A Largely Facetious Work called "Mathematising Musical Aesthetics"

Mathematising Musical Aesthetics

I have a theory about the aesthetics of music. I postulate that genre-independent musical quality (Q) can be usefully understood in terms of the relationship between two only indirectly aesthetic variables, musical complexity (C) and musical difficulty (D) (I will explain how I quantify these soon), according to the following equation:

Q = (C−25)² ×1/D

I came to the specific form of this equation just by seeking out the right shape of graph, with appropriate units. This is what the function looks like in 3-D graph form, with Q represented on the vertical axis, with x substituted for C and z substituted for D:

The way I define musical difficulty is solely in terms of the number of listens it takes to appreciate the music to its maximum extent by an ideal listener in ideal conditions. I define an “ideal listener” imprecisely as someone with a ‘good ear’ accustomed to listening to all genres of music. “Ideal conditions” means the ‘listenings’ are not clustered together, and the listening occurs on the same medium each time, preferably a high-quality audio system. The units of D are dictated by our desire to preserve the structure of the function for numerical values -1.0 to 1.0. A musical difficulty value (D value) of 1 listen required for full appreciation by ideal listener gets the numerical value -1.0; a D value of 2 listens required for full appreciation by ideal listener gets the numerical value -0.5; a D value of 3 listens required for full appreciation by ideal listener gets the numerical value 0 (which we can adjust to 0.01 for the sake of avoiding mathematical difficulties (incidentally, the abrupt jump of the graph works fine because the D value itself abruptly jumps)); a D value of 4 listens required for full appreciation by ideal listener gets the numerical value 0.5; and a D value of 5 listens required for full appreciation by ideal listener gets the numerical value 1.

I would claim that the vast majority of pop music on the charts (“fairy floss music”) gets a D value of 1 listen (and thus a numerical value of -1.0). These songs are immediately ‘catchy’ because you’ve heard hundreds of very similar songs before. Highly repetitive ambient or minimalist music (much of the music of Steve Reich or Phillip Glass, or lounge music, or ‘chillhop’ or any similar such anodyne genres etc) also gets a D value of 1 listen. It’s not useful to give examples of music which has a D value of 2 listens, because there is so much of it. I would say that music with this difficulty value ranges from poppy indie music and slightly more difficult pop music (many Beatles songs, for example) to extremely dissonant music with no depth, like the music of Schoenberg. Schoenberg’s music is some of the worst music in existence, and that’s reflected in the 3-D graph, because I believe it sits somewhere near the steeply sloping bottom of the right-hand side of the half of the graph nearest to us. Probably the majority of Radiohead’s discography has a D value of 3 listens (some gets a value of 2 listens and some gets a value of 4 listens). Other slightly more interesting rock, ‘indie’, folk and prog-metal music has a value of 3 listens. Perhaps some ‘classical’ music (an immensely broad genre, encompassing, on my reckoning, Medieval, Baroque, Classical, Romantic and Impressionist) – perhaps Mozart, perhaps Tschaikovsky, perhaps Vivaldi – does also, and definitely some jazz. I would say that the majority of well-known ‘classical music’ from all the most acclaimed and famous composers has a value of 4 listens. Much of J.S. Bach’s music has this difficulty, as does much of Beethoven’s and Strauss’ and Berlioz’s and Schubert’s and Debussy’s and Borodin’s and Mussorgsky’s and Shostakovich’s and Sibelius’ and Stravinsky’s and Rachmaninoff’s (and so on). Most of Gustav Mahler’s music, on the other hand, has a value of 5 listens, I would say.

The variable C is vague and clearly can’t be made wholly objective. It can be understood as a function of: the number of instruments involved in the music; the unusalness of key, modes, harmonies and cadences; the unusualness of timbre and textures and tone colour; and the unusualness of time signature and rhythms. We can formalise this according to the following, very simple equation: C = N + U(k) + U (t) + U (r), which reduces to: C = N + U(k + t +r).  

