approach of fretboard visualization
That's the problem with a
guitar you
can't see it as easy, especially at first, as with a keyboard in the key of C _ _ other than dots that make up a block pattern.
I'm going to stick to the theme of Key Of C Major (because that is the obvious choice and it will keep in line with where we have begun) so your posted equations are relevant...
Transposing to C is relevant because the notes on a keyboard are laid out in a row and all white keys with sharps and flats highlighted in black. Now you
can see what you hear that's going on
and why, when you move a note or play a different scale.
This is why with a fretted instrument where your playing movable patterns up and down the neck to change keys they throw out the note letters and go with the notes interval numbers. A keyboard is different because there are no movable patterns, every scale is a different pattern or form in every key. But even they are still thinking numbers.
I had a friend who played for Etta James for a while years ago and he told me 1 time that when she would rehearse her band she would tell him sometimes try a b7... chord instead here and try a .... chord here. She did not play any instrument but still she knew jazz voiced chords and what she would tell him to try he said was right. Said he learned a lot from her.

Why did she know what chords to substitute and where?

If you major in music in collage, no matter what instrument you major with, they
make you minor in piano.
(At least they used to.

)
When I look at the fretboard of a guitar I don't see the block dot pattern of the scale and chords but instead I see numbers. The 1, b3'rd, 5, b7, all those numbers move and stay the same with the pattern so it's not a lot to learn. 1 scale, 1 set of numbers. It's the blue print for the fretboard both layout
and parts values. You
can't be thinking, A, F#, Bb, D, etc... All those letters change in
every scale. Now that is a lot to memorize.

All dots tell you is that your playing notes that are in the scale, but that's all they tell you. I've had guitar teachers who when they wrote out a scale or chord in tab they used the interval numbers where the dots would go instead.
I'm trying to imagine what it's like to have NO music theory, but still want to apply the modes to your playing
For the moment,,,throw the music theory out the window,
I don't want to go over anyone's head.......
Sorry, and I mean this with all respect, but that dog won't hunt. What I've posted are the first steps in the most basic theory, it don't get any easier than this.
Think about it, All the modes are built on this, it's where they come from. It's how they found them. Major and minor 3'rds, nothing more.
You can't do algebra (Jazz) or calculus (BeBop) if you can't add, subtract, multiply and divide.
Once you try and get past the major and minor block pattern scales on guitar you've just stepped up in harmony and there's
no way to organize it any more to be able to apply it very well if at all.
Please help me make it more digestible for the guys who are looking for a quick way to add some spice to their playing.
Ringo Starr, "Got to pay your dues if you wanna sing the blues and you know it don't come easy"
Of course everyone teaches a little differently, and we are teaching the same thing,,,just from different angles
Well yes and no.
What your showing us is right as far as I understand music (which is not really all that much) but IMO you need to show both sides of the coin. That's what I was trying to add. I don't think it muddied up the waters by adding it.
I'm enjoying what your posting and will keep reading this thread with interest my friend. I'll bow out now and leave the full direction and pace of this to you as it's in good hands.

Brad
