Graphing the Glissando in "Rhapsody in Blue"
George Gershwin's "Rhapsody in Blue" famously opens with a trilling clarinet that rockets up more than two octaves in what's known as a "glissando," in which an instrument's pitch bends across a span of notes in a continuous climb or descent rather than stopping at discrete frequencies. Here's an example from the Columbia Symphony Orchestra, which probably sounds familiar:
The modern score allows for two beats to span the interval at Molto moderato--a tempo of 80, so 2⅔ seconds--but the solo clarinetist is allowed the artistic liberty to draw out the bend, and more importantly, what path to take from the low F to the high B♭. In data visualization terms, he or she can choose both the duration of the tween and the easing formula.
I've always wanted to compare different performances to see what those easing formulas look like. After playing around with about a dozen pitch-detection libraries, I put together this comparison of five glissandi from different renditions I found on YouTube:
As you can see, all five musicians front-loaded the ascent to at least some extent and then glided smoothly into harbor. This most closely resembles the easePolyOut formula in D3, though the curves you see here are not nearly as clean--nor should they be.
This may have to do with the mechanics of bending a pitch, which is easier on a reed instrument than on brass--though this guy pulls it off masterfully on a piccolo trumpet. Miles Davis also bends a short distance in the opening passage of "My Funny Valentine" in his Live at the Lincoln Center performance in 1964. (If you don't own that album, which features Herbie Hancock, Ron Carter, Tony Williams, and George Coleman, buy it immediately.) Though as a longtime trumpet player, I can attest that you can cheat a little by gradually pushing down the valves.
Curiously, the original score for the 1924 debut of "Rhapsody in Blue" does not specify a glissando:
But a grainy recording of that performance--which was allegedly attended by Sergei Rachmaninoff--suggests it was always part of Gershwin's vision. The orchestration was finished at the last minute, to the point that, at least according to lore, "The work’s significant piano solo was incomplete in this hasty arrangement. [...] At one point, a full page of blank sheet music was inserted in place of a piano solo, with a note for the conductor to 'wait for [Gershwin’s] nod'– when George would signal the band at the end of his solo."
I went into this diversion knowing nothing about how audio is encoded and came out knowing just a little bit. Any given note in a musical recording has many different frequencies--known as "partial tones" or "overtones"--which are responsible for the timbre. This is why a clarinet and a trumpet can play the same note but sound distinctly different.
Distinguishing the fundamental frequency from the partial tones is not simple. There are several different algorithms that attempt this with varying success. The most sophisticated open-source library I found is called "aubio", which is written in C but has a well-documented Python wrapper. (I prefer to use Node for everything, but that binding is not maintained.)
You can see my tests of each algorithm in the "Pitch Detection Tests" Python notebook in the repo. The best, by far, is the default algorithm, called "yinfft," which gets most of the notes in a given recording correct. For example, here's what it returns for the glissando from the Columbia Symphony Orchestra when you split it up into little pieces that are about 0.23 seconds each. (WAV files typically have 44,100 frames per second, which the code evaluates in chunks of 1024).
The basic shape is intact, but there are several outliers. To remedy this, I wrote a simple algorithm to guess the appropriate frequency of any note significantly distant from its neighbors, which is detailed in the README. It's not perfect, but it reins in the mistakes reasonably well without overfitting the curve too much. By all means, feel free to hack on it and send me a pull request.