I will use the music of post-Pablo Honey Radiohead, which I rank (overall) at a level of intermediate commplexity, as the test case for this equation. The N value  may be understood as the log₂[average number of players on each piece of music]. For Radiohead, the average number of players on each song is 5, so that figure is  approximately 2.32. (Meanwhile, symphonies involve an average of 90  players, meaning the N value for the average symphony is 6.49.) For the other values in the equation, there is no means of precise quantification. However, our desire to have post-Pablo Honey Radiohead ranking overall at a level of intermediate complexity gives us a yardstick for imprecise quantification.  I would say that on a scale where Schoenberg ranks 10 and Britney Spears ranks 1, post-Pablo Honey Radiohead on average has a U(k) value of 5; that, on a scale where Hector Berlioz’s Symphonie Fantastique ranks 10 and My Chemical Romance ranks 1, post-Pablo Honey Radiohead on average has a U(t) value of 8; and that, on a scale where Steve Reich’s “Clapping Music” or The Rite of Spring ranks 10 and Chris Brown ranks 1, post-Pablo Honey Radiohead on average has a U(r) value of 7. This would give it an overall C value of 22.32, which seems about right. Combined with an average D value of 3 listens, yielding a numerical value of 0.01 on the z axis of the graph, this would place it on the top half of the graph, in the trough, and give Radiohead's post-Pablo Honey music an overall score of approximately 79.1

Because of the limited number of instruments, and the predictibality of key, time signature and tone colour, almost all pop music (music which makes it to the charts) would get a complexity value around 10 (or even lower). Meanwhile, all the music of my favourite composers – Tschaikovksy, Shostakovich, Dvorak, Sibelius, Rachmaninoff, Berlioz, Mahler – would get a complexity level of at least 32. Their best music would score at least 220.

One notable thing about my function is that you can basically ignore the far left-hand sector of the 3-D graph, since I don’t think there is any music which requires many listens to fully appreciate but is not highly complex. On the other hand, I do believe there is highly complex music which does not have a high degree of musical difficulty in my sense. This is music which is complex but just totally ugly and atonal – in which the beauty of the melodies and harmonies don’t reveal themselves over the course of multiple listens precisely because there are none. A lot of contemporary classical Postmodern stuff is like this, or really really shitty experimental jazz.

The reason that I like this mathematisation of musical aesthetics is, of course, that my equation fairly well ‘explains’ my own musical preferences, which I have previously laid out in the constantly updated document called “Lists”, published in June of last year. Here are my own musical preferences:

Rank 1:

Antonin Dvorak
Arvo Part

Augie March (Strange Bird is their best album by a significant margin)

Beirut

Dimitri Shostakovich

Fela Kuti

Porcupine Tree (a really wonderful prog band (though I almost exclusively listen to just three of the albums in their extensive discography, In Absentia, Deadwing and Fear of a Blank Planet (these albums all have Gavin Harrison as drummer (an out-of-this-world musician), and showcase Steven Wilson's electric guitar range and atmospheric composition at its finest (unfortunately, Wilson’s lyrics are uniformly terrible))))

Pyotr Illyich Tchaikovsky

Radiohead (obviously)

Sergei Rachmaninoff (my favourite composer (sends me into convulsions) (as you can see, I really like the Russian Romantics))

The Shins (best indie-rock band (Chutes Too Narrow is a masterpiece of the genre))

Sufjan Stevens (I adore The Age of Adz, Illinois and Michigan, and I think Carrie and Lowell is a terrific album also)

Yann Tiersen

XTC (first got into XTC when I was 5, through my dad)
Rank 2:
Alexander Borodin

Andrew Bird
Aphex Twin

Arcade Fire (Funeral is the best album by a significant margin)

The Beach Boys (/Brian Wilson (Smile is a brilliant album))
The Beatles

Bjork

Brian Eno (I like Another Green World best – not that interested in the ambient stuff for which he is most famous)
The Cat Empire
The Clash

Crowded House
Dan Kelly
David Bowie (70s stuff and Blackstar)
Devotchka (only Little Miss Sunshine soundtrack)
Elvis Costello

Felix Mendelssohn
Gustav Mahler

Hector Berlioz
Igor Stravinsky

Jimi Hendrix

Jean Sibelius

Johannes Brahms

Johann Sebastian Bach

King Crimson
Kraftwerk

Led Zeppelin

Ludwig van Beethoven

Max Richter

Mighty Sparrow
Miles Davis
Nick Cave and the Bad Seeds
Nikolai Rimsky-Korsakov
Paul Kelly

Percy Grainger (for Horkstow Grange)

Pink Floyd

Richard Strauss

Sergei Prokofiev
Steven Wilson

The Smiths

Tame Impala 

Tom Waits
Wilco
Yello
Rank 3:
Blur
Boy and Bear                                                      

Broken Bells
Cloud Control
Doves
Gotye (favourite song is "State of the Art")
Josh Pyke
Kate Bush
Madness
Mental as Anything
Modest Mouse
Modest Mussorgsky

Pixies

R.E.M
Sigur Ros
Something for Kate