Doubling the frequency of a note certainly changes it. The ear hears a higher-pitched note. But there is something in the essence of the note that does not change, a character that stays consistent through the octaves.
This allows a process called octave reduction. When you’re working with notes as ratios, it’s convenient to multiply or divide the raw ratio by 2, as many times as is necessary to bring it into the same octave as the tonic.
This note is actually an octave plus a fifth above the tonic. Now divide by 2 and you have 3/2, one and a half times the original frequency, and just a fifth above. 3-2
The reference frequency is 1, the octave is 2, so what you want to achieve with octave reduction is a ratio, or fraction, between 1 and 2.
These are the beginnings of a scale, a collection of notes within a single octave. Such a scale can be repeated up and down the octaves to cover the whole range of hearing.
Next: The Major Third