When we look at the overtone analysis, the second overtone is three times the base frequency. When we normalize this back into the first octave, it produces a note with the frequency ratio of 3/2. This is almost as harmonious as the octave, which had a frequency ratio of exactly 2. In the original 8-step scale, this was the 5th step; the interval is called a fifth for this historical reason. It is also called a dominant. Looking at the names of our notes, this is "E", the 7th step of the more modern 12-step scale.

This pitch has an interesting mathematical property. When we look at the 12-step tuning, we see that numbers like 1, 2, 3, 4, and 6 divide the 12-step octave evenly. However, numbers like 5 and 7 don't divide the octave evenly. This leads to an interesting cycle of notes that are separated by seven steps: A, E, B, F#, C#, .... We can see this clearly by writing the notes around the outside of a circle, and walking around the circle in groups of seven pitches. This is called the Circle of Fifths because we see all 12 pitches by stepping through the names in intervals of a fifth.

This also works for the 5th step of the 12-step scale; the interval is called a fourth in the old 8-step scale. Looking at our note names, it is the "D". If we use this interval, we create a Circle of Fourths.

Write two loops to step around the names of notes in steps of 7 and steps of 5. You can use something like range( 0, 12*7, 7 ) or range( 0, 12*5, 5 ) to get the steps, s. You can then use names[s % 12] to get the specific names for each pitch.

You'll know these both work when you see that the two sequences are the same things in opposite orders.

Circle of Fifths Pitches. Develop a loop similar to the one in the overtones exercise; use multipliers based on 3/2: 3/2, 6/2, 9/2, .... to compute the 12 pitches around the circle of fifths. You'll need to compute the highest power of 2, using Equation 39.2, “Highest Power of 2, p ₂”, and normalize the pitches into the first octave using Equation 39.3, “First Octave Pitch”. Save these first octave values in a list, indexed by s % 12; you don't need to sort a list, since the pitch can be computed directly from the step.

Check Your Results. Using this method, you'll find that "G" could be defined as 49.55 Hz. The overtones suggested 48.125 Hz. The equal temperament suggested 48.99 Hz.