From the intervals we get many octaves of pitches in the twelve-tone system. Ignoring rhythm, chords and scales are just musical objects. Octave equivalence brings trillions of possible objects down to hundreds of thousands.
Chromatic Scale to 88-Key Piano
Using intervals, we can start from one note (the A string on a violin, in the previous example) and generate the entire chromatic scale. We can compute the pitches for over seven octaves of notes on an 88-key piano.
If we ignore rhythm and we ignore the difference between chords with notes played sequentially (arpeggios) versus simultaneously, and treat scales the same way, as a simple stack of notes, we can treat both chords or scales uniformly (for the purpose of analysis) as something called musical objects.
Two musical objects. Top: C Major Triad. Bottom: A Blues Minor Pentatonic.
Naive Object Count
Mathematically, there are over 300 trillion trillion musical objects that could be played on an 88-key piano (if you had 88 fingers). With ten very long fingers, that number would be 4.5 trillion chords or scales you might play, with over 20 trillion trillion possible combinations of two chords or scales.
How do we get from a realm of nearly infinite possibilities to a place that sounds remotely like music? All without skipping any useful possibilities? Where do we even start?
Human perception of music is based on the perception of the octave, when two notes that are clearly not the same pitch sound the same but one is clearly “higher” and one is “lower.”
The first bars of “Joy to the World” demonstrate how a melody can leave a “home” note, take a journey, and return home to the “same” note where the melody started, but down an octave.
This “sameness” relationship is enough to bring the number of possible objects to analyze down into the hundreds of thousands, and the number of potential relationships between objects down into the billions. (Again, this is still completely exhaustive—no chord or scale left behind. Later we will bring the number of objects down into the tens of thousands and the number of possible relationships into the millions.)
In the next section we will discuss how to reduce this number even further by grouping objects together, again without eliminating any possibilities.