This Music Trick Shouldn't Work

But it does. What's up with that?

One thing I've always believed is that music theory is more than just a toolbox. It's useful, sure, but it's also an art form, a way of constructing new kinds of beauty and meaning, and when I get the chance, I like to try to show that side of it. There's lots of great examples, but one of my favorites has always been Jack Douthett's infamous Cube Dance, a deceptively simple map of triads that both hides a ton of insight about harmonic relationships and also just looks really frickin' cool!

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[tone]

Some additional thoughts/corrections:

1) Do I actually think 19-tone equal temperament is the future of Western tuning? Nah. The argument is mathematically reasonable but the practical implications seem daunting enough for relatively little gain that I suspect we'll just coast on 12 notes until we move on from equal temperament entirely. But I thought it was interesting how the system relies so heavily on current musical norms and breaks immediately if you change basically anything, and 19-tone is a good way to demonstrate that.

2) I couldn't find a way to fit this into the script, but another way to think about how the augmented triad can replace all the other bridges from A to its various 2-step neighbors is that, no matter which bridge chord we used, the process of getting to any of those minor triads _always_ involved raising E up to F. And since you can do your voiceleading work in any order, we can just put that step first and boom, A augmented every time.

3) To be a bit more explicit about the process of combining moves to measure distance, it might seem like there are so many options that it'd be impractically difficult to find the shortest path, but that's actually not true, thanks to one little fact: The two-steps moves are only useful for moving between cycles. Anything else can be done with just the one-step moves. That makes the process a lot easier. If the two chords are in the same cycle, just do parallels and mediants, and if you're hopping between cycles, use whichever of the two-steps will put you closest to your target. One of the beautiful things about this system is that it's fully navigable with a basic greedy algorithm.

4) Those of you already familiar with Cube Dance and Neo-Riemannian Theory in general may be wondering why I bothered including the other bridge chords at all, as they exist nowhere in Steinbach and Douthett's original paper or in Cohn's descriptions of it in Audacious Euphony. The short answer is that last time I made a video about Cube Dance, I followed Cohn's basic approach, and I got a lot of people asking how diminished triads fit in, so this time I wanted to explain why they don't. They work perfectly well as secondary bridges, but their function is superceded by the augmented chords, and I thought it'd be good to show that.

5) I suppose I didn't mention the major(b5) chord, so there's still one possibly-reachable triad left out of this analysis, but uh… I don't care about the major(b5) chord. Sorry. It's useless here anyway.

6) Did you know that the original Douthett/Steinbach version of Cube Dance is different from the one Cohn uses in Audacious Euphony!? Not structurally, but it's mirrored, so the bit where I talk about the clock numbers had to be rerecorded 'cause I wrote the script based on Cohn's version, used Douthett's version as a reference while filming, and didn't notice the difference until it was too late. This means you lose Cohn's mnemonic observation that, in his arrangement, the clock numbers represent the sum of the pitch classes in each chord mod 12, but I wasn't gonna talk about that anyway and also you have to be pretty familiar with pitch classes for that to be useful.

[gothiclb]

You've heard of the Circle of Fifths. Get ready for the Cube of Augmented Triads

[Turn.Colors]

These are my absolute favorite kind of 12tone videos. Don't get me wrong, I love the song analyses, but this channel is so great at expanding my understanding of music theory, especially on niche topics like Cube Dance. As a lifelong music nerd and very, very novice composer, these theory videos truly challenge and elevate my thinking. It's awesome. Thanks, y'all.

[morismateljan]

If you squint really hard you can see Coltrane's ghost smiling.

[davidbowmann]

The Term "Nebenverwandt" rather means something like secondary related.

Neben = Next to

Verwandt = Related

It is a compound word, but it is quite old and has no use outside of a few very specific areas of application, such as music theory.

[Datamining]

I feel like the words "Hamming distance" should come up in this work.

[RipRoaringGarage]

Youre entering what used to be my territory. This is Group Theroy, in Abstract Algebra.
Add some Hyperbolic Geom, and some non-Euclidean stuff and you get music.

