In the following considerations, we are focused on the frequency related relationships between two tones because these relationships are particularly interesting for music.
In music, two tones that sound pleasing when played together are considered consonant. On the other hand, two tones that sound unpleasing when played together are considered dissonant. The measure for the consonance or dissonance, that we hear when two tones are played together, is related to the distance between the frequencies of these tones.
If we have two tones whose frequencies are f1 and f2, the distance between these frequencies is f2 / f1. The important thing here is that the distance between the two frequencies is not measured as the difference between the frequencies, but as the ratio between them.
Certainly, the most consonant are two tones of the same frequency i.e. when the distance between their frequencies is 1. For instance, such a case occurs when two singers are singing together exactly at the same frequency.
When we listen to two tones of different frequencies, the most consonant case is when the distance between their frequencies is 2. This ratio of the frequencies (f2 / f1 = 2) is very important and has its own name, octave. It leads us to an important principle which is called the octave equivalence.
A human ear tends to hear a tone of any frequency and another one of doubled that frequency as similar. A human ear can easily distinguish between these tones when they are played one after the other and perceive that while the second tone is higher than the first one, they still sound somehow equivalent or related to each other. This psycho-acoustic phenomenon forms the basis upon which music is built and is called the octave equivalence.
The following chapters will emphasize on the distinction between physics and music: while physicists care for measuring frequencies and consider any frequency as equally important, musicians are most interested in certain frequencies. They assign labels to these frequencies and, as we shall see, may even use several different labels for the same frequency.