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Here are the first few chords of "City of New Orleans" as performed by Arlo Guthrie:

F# C# F# D#m B F#

F# C# F# D#m B F#

D#m A#m C# G#

If I lay out the chords in a roman numeral chart, starting at F#, I get:

| I | ii | iii | IV | V | vi | vii⁰ |
|----+-----+-----+----+----+-----+------|
| F# | G#m | A#m | B | C# | D#m | Fdim |

So, in the song, he plays a G#. However, the chart predicts a G#m, which is an exception to the chart that sounds good.

Where can I read about these "exceptions"?

Thanks!

The chords you've listed in your table are what we call the diatonic triads of a key. "Diatonic" basically just means "within the key," so the diatonic triads are what are created when we only use the pitches in a key's key signature.

(Quick note related to key signatures: your vii° chord will actually be E♯°, not F°, because E♯ is the seventh scale degree in F♯ major. For more on this concept, see Why do notes have multiple names?)

The opposite of diatonic is chromatic. Put another way, a chromatic pitch is a pitch that is not in the key's key signature. The G♯-major chord in question has B♯ as its chordal third, which is not in the key signature of F♯ (that would be B♮).

So whenever we have a chromatic chord like this, we want to try and understand how it functions. In many cases, the appearance of an unexpected major triad can be understood as being a temporary V (or "dominant") of another chord. In this case, the G♯-major chord can best be understood as a temporary V of the C♯ that comes before it. We say that this G♯-major chord is V/V (read "five of five") because it is V of C♯, which is itself V of the tonic F♯. We also call this chord a secondary dominant or applied dominant.

Typically the secondary dominant chord appears before its resolution; in other words, it's more normal for the G♯-major chord to resolve to C♯ major. But sometimes we have the applied dominant come after the temporary tonic, too. (It's unclear to me where this G♯ moves; it might go back to C♯!)

As for another "exception," if you ever encounter a minor chord where you're not expecting it, it's often a result of mode mixture (some use the term "borrowed chords"). (See How do I correctly borrow chords from one key to another? for one possible discussion of mode mixture. You also might enjoy What are the different ways of adding non-diatonic chords?)

TL;DR: That G♯ chord is understood as a temporary V/C♯. Since C♯ is itself V of F♯, we label this chord V/V.

The Ⅱ chord can serve mostly two roles:

• 
  

  A secondary dominant, as said by Richard. This would be clearly the intepretation to go if the two last chords were turned around:

  
  
  

  ⅵ | ⅲ | Ⅴ/Ⅴ | Ⅴ
  

  
  
  

  I wouldn't interpret the G♯ as a secondary dominant here though, if it doesn't really depend on the Ⅴ chord.

  
  



• As a borrowed chord from the Lydian mode. The F♯-Lydian scale is F♯-G♯-A♯-B♯-C♯-D♯-E♯-. I does contain all the third line as diatonic chords. Or you might say that this entire section modulates to d♯-Dorian.

You've fallen into the common trap of seeing a scale, and the triads that can be formed from it, as a restriction.

Yes, there is a set of chords that can be constructed from the notes of a scale. But that's no more than a mildly interesting fact. A song in a given key will use some of the diatonic chords a lot, others not so often. And other, non-diatonic chords will also very likely appear. The diatonic, scale-derived chords are not a restriction, not the chords that you MAY use. They're just the diatonic chords.

For instance, consider the C major key. C, Dm, F, G, Am will probably be used a lot. But D7, C#dim, Bb, Fm are more likely to be used than the (diatonic) B dim. And, if you're playing blues-based styles, C7, F7 and G7 are the basic chords, two of which are non-diatonic!

THE SCALE IS A FRAMEWORK, NOT A RESTRICTION.

Once you absorb this, a lot of the 'borrowing' thing becomes unnecessary. Where a chromatic chord IS used as a gateway to a modulation, there's sense in calling it borrowed. An Fm chord in C major MIGHT be a pre-echo of a Fm7, Bb7, Eb modulation to Eb. So calling it 'borrowed from Eb major' is useful. But C C7, F, Fm, G7, C at the end of 'When the Saints' doesn't go anywhere. Just let C7 and Fm be chromatic chords. 'Borrowing' them doesn't help.