D-F♯-A-C♯ I7
E-G-B-D ii7
F♯-A-C♯-E iiI7
G-B-D-F♯ IV7
A-C♯-E-G V7
B-D-F♯-A vi7
C♯-E-G-B vii7
Look at the cyclic permutations of (say) ii7 (E-G-B-D). These are: G-B-D-E; B-D-E-G; and D-E-G-B. These are a sixth wide, and have two notes next to each other, the D-E. The first is major, the second diminished (two semitones), and the third can be described as a sus2 (D-E-G) with an add 7 (B).
For a long time classical musicians stopped at seventh chords, with an occasional foray into a ninth as a stunt. Jazz musicians, however, started with sevenths and worked upwards, notionally to five note chords (ninths), six note chords (elevenths) and seven note chords (thirteenths). A jazz pianist or guitarist thinks nothing of playing D♭13, which is
Furthermore we can shift the fifths, sevenths, ninths and so on, up a sharp or down a flat, to get truly wonderful monstrosities such as D♭13: D♭(1)-F(3)-A♭(5)-C (7)-E&flat(9)-G♭(11)-B♭(13).
We can’t play the full-fledged Triadic D♭13 on the guitar, or with a string quartet, and it would need some skilful orchestration to be heard if played by an orchestra. Even if we could, we would only do so very rarely. It’s a mess. As are full-fledged elevenths.
Composers and songwriters know that chords extending above sevenths are a special effect. (Hindemith says as much in an aside in his book on Harmony.) They may want, say, the effect of the root and the eleventh (fourth an octave up), with the third to indicate that the chord is “really” a minor, and a flat (aka dominant) seventh, because that flavours the chord, but the fifth and the ninth don’t do anything musically useful, and are just clutter. So they write the root, third, seventh and eleventh, and everyone calls it an “eleventh chord”.
Suppose we want the special effect of a sharp nine against the eleventh? Write the root, sharp nine and eleventh. How about the third? Sharp nines are flat thirds an octave up, and that sounds messy, so let’s leave out the third. Dominant sevenths are a special effect of their own that will distract from the one we want, so let’s leave out the seventh. Let’s put in the fifth so that the chord doesn’t sound too thin. So that’s root, fifth, sharp nine and eleventh. Which is also called an eleventh chord, strictly an “eleven sharp nine” chord.
Nobody plays or writes full-fledged triadic extended chords. They play or writerandom carefully-chosen groupings of different notes spreading over two octaves. (Playing the same note an octave apart doesn’t “extend” the chord.) And no group of instrumentalists does this more than guitarists - and any other ensemble with less than five players.
Look at the cyclic permutations of (say) ii7 (E-G-B-D). These are: G-B-D-E; B-D-E-G; and D-E-G-B. These are a sixth wide, and have two notes next to each other, the D-E. The first is major, the second diminished (two semitones), and the third can be described as a sus2 (D-E-G) with an add 7 (B).
For a long time classical musicians stopped at seventh chords, with an occasional foray into a ninth as a stunt. Jazz musicians, however, started with sevenths and worked upwards, notionally to five note chords (ninths), six note chords (elevenths) and seven note chords (thirteenths). A jazz pianist or guitarist thinks nothing of playing D♭13, which is
Furthermore we can shift the fifths, sevenths, ninths and so on, up a sharp or down a flat, to get truly wonderful monstrosities such as D♭13: D♭(1)-F(3)-A♭(5)-C (7)-E&flat(9)-G♭(11)-B♭(13).
We can’t play the full-fledged Triadic D♭13 on the guitar, or with a string quartet, and it would need some skilful orchestration to be heard if played by an orchestra. Even if we could, we would only do so very rarely. It’s a mess. As are full-fledged elevenths.
Composers and songwriters know that chords extending above sevenths are a special effect. (Hindemith says as much in an aside in his book on Harmony.) They may want, say, the effect of the root and the eleventh (fourth an octave up), with the third to indicate that the chord is “really” a minor, and a flat (aka dominant) seventh, because that flavours the chord, but the fifth and the ninth don’t do anything musically useful, and are just clutter. So they write the root, third, seventh and eleventh, and everyone calls it an “eleventh chord”.
Suppose we want the special effect of a sharp nine against the eleventh? Write the root, sharp nine and eleventh. How about the third? Sharp nines are flat thirds an octave up, and that sounds messy, so let’s leave out the third. Dominant sevenths are a special effect of their own that will distract from the one we want, so let’s leave out the seventh. Let’s put in the fifth so that the chord doesn’t sound too thin. So that’s root, fifth, sharp nine and eleventh. Which is also called an eleventh chord, strictly an “eleven sharp nine” chord.
Nobody plays or writes full-fledged triadic extended chords. They play or write
NY: 1) It's common to refer to the 'cyclic permutations' as inversions; 2) in your descriptions of their qualities you are applying a Baroque era Thoroughbass style of analysis whereby the inverals are always analysed relative to the bass note as opposed to a notional 'root note'. The baroque figurations translated into inversions are: 7: Root position, 6_5: First Inversion, 4_3: 2nd inversion, 2: Third Inversion. Note that these figurations are incomplete - the full figurations would be: 1 3 5 7, 1 3 5 6, 1 3 4 6 and 1 2 4 6 for each inversion respectively. It's true that you could describe each of these using modern chord symbols but remember that these are probably 20th century inventions.; 3) This statement is arguable: 'For a long time classical musicians stopped at seventh chords, with an occasional foray into a ninth as a stunt' but it depends on how you define 'extended chords'. If you mean a literal stack of 3rds including every interval up to the named number (e.g. 1 3 5 7 9 11 13 for a 13th chord, then it's true), however if you consider a common baroque harmony like Bo7/C which produces a sound that could be described as Cma9sus4b13 (1 7 9 11 b13) an extended chord, then it's not true; 4) chord make a lot more sense when you think about them being used to harmonise and existing melody - it's quite common in Jazz standards for the wacky chord names to be caused in part by the interaction of the melody note above a fairly conventional harmony. When the melody note lingers on the 1, 3, 5 of a chord, the effect is fairly stable, but when it lingers on the 2(9) or 4(11) it creates more tension. The 7th is interesing since it is a dissonance which has become normalised, but a major 7th is still practicularly dissonant. The 6(13) is also interesting since it's more consonant than the 7th, but isn't considered part of the basic triad however, if you consider 1st inversion chords we can argue that a 1 3 5 6 is implying a different chord with a root a 3rd lower, and that therefore the 6 is a chord tone and a consonance. Diminished and augmented intervals are always considered dissonances; 4) chords built in successive 3rds beyond 9ths aren't super useful since they don't sound that amazing (as you point out). The thing to remember is this discussion of theoretical 13th chords was written long after the music they describe was written. Conclusion: you're not exactly wrong but I've come to the conclusion that this kind almost mathematical approach does not describe the compositional though processes of the originators of the music and while it is interesting as an approach for future music (consider Allan Holdsworths "Telephone Book from Hell") it's a dead end when it comes to understanding most non-avant garde music of the past (including most Jazz). In my opinion, understanding pre-20th century classical music is futile without considering the horizontal melodic element of counterpoint (harmony originating as a by-product of stacking suitable melodies)
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