Why 13th Chords

"gives the chord a very different direction it wants to go"

I struggle with this every time I write songs. How exactly do you know which direction you should go, and how to transition from verse to chorus (i.e. how to pick good transition chords/notes). Currently, I do this by ear, whatever sounds good, but I know there is a better way.

That's one of the reasons that a 9th chord inverted, were the 9th is on the bass, was frowned upon in 19th Century music theory. They argued that the 9th in the bass is a 2nd, not a 9th.

To give you an idea of how conservative people were regarding music, even in the beginning of the 20th Century, the Vienna Music Society rejected playing Schoenberg's Verklärte Nacht at first on the ground that it had inverted 9th chords [1]. See [2] for an example, it's the first chord in the second bar.

[1] See "reception" in http://en.wikipedia.org/wiki/Verklärte_Nacht

[2] http://www.youtube.com/watch?v=_Ic3ZLj5RPw#t=3m12s

These chords progress in units of 2. So you start with your triad (1st, 3rd, 5th). For the 7th chord you add the 7th, naturally enough. The next chord is then a 9th chord, so you add a 9th interval, but it wraps around modulo 8, so it's really a 2nd interval (since intervals are 1-indexed). Next it's an 11th chord, but the interval is really a 4th, and so on. The idea is to think of them spread out though, as successive thirds.

No, there really isn't a better way. All the rules behind music theory are based on what sounds good, not the other way around.

What you can do, however, is push yourself to try out new harmonic structures. You can make your chords more complex, you can delay resolution much longer, you can shift keys.

There are also other musical idioms like polytonality, alternate scales, etc. that you can use to expand you horizons.

But ultimately, you still should pick something that sounds right to you.

I was puzzling over the same question. Some of these words are semantically overloaded, and as far as I can tell there's no single interpretation that makes it all "work out" the way you would require if this were math. There are two meanings to "adding 7":

1. adding 7 arithmetically to obtain the same note an octave higher;

2. adding in a 7th note to whatever chord you are playing.

But the way the chords are named isn't what you'd guess from those two meanings. If it were, you'd add a 6th to C major to get C6 (that part is true) and then a 7th to C6 to get C13 (that part is not true). Instead, there's yet a third way the notes are "added", namely cumulatively in the following sequence (I've cribbed most of this from the OP):

  C      C+E+G
  C7     C+E+G+B♭ 
  C9     C+E+G+B♭+D 
  C11    C+E+G+B♭+D+F 
  C13    C+E+G+B♭+D+F+A

(The author's argument is that while the chords are theoretically defined to include all these notes equally, some notes - especially B♭ - are more important than others. So even if you drop E,G,D,F and only play C+B♭+A, that's still arguably C13. He's saying that's the essence of the chord.)

I suppose some arbitrariness is inevitable, because there are only so many letters and numbers to go around, and more chords with a plausible claim to the "best" names -- which, if you know regular expressions, would be something like [A-G][1-9]+ -- than there are names available. One beautiful thing about music is that it isn't math (we have math for that). It seems to overlap with math and then mixes everything up in mind-blowing ways, at which point all you can do is feel. Well, memorize and feel.

Does anyone know historically how C13 got to mean "C+E+G+B♭+D+F+A" rather than "C+E+G+B♭+A", which if you consider things only symbolically, seems more likely? Obviously, it must be because that's what people were playing - but who? Presumably jazz musicians? When was this name settled upon?