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I notice that 4 chord songs almost never use the iii chord or the viidim chord. Do these chords not fit mathematically (into the circle of 5hts)?

3-chord history https://en.wikipedia.org/wiki/Three-chord_song

Circle of 5ths and the 4-chord song mathematically speaking https://www.classicfm.com/discover-music/music-theory/four-chords-every-pop-song/

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    By the way, have you ever checked out Hook Theory? hookpad.hooktheory.com
    – Giovanni
    Jun 8 at 12:03
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    By "VII," do you mean (in C major) a B diminished triad or a B-flat major triad?
    – Richard
    Jun 8 at 12:06
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    What do people do with this math mysticism stuff? Make music? How? Jun 8 at 13:14
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    i-iii-IV-V is very often used .... I'm afraid many Pop singers simply don't know more chords the othe most used 4 chords ;) Jun 8 at 13:55
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    @piiperiReinstateMonica when you make music, you make math as well, no matter if you know it or not. The other approach is pretty much legitimate, as long as people want to listen to the result.
    – fraxinus
    Jun 8 at 18:37
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Strictly speaking, you can use any chord in a 4-chord progression, but you're looking for a spiral of chords that have a strong function that leads from one to the next. All of the chords work mathematically thanks to equal temperament.

The iii chord doesn't have a clear function because it shares so many notes with the I and V; however, they're more often used in an altered form as a chromatic mediant or a V/vi.

My guess is that the vii° chord is used less often in pop because the diminished interval needs to resolve quickly and makes the tonic sound too resolved. vii°6 to I6 is more common.

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What are you basing your 4 chord theory on? Can you provide a list of songs? I would say that the iii is one of the most common chords used but that may depend on the genre of music. Also, the vii is a substitute for the V7 and is probably just as common as the V7 for this reason.

The circle progression is constructed by walking through the diatonic chords by 4ths (while staying in key). This is NOT the same thing as the circle of 4ths or 5ths which moves you out of key. The circle progression is...

I --> IV --> vii --> iii --> vi --> ii --> V --> I

You can play this with all 7th chords in the key and you will immediately notice the overlap.

Poly chords and substitutions can be used to relate these chords. For example, the I maj7 can be created by "adding" together the I triad and the iii triad, I + iii = I maj7. For this reason they can be substituted for each other. The same is true for the V7, V + vii = V7, and for all other pairs of triads separated by a 3rd.

Rather than say the vii never shows up think of how it is related to the V7 and see that it's always there. In fact, there is a cadence in classical music (perhaps earlier than classical) called the leading tone cadence, vii --> I, which has the classic 7->8 and 4->3 motion in it.

Other substitutions include replacing a Maj chord with its relative minor, the vi can replace the I and vice verse.

With this in mind it is easy to see that the entire circle progression can be distilled to the I, IV, V. I'd say if you are using 4 chords instead of 3 you missed a substitution. That's not to say that the other chords are useless. They create more interesting harmonies and movement. But in fact, you can harmonize a diatonic melody with just 3 chords.

But to answer your title question directly. None. Why should any chord be avoided? If you want to harmonize a diatonic melody then any group the 3 chords that covers the scale should work and by using substitutions they may all be equivalent to the I, IV, V in some sense.

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    @JosephDoggie No. C7 is either a note, the next highest C on a piano; or it is a C major chord with an added 7th so it contains the notes C-E-G-Bb. While VII is Roman numeral chord notation - VII means that the root note of the chord is the 7th note in the scale you are playing in. VII (uppercase) is a major chord, vii (lowercase) is a minor chord. Jun 8 at 21:17
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    If you played a progression in the key of 'C' then the V chord would be a 'G'. You would be looking for 'G7'. Jun 8 at 21:55
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    I don't think that's what he's asking about w/r to the C7.
    – user50691
    Jun 8 at 22:24
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    @j-g-faustus: To further confuse things, it's common to use a diminished vii chord as a substitute for a V7; in F major, such a chord would be E-G-Bb, the E and Bb of which are often the most functionally important notes in a C7 chord (since the tritone between them creates tension, which can be resolved by moving the E to F and the Bb to A).
    – supercat
    Jun 8 at 22:59
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    'You can harmonise a diatonic melody with just 3 chords'. True, go to most bars in Spain or Canaries, and you can hear just that...
    – Tim
    Jun 9 at 13:57
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iii isn't that uncommon. viidim lacks individual identity because it sounds so much like V7. There's more to harmony than the circle of 5ths. Other frequent visitors to pop/rock music are bVII, iv, bIII, III....

