I bet you enjoy music? Music is just particular types of changing air pressure (sound waves) that are received by our ears and interpreted by our brains. These changes in air pressure we call music are something everyone enjoys. The reason is that these changes in air pressure contain lots of mathematical form, which your subconscious brain analyses, identifies, and creates feeling in response to. You love music because your subconscious mind loves math. More specifically—
Symmetry is when things are identical, save for some change in their location in time or space, such as a repeating pattern or reflection. Music is fundamentally about symmetry, with repeating patterns operating at many different time scales — from that of waves repeating thousands of times per second to form musical notes, to repeating melodic and rhythmical structures whose single units can last for minutes. It’s pretty amazing that we’re able to pick out such detail from the air entering our ears, and find all sorts of different symmetries which we only become aware of as our humanised experience of a piece of music.
Rhythms and melodies are great, but what really gets me are chords – when several notes are played at the same time, with particular ratios between their frequency values. We’re all experts at picking out how these frequency values interrelate, and there is a broad consensus in Western listeners about which ratios sound nice, and which don’t. In general, it’s the simpler ratios like 2:3 (the perfect fifth) and 5:4 (the major third) that sound the best to us (1:2 is so basic we hear the notes as the same, just shifted by an octave). When your subconscious mind detects some of these in the sound coming into your ears, it gets pretty excited – you massive geek! Ratios in rhythmical structure can also be very pleasing, with different rhythms interplaying as they weave in and out of each other.
You have probably heard of musical scales, or the key, in which a piece of music is played, which limits which of the twelve notes of the octave can be used (in a standard Western tuning system). When a piece of music is being played in a particular scale, and a note from another scale is thrown in, it sounds wrong to us. Our sensitivity to scale goes much further, so that we have expectations of what notes melodies should start and finish on, and which notes should be emphasised more, for each different scale – all tools for composers and producers to communicate with as they manipulate our musical expectations. The subtlety of our feeling for scales and expectations comes from a subconscious probabilistic model for music that we all learn via exposure to music at a young age – we hear how often particular notes occur relative to the rest of the notes in a melody, and use this along with how each note is emphasised, to construct a hierarchy of notes from the most to the least important, with a different hierarchy for every different scale. From this understanding we are able to subconsciously latch onto a melody within a few notes, and use it to inform our experience of the piece of music. I find this very surprising given that it seems like a very complex task, but for some reason it comes naturally to your geekish subconscious mind, and makes you feel the way you do in response to different pieces of music.
5. Infinite Series
An infinite series is the sum of a never-ending sequence of numbers, where each is defined by a simple rule that can be applied in an iterative manner. An example of this occurs whenever we hear a musical note, because it is one type of infinite series, called the harmonic series, that our brains analyse to give us the experience of musical timbre – how an instrument or object sounds, and the reason we can tell a cello from an oboe from a human voice, even if they’re all producing the same note. Any note we hear is primarily based on the fundamental frequency, and this is the lowest frequency at which the instrument, vocal chord, or string, for example, is vibrating. But this is not the only frequency of vibration occurring, because every multiple of the fundamental frequency – 1x, 2x, 3x, 4x and so on, occurs simultaneously, in the form of an infinite series, whose movements are summed to create a complex sound wave that we can interpret to tell us, from experience, what it was that made that sound – the harmonic series acts as a sort of musical fingerprint. Electronic music is a realm where we delve into the potential of overtones, where pieces of music can be as much about how they sound, and how overtones differ from the perfect harmonic series, as what notes or rhythms they contain. Instruments or sound sources that seem inharmonic, where the note being played isn’t clear, sound like this because the overtones do not relate to the fundamental frequency according to the rules of the harmonic series. Our appreciation of the intricacies of sound design are heavily influenced by our subconscious awareness of the harmonic series.