle accompanying them in the background. ...

College classrooms can sometimes become awkwardly dead silence. This happens often when I was in Shanghai. The cringe culture can explain it a little. But I think all of us would be better off if we can shed that away. Here’s a new podcast report by Ariana Grande, snippet: The world had a difficult time with earnestness… We should be able to be happy and not be corny or cringe or classified to be something like that. Being passionate/happy/excited about something, loving people. What else do you want to me be, sitting here numb? ...