Why do certain songs sound so good? Why are there simple tunes that are so catchy? How can we tell immediately that some sounds just don’t go well together? Of course, there are many factors that contribute to how music is interpreted by human ears, but much can be understood by seeing the networks behind music.

In this article on combinatorial music theory, Andrew Duncan first represents the basic music scale, a 12-tone, equally-tempered scale as a graph. Each of the 12 nodes represents one of the 12 notes in an octave. The most basic graph Duncan includes is one where there are edges between adjacent notes, or notes that are one increment in pitch apart from each other. (In physics, we see that the pitch and the frequencies of wavelengths of the pitch are related, but that the increments between frequencies are linear only on a logarithmic scale.)

Duncan also discusses the appearance of a network on the fretboard of guitars and other stringed instruments. Musicians know that there are not online connections between the notes along each string (adjacent nodes), but between nodes on strings below, on certain increments, or frets. Scales played as paths on a network on the fingerboard can be shifted to any node, and be played the same, harmonically speaking.

Going back to the music note network, in a more complex graph, we would have edges that are directed and weighted. The most basic scales are the major and minor scales. Take the C major scale, for instance. This is the sequence C-D-E-F-G-A-B-C, and we can draw the edges between said nodes, so that a path is made for that major scale. Additionally, we can draw edges from C to each of the nodes in the C major scale. These are the major 2nd, major 3rd, perfect 4th, perfect 5th, major 6th, and major 7th intervals, which define the weights for each edge. Furthermore, the relationships between notes are not reciprocated symmetrically. Take the connection C-E, for example. This is a major 3rd in the C major scale, but E-C would be an interval on a different scale, a minor 6th on the E minor scale. So edge C-E is weighted differently than E-C. Duncan creates separate graphs that represent each of these intervals, to demonstrate the symmetries in music.

The human ear is more attracted to certain intervals in music. For example, the beginning interval in the wedding march song is a perfect 4th. Chords, also, are more pleasant in sound. The three-note NBC tune is in fact the three notes of a major chord, the 1st, 3rd, and 5th, with the order simply switched. The intro to The Postal Service’s song, “Natural Anthem” is the minor 3rd drawn out and repeated multiple times. We can give greater weights to these intervals than the weights of, for example, the intervals C-C# or C-F# (the tritone). Played simultaneously, these intervals are dissonance to our ears.

We can think of songs, or at least melodies, as paths between nodes on a new graph. Edges are more likely to be formed between nodes with strong ties, that is, notes with intervals with greater weights. Each node, in a way, has a valuation for each other node in the network’s graph, as each note will tend toward certain other notes in music. Because of this, we find many songs that sound similar. (A teacher once said that Green Day songs are all the same progressions, simply in different keys, which, of course, does not lessen their values any…)

Finally, for the entertainment of the readers (and myself earlier), here is another link. In this online program called Graph Theory, Jason Freeman creates a graph of 61 small scores played on the cello. Each node on this graph represents a small segment of notes played on the cello. These are connected, forming edges, to other nodes with which they are compatible. The user starts with one node, and can select a node to which it has an edge, and so on, creating a simple interlude on the cello. The mini composition can then be played back. And here, we have it— music, a song created through paths on a network.