Relating Linear Algebra to Soundcloud

But in a weird way..

My Soundcloud playlist titled Fog is a gloomy amalgamation of songs that starts off melancholy and disjoint and dips you in a grim black muck that you can’t shake off. The tempo begins to rise awkwardly and you don’t feel reassured. You’re in an uncertain medium that induces self-doubt and pensiveness. The songs are heavy and lack vocals. But soon the uneasiness dissipates as the quickening tempo of the tunes creates a smooth transition into upbeat vocals. But the fog has yet to clear when you reach the end; you’re at the last song listening to a man moaning in what seems like Portuguese in a gypsy folk song you would hear on a train ride through Kazakhstan.

When I listen to Fog, I awe at the fact that such a playlist can be so eclectic in terms of its genres but can contain songs that all induce similar feelings. I mostly listen to it while I’m doing Linear Algebra, so I can’t help but relate the playlist to the concepts I learn. Because Soundcloud calls playlists “sets”, it’s difficult not to think of math. So let’s begin.

A set is made up of vectors, and we can think of these vectors to be songs. A vector has a magnitude and a direction. A song does too; a song can induce very specific feelings of different strengths within a listener. We can imagine a song plotted on a 2-dimensional plane with the axes being emotions — a huge range of feelings! But what if we were to combine bits and pieces of songs to induce even more powerful emotions?

This leads us to the span of a set. We define the span of a non-empty set to be the collection of all linear combinations of the songs in it. We aren’t making mashups when we’re talking about linear combinations nor are we looking at how the order of songs in a set makes us feel. We’re specifically looking at all the music and sounds that come to mind if we take elements from certain songs and mix them with others. Questions that come to my mind are: how many flavors does a song have? Can a song reach multiple genres? If so, which ones? When we have a linear combination of songs, how many sounds, notes, etc. can we feel?

A set can span a wide variety of genres and not necessarily contain songs from them, and that’s because each artist can be influenced by musicians outside his or her genre. Since we can see each song as a work influenced by multiple genres, containing similar vocals, instruments, and tunes, it’s possible to argue that a Soundcloud set can be broken down to a subset of a linearly independent genres of music. These genres of music cannot be expressed as linear combinations one another and also span our Soundcloud set. In other words, the linear combination of these genres is able to achieve the same feelings as the linear combination of the original set — their spans are the same!

As an example of how a song can be broken down into two “non-overlapping” genres, Beethoven and Rakim produced two vastly different types of music, but both influences were seen in Nas’ song titled “I Can”. The “doodle-doodle-doo” from Fur Elise echoes in the background while the artist gives a ice-grilled rap sermon echoing Rakim’s diamond-sharp style.

And now I guess my humanistic thought processing has lead me to a more technical path, because now I feel like I hit a dead end in my comparisons. Rather than looking at a song and its influence on our emotions, what if we looked at its components? Instead of being limited to two dimensions, what if we broke down a song into its different characteristics, like beats per minute and acoustics and range of notes and hundreds of others?

Well now I feel like you can do a lot more with the songs if you were to have some sort of scale for the characteristics, because once you have a ton of vectors (songs) in 100+ Dimensional space, you can actually make really interesting comparisons. You would also be able to have a stronger analysis of the different emotions songs can induce, assuming a song can produce a generally agreed-upon emotion. (It can.) You could run statistical tests on sad music and look deeper the fact that sad songs generally happen to be in the minor key, maybe understanding the common instruments, vocals, beats, that evoke such emotions. That would be an awesome way to apply big data analytics.

Aaaaaand I think I’ve just hit Pandora.

Once you have all these vectors and their variables, if you have some gnarly computational power, you can generate a playlist of songs that are close in distance from the input. And when you add some flavor after you searched by adding another artist, composer, whatever, you’re probably just taking the average between those two vectors. When Pandora tells you, “You’re listening to this song because…”, it could be describing the specific characteristics that are within a close range to those of the song you originally searched. That’s awesome. Linear algebra’s awesome.