unconnected tonnetz chords

Question

In terms of Tonnetz analysis, is there any special significance to a chord that does not form a connected graph when using a Tonnetz diagram? For example an augmented 7th chord ( as in C-E-G#-Bb ) is connected for C,E and G#, but the 7th, Bb, it is disconnected.

From that we can see that Bb is not a [3rd, flat 3rd, 4th, 5th, 6th, or flat 6th/sharp 5th] of any of the notes C, E or G#. (Or inversely, using Bb as the root.)

Is there anything more we can learn from this about an augmented 7th, or other chords that don't form a connected graph?

On a side note: Just tested all 495 4-note combinations and 132 of them are not connected. And of the 220 3-note combinations 96 are not.

Answer 1

All diatonic collections of notes built out of thirds will connect on a Tonnetz diagram. However, a whole lot of other things will also connect on a Tonnetz diagram. What is more interesting is that if something doesn't connect, it means it's not a diatonic construction of thirds. Due to the relatively small number of possible chords that fit that criteria - major, minor, diminished, mi7b5, maj7, unaltered dominant, etc. - this is fairly easy to test. Some examples:

Things that don't connect:
C E G# Bb - Not diatonic (you can't build G# and Bb into a diatonic scale)
C E Gb B - Not Diatonic (Gb and B natural similarly impossible)
C F G Bb - Diatonic but not built out of thirds

Things that connect:
C E G B - diatonic
C E G B D F# A - diatonic
C Eb Gb Bbb - Not diatonic 
C E G Bb Db - Not diatonic but close
C F Bb Eb Ab - Literally just a bunch of fourths
C Db D Eb E F F# G Ab A Bb B - Every note*

*It won't connect in order, but it'll connect.

Things get a bit strange if you don't accept chromatic spellings. It's impossible, for example, to create a Tonnetz diagram that spells every diminished chord correctly simultaneously if you assign each space only one value. Indeed, anything with a tritone will be problematic in this arrangement.

Overall I'm not sure what value you can get out of this analysis. If a chord is an intelligible chord and it connects you can say that it's probably diatonic in origin, but it might not be. And of course there are a very large number of nonsensical note combinations that connect. Something not connecting tells you a lot more about a combination of notes, however; no diatonic collection of notes separated by thirds won't connect. Yes, the logic behind this is fairly straightforward -- both diatonically occurring types of thirds (major and minor) are available -- but that's about as in-depth as this gets, I think.

It is also worth noting that there are much easier ways to determine if something is a diatonic construction of thirds or not: Any combination of accidentals falling out of order (FCGDAEB, BEADGCF) instantly makes something impossible in a diatonic system. By extension, there are a relatively narrow range of possible intervals.

The properties intrinsic to this formation are built entirely out of the fact that diatonic scales are simply a series of successive perfect fourths/fifths.

Answer 2

I think that you don't succeed to interpret C-Bb on the tonnetz because this interval and more generally chords with 7th need the extra dimension "7" in the tonnetz. The interval C-Bb is the frequency ratio 7/4. The "traditional tonnetz" imagined by Euler represent musical intervals as ratio of frequencies equal to rational numbers build only with the prime numbers 2 (octave), 3 (fifth), 5 (third). This is not enough to analyse music: we need the prime number 7 (and may be 11,13,..) , so we need to add a dimension in the Euler lattice called "tonnetz". More more explanations, see http://medias.ircam.fr/x012745 or https://hal.archives-ouvertes.fr/hal-01119499