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.