The most popular music video of last year is, arguably, Gangnam Style by the famous Korean artist, Ψ. One of the noteworthy features is the presence of ‘accumulation points’ in the music; these occur at 1:07 and 2:29 in the video:
Also, the notes in the song are well-ordered, meaning that every set of notes has an earliest element. Indeed, since there are two limit points, the song has order type 2ω + n, where n is the (finite) number of notes after 2:29 in the video.
The night before last, this inspired me to produce music corresponding to a larger infinite ordinal. ω^2 limit points actually sound effective, but when you approach larger limit ordinals such as ω^3 and ω^4, it degenerates into an absolute cacophany. As a proof of concept, I decided to produce the sound of ω^4.
The scale above represents the entire duration (3:39) of the music, with a linear scale showing certain ordinals marked on it. The first clap you can hear is ‘1’, followed shortly by ‘2’, ‘3’, and all of the other positive integers before reaching a limit point at ω. The entire process is repeated again and again, resulting in limits of limits, such as ω^2 (0:10 in the sound file) and ω^3 (0:46). The music ends at ω^4, although it is possible in principle to produce ordinal music corresponding to any countable ordinal. Most pieces of popular music merely have finite ordinals, with Gangnam Style being a rare exception.
Performing an infinite number of operations in a finite amount of time (such as producing music corresponding to ω) is a transfinite process known as a supertask. For instance, a computer capable of doing an elementary operation in 1/2 of a second, and the next in 1/4 of a second, and the third in 1/8 of a second, can effectively solve the halting problem in 1 second. Such a computer is known as a Zeno machine, since it resembles Zeno’s paradox.