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A loosely sorted and lightly annotated catalogue of through the music posts on this blog.

Having been playing Stravinsky’s Four Études for Orchestra, my main reaction has been: what? Here’s some background. Despite the title, these were never orchestral studies. Three are orchestrations of his Three Pieces for String Quartet (1914). The fourth, Madrid, began as an Étude for Pianola (1917), written for a player piano. Its score needs six staves, three pianists’ worth of notes, because no human was supposed to play it. In 1928 Stravinsky gave all four titles and an orchestra; Ansermet premiered them in Berlin in 1930. So we’re playing music written for four soloists and a machine. ...

Notes on Chapter 2 of Professor MacKenzie’s Mechanizing Proof: why anyone wanted to prove programs correct.

Prof Ortner gave a talk on Tuesday about his recent working paper, “Leader-Driven Collusion” (19 August 2026). The paper finds that a patient cartel leader’s best scheme opens with gradually rising prices before settling at the monopoly price. The method is as interesting as the result. In place of the recursive machinery of Abreu, Pearce and Stacchetti (APS, 1990), the field’s standard tool, the paper uses Lagrangian duality. One constraint, two readings. In repeated Bertrand with zero cost, a firm that undercuts takes the whole market today and earns nothing thereafter. A fringe firm therefore stays in the cartel only if ...

This note came out of the first sessions of a PhD course in monetary economics. The opening question was what “fiat” money is. The word is Latin for “let it be done”; it has nothing to do with the car. Fiat money is an object that is intrinsically worthless and inconvertible: it gives no utility, has no use in production, and cannot be redeemed for anything that does (Wallace, 1980). That definition is the source of the difficulty. In Arrow–Debreu general equilibrium, markets are complete and all trades for all dates and states clear at date 0 under a single budget constraint. An intermediate object has no function there, and its equilibrium price is zero. ...

Eggertsson and Woodford’s 2003 paper was written when Japan’s policy rate had sat near zero for years. It starts from a simple fact: nominal interest rates cannot fall much below zero. A central bank conducts policy by setting a short-term nominal rate. In the US this is the federal funds rate, the overnight rate at which banks lend reserves to each other. Lower rates make borrowing cheaper and saving less attractive, so spending rises. ...

Banks lend against a firm’s liquidation value, what its buildings, inventory and receivables would fetch if sold separately, and monitor it through EBITDA-based covenants. EBITDA is Earnings Before Interest, Taxes, Depreciation and Amortization. That suits a deposit-funded institution, whose loans must be safe and easy to value. It fails firms like software companies, whose continuation value far exceeds their liquidation value: you can’t sell the software engineer. Even after default a bank cannot capture that value: the Bank Holding Company Act limits it to 5% of a non-financial firm’s voting shares, so it cannot take control and sell the business intact. ...

Dawn. Kashchei’s soul is destroyed, the spell breaks, and the captives come back to life. A solo horn plays a folk tune over quiet strings, the most famous tune in the ballet. It passes through the orchestra and grows each time. Then the brass take it in 7/4, every note given its own weight, and the suite ends in a blaze of B major. The 7/4 is the trick. The tune began in a lilting triple meter. Stretched into seven beats, it stops being a song and becomes a procession. Same tune, new body. ...

After the Infernal Dance, a lullaby. The monsters have collapsed, and the Firebird puts Kashchei to sleep. The bassoon gets the tune, a lazy, swaying one, over rocking harp and hushed strings. The bassoon is usually the orchestra’s comedian. Here it sings, low and alone, and it’s one of the loveliest things in the piece. The movement doesn’t end. It thins to a quiet string tremolo and hangs there, and the Finale grows out of it without a break. Kashchei sleeps; we wait. ...

The Infernal Dance opens with one enormous chord from the whole orchestra. Anyone who dozed off during the Khorovod is now awake. The plot: the Firebird enchants Kashchei and his monsters and makes them dance until they collapse. The music is built on syncopation. The accents land off the beat, the low brass and bassoons stamp out the theme, and the xylophone rattles on top. I’ve complained before about how hard it is to play together. ...