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Playing lower notes in 1st position below the root?

You have to visualize the scale or arpeggio you are working on as extending infinitely through octaves both above and below where you currently are. The visual pattern on the fingerboard is the same.

So on a standard-tuned 4-string, in D major, that F# at the 2nd fret, E string, is still the 3rd of your chord. It’s relationship to the D at 5th fret, A string, is exactly the same as the relationship between F# at 4th fret, D string and the D at 7th fret, G string, albeit one octave lower.

Just as you noted that the open A, or A at 5th fret, E string, is the 5th of D major just as much as the A at the 7th fret, D string is. (aka 2nd fret, G string; 12th fret A string; or 17th fret E string - those are all the same A note).

Another way to visualize it is to explore the D major scale starting at the 7th fret, G string, descending, then transpose that pattern one octave down.
 
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Another way to visualize it is to explore the D major scale starting at the 7th fret, G string, descending, then transpose that pattern one octave down.

Another way of stating the above: once you've played the D Major Scale descending, from "G7" to "A5" (where your middle finger should land), put your pinky on "A5" and play another descending D major scale, going as far as you can.
 
Another way of stating the above: once you've played the D Major Scale descending, from "G7" to "A5" (where your middle finger should land), put your pinky on "A5" and play another descending D major scale, going as far as you can.
“A5” and “G7” are confusing, even to me, and I know my fingerboard pretty well. Aside from the fact that each of those are names for chords, I don’t know of any G at the 7th fret. What did you mean? Did you accidentally transpose “A7” and “G5”?
 
I understand my scales and arpeggios starting at the root and ascending or descending and ending at the root. I also know the 5th is the same fret just a lower string. But how do I think about playing the notes a string lower? Like if I am playing in the Key of D in the first position that is the 5th fret of the A. But let's say I am playing the open E, the F# A, and B, all on the E string. My brain isn't wrapping around what is happening here interval or pattern/shape-wise...
Spend a couple of days playing the scales you know on one string. While doing this repeatedly, remember the distances between the intervals. It will put things in perspective to playing up the neck and connecting positions. By doing this the same applies as you play lower into the scale.
 
When you are in a specific key you need to know the notes in the key. You also need to know your fret board; in other words you need to know specifically where the notes are.

So in the key of D Major, the notes are D, E, F#, G, A, B, C# The way I think about it is by exception. F# and C# are the exceptions and all the other notes are natural.

The brain starts to struggle when it has to hold more than about three things at a time in working memory. You can handle four, but it becomes increasingly challenging. So at some point you might think of the exceptions as the number of naturals rather than the number of accidentals. For examples in F# Major, all notes are sharp except B; I.E. B natural is the exception.

In the key of D Major, D is home or the root of the key. The note names go in in circle as pitch increases or decrease away from the root. For example consider the bold D in the follow sequence the open D string: [E, F#, G, A, B, C#, D, E, F#, G, A, B, C#, D, E, F#, G, A, B, C#, D] These are the diatonic notes you can play on a 4-string, 20-fret instrument that are above and below the open D string.

It takes lot's of practice and you learn the key center in a number of ways. For example: 1. You learn to spell the key using accidentals, I.E. D Major has F# and C#. 2. You learn to hear the shape of the scale from different starting points, and how these starting points relate to the root. 3. You learn where the notes are on your instrument and learn finger patterns or boxes associated with various scales and modes. Ultimately all of these different ways of thinking should become automatic, so you can think in a key center in real time while you are playing.

If you are taking a formal theory class, one of the first things you will learn is how to build major and minor scales by interval. For example the shape of the major scale is (1-2) whole step, (2-3) whole step, (3-4) half step, (4-5) whole step, (5-6) whole step, (6-7) whole step, (7-8) half step. You will also learn how the intervals relate to the root. For example the intervals of the major scale are Root, Major 2nd, Major 3rd, Perfect 4th, Perfect 5th, Major 6th, Major 7th.
 
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Here's an exercise to help

1. pick a major scale key center. Find with the lowest possible note in the key that can be played on your bass. This is your 'floor' note. Identify the step of the scale tone.

For example, if we choose C as the key, the 'floor' is F (first fret on E), and F is the 4th of C major

2. Play the root of the scale, next drop down to the 'floor' note, then play every scale tone up to the tonic. Say the note names out loud as you play them. Then, play the same thing, saying the interval numbers as you play them.

In our Example we'd say " C, F, G, A, B, C"; then " 1, 4, 5 , 6, 7, 1"

3. Repeat step 2 but 'raise' the floor to the next available note
" C, G, A, B, C" ; " 5, 6, 7, 1"

4. repeat , raising the floor again and again until you end up on the 7th, just below the root.

this exercise teaches you the sound of dropping down to various intervals from the tonic,
and what he run back up sounds like, and the way that feels under your fingers.

You should be able to navigate to the tonic from any note above or below.
 
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Here's an exercise to help

1. pick a major scale key center. Find with the lowest possible note in the key that can be played on your bass. This is your 'floor' note. Identify the step of the scale tone.

For example, if we choose C as the key, the 'floor' is F (first fret on E), and F is the 4th of C major

2. Play the root of the scale, next drop down to the 'floor' note, then play every scale tone up to the tonic. Say the note names out loud as you play them. Then, play the same thing, saying the interval numbers as you play them.

In our Example we'd say " C, F, G, A, B, C"; then " 1, 4, 5 , 6, 7, 1"

3. Repeat step 2 but 'raise' the floor to the next available note
" C, G, A, B, C" ; " 5, 6, 7, 1"

4. repeat , raising the floor again and again until you end up on the 7th, just below the root.

this exercise teaches you the sound of dropping down to various intervals from the tonic,
and what he run back up sounds like, and the way that feels under your fingers.

You should be able to navigate to the tonic from any note above or below.

Yes this exercise is excellent because it teaches you the sound of each note of the scale in relation to the tonic.

Learning a scale (or mode) is not learning a pattern on the fingerboard, it is learning a particular sound, how to sing it, how to recognize it aurally, then how to name those sounds and where to play them on the fingerboard.

The pattern recipe is a shortcut that ends up in people being locked up as described by the OP.
 
Here's an exercise to help

1. pick a major scale key center. Find with the lowest possible note in the key that can be played on your bass. This is your 'floor' note. Identify the step of the scale tone.

For example, if we choose C as the key, the 'floor' is F (first fret on E), and F is the 4th of C major

2. Play the root of the scale, next drop down to the 'floor' note, then play every scale tone up to the tonic. Say the note names out loud as you play them. Then, play the same thing, saying the interval numbers as you play them.

In our Example we'd say " C, F, G, A, B, C"; then " 1, 4, 5 , 6, 7, 1"

3. Repeat step 2 but 'raise' the floor to the next available note
" C, G, A, B, C" ; " 5, 6, 7, 1"

4. repeat , raising the floor again and again until you end up on the 7th, just below the root.

this exercise teaches you the sound of dropping down to various intervals from the tonic,
and what he run back up sounds like, and the way that feels under your fingers.

You should be able to navigate to the tonic from any note above or below.
The “floor” for C major is the open “E”, not “F”.