Why can harmonics be a different note than their fundamental frequency?

Question

I was under the impression for the longest time that when you hear a harmonic on a string, its basically a sum of different resonating frequencies, which are all INTEGER multiples of the base frequency.

However I was watching this video at time stamp 1:00 https://www.youtube.com/watch?v=FqMxmfjCk4U, and I was surprised to find that the video creator claimed that the 3rd, 5th, 6th, and 7th harmonics are all not the same note as the fundamental frequency.

Why might this be the case?

Where my understanding breaks down:

Given a string $L$ whose vibration has fundamental frequency $f$:

I believe (wrongly) that the $N^{th}$ harmonic (played by muffling the string at a location $\frac{1}{N} \times L$ where L is the length of the string) consists of all vibrations with frequencies that are integer multiples of $Nf$.

That is when playing the string open you hear vibrations with frequencies $f, 2f, 3f, .... $ and the $N^{th}$ harmonic (in some combination weighted by intensity that depends on the physical characteristics of the string) of $Nf, 2Nf, 3Nf...$ now if I understand correctly if a note appears at a particular frequency $f$, then $2f$, $3f$, etc... are all the SAME note, just at higher octaves. So I would be under the impression the harmonics are all going to be the same NOTE as the fundamental frequency, just of higher octaves and weaker loudness.

Answer 1

When you go up by one octave, the frequency doubles. When you go up by another octave, the frequency doubles again. So that means that $f$, $2f$, $4f$, $8f$, etc. are all the same note in a different octave.

If we look at that list, we immediately see that $3f$ is not present! Lets say that $f$ corresponds to $A_4$ or something, such that $2f$ corresponds to $A_5$ and $4f$ to $A_6$. This means that the harmonic $3f$ will be a note somewhere in between the notes $A_5$ and $A_6$ - it will not also be an $A$.