Where we learn that it’s all about being super.
You can read – or listen to me read to you – the first essay on harmony: Let’s Get Into Harmony
Let me set the stage for the way that harmony is everywhere because of progression.
A mathematical progression is a series of quantities that get added together in a predictable pattern, have an ongoing correlation, and then that series ends at some defined destination. Or, it goes on and on into infinity, ever growing and getting bigger and maybe even stronger, never ending. But it sure is moving the whole time cuz it’s, you know, progressing.
As I put in Let’s Get Into Harmony, “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.” What puts the harmonic in harmonic progression is the ongoing specific measurement (aka ratio) of how those parts in that series are in relation to each other.
So yeah, surprise surprise, in harmony
it’s all about the steady relationship between the parts.
Can you imagine watching a film without a plot or series of related events and just see random scenes? So, the script writer has to ensure a series of related events or sequences of scenes in which the film would be created so that it makes sense to the audience. The harmonic formulae can also be used by scientists to conclude the value of their experiments. For example, to establish the degree at which water boils each time the temperature is changed with the same value. It is also used in the music industry to establish theories on sounds and to closely study them.
CueMath – Harmonic Progression in real life
The math behind harmony and its presence in music reflects all the way back into Pythagorean times.
[The Pythagoreans] showed that music provided a bridge between quantity and quality, between mathematics – measurable aspects of music – and subjective experience. Musical intervals could be both heard consciously and expressed mathematically. “Science and Spiritual Practices: Transformative Experiences and Their Effects on Our Bodies, Brains, and Health” by Rupert Sheldrake
(pg. 126)
The things that have been found on the path of harmonic studies show up in our everyday world, but they also stretch all the way up into the cosmos. Over four centuries ago(!) mathematician Johannes Kepler wrote a book of his studies on planetary motion(!!), called Harmonices Mundi.
…Kepler, in later life, became convinced that the planets play music along their paths. In his 1619 book Harmony of the World he derived the planet’s tunes and concluded that “the Earth sings Mi-Fa-Mi.”
“Lost in Math: How Beauty Leads Physics Astray” by Sabine Hossenfelder
(pg. 19)
You know, during those pandemic times, when I was searching for any and all math resources of info online, hoping for a lecture vs a class, I found a workshop series on Distributed Solutions to Complex Societal Problems from IMSI, and it was freeeee. It was mid-2021 and I was reaching out to my go-to arena of making sense of all the things (aka, math) and just the title of that workshop energized me. Alas, the lectures were waywayway over my head so I ended up only paying half-attention to it, but I attended all of them because they did fulfill some sort of input-of-new-information need that I had. And actually, one of my takeaways from this class was that there were – and are – so many prominent French mathematicians out there – who knew?
Jean-Baptiste Joseph Fourier was a mathematician quite prominent at the turn of the 19th century with his work within harmonic analysis, along with mathematician Augustin-Louis Cauchy and his work in wave theory. But before Fourier and Cauchy, there was Pierre Simon Laplace. A portion of his work was in celestial and terrestrial mechanics (!!!), achievements “that foreshadowed much of the important French work in applied mathematics.”
Can you even imagine – in fact, yes, let’s picture this together – let’s think about our world back in the late 18th century, and particularly in France as it was on the precipice of a flippin’ revolution. And here was this Laplace guy, yapping about this thing, “celestial mechanics,” something he actually coined the term for. I find this to be W I L D.
For Laplace, celestial mechanics and probability were complimentary instruments that implemented a unified vision of a fully determined universe. Celestial mechanics vindicated the Newtonian system of the world. Probability was the measure, not of the operations of chance in nature, for there are none, but of human ignorance of causes, which was to be reduced to virtual certainty by calculation.
“The Princeton Companion to Mathematics”
(pg. 753)
As I learned when I was attending those online workshops during that most recent pandemy, the mathematicians of modern day France – as well as nearby Belgium – continue to do significant work in harmonics and the closely related field of wavelet theory. From Belgian mathematician Dr. Ingrid Daubechies, who just became the Wolf Prize Laureate in Mathematics 2023, we now have the “Daubechies’ wavelets”.
Daubechies has also made unparalleled contributions to developing real-world applications of harmonic analysis, introducing sophisticated image-processing techniques to fields ranging from art to evolutionary biology and beyond.
