Let’s Get Into Harmony

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Where we learn that harmony is truly a team sport.

Who doesn’t like harmony? Like, are there people out there who hear something quantifiably harmonic and they make a face like they just bit into something gross? Is there anyone here who has a misophonia for harmony?

I love all things harmonic so much so that I have a tattoo of a mathematically-accurate visualization of the harmonic series. I first saw this particular visual in a book called “Beautiful Geometry” by Eli Maor and Eugen Jost, with this explanation: 

Imagine stacking n identical domino tiles one on top of the other, but with each tile offset with respect to the one below it according to sequence ½, ¼, ⅙, ⅛, …, ½n (taking the length of each tile to be 1).

Beautiful Harmony
pg 103
visual representation of harmony using alternating colors of black and purple rectangles in a stack, with the long edges flush with each other and the shorter edges staggered at mathematically precise ratios to each other.

It looks super tippy, right? But it’s not gonna tip over, even if you continue to add more rectangular tiles, as long as you obey the established mathematical sequencing as you’d add each of those additional tiles; the center of gravity does not extend beyond the bottom tile. I’ll even go as far to say that that bottom tile is a great representation of the trust that is required for collaborative harmony to exist (more on this later). Personally, I’d be hesitant to continue to test this sequencing IRL with anything precious or breakable. All that being said, this harmonic representation is finite; it diverges to infinity.

Harmony is not just about balance,
it’s actually a calculated balance of a few pieces together
that are of near equal size / shape / volume.

Harmony elicits not only an emotional satisfaction for me but a physical one as well. You know how your body physically reacts to being nervous, or excited, or relieved, when something that is initially an emotional reaction becomes a physical reaction because it churns up endorphins like adrenaline, or maybe it spikes-up your dopamine level for a little bit? 

Well, that’s legit what happens for me when I witness harmony in the wild – in music, in architecture, even in moments when a well-coordinated project schedule has tasks that are being worked on concurrently and everyone involved delivers their piece at the exact right moment to combine with the other pieces. You might even see me levitate.

The thing is, I know that it’s math that makes that goodness happen. When something is harmonic, that means that there is a progression of sounds (or moments) where the vibrations (or movements) are at specific ratios to each other. It’s pleasing to my soul because of that feeling of balance.

The holy grail of successful harmony is when it is far reaching. It’s the reason you work towards attaining it as a team. It’s one thing for the barbershop quartet to feel good in that moment when they lock into their precise harmonic sound, but it’s a whole other thing when that harmony has a positive effect on the quartet’s audience.

The ultimate joy in harmony is when it’s mutually beneficial.

Think about what harmony might look like in your family, or in your social group, or at work. If you and your co-workers build one thing together – it could be a product or maybe it’s a service – and you deliver it to your audience (your client, your customer), the greatest success is when your team and your audience get (hear, see, feel) that harmonic balance from the collaboration of parts. It’s a balanced joining of complementary parts.

In the early 1700’s, the Swiss mathematician Leonhard Euler was kicking mathematical butt all over the place (well, mainly in St. Petersburg and then in Berlin); he’s one of the most accomplished mathematicians in history and he spent some time focused on the mathematics behind consonance and dissonance and their relevant ratios. The arithmetic function that he developed inserted fractions that represent ratios of musical intervals – that distance in pitch between tones, that distance between one note to another note in a typical 12 note scale. These are the combination of notes that make up a chord.

With that, Euler was able to create a formula for a numeric value that could be attached to different types of chords, like an octave (value = 2) or a major third (value = 7), or a minor third (value = 8), etc. Euler called this the “degree of agreeableness”, where the smaller the numeric value, the more pleasing the chord sounded.

Harmony is a very exacting – and pleasing – combination, a combination that needs to maintain its balance and be in accordance with all the other parts in that combo.

When you’ve been a member of a team, in sports or in the workplace, the words harmony and sync may have been used interchangeably. Though the outcome of each seems to be the same – a collective of parts coming together to create one ‘thing’ – harmony and sync are actually not synonyms, and that difference is actually an important distinction.

Harmony is intentionally created and it is composed of different, though perhaps similar, parts. Sync on the other hand is not deliberately created, it appears to be more spontaneously created, and there’s an exacting rhythm to its creation, and when sync shows up all signs of individuality goes away…AND you can bet your sweet bippy that I’ll be writing a math essay in a few weeks that focuses on sync.

