Chords provide a background against which the melody is set.
Western chords start with Triadic harmony and get more complicated from there.
We start with… Triads. A Triad is a three-note chord. The simplest are fifths: the base note, the one a third in the scale above it, and the one a third above that. The triadic fifths of C-major are
C-E-G I
D-F-A ii
E-G-B iii
F-A-C IV
G-B-D V
A-C-E vi
B-D-F viio
Lower case indicates minor chords, upper case indicates major chords, the 'o' indicates a diminished chord. Minor chords have three semitones at the bottom (D-E-F is Tone-Semitone), and two Tones at the top (F-G-A is Tone-Tone). Major chords are the other way round. Diminished chords have two sets of three semitones (B-D-F is Semitone-Tone-Tone-Semitone). Augmented chords have two sets of four semitones (C-E-G♯ is Tone-Tone-Tone-Tone).
What happens if we play (say) E-G-C (in that order on the piano)? Now the interval at the bottom is minor, not major.
Flipping the notes of those triads around, we get the so-called Neapolitan Sixth chords
E-G-C I 6 (minor)
F-A-D II 6 (major)
G-B-E III 6 (major)
A-C-F IV 6 (minor / sort of diminished-ish)
B-D-G V 6 (minor)
C-E-A VI 6 (major)
D-F-B vii6 (minor)
Flip once more, we get the 6-4 triads
G-C-E I 46 (major)
A-D-F II 46 (minor)
B-E-G III 46 (minor)
C-F-A IV 46 (major)
D-G-B V 46 (major)
E-A-C VI 46 (minor)
F-B-D vii 46 (minor / sort of diminished-ish)
The 6-4 triads get their major or minor flavour from the top of the triad, rather than the bottom, as with the fifth and Neapolitan sixth triads.
All these chords have the property that adding another note a third above the top one just produces the bass note an octave higher. A-D-F goes to A-D-F-A. This is because in the weird arithmetic of notes, a sixth plus a third is an eighth. So these inversions are a Triadic dead-end - though we can add whatever note we want to any of them, and later on, we will.
The idea of the root of a triad was invented to explain why it is that C-E-G, E-G-C and G-C-E are all I chords in C even though they have different bass (bottom) notes. The root of a chord is the note that would be in the bass, if it was re-arranged as a series of ascending triads, filling in any missing notes and allowing for modifications.
Simple enough, surely?
Bassists play the root note, so the rest of us don’t have to.
Classical harmony theory loves these inverted triads, jazzers barely know they exist.
D-F-A ii
E-G-B iii
F-A-C IV
G-B-D V
A-C-E vi
B-D-F viio
Lower case indicates minor chords, upper case indicates major chords, the 'o' indicates a diminished chord. Minor chords have three semitones at the bottom (D-E-F is Tone-Semitone), and two Tones at the top (F-G-A is Tone-Tone). Major chords are the other way round. Diminished chords have two sets of three semitones (B-D-F is Semitone-Tone-Tone-Semitone). Augmented chords have two sets of four semitones (C-E-G♯ is Tone-Tone-Tone-Tone).
What happens if we play (say) E-G-C (in that order on the piano)? Now the interval at the bottom is minor, not major.
Flipping the notes of those triads around, we get the so-called Neapolitan Sixth chords
E-G-C I 6 (minor)
F-A-D II 6 (major)
G-B-E III 6 (major)
A-C-F IV 6 (minor / sort of diminished-ish)
B-D-G V 6 (minor)
C-E-A VI 6 (major)
D-F-B vii6 (minor)
Flip once more, we get the 6-4 triads
G-C-E I 46 (major)
A-D-F II 46 (minor)
B-E-G III 46 (minor)
C-F-A IV 46 (major)
D-G-B V 46 (major)
E-A-C VI 46 (minor)
F-B-D vii 46 (minor / sort of diminished-ish)
The 6-4 triads get their major or minor flavour from the top of the triad, rather than the bottom, as with the fifth and Neapolitan sixth triads.
All these chords have the property that adding another note a third above the top one just produces the bass note an octave higher. A-D-F goes to A-D-F-A. This is because in the weird arithmetic of notes, a sixth plus a third is an eighth. So these inversions are a Triadic dead-end - though we can add whatever note we want to any of them, and later on, we will.
The idea of the root of a triad was invented to explain why it is that C-E-G, E-G-C and G-C-E are all I chords in C even though they have different bass (bottom) notes. The root of a chord is the note that would be in the bass, if it was re-arranged as a series of ascending triads, filling in any missing notes and allowing for modifications.
Simple enough, surely?
Bassists play the root note, so the rest of us don’t have to.
Classical harmony theory loves these inverted triads, jazzers barely know they exist.
NY: 1) 'triadic fifths' is not a term I've heard before; 2) 1st inversion/6_3 chords are not called Neopolitan 6ths (this would be a 1st inversion of a bII, so Db/F in the key of C, which effectively functions as a kind of IV chord); 3) your description of their chord qualities in inversion in inconsistent - with the 1st inversions you are analysing them as if the bass was their root, but with the 2nd inversions you are analysing them as if their middle note was their root - this is mixing up the Baroque style Thoroughbass style of Analysis with the Rameau Fundamental Bass style of analysis; 4) adding additional interations of the triad notes above o (or below) is called 'doubling' - orchestral voicings of triads will commonly span more than one octave; 5) the identification of a 'root' note is attributed to Rameau in the 18th century - it's questionable whether earlier musicians like Bach thought this way; 6) It's quite common for multiple instrument to double root notes in multiple octaves, even in pop such as Queen where you might hear John playing the root and Freddie playing the same notes as octave on piano; 7) Jazz musicians are aware of pedal point and slash chords which include but are not limited to triadic inversions. It's true that they are slightly less prominant since the bassline was a fixed compositional feature of common practice era classical music, whereas the bass tends to be improvised in Jazz music with the chords and melody providing the compositional back bone (there are exceptions though in certain patterns like endings and so forth).
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