I came up with this game, which can also be thought of as a puzzle, to learn how to find the frequencies of all the notes using only two operations: multiply and divide.
For example, we start from middle C. We know that if we multiply its frequency by 2, we reach the C of the next octave. And if we multiply it by 3 and divide by 2 (that is, if we take three halves), we reach its fifth: a G.
With this node system (the tutorial is in the attached YouTube video), you will be able to think through and memorize the strategy for finding the notes that appear on the piano above. The goal is not just to complete it quickly, but to do it with the fewest nodes possible.
Are you up for it? Collect the 12 Notes by Their Frequency
The trick to memorizing them all: ÷n and then ×(n−1).
Nodes | Length | Interval | From C |
|---|---|---|---|
÷2 | 1/2 | octave | C |
÷3 ×2 | 2/3 | fifth | G |
÷4 ×3 | 3/4 | fourth | F |
÷5 ×4 | 4/5 | major third | E |
÷6 ×5 | 5/6 | minor third | Eb |
It is the same staircase: 1/2, 2/3, 3/4, 4/5, 5/6. The larger the number, the smaller the interval.
Chaining means adding intervals. Making one node after another multiplies the fractions, and that adds the intervals:
Intervals | Fractions | Result | From C |
|---|---|---|---|
fifth + fifth | 2/3 × 2/3 | 4/9, ninth | D |
fourth + fourth | 3/4 × 3/4 | 9/16, minor seventh | Bb |
major third + major third | 4/5 × 4/5 | 16/25, augmented fifth | G# |
fifth + fourth | 2/3 × 3/4 | 1/2, octave | C |
major third + minor third | 4/5 × 5/6 | 2/3, fifth | G |
fourth + major third | 3/4 × 4/5 | 3/5, major sixth | A |
fifth + major third | 2/3 × 4/5 | 8/15, major seventh | B |
The two shortcuts that pay off most in the game:
Major sixth (A): ÷5 ×3. The ×4 and ÷4 in between cancel out.
Minor sixth (Ab): ×5 ÷4 ÷2, which is a major third down and then up an octave.
A nice fact for the book: three major thirds (4/5 × 4/5 × 4/5 = 64/125) almost reach the octave, but fall 41 cents short. That is why C, E, and G# do not close exactly on C.