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This is probably a stupid and naive question but:

If I play the Cmaj chord on the C major scale I get the notes C-E-G. Now if I play the Cmaj chord on the C minor scale I get the notes C-Eb-G.

So does the C major chord have the notes C-E-G or C-Eb-G?

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    C Eb G is not a C major chord. It’s a C minor chord. The only thing that determines the name of the chord you are playing is the notes you play. Any time you play C Eb G you are playing a C minor chord no matter what scale is being used at the time. Any time you play C E G you are playing a C major chord no matter what scale is being used at the time. You don’t play chords “on” scales. You just play chords. – Todd Wilcox Jan 10 '18 at 5:06
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If I play the Cmaj chord on the C major scale

As Todd says in the comment, You don’t play chords “on” scales. As we said in How to figure out which scale to play a specific chord in?, when you play a chord, you don't have to know what scale to play it in.

Generally, when you're playing - you don't think what chord to play, then think about what chord you're playing. That's thinking backwards!

What you might do, though, is think the other way round: FIRST consider the scale you're playing on (or, as we'd more normally say, what KEY you're playing in), THEN consider what chord you would build on particular notes.

So you might think - "Hey, I'm in C minor - if I want to build a chord on the note C, what chord would it be? Aha, it would be the C minor chord".

Or: "Hey, I'm in C Major - if I want to build a chord on the note C, what chord would it be? Aha, it would be the C Major chord".

But you wouldn't usually think "I want to play a C Major chord... now what scale should I play it in?". That doesn't really make sense.

"Chords have different notes on different scales" is not true. (It's sometimes true that we might give the same chord different names in different keys though).

But "different scales (or keys) have different chords" is true.

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    @ToothyRel If people call a chord just 'F', that (almost) always means F Major. So F add4 would be F major with the added fourth. If people want a minor chord, they'll call it F minor, F min, or Fm. There are other ways to denote a minor chord (e.g. sometimes with lower case letters) but the rule is: if someone wants a minor, or diminished, or other 'non-major' chord, they'll ask for it. Otherwise, assume major! – topo morto Jan 10 '18 at 23:09
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    @ToothyRel Any Major chord is always the root note, then the note 4 semitones above that, then the note 3 semitones above that. Usually when I actually play a major chord, that's all I need to remember. What instrument are you learning? – topo morto Jan 11 '18 at 7:44
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    @ToothyRel What he's doing is explaining in a very slow and step-by--step way how to work out what notes are in a chord. When he talks about constructing the major scale, the only reason he says that is because he's explaining that elements of chord names represent 'deviations' from the intervals in the major scale. – topo morto Jan 11 '18 at 21:20
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    So in your original question here, when you said Now if I play the Cmaj chord on the C minor scale I get the notes C-Eb-G. The problem there is that you're not actually following his advice in step 2. You're constructing the minor scale, but he doesn't say you can do that! He says construct the Major scale - ALWAYS - for the purposes of working out what the name of a chord means. – topo morto Jan 11 '18 at 21:23
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    @ToothyRel so I guess the question is - why were you thinking that in step 2 you could 'construct the minor scale' instead? Is it because you'd learned that there are major and minor scales and you thought you might be able to swap them? – topo morto Jan 11 '18 at 21:25
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The problem here is that the chord you called Cmaj when in C minor is actually Cmin.

The tonic chord takes on the quality of the scale, so the tonic triad of a major key is a major chord & the tonic triad of a minor key is a minor chord. Things get more difficult with the other modes, but:

  • major key = major tonic chord,
  • minor key = minor tonic chord.

e.g. The chord Cmaj is always C-E-G:

Cmaj - in C major     Cmaj - in C minor

The chord Cmin is always C-E♭-G:

Cmin

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A chord name depends on the quality minor, major, dominant, .. and the tonal center scale mixolydian, regardless of the functions of the notes in the scale you are playing. In other words, a chord's name is context-free.

General flow of determining a chord's name:

  1. Pick the quality that fits best with the chord
  2. Alter, suspend, omit notes from that quality
  3. Add and alter notes that have no functional equivalent in the quality, based on the mixolydian scale

Example:

Suppose you are playing a song in Superlocrian in the key of C and the following chord gets played:

C-Db-Eb-Ab-Bb

Its interval set:

0,1,3,8,10

Its pitch classes in the context of the scale:

1-b9-b3-b13-b7

Its pitch classes (context-free, based on mixolydian)

1-b9-b3-#5-b7

Step 1: we find a best fit on the Minor7 quality 1-b3-5-b7

Step 2: we alter the fifth (5 -> 5#)

Step 3: we add the altered 9th (9b)

Chord name: Cmin7b9#5

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