Here is your answer. If your brain is wired only to call a note C of F# you run into problems later. I will call these names "absolute pitch names", yes you need them, but if you think of the (same) notes as "root" or "sharp fourth" you will do yourself a big favour, music will open it's doors for you. You see it's the ROLE of the note that is critical to it's sound. Thus C will have that homely feel when playing a C triad chord, but if its context is in a Db chord it has a different role, making it more clashy. The terms of these concepts do change, but I call them "relative pitch names" as they same "absolute pitch" (F# Db A etc) gets a different name according to its role relartive to the chord.
Now to the numbers. Yes you can use numbers and they have the advantage of letting you know where the note is in a scale. There are other terms, we will come to these in a minute. First, if you list the notes of a major scale from C to B, you get the numbers 1 to 7, after this 8 -14. Notice this. If you subtract 7 from any note above the octave you get it's equivelent an octave down. So, for example in the C major scale a D can be a second tone, above the octave it is a ninth. Subtract 7 from 9 and you get a second. So when the brain starts worrying about exactly what a thirteenth is, subtract seven and you get 6. It's a sixth an octave above. No one goes to the third or fourth octaves like this, but if you did, you would subtract 14, 21 etc. Don't bother.
Now, back to relative pitch names. The list of terms for identifying the roles of notes within a chord, can vary and the classical term for these notes are very arty farty latinised, but this does NOT make the concept harder, make sure your brain does not panic, there is no need. Somewhat incorrectly but very usefully, you can also associate the note after the name of mode it is the root of I will put these terms in brackets here.
Numbering the notes of a major scale is one way but there are other terms:
1 Root, Fundamental, Tonic (Ionian)
2 Second, Supertonic (Dorian)
3 Major third Mediant (or minor third if it is a minor scale)
4 Fourth, Subdominant or "perfect fourth"
5 Fifth, Perfect fifth or Dominant
6 Submediant or 6th
7 Leading tone, major 7th,
8 Octave
Don't worry at this stage about why the complicated names are what they are, you just need to know that a sixth can be called a "submediant" by very strange intellectual people ( 🙁 ). What you need at the moment is a quick and easy system for yourself. Using the numbers is probably the best.
Now so far we have only talked about the notes of a major scale. What about the other notes? In the key of C, the white notes on a piano are the C major scale notes and the black notes are the awkward ones to fit in. Talking in C, the first semitone above C can be called a C sharp (C#), or a D flat (Db). This again is according to it's role. Similarly, each black note which is "outside" the C major scale, can have two names. There are 12 tones involved if you include the blakc notes (often called "accidentals". I list them below>
Root
C#/Db
D
D#/Eb
E
F
F#/Gb
G
G#/Ab
A
A#Bb
B
C
Lets take another look at this list. You have the scale tones 1-7. You can flatten or sharpen some of these. Why only some? Well if you sharpen an E (when in the key of C) it simply becomes an F. Same with B, B# is almost always written as C. Now what happens beyond the octave? This bit is important. Yes you have ninths, tenths, 11ths, 12ths etc, but for reasons I shall not go into depth with here,almost universally this list is SIMPLIFIED (Whew!). The intervals you will encounter in chord notation are called "Chord tensions".: They are:
b9 (try subtract 7)
#9
#11/ b5 (never called a b12th for unkown reasons)
#5/ b13
b7
Now it takes a time to bed all this in. Here is the summary version:
Number the notes of your scale 1-7. at the 8th tone your at the octave which is the same as the root, all notes from here you can subtract 7 to find its partner note in the first octave.
The inbetween notes that are used as accidentals are (when in C major) the black notes on the piano. The main ones to worry about are the b9, the #9. the #11/ b5 (same note), the #5/b13 (same note) and of course the b7th.
All these notes can be called by different exotic names but all you need at the moment is to understand the number system and gteh commonest terms - as I have described. You might encounter different names for the same concept but if you have the number system cracked, this isa huge srtep forward.
