Has anyone ever come across the Tonnetz (German for ‘note-net’)?
It goes back to Euler in the 18th century.
As a visual learner, I find this kind of thing fascinating 
I’ve been a huge fan of the Circle of 5ths since I came across it, but this seems to contain so much more in terms of types of chords and their relationships.
Here’s the Wikipedia page 
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Thank you for sharing. Really worth analyzing; plenty of food for thoughts…
That is awesome, I’m definitely going to investigate. A quick look around led me to this:
I wonder when Justin will teach this in his music theory course? 
I have come across this and even seen a couple of composing tools and even musical keyboards based on this or similar principles. I’ve not studied it too much though.
I am too (I actually think most people are). But, whilst it’s probably very useful in some ways, and certainly very clever and interesting, I don’t personally find it particularly compelling.
It’s very information dense for one, which means you have to go searching for the information you are looking for. It’s certainly far too much data to be useful as a mnemonic, as the circle of fifths can (I can keep the CoF in my head).
And, whilst it shows many relationships, I also find it does nothing to really explain anything.
The analogy I would present to @brianlarsen is bell ringing diagrams:
Blue line diagrams and circle-of work diagrams are extremely useful.
This “blue circle” diagram is pretty, and clever. But I find it utterly useless, and I don’t know anyone who actually uses them:
A significant part of my job is trying to distil technical, operational, and business concepts into diagrams that can be used to communicate how stuff works to people with a short attention span and who have very little understanding of the subject (like explaining to CxOs how their business runs 
).
This is not the sort of diagram I would use (unless the aim was to say “look how complicated this is”).
I’m not saying it’s useless. But I suspect its main utility is a tool to assist composing, which I don’t do a lot of. I don’t see it’s particularly useful as a theory learning tool, or something you would use in day-to-day music playing/learning.
Cheers,
Keith
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I think this is what grabbed my attention initially
I love it when I can see how things are connected (esp. if they appear to be unrelated at first glance).
This was my first thought. I had recently noticed how some songs have chord progressions where only single notes are changed to arrive at the next chord (I think it was Radiohead).
I also like that you see all the notes that make up the chords, (although I really should have internalised that by now )
There’s no ‘new information’ here, but it is wonderfully displayed.
Yes you do…
Me! 
Well, not quite.
I try to keep my ‘learning new methods’ to a bare minimum and when I’m forced to, I use the traditional blue line method (although I always turn the book or page sideways to view the ‘mountains and valleys’ in the landscape format 
)
I always hated the bob and single diagrams, finding them difficult to understand and much easier just to learn by being told what you do where. Then I look at the picture.
When I came across the circular diagram, my first thought was: “This makes sense! Why don’t they show you this in the first place?”
Horses for courses…
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You, sir, are a pyschopath!
Yes, agreed here. Although I don’t find the Blue Circle diagrams helps at all with this. I prefer a modified circle of work diagram.
I look at the Tonnetz diagram as a bit like the Periodic Table of Elements for Chemistry, although the Periodic Table is a bit more fundamental and explanatory IMO. Most people don’t learn and internalise the whole Periodic Table (and all the accompanying detail) but just use it as a reference. I see the Tonnetz diagram as a useful reference in certain situations.
Cheers,
Keith
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YES!
And I loved the layout and structure of it when I was learning chemistry, both in school and at uni. I did use it as a learning tool, although lazy me would only learn the useful bits.
Long time ago now; all I recall is ‘He’ 
… or is that Hehe?
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A truly fascinating diagram Brian. I searched for familiar ground using the “guide notes” and manage to find the C Major scale and associated diatonic chords. Once I found that and could see the relationship of how this is laid out the rest of the diagram made sense. A lot to absorb but quite a handy reference tool, if you’re struggling to find chords tones etc. 
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Yes, it is interesting.
It’s based on the fact that both minor and major chords are made up of intervals of 3, 4 and 5 semitones. Incidentally, a triangle with sides of those lengths is right-angled, whereas the Tonnetz uses equilateral triangles, probably for aesthetic reasons. The 5-semitone intervals are on horizontal lines, the 4-semitone intervals run NW-SE, and the 3-semitone intervals run SW-NE, so to speak. It’s nice how it all fits together. All three axes will repeat, each at a different rate, so the whole thing could probably be rendered unbounded on a torus, with the horizontal lines running in what I think is called the toroidal (i.e., the longest) direction.
