Lesson 6
The grid
Back where you started, in a piano roll, with nothing left unexplained. Every property of this grid was derived somewhere in the last five lessons, and the point of this one is to collect them.
Where each part of it came from
The rows are evenly spaced because pitch is logarithmic in frequency, from lesson 1. Equal distances on screen are equal musical distances, even though the frequencies double every twelve rows.
There are twelve of them per octave because twelve pure fifths almost close the circle, from lesson 4, and because seven-twelfths is an unreasonably good approximation to the fifth, from lesson 5.
Each row is slightly out of tune, deliberately, so that all twelve keys work equally well. Also lesson 5.
Some pairs of rows sound smooth together because their partials coincide rather than beat, from lesson 3, and those partials are there because a string cannot vibrate at one frequency, from lesson 2.
Which makes intervals arithmetic
MIDI numbers each note as an integer, and because the rows are evenly spaced in pitch, the distance between two notes is the difference between two integers. No lookup, no spelling rules, no clefs. Middle C is 60; the G above it is 67; the interval is 7, and 7 is a fifth everywhere on the keyboard, in every key, forever.
That is the property equal temperament was bought with, and it is why transposition in a DAW is addition. Select everything, add 3, and the music arrives intact in a new key. On an instrument tuned to pure ratios that operation does not exist.
One octave of the roll, annotated
The root is fixed at middle C. Click a row to sound it against the root.
- Interval
- 7 semitones
- Subtraction
- 67 - 60 = 7
- Standing in for
- 3:2
- Off by
- -1.96c
The bars are roughness, from the same model that drew the curve in lesson 3. Read them down the column and the grid stops looking arbitrary: the short bars are the octave, the fifth, the fourth and the thirds, which is to say the intervals whose partials line up. The rows are evenly spaced because pitch is logarithmic, from lesson 1. The numbers on them are integers because twelve fifths nearly closed the circle, from lesson 4. Everything on this page was derived somewhere earlier.
What the numbers are telling you
The roughness bars are the closing of the loop. They come from the same model that drew the curve in lesson 3, evaluated at each row. Read them down the column and the grid stops looking like an arbitrary ladder: the short bars land on the octave, the fifth, the fourth and the thirds, and they are short for a reason you can now state precisely, which is that those intervals put partials on top of each other instead of near each other.
The off by column is what tempering cost at that interval. Note where it is largest. The tritone and the thirds carry most of the compromise; the fifth and fourth barely notice it.
What this course did not do
It gave you no chords, no scales beyond the raw twelve, no keys, no progressions. That was deliberate: everything above is derivable from acoustics and arithmetic, and the moment we start talking about why certain chords follow other chords, we leave that territory and enter one where convention does real work.
A second course goes there. Triads as harmonic-series fragments, the diatonic set as a maximally even seven of twelve, the circle of fifths as a consequence of the generator rather than a mnemonic, and voice leading as minimal motion. Where those arguments stop being physics, they will say so, as this one has.