Equal temperament has nothing to do with changing octaves; equal temperament keeps the exact octave interval present in older tunings. The purpose of equal temperament is to allow you to change key freely on a fixed tuning chromatic instrument without retuning it.Cork wrote:The "solution" was to compromise the tuning of the scale, so that the lowest instruments could match the tuning of the highest instruments, and that is the tuning known as equal temperament. Without equal temperament, for instance, there could be no such instrument as a piano, with it's several octaves, and, similarly, there could be no such thing as an orchestra.
On the other hand, pianos are not tuned in ideal equal temperament, precisely because it does not sound good across octaves on the piano. Good orchestral strings and winds do not use equal temperament at all, but are expected to make small tuning adjustments on the fly to bring the note they are playing in tune with an ideal chord. Both orchestras and pianos existed for centuries before equal temperament was invented.