This is my contribution to the First Climate Message Video Festival, an online event organized by vocalist, composer, and teacher Warren Senders. The idea is to record yourself making a little music, then stop and say a short message urging the listener to learn about climate change, and then continue with the music. If you’re involved in music you can participate too. See the Facebook Event for the video festival and the project website.
The Tonnetz is a graphical depiction of chord relationships that was developed by 19th-century music theorists building on work by the mathematician Leonhard Euler. If you pick any major or minor triad, you can use the Tonnetz to quickly find all of the triads that share one or two of its tones. If you’re building a chord progression, you can use the Tonnetz to find extended sequences of chords that are connected by common tones.
When I first came across the Tonnetz I found it amazingly elegant but also a bit mysterious: “Yeah, it works,” I thought, “But how on earth did anyone come up with this?” In this post I’d like to share a step-by-step process for building a Tonnetz. This is not necessarily the fastest way to construct the diagram, and I’m neither claiming that this approach is original nor that it matches the process that was used historically. This is just a way that made sense to me in my own study and I hope it might help the reader gain insight into how the Tonnetz works.
Step 1. We’ll start by drawing a simple grid:
Step 2. Next we’ll turn our grid into a representation of musical space by placing a note at each grid intersection point. If we were building a so-called just-intonation lattice, we would represent notes by their precise frequency ratios, but here we are going to use scale degrees from an equal-tempered octave: 1, ♭2, 2, ♭3, 3, 4, ♯4, 5, ♭6, 6, ♭7, 7. We’ll lay the notes out so that stepping right in the grid corresponds to ascending by a perfect fifth and stepping up corresponds to ascending by a major third. So, if we put 1 at an arbitrary intersection, we must put 5 to its right, and 3 above it. Now let’s look at 5 — what should its neighbors be? Following the rules, we must put 2 to its right and 7 above it. Continuing like this we fill up the whole grid. Each horizontal line (reading from left to right) is a sequence of ascending fifths and each vertical line (reading from bottom to top) is a sequence of ascending major thirds:
Step 3. Next we’ll add diagonal lines to our grid. These lines represent the minor third / major sixth relationship. Starting at 1, for example, and taking a diagonal hop down and to the right, we end up at ♭3, a minor third higher in musical space. Going in the other direction graphically (up and to the left) we land at 6, a minor third lower or a major sixth higher in musical space:
Step 4. Notice that each of the triangles formed by adding the diagonals is in fact a triad in the musical sense. Take a look at 1, for example: above it we have 3 and to its right we have 5. All three points are now connected by an edge — together, these edges form a right triangle that we can take to represent the major triad rooted at 1. We can also find an upside-down right triangle in the points 1, ♭3, 5, representing the minor triad rooted at 1. Let’s label all the chord-triangles in the diagram using Roman numeral notation: the 1, 3, 5 triangle gets the label I and the 1, ♭3, 5 triangle gets the label i:
Step 5. Notice that if two triangles share an edge, the corresponding chords have two tones in common. If two triangles share a single point (i.e. they meet at a single intersection in the grid), the corresponding chords have one tone in common. In preparation to make these relationships more explicit, we’ll remove the black grid lines, leaving only the chord-triangles in place:
Step 6. Finally we’ll draw new lines connecting all of the chords that have two common tones. This will create a background pattern of rectangular tiles. We can see that any six triangles that share a rectangle represent six chords that share a common tone — the tone written in the center of the rectangle. The diagram can be prettified by stretching the rectangular tiles into hexagonal ones, but we’ll leave that for next time.
Here’s a bigger version:
PS. The diagrams here have had limited proof-reading so far — if you catch a typo, please let me know.
I like musical exercises that are extremely simple but unexpectedly challenging. One such exercise in the domain of rhythm is to set your metronome really slow — 10 or 20 beats per minute — and try to tap along. We often associate difficulty with high speeds but high speeds present more of a motor challenge than a perceptual one. In the case of very low speeds it’s difficult to tap accurately because it’s difficult to feel the beat as such — the very slow metronome clicks may seem like disconnected events instead of being part of a continuous rhythmic thread that we can follow. (Well, that’s my own subjective account of the difficulty, but it turns out that tapping has been the focus of extensive research in psychology — see for example a paper by B. Repp titled Sensorimotor Synchronization: A review of the tapping literature.)
When you try to tap at 10 beats per minute you might find yourself playing a game of chicken with the metronome. You don’t want to tap too early, so you wait, and wait some more, and then you suddenly hear the metronome click and you rush to tap. If your reflexes are fast enough you can make it seem like you’re tapping in sync with the metronome but really what you’re doing is tapping in response to it, and each tap is ever so slightly late. One way to break out of this pattern is with plain mental fortitude: try to make a clear decision to tap and then execute it completely without second guessing, but if you’re way too late and you hear the metronome click before you’ve even decided to tap, just skip that beat, don’t rush to tap in response. Another way to break out of the pattern is to change the exercise so you aim to tap slightly before each beat instead of right on the beat. This way you can’t use the sound of the metronome as a cue. An advantage of practicing this way is that you’ll notice the brief duration between your tap and the following metronome click, and you can take this as feedback. Try to make those durations as consistent as possible so that you’re always anticipating the beat by the same amount each time — easier said than done.
