A particular musical sound consists of the fundamental pitch, plus
a sequence of *overtones* of higher frequency, but
lower power. The distribution of power among these overtones determines
the kind of instrument we hear. We can call the overtones the spectrum
of frequencies created by an instrument. A violin's frequency spectrum
is distinct from the frequency spectrum of a clarinet. The overtones are
usually integer multiples of the base frequency. When any instrument
plays an A at 440 Hz, it also plays A's at 880 Hz, 1760 Hz, 3520 Hz, and
on to higher and higher frequencies. While we are not often consciously
aware of these overtones, they are profound, and determine the pitches
that we find harmonious and discordant.

If we expand the frequency spectrum through the first 24
overtones, we find almost all of the musical pitches in our equal
tempered scale. Some pitches (the octaves, for example) match precisely,
while other pitches don't match very well at all. This is a spread of
almost five octaves of overtones, about the limit of human
hearing.

Even if we use a low base frequency,
*b*
, of
27.5 Hz, it isn't easy to compare the pitches for the top overtone,
*b*
×24, with a lower overtone like
*b*
×8: they're in two different octaves. However, we
divide each frequency by a power of 2, which will normalize it into the
lowest octave. Once we have the lowest octave version of each overtone
pitch, we can compare them against the equal temperament pitch for the
same octave.

The following equation computes the highest power of 2,
*p*
_{2}, less than or equal to
some frequency,
*f*
compared against our base
frequency,
*b*
, of 27.5 Hz.

**Equation 39.2. Highest Power of 2,
***p*
_{2}

Given this highest power of highest power of 2,
*p*
_{2}, we can normalize a
frequency by this simple division. This will create what we'll call the
*first octave pitch*,
*f*
_{0}.

**Equation 39.3. First Octave Pitch**

The list of first octave pitches arrives in a peculiar order.
You'll need to collect the values into a list and sort that list. You
can then produce a table showing the 12 pitches of a scale using the
equal temperament and the overtones method. They don't match precisely,
which leads us to an interesting musical question of which sounds
“better” to most listeners.

**Overtone Pitches. **Develop a loop to multiply the base frequency of 27.5 Hz by
values from 3 to 24, compute the highest power of 2 required to
normalize this back into the first octave,
*p*
_{2}, and compute the first
octave values,
*f*
_{0}. Save
these first octave values in a list, sort it, and produce a report
comparing these values with the closest matching equal temperament
values.

Note that you will be using 22 overtone multipliers to compute
twelve scale values. You will need to discard duplicates from your list
of overtone frequencies.

**Check Your Results. **You should find that the 6th overtone is 192.5 Hz, which
noralizes to 48.125 in the fist octave. The nearest comparable
equal-tempered pitch is 48.99 Hz. This is an audible difference to
some people; the threshold for most people to say something sounds
wrong is a ratio of 1.029, these two differ by 1.018.