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The seventh is a distance of seven letters.
C(1) - D(2) - E(3) - F(4) - G(5) - A(6) - B(7)
C-B is a seventh, and D-C, or G-F, and also C-Bb, or D-C#, etc.
The sevenh is an interval with a major. minor, diminished and augmented variation.
Respectively a major, minor, diminished and augmented seventh
In "normal" - not theoratical - live, seconds, thirds and sixths are major or minor. The diminished and augmented versions are more theoretical. It's "normal" to ask a piano player to play a minor third, it is very uncommon to ask to play an augmented second.
The octave has also been discussed in the first part of this music theory course.
An octave is a distance of eight letters.
C-C' is an octave. The apostrophe at the second C indicates that the second C is an octave higher. The same way D-D' is an octave, And F#-F#', but also D-Db', G-G#', etc.
The octave can be perfect, dimished or augmented. The same system as with all intervals applies.
A special one is the "double octave". A double octave has a distance of two octaves:

Respectively a perfect, diminished, augmented and double octave