Is the Dmaj chord made from the first, third, and fifth notes of the D major Scale or the first, third, and fifth notes of the C-Dorian mode scale?
It is made from the 1st, 3rd and 5th of the Dmajor scale.
Notes D, F#, A
The 1st, 3rd and 5th notes of C Dorian are;
Notes C, Eb, G
The root chord you get from C Dorian is a C minor chord.
C Dorian is not related to the Dmajor scale, nor is a D major chord found in the harmony of the C Dorian scale. The D chord you find in the harmony of the C Dorian scale is a D minor;
Notes D, F, A
The D chord you find in C Dorian is aligned with the D Phrygian scale, which is a scale with a minor 3rd.
With key OR modal analysis, you can get a D major chord in some places other than the root of a Dmajor scale. eg, the IV chord built from the A major scale is D major. But this may be of not too much use in aiding your clarity at this stage.
If you want to develop the question to outline your current understanding then that may help. Chords can be thought of as being built off of modes, but it’s probably best to first learn the concept of a key and the chords you get within it. Modal analysis, while fairly fundamental to some music, is kind of a layer of abstraction over basic diatonic harmony and it’s probably best to start with key first. Essentially, chords are not really built off of the modes, they are built off of the key, modes are a way of visualising a particular position in diatonic harmony with a new scale specific to the root of that degree when compared to the key.
They all merge together at some point, and modal theory extends much further than purely key based, but your question makes me think you may have skipped a step, understanding the chords you get from a key in diatonic harmony.
A D major chord is the tonic chord of the key of D but it is not only made from the D major scale. It however is not made from a C Dorian scale, whose D chord (ii) is D minor.
Thinking just in terms of major scales, a D major chord, D F# A, can be built as the I chord in D, the IV chord in A and the V chord in G. That is not even counting the minor keys and modes (for example the III chord in B minor, the II chord in C Lydian, etc.) A single chord like D major can exist and function in a multitude of keys.
The D major chord comes from the key D major, which has the key signature of two sharps - F♯ and C♯. Thus the notes that make up the root triad are D, F♯ and A.
C Dorian has the parent key of B♭. That has the key signature of two flats - B♭ and E♭. Meaning the root chord from C Dorian is made up from C, E♭ and G.
There is no relationship between the two, as the notes from one triad are completely different from those of the other. And they will sound very different. D major being major, C minor being minor at very least!
It would seem the question is asked from a false premise! Also please note that the 'Dorian of C' is not the same as 'C Dorian'. Maybe that's where the confusion lies.
For what it's worth, the D major chord can be made from any diatonic scale with a key signature of one, two, or three sharps, so:
- G major
- A Dorian
- B Phrygian
- C Lydian
- D Mixolydian
- E minor
- F♯ Locrian
- D major
- E Dorian
- F♯ Phrygian
- G Lydian
- A Mixolydian
- B minor
- C♯ Locrian
- A major
- B Dorian
- C♯ Phrygian
- D Lydian
- E Mixolydian
- F♯ minor
- G♯ Locrian
Key signatures with four or more sharps have D♯, so they lack the root of the D major chord, and key signatures with fewer than one sharp (that is, the natural key signature and all of the flat-side key signatures) lack F♯, so they lack the third. D major chords can or course appear in music using these key signatures, but only as a result of chromatic alteration.
I will also mention that the question suggests a possible misunderstanding of the convention for identifying various modes. I suspect that by "C Dorian" you meant "the Dorian mode that has the same key signature as C major." As you've no doubt inferred by now from the comments and other answers, that's not how it works. Instead, the mode is named after its home key, so "the Dorian mode that has the same key signature as C major" is called "D Dorian." (But even still, if you check the list above, you'll see that D Dorian isn't there, which is because the third degree of the D Dorian scale is F natural rather than F♯.)
To complete other answers, and to give some theoretical background behind this (you don't have to understand this stuff, only if you're interested!):
A chord is a collection of multiple notes grouped in a meaningful way. A chord with 3 notes is a trichord. A triad is a trichord build on successive thirds.
Triads are so prevalent so we use the term "chord" and "triad" interchangeably.
So a triad is build on successive thirds, and a Major triad (or Major chord) is build with a major third, and a minor third on top of that. Alternatively, you can think of it as a root, a major third on top of the root, and a perfect 5th on top of the root.
So if you take your root, D, and add a major third you get F#, and if you add a minor third on top of that (or a perfect 5th on top of D), you get an A.
These notes are also the 1, 3 and 5 of the D Major scale. This will be true for any key - the Major chord is always build using one major and one minor third, and it is always the 1, 3 and 5 of the major scale.