Blimey, I just checked my use of ‘toroidal’ on the Wikipedia page for the torus, and guess what’s illustrated there - the Tonnetz!
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It’s all rather exciting isn’t it.
I think I may have seen small chunks of this but not the entire system.
And … ooohheeeh … look what I found!!!
C → G → D → A → E → B → F# → C#
From here I alter the sharps to flats which is a more coomonly used version of the enharmonic equivalent notes.
Ab → Eb → Bb → F → C
You beauty!
It’s my old friend, the Circle of Fifths.
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Yes, that would appear on all horizontal lines if they were continued far enough. The middle line in your selection has most; it would have a C on the left if it went further.
Note also that if you drop vertically from any point to the points directly below, you find the chromatic scale.
I am still a little unclear about why only sharps are shown, and not flats too. In one version of it I even saw that double sharps were shown, which seems to verge into muddying rather than clarifying.
More thoughts…
Some of the complex chords can be quite easily pointed out in the Tonnetz. Take a Cmaj13, for example. Theoretically, that has C E G B D F# A - too many notes for the ordinary guitar, so at least one has to be dropped, and I think most fingerings drop the F# (the 11th), don’t they? But regardless of the guitar’s limitations, see how easy it is to spot in the Tonnetz:
Unfortunately, that’s just one form of the 13th. There are a few others and they don’t all fit quite so neatly.
PS: I am not so sure about this now. That F# bothers me.
The F# would be F natural in that chord @MarkPeters
Indeed. - Pythagorean triples all the way!!
Major chords
Root → Major 3rd → Perfect 5th → Root
Root → 4 semitones → 3 semitones → 5 semitones
Minor chords
Root → Minor 3rd → Perfect 5th → Root
Root → 3 semitones → 4 semitones → 5 semitones
Major — — — — — — Minor
—
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Yes, so actually no 13ths really fit the zigzag pattern 
. The zigzag only works up to the major 9th. So chords can be found in connected sets, but the shapes won’t necessarily be neat. That’s a shame, I thought I was on to something.
for the F note of C13, look behind you …
This fits with the view of the full major (Ionian) scale I displayed on the Circle of Fifths here. And the other modes of the major scale that follow in subsequent posts of that topic.
Which all match with the modes diagrams below the main body of the Tonnetz in post #1 above.
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That’s true, and of course it would give us the notes of the 11th as well, C E G B D F, though they are also out of sequence in the pattern. So it is slightly compromised as a teaching aid, by having to explain that after 9ths you need to add notes from the other direction.
Am I right in thinking the zigzag pattern actually works better for Am13?
No change of direction required.
@brianlarsen I used to love doing Maths problems, but A level maths and university engineering maths are a long way back. 
As a (still) practicing engineer I’m into simple solutions that work and that I can explain to people. The joy of a good maths problem doesn’t get much of a look-in in my job😊.
Musically, I think theory helps one’s playing, but you should learn the theory you need when you need it.
Combining the above 2 paragraphs: I think this would be difficult to understand and explain to most people; and it’s probably more than most people are going to need.
I think the circle of 5ths is a more digestible tool for most people (@Richard_close2u ‘s coloured one helps with understanding). The linear nature of a piano keyboard can also help with intervals, chords and scales; as does the pattern like nature of a guitar or bass fretboard (if you know where the 1 is you can easily find the other notes of the scale).
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I keep running into it all the time. Facebook video algoritm, youtube video algoritm. Last week there was a great one explaing it…and of course I lost it 
Putting a construct in a graphic and understanding it is level 1
actually using it as a tool to help you analyse songs or write your own is level 2.
I’m still on lvl 1 on this one 
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This is brilliant. Where can I buy a wall poster to put next to my Periodic Table?
If you want to play around with this, check out the github project. You can play notes and chords from your keyboard using asdfghjkl for the white notes and wetyuo for black notes. They show up on the tonnetz grid diagram.
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