[pawelmiechowiecki]

Funny fact. In English words "note" and "tone" looks very similar and sometimes they are used interchangeably. But the word "note" comes from "notation". Enharmonics C# and Db are not the same "notes" - they are notated diferently - but they are the same tone. The tone is related to their pitch.

[craigthomson]

I remember seeing a video with Pat Martino talking about exactly this thing of how the augmented triad can collapse in all these different directions to give you all the major and minor triads. So major and minor can be derived from just the four augmented triads.
The counterpart to this is that all the dominant 7 chords can be reached with half step voice leading from the three distinct diminished 7 chords by lowering any one note of the chord in turn. (Raising a note instead gets you to the min6/half diminished inversions, depending how you look at it)
It's a fun way to look at it, that from a certain view these 'exotic' chords can actually be seen as the fundamental building blocks that functional harmony emerges from.

[Rubrickety]

“…and that’s why Surfin’ Safari is so catchy.”

[rienpost]

Have you ever considered putting your music sheets up for sale? They are works of art.

[beeble]

I feel like I'm watching somebody rediscovering group theory without realising they're rediscovering group theory.

[MCDbits]

You randomly throw a Feynman diagram in beautifully. What's the Venn diagram overlap of quantum physics geeks and music theory geeks?

[bsorofman]

I'm glad Cube Dance came back up because it's been living rent free in my head since you last brought it up.

[anthonywestbrook]

I love following along to your videos with my isomorphic keyboard in front of me (specifically the Striso, which uses the DCompose layout). Patterns that never made sense to me on a piano nor on sheet music become obvious. I mean, the negative transformation literally rotates 180 degrees around the root (also, negative harmony is just flipping the board around 180 degrees rotated around the spot between the root and its fifth — I love this thing).

[renatoaj]

i smiled so wide when you drew cat-dog, such a classic

[ThatOneGuyRAR]

Xenakis took the phrase “cube dance” in a totally different direction

[Gnurklesquimp]

Regarding 12EDO being a goldy locks zone for this concept, you'll find other EDO tunings also have amazing relationships enabled by their particular evenly spaced intervals. I particularly love more granular ones, but go very granular and you sort of lose what makes EDO so special to me. 22 is my favorite sweet spot where you can still clearly hear the angular and symmetrical nature of EDO, the chromatic motion is REALLY nice. Conversely, you'll run into problems when relying on 12EDO tendencies, you can sort of drift far from home with more granular tunings, a bit like how we do with just intonation.

You go through all these chords and it sounds a bit odd but smooth, then you wanna return to the beginning and... we're suddenly a 22nd of an octave lower or higher.
With long progressions, that gets obscured and it can be a cool way to subtly get brighter or darker, but with shorter ones that jump all over the place, it can be very jarring.

Also notice 21EDO, for example, is 3x7, whereas 12 is both 2x6 as well as 3x4, this ruins some of the interval cycles/stacks and general symmetry devices we might otherwise use, but this can also potentially be used to move back into place when already offset. And, of course, we get that unique 7EDO effectively embedded within 21EDO, a very cool sound.

And yes, that does mean that if you made a piece entirely in whole-tone, for example, you've written in 6EDO.

[mariozehren]

Quick info on this "Nebenverwandt" thing as I'm German:
It doesn't just mean related. That would be the term "verwandt" alone, without the "Neben-".
The compound "Nebenverwandt" isn't really used in natural language outside of this specific case in music theory, so I don't know its exact origins. But typically in contexts like these, the prefix "Neben-" means something like "side" or "secondary" as the opposite of "main" (Haupt-). So for example, you have "Hauptstraße" (main street) and "Nebenstraße" (side street).
So "Nebenverwandt" implies that the chords aren't related in the "main" or direct way but rather that they are "side related" if you can say that, or related in a secondary way.

[scottishgentlemen]

This hurt my brain. I'm going to revisit it after I have coffee.