This is a very good explanation - and I agree my methods were not nearly as scientific. However, I have a decent feel for the normal subtle embouchure adjustments per note when I am able to wrangle it to the +/- 5 cents range. I wasn't trying to hold the same embouchure for each note, this was with the normal amount of adjustments I've been used to making - and still quite far off.
It was definitely audible to the listener though - as my instructor noted the intonation of something I worked on yesterday was indeed all over the place.
As a side test, I tried another tuning test today with the same old reed (no new reed yet, have some soaking for later testing). The tuning test I chose was the one where you tune the horn to itself - Low Bb, overtone to Middle Bb, then tune the overtone to the Bb. I wasn't even able to get them in tune. With my mouthpiece almost completely off the cork, they were about 10 cents apart still.
More to come once reeds are tried.
I checked out some of your recordings. So please don't take my comments personally, but as constructive criticism.
Like everybody, I was a beginner once. I have taught many, many beginners, including my own children. What I'm hearing is very typical of a beginner who has not gained enough experience to control the pitch. There is nothing wrong with your horn or your mouthpiece. It's obvious to me that you're biting to get the note to speak, so this uneven bite pressure results in the pitch going all over the place, which is very common with beginners. Air stream direction and the shape of the inside of your mouth also affect intonation. But that aspect is more difficult to describe and to control. Thankfully that part usually works itself out naturally. But if reducing biting doesn't work after a few weeks, we'll have to address this aspect.
The solution is don't bite, at all, ever (except maybe super high altissimo). Support the bottom of the reed with chin muscles, not the jaw. When you do this, it's going to be really hard to get the notes to speak until you build up your embouchure. So switch to a softer reed and pull out until you get it under control. Do long tones with a relaxed jaw, but firm chin muscle support. It will be uneven and unsteady at first. But over the course of a month, things will start to come together. Unfortunately there are no shortcuts. Keep working on other things and having fun playing tunes as well. But the long tones right now are critical to developing your chops and will be your foundation going forward. But you have to be very patient. Beginners can often take up to a year or more to get their sound under control, but it's much less if they really focus on it.
I strongly advise beginners to avoid synthetics. They're easy to get a sound out of and are low maintenance, but they respond strangely and unevenly and take some skill to control. Get some soft cane reeds. Go back to synthetic if you prefer them AFTER you get your tone under control and learn the basics.
For a step by step approach, play a C, very relaxed, and move the mouthpiece until it's in tune with a drone or some other pitch reference you can hear. Play this note steady and in tune for 8-12 beats if you can. Now repeat with B and keep working your way down to low F (keep the mouthpiece where it was for C). Now return to C and work your way up to middle F. Don't move on to the next note until you can play the current one in tune. In your next practice session go a little lower and a little higher. In a week or two, you should be able to play every note in tune.
I encourage others giving advice in this thread to listen to your recordings before doing too much speculation and guesswork.