I strongly recommend you to look at a piano keyboard when understanding this. In fact I strongly recommend you learn a little basic piano alongside sax as this will enable your visual mind to engage. You cannot see your fingers when playing a sax and if you did you would find thart the keywork is much more confusing than a piano keyboard.
Now, having laid this ground work, your ready to understand WHY this is so important:
Imagine your learning to drive a car. To begin with, there is soooo much to think about: Was the indicator on the right or left of the sterring column? Oh no! you have gthe window wipers going and it's a Sunny day, but hey, does the old lady crfossing the street not come first? Where exactly is fourth gear? What about the mirrors, oops that is the radio, not the heater etc. When a learner brain first starts driving, they have to drop certain tasks as the mind is not yet habituated to doing essential things - like stopping. Thus a learner might not look in the mirrors and this may become a bad habit that is later mission critical.
Now imagine a learner musician being presented by a piece of basic music. They know the first note is a C and they know the second note of the piece is an F and so forth. They understand the timings and can produce the piece on their sax. Voila! Job done"! Congrats! They then tackle another piece, perhaps a bit more complex, the achieve this too and after a year or so can play a few pieces so that they sound good, they may even memorise the tunes. But.....
When they begun, they dropped analysing the piece in terms of relative pitch names (root, third etc), it simply was not necessary. Why do you need to know?
If you proceed by simply learning tunes what happens is you gain a library of licks and phrases but none of this is TRANSFERABLE. Your brain will become like an office where all the files are piled up in random heaps and nothing can be filed. If on the other hand, when you learnt to play every bar, you took additional time to identify every note not only by it's absolute name (f# Ab etc), but also by it's role in the chord, 5th, 6th b9th), this WOULD slow you down, BUT only initially, it is only a matter of months to aquire this habit and make it trivial. Sometimes I think of this like a soldier doing mountain climbling with a pack of rocks on his back. At first the rocks are really cumbersome, but later they become light as a feather.
If a player does this - ensures that for every bar they read or concieve is identified not only in terms of absolute pitch names but also in terms of relative pitch names, this enables the brain to store musical materials differently. This is why musicians learn to take simple phrases - like a triad for example -around cyles. it forces the brain to understand gthe roles of notes and it encourages a good internal filing system. This is the route to full musical understanding.
Here I use an idea which I call "ownership". Lets say I wanted to "own" a major triad. To do this i insist to myself that i must be able to play it in 12 keys, up and down, through any cycle, at any speed, knowing the absolute note names of each triad. This is a LOT of work. It takes many months. However, once you have achieved this, you can play SOMETHING over any Major chord that comes your way, this is a very big deal, for two reasons. Firstly you will find that doing only this is very effective in all major chords. Secondly you have given yourself 3 notes that you "own" (can deliver) in any major chord. Since there are only 12 notes available in any chord, you are already a quarter of ghte way there with your impro - this is a big deal. By working of this concept, you are now thinking three notes at a time rather than simply one then the next. To this structure you can then begin adding tones - again by ownership - that is by ensuring you can deliver that tone in any key -instantly. Suppose you were to achieve ownership of a sixth (13th). You now can work 4 tones in your impro. As my former sax teacher once said. A single note is a Big thing, it can change everyrthing. Using these "owned" tones in C gives you C E G A. That is most of a pentatonic scale. Add the 2nd/9th and you have conquered the pentatonic scale.
Thinking in this way, it is possible to add "owned" tones in a structured way, until you have all 12 tones availabvle over ANY chord.
For the most part tones can be regarded into three groups
1] The diatonic chord tones
2] 2nd, Fourths and sixths. (In C the white keys not involved in the triad but still part of gthe major scale)
3] the black notes or accidentals.
Think of group 1 as your safe tones, they are the ones you can land on. Group 2 are sort of intermediate, they create a sort of floaty feel somewhat unresolved, but not fully spikey and "out". The third group create tensions. They can be used anywhere and are an important color to your palette, but they require resolving to a chord tone in either gthe bar your in or the next.
I have gone a little further here that I intended. But I hope it helps. Spend time to absorb this fully. Ask questions here and I will reply.
Z