When the frequency of a note is twice the frequency of another, the notes are a(n) ______________ apart.

When the frequency of one note is twice the frequency of another note, the notes are an octave apart.

To understand why, let's start with the basics of musical notes. In Western music, we use a system that divides an octave into twelve equal intervals called semitones. Each semitone represents a half step or half-tone difference in pitch.

The relationship between the frequencies of two notes can be calculated using the formula:
Frequency2 = Frequency1 * (2^(n/12)),
where Frequency2 is the frequency of the second note, Frequency1 is the frequency of the first note, and n is the number of semitones between the two notes.

In this case, when the frequency of one note is twice the frequency of another note, we can set up the equation:
Frequency2 = 2 * Frequency1.
By substituting this into the formula above, we have:
2 * Frequency1 = Frequency1 * (2^(n/12)).

We can cancel out the Frequency1 on both sides of the equation:
2 = 2^(n/12).

To solve for n, we take the logarithm base 2 of both sides of the equation:
log2(2) = log2(2^(n/12)),
1 = (n/12) * log2(2).
Since log2(2) = 1, we get:
1 = (n/12),
n = 12.

So we find that when the frequency of a note is twice the frequency of another note, the two notes are 12 semitones or one octave apart.