From an announcement from The Wolf Foundation
Daubechies’s most important contribution is her introduction in 1988 of smooth compactly supported orthonormal wavelet bases. These bases revolutionized signal processing, leading to highly efficient methods for digitizing, storing, compressing, and analyzing data, such as audio and video signals, computed tomography, and magnetic resonance imaging. The compact support of these wavelets made it possible to digitize a signal in time linearly dependent on the length of the signal.
Though what is known as “wavelet theory” was further developed in just the past 40 years, you can totally trace it back a hundred years ago in the study of sound waves, the frequencies that they emit, and their mathematical decomposition, all of which is known as the study of harmonic analysis.
Oooh, wait, do you know what a superposition is?
It’s, well, I gotta say – it really is super, in the truest sense of that word.
I mentioned the French mathematician Fourier just a few sentences back and he put together a little something called the Fourier series, the sum of an infinite series of the expansion of a signal’s repeating pattern and which are mathematical representations of wave functions, known musically as sound waves.
Whenever I fall into this thrum reading about sound waves and harmonic analysis, I can’t help but think of my love for those other waves – yes, indeed, the ones in the ocean. There’s a harmony that exists within the ocean as well, dontcha think? Sure, it’s a harmony that is much more fleeting, it isn’t’ constant cuz, well, it’s ever changing and moving and as you (of all people!) well know by now – harmony is all about a balance of individual parts, parts that have a precise ratio of their expanse to each other. When there’s a rhythmic swell of those waves, when you’re physically in it- maybe you’re body surfing or you could be laying on your back and in total trust of the role that the tide is playing – it can feel very peacefully harmonic.
BUT but but, here’s the ‘super’ thing of all of this. The Fourier series is not just any ol’ series of wave functions marching along in a sequence, nuh uh. The Fourier series is a superposition of wave functions, a gorgeous harmony of sound wave frequencies.
The idea is that a general sound wave is a superposition of harmonics, the way the sound of a symphony is a superposition of the harmonics corresponding to the notes played by various instruments.
“Love and Math: The Heart of Hidden Reality” by Edward Frenkel
(pg. 80)
So there we are, doing that whole harmonic analysis thing, studying those sound waves and their frequencies and how those sound waves then affect each other when they bump up on each other… much like those ocean waves looking for that balance of all its elements. And then that moment happens, that Fourier moment, where all the elements are trusting each other, and it’s friggin’ super, yo. It’s top level super, in fact. It’s a superposition, a glorious series of multiple harmonic sequences, where each of the many participants are – somehow – not limiting nor disrupting and certainly not damaging any of the others.
Phew. I gotta take a knee for a moment, and just revel in all that superness.
No doubt this experience of connection and unity is a major reason for the use of chanting and singing in practically all traditional societies, communities, and religions. And it is probably one of the main reasons why so many people join church or community choirs in the modern world. These are voluntary activities, and people would not take part unless they derived some benefit. Indeed, scientific surveys of people who sang in choirs found that most said singing together made them feel better and contributed to their mental and emotional well-being.
“Science and Spiritual Practices: Transformative Experiences and Their Effects on Our Bodies, Brains, and Health” by Rupert Sheldrake
(pg. 255)
See! If singing together has a proven physiological benefit, I can barely comprehend (and would love to
know how to mathematically calculate) when that singing reaches the status of harmonic balance.
My tattoo of a harmonic progression is prominent; I am looking at it as I type, right there on my forearm reaching up onto the top of my hand. It’s a constant reminder of what can be possible if all involved parts can trust what sometimes might feel like and even, at first, look like the defiance of gravity. The key to that balance is all about centering the gravity.
I so wholeheartedly believe that harmony – true mathematical harmony – is the ultimate stance in life to always be striving for. It’s comforting to confirm that this concept has been thought about, written about, and shared for so many centuries throughout so many cultures.
Yin and yang are universal symbols of harmony… Yang is about domination. Yin is about collaboration… The old script oozes yang. It has left yin out of the narrative. Today we urgently need to yin our yang and get our harmony back…Yin and yang ground us in our humanity, in equal measure. Both are essential to thrive in flux.
“Flux: 8 Superpowers For Thriving in Constant Change” by April Rinne
(pg. 167 & 170)
You’ve certainly heard some variation of the adage, “Let’s not just survive, let’s thrive.” I know that you can do that, as can I, by harnessing our individual keys of strength, monitoring the frequencies that we each emit from those individual keys, anticipating how to measure our complimentary ratios related to each other based on probability factors from humans acting like humans do, and forming our own community-based harmonic groupings so that we can all come together in one great superposition of harmonic balance.
Thanks math, you’re the best.