Of course you can greatly improve your accuracy by subdividing the beat in your mind, and actually counting to yourself “One and two and three and four…” as you might do in standard musical practice. But that’s not the point of this exercise — the point is to see how close you can get to perceiving the slow metronome clicks as a fundamental beat without relying on a faster beat that you’re tracking internally. So give it a shot without counting.
In experimenting with this ultra-slow rhythmic practice I’ve noticed that it can make an interesting sort of meditation. Tapping is useful from a meditative standpoint since it requires attention, but it doesn’t really require thought — it forces you to stay present and not zone out. For a meditative exercise that combines pitch and rhythm try this: turn on a drone (electric tanpura, etc.) and keep singing its pitch on a steady, continuous “aaah” vowel; meanwhile turn on a metronome at 10 BPS and tap to it as you sing.
If you’re like me you’ll probably want to get the tapping right and you’ll experience a little bit of frustration every time you tap too early or too late. Can you isolate that frustration, notice any ways that it might manifest physically, and learn to dissolve it, even as you keep making mistakes? And when you get it right can you move on without becoming distracted?
When I turn on the news I very rarely hear political issue reporting. What I hear is political strategy reporting, and I don’t know what to make of it.
Let’s say there’s a candidate running in some election with lots of Latino voters. Instead of asking “Why would a Latino voter prefer this candidate’s platform?” journalists are more likely to ask “What is this candidate doing to reach out to Latino voters?” These are different questions. The first question requires an analysis of the candidate’s record and policies as they relate to the concerns of a particular group of voters. The second question only requires a reiteration of what the candidate has recently done or said in effort to look appealing to that group as part of an explicit campaign strategy.
I’ve never planned to run for political office, yet in listening to the news over the past couple of decades I’ve received thousands of hours of political strategy coaching from pundits eagerly describing just what a candidate would need to do to convince someone to vote for them. Why should I care?
One could argue that to be an informed voter we must understand the tactics of persuasion employed in political races so we won’t be fooled. And one could argue that a candidate’s strategy reflects something about his or her character — that in analyzing a candidate’s strategy we learn who the candidate is, how they think, and what matters to them, and that in studying how voters respond to campaign strategies we learn about the mood and sentiment of the country. These are fair points but they don’t address the disaster of political journalism – that in focusing on the endless intricacies of strategy, we lose track of substance.
Strategy is easier than substance and safer than substance to discuss. A journalist striving to remain impartial never needs to endorse or oppose a candidate’s position, or pass anything resembling judgment on it. The journalist only needs to ask how the candidate’s recent gestures are likely to be interpreted by voters, a question that can be explored through the endlessly bountiful mechanisms of gossip and polling – certain to fill as much airtime as necessary.
And I, as a voter, am left with little information to use in selecting a candidate, but a whole lot of information I might be able to use in crafting an image that would make me appealing to Latino voters (or black voters, or elderly voters, or young voters, or whatever) if I decided I wanted to enter politics. No need to pay a campaign strategist, I could just turn on the radio and do what they tell me.
Just today I listened to my local radio station WGBH and got some advice I could use if I were Bill Clinton and I were wondering whether my involvement in Hillary’s presidential campaign would be an asset or a hindrance. And I got some great advice I could use if I were Chris Christie and I wanted to know whether voters considered my presidential bid to be over. But since I am neither of these people I’m still struggling to find any practical application for the generous strategy guidance I spent a chunk of my morning receiving.
As for journalistic impartiality, I don’t consider strategy reporting impartial at all. Journalists who focus on strategy reporting are casting partisan votes in favor of strategy over substance as the thing that’s worth our time.
Imagine some aliens descend on earth looking for intelligent life; the self-dubbed homo sapiens proves disappointing in that regard, but we turn out to be tasty.
Imagine the aliens have such cognitive sophistication that they can hold thousands of simultaneous conversations at rates thousands of times faster than the fastest human speech, never losing their places. From their vantage point, all of human behavior – anything we might do or say – is predictable and boring, and yet they adore the succulence of our flesh.
We can’t converse with them, since we can’t formulate thoughts that match the complexity of even their most vacuous chit-chat, and yet they do understand our own grunts and gesticulations – in fact, they can anticipate these grunts with stunning precision. To them we are robots acting on a discernible program. We might proudly present the greatest achievements of our science, our literature, our music, and to them it all seems as insect architecture might seem to us.
These aliens notice how we pollute our habitat, how we slaughter and enslave our peers, and how we eat other animals we deem inferior and expendable. How then could we persuade the aliens that while we can be ground into delicious burgers we should rather be allowed to live, even to be recognized as members of their moral community?
If we had no hope of befriending them as equals, and if we could offer them nothing new in the domain of information, perhaps our best argument would simply be that we’re alive, that we feel pain, that in our capacity to suffer, we are like them. But that’s not a logical argument, it’s an appeal to empathy, and how could we expect these aliens to relate to creatures as primitive as we are, particularly when the aliens are hungry and the smell of our flesh on the grill makes such a persuasive case against compassion and for exploitation. How – tell me – how could a human persuade a peckish alien taxonomist not to classify all humanity as a resource to be tapped for nutrition and enjoyment?
I suppose we might still carry out the hope that empathy – as a phenomenon – can extend across species and types of mind and can be entertained even on an empty stomach. Our treatment of other animals on earth is an opportunity to affirm or destroy that hope, no matter whether the aliens here discussed are mere figments of a thought experiment.