Now, take into account that harmony is about balance – that then means that it is totally knock-overable, be it by a strong wind or a bee landing on the nose of the baritone in the aforementioned barbershop quartet which then causes them to become distracted and their vocals either go out of key or stop completely as they swat and/or yelp that bee away. 

On a team that is considered to be harmonic, if one member leaves and is replaced by someone else who moves in a different way, someone whose vibration is not the same as that previous team member, then harmony has now turned into dissonance.

So how do you preserve harmony?

Well, it needs almost constant attention to maintain its existence. It definitely needs entire group coordination. On rare occasions, one individual might be able to facilitate harmony into existence – meaning, an entity that is not an active part of the harmony – but that’d be an exhaustive and never-ending chore for one person to keep alive and kicking. 

In day to day life, I strive for harmony over sync when working with humans in a group environment. I love that harmony allows for individuality to not only exist but it’s actually a requirement in the first place. It’s that individuality that needs to be very present in order for complete balance of the whole to be achieved. 

A few years back there was some hoopla in the business world around harmony in the workplace, most notably in the Harvard Business Review article aptly titled “Too Much Harmony Can Kill Creativity.” Basically, the word harmony was being presented as a negative, and it was associated with giver-upper compliance. They summarized that if you – as a leader – opined that there was harmony within your team and their work, that that actually meant that there was a suppression of varying ideas and voices, and that leadership needed to “fight harmony, inject some tension into your teams and organizations, and embrace a moderate amount of conflict.”

Here’s how I saw that missive play out IRL: A large portion of my client base are founders of start-ups, and letmetellya, there was a period of time that I involuntarily twitched every time someone said they wanted to be disruptive, that they wanted to DISRUPT the norm of whatever industry they were looking to conquer. Let’s shake things up! Patooey to harmony!

And I’d always respond to those declarations with my own factoids, specifically calling out that one of the existing components of harmony is tension. Just like you can’t get to a rainbow without experiencing the rain, there’s a requirement for tension to be present on the path to harmony. It’s an intentional tension. Just ask Euler (and Pythagoras , and Oresme, and we can’t forget Fourier and all the other mathematicians who studied harmony).

Think of how this presents in the musical world: as with any string instrument, like a violin or a guitar, tension is produced in a variety of ways depending upon where the bridge that elevates the strings above the fret is placed or where a finger can be applied to a string.

With my clients, I’d welcome & encourage their search for tension in the quest to reach harmony while also stressing the point that disrupting for the sake of proving that you can disrupt isn’t helpful, it just creates dissonance and, quite honestly, makes the instigator look like they are making a weird power move. That type of disrupter is now pushing that bottom tile – the one I referenced earlier – that tile that represents trust between the parts.

If your team’s goal is to actually explore dissonance, change up the points of tension, and use Euler’s formula and try out different chords, then that’s fabulous.

Just keep this in mind: to harmonically reach your team’s common end goal, you can’t alter just one of the chords, one of the ingredients, and then not touch any of the other parts. You need to reach that “degree of agreeableness”.

As with anything where perfect balance is a key element, the amount of possibly disruptive variables seems to multiply exponentially over time. It’s not always realistic to expect harmony to be sustainable, just as you can’t expect all of your peers to forevermore be on your team or to stay at your workplace. It’s quite likely that the amount of pieces (individuals, objects) increase or expand. Luckily, there’s a little something called modulation which, if it’s not used in the extreme, might be the path to take if you’re needing to reform what’s headed into dissonance.

In Simone Weil’s essay, The Pythagorean Doctrine, which can be read in this collection of her essays, she writes:

Harmony is proportion. It is also the unity of contraries.

The Pythagorean Doctrine by Simone Weil
~pg. 198

Well. I couldn’t have summed that up better myself. Wish you were still alive, Simone, it’d be wild to get to have a conversation with you.

The only thing that I think is weird is that for something so nice, so pleasing (to me) — why is the word harm the first four letters of harmony? 🤓 Hullo, can I get an etymologist to explain this to me, please?

SO, here’s a recap of some of the questions from this essay (the ones that aren’t rhetorical). If you have any answers or related info, please do comment below.

  1. Who doesn’t like harmony?
  2. Who here has a misophonia for harmony?
  3. Is it possible to have harmony as a repeatable pattern that essentially interlocks with itself over and over and over and etc? [this question is from one of the additional info asterisk]
  4. Why is the word harm the first four letters of harmony?

Thanks, math. You’re the best.

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