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Circle of 5ths and the 4-chord song mathematically speaking https://www.classicfm.com/discover-music/music-theory/four-chords-every-pop-song/

The way you worded your question sort of implies that only the last four chords of the circle of fifths progression would be used for a four chord progression. So, let's first list out that progression...

7    6    5   (4    3    2    1)
IV   viio iii  vi   ii   V    I

...viio and iii are more than four steps back so supposedly they would be excluded. Keep in mind it a circle, there isn't a start or end, so maybe you're thinking of the section ii V I IV.

There really isn't any math involved with this and I don't think the linke article does a good job of explain the music theory about this stuff.

Let's first list out the Axis of Awesome chords and the first four of Pachelbel's Canon...

I V vi IV

I V vi iii

First, neither of those is a segment of the circle of fifth progression.

The second thing is the notion that iii is not used because of some connection to the circle of fifths progression don't really make sense when the article goes on to use Pachelbel's Canon as an example, because it actually uses the iii chord.

The problem comes from trying to explain too much through the literal circle of fifths progression. Instead of that, a better fundamental concept to work with is roots by descending/ascending perfect fifth. When roots by descending/ascending perfect fifth are combined with the idea of harmonic sequences, it goes a long way to explain a lot of harmony, harmony that has been used for the last 500 years! Add in relative major/minor chords (which the article does mention) and you have the bulk of most chord progressions.

Using brackets to enclose the roots by fifths we can see those fundamental units and how they are repeated using harmonic sequencing...

|: [I V] vi [IV :|: I] ...

Considering the Axis of Awesome usually is repeated, we have I V and IV I as two ascending fifth progressions. This is literally a harmonic sequence, because the I is elided, the sequence would be [IV I][I V], but the I isn't repeated so it's three ascending fifths IV I V. The vi can be called a deceptive progression (like the deceptive cadence), or it's the relative minor of IV.

[I V][vi iii]

The Pachebel progression is very obviously built of ascending fifths using harmonic sequences. The is a common minor analog to that [i V][♭iii ♭VII].

Other super common progression using this process are:

[I IV][V I] two descending fifths, sequenced by ascending fifth

[I V][♭VII IV] two ascending fifths, sequenced by descending whole step

[I V][VI I] two ascending fifths, sequenced by descending fifth

Notice that the last one is the common blues turnaround and it is just the progression/sequence inversion of I IV V I. Some people get themselves all worked up over the retrogression of V IV in the blues, but when you view progressions in terms of progressions by fifth and harmonic sequence, things all fall into place nicely.

You don't need to worry about chords to avoid. Try thinking positively about the fundamental relationships that underlie common progressions. Add to the stuff above borrowed chords and the tritone substitution/augemented sixth chords and you have a very nearly complete picture of tonal harmony. At that point you should have solid harmony and any supposed "avoid" chord, becomes either a potential source of surprise/uniqueness to exploit, or can be made to work through means like repetition, harmonic sequence, or modulation.

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There are two relevant facts here that often get merged together, but can be separated...

  • In Major tonality, the chords I, IV, V and vi make a strong group. (You can look at reasons why this is from a number of different perspectives).
  • Harmonically simple music often features progressions featuring a small number of chords, which may indeed be 3 or 4.

When you combine these facts, you can see why it is quite common to find songs that have only I, IV, V and vi, or only I, IV, and V; and it's unsurprising that there are a number of common particular progressions that are based around those chords.

However: there's no reason to only use certain chords in a 4-chord progression. In fact, repeating a chord progression round and round can give it a coherence and sense of 'resolved expectations' without that needing to solely come from the harmonic relationships between the chords. And there's certainly no need to only use I, IV, V and vi, or only I, IV, and V: for a start, consider minor harmony, which wouldn't use a major I.

Which chords should be avoided in a four-chord progression (mathematically)?

None need be avoided. Unless by "four-chord" progression you actually mean four particular chords - in which case, of course, avoid chords other than those!

I notice that 4 chord songs almost never use the iii chord or the VII chord.

As per Laurence's answer, iii isn't that uncommon, and vii° is very close to being an inversion/voicing of the very common V7 - though you could also consider that it's rather dissonant.

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The simplest answer would be that the iii chord and the vii chord are indeed the farthest away from the I chord in the circle of fifths of a major scale. I suspect this is part of the reason they are uncommon, but there are probably other factors as well.

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    I think is less common than in pop music, though. (Not in some other genres, of course.) So arguing with distance in the circle of fifths alone is quite dubious. Jun 8 at 11:04
  • @leftaroundabout Distance in the circle of fifths still places iii and vii, which I'm willing to bet are the least common chords used in pop music, the furthest from I. But as I said, there are probably other factors as well. Jun 11 at 8:22

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