Challenge: Explain a musical concept through sound and visualization only, without words.
Think of this as the “Myst” school of music. Remember it? A wonderful and expansive computer game from 1993 that featured no people and no words: just vibes. It was quiet, introspective, and beautiful.
Half the gameplay was just wandering around wondering what you’re supposed to be doing. It was immensely pleasurable, and introduced me to the autodidact's motto:
Sometimes identifying the puzzle… is the puzzle1.
Primer
This is a follow-up to the video I posted last week, titled "⬛️ 🟦 🟥 🟩 🟪"
That video consisted of four minutes of pulses and tones accompanied only by the above emojis, an oscilloscope, and two mathematical operators (a colon and a slash).
This was an attempt at the above “no words, just vibes” challenge (in this case with literal vibes). As for the concept to explain, I had given myself a specific assignment:
Assignment:
- Demonstrate how prime numbers relate to pulse and tone.
- Use three of them to build a familiar scale from scratch.
I deliberately left the text of the post blank. Because after waxing poetic for two thousand words on the joy of following an autodidact's path, I thought it might be interesting to give some space for that to maybe happen naturally.
Curious if anyone tried to solve “the puzzle of identifying the puzzle?” I imagine that musicians trained in music theory may have got the gist, but I’m particularly curious if any non-musicians were able to make sense of it.
I did receive one comment that it all looked "like Tetris on acid," which made me very happy and is, I think, one type of solution to the puzzle.
Now that I've revealed the assignment, I'll share my own process and result.
Step one: if “words” are not allowed, am I using a language?
A Language For Music
For this assignment (and in my like, life), I need a written musical language of symbols. Hopefully it will meet all the criteria of a real language: it will contain vocabulary, grammar, syntax. It will have users (past or present), and the capacity for narrative.
Additionally, I’m imagining a language with two qualities that most spoken languages don’t have:
It is international — the meaning of its symbols can be understood by anyone, specific to no particular culture, geographical region, or user group.
It is universal — it can be used with any music, from any time or place.
Music itself is international and universal, and I’m looking for a language to match.
Bonus: A logic-based language with a built-in pedagogical progression would really help with this assignment. I want to start from scratch and gradually build to more complexity.
I want a language that can teach music through immersion. Demonstrations could be followed without translation — only direct correlation between symbol and meaning.
Lessons in this language would take the form of a narrative, explaining the science of music while simultaneously explaining the language itself.
There is a language that I strongly suspect will fit the bill:
Language: Math
Mathematics is a written language — it contains a vocabulary of symbols and visual elements that have meaning. It contains grammar, syntax, users, and narrative.
It is perfect for this particular challenge. They say "music is math," and they are right. I think of math as the "machine code" underneath the science of music — the lowest level programming language containing all its building blocks.
Math is unquestionably an international language. It is non-verbal — "spoken" only through translation. A Japanese speaker and a French speaker will read an equation out loud in two completely different ways, and yet both take away exactly the same meaning. I want a language for music that operates this same way.
As to whether math is a universal language (applicable to music of any time or place), I propose that it is. I strongly believe you can understand any music by stripping it down to its maths. But I realize I'm going to have to prove this to myself through experimentation. This assignment is one of those experiments.
Math is logic-based, and pedagogical. It has the ability to teach itself… and even prove itself! It is beautiful and elegant.
It can be dry, though, especially in the particular ways it applies to music. It would be nice to have some extra vocabulary to make it a little more approachable, efficient, eye-pleasing, and fun:
Language: Feelings 🙂
Readers of this publication know that I hold emojis in high regard. I find they take the edge off of dry or dense concepts. They can certainly be overdone, but used thoughtfully, they can help consolidate information and make the whole text friendlier and easier to "see."
Math can be quite intimidating on the page, especially for musicians — I imagine seasoning it with characters that add flavor and help visually organize. Why always use "xyz" when we have "🍓🦠👽" available to us? (I'm kidding. Sort of!)
Emoji is not a language. It is an ever-expanding vocabulary that can augment a language. And it has an impressive ability to communicate actions, impressions, and feelings — without words.
What about "Music Theory"?
Music theory is also a language. It has an extensive vocabulary of symbols and visual elements. Some of its symbols did originate as letters or words in various spoken languages, but their use in music theory has separated them from their mother tongues and repurposed their meanings.
Music theory has grammar and syntax, and can be used to communicate narrative and demonstrate concepts. It is especially good at giving instructions: it can tell a musician what buttons to push on a clarinet, and precisely when.
However: by definition, languages also have users (either past or present). And for me, this is where music theory gets wrapped in quotation marks and turned into "music theory." Because what today is widely taught as "music theory" is neither international or universal. It evolved organically over the past several hundred years for a subset of users, with a subset of musical applications. In fact:
A full realized, universal, and international language of music theory is an unrealized dream.
What we do have is quote, unquote "music theory." Like an indispensable but bloated piece of software, gradually versioning itself up year after year, we use workarounds and hacks in an attempt to use it with broader and broader musical applications — rarely addressing the design of its operating system or the limitations of the underlying code.
This is the written language of music as taught in piano method books, at the Juilliard School, and by Rick Beato. It is the subject of doctoral theses and orchestration textbooks. It's great for writing about 19th century romantic orchestral music, and okay for writing about 20th century pop music.
J.S. Bach and Scott Joplin broke all sorts of its rules. Jacob Collier eats it on toast three meals a day. Harry Partch put it out with the trash, and Billy Corgan straight up ignored it.
Arnold Schoenberg even thought that if he rewrote the operating system, he could use the software to program the user.
In summary: I could certainly try to use Photoshop to write a novel, and I could try to use "music theory" to complete this assignment. I might even succeed. But it wasn’t designed for the job.
For this assignment I went with pure mathematics, and when useful and appropriate, I peppered in some emojis.
If this is math, where are all the numerals? 🔢
I've chosen to use numerals to represent discreet speeds. For example, "60 beats per minute" or "200 Hz." Numbers in numeral form will express how fast the air is vibrating, as a specific value.
▶️ When that rate is slow enough (e.g. at 120 bpm), you hear each pulse.
⏩ When the rate is fast enough (e.g. at 262 Hz), you hear a tone.
Although our sensation of tone and our sensation of pulse feel and sound like different phenomena, the nature of the vibrations are exactly the same: they’re just moving at different speeds.
🧬 I'm using an oscilloscope to demonstrate this. It visualizes the air pressure vertically, and time horizontally. Here’s a clip of a pulse “bending up” into a tone in the oscilloscope:
Reference rates exist (but we don't need to care)
Now that I've explained how I'm using numerals in this video, you might have noticed: There are no numerals in the video. 🙄
That's because for this assignment I only need to demonstrate musical concepts in the abstract, and thus every specific rate that occurs along the way need not concern us (apart from maybe the first one).
The method for staying in the abstract is to set two underlying reference rates, and then let them go unnamed:
⏩ - a reference rate for tone, which is faster (in Hz)
▶️ - a reference rate for pulse, which is slower (in bpm)
Now all symbols used can be in service of demonstrating relationships.
And in the language of mathematics, musical relationships are built with prime numbers and take the form of rational numbers.
🟦 🟥 🟩 Music in 3D
My assignment is to introduce prime numbers as building blocks of pulse and tone. Although there are an infinite number of primes, you can analyze 99.9% of existing music from all of human history with just the first five.
And I'm only going to use three!
I could easily just use the numerals (2, 3, and 5), but I don't want to get them confused with any discreet values. These are meant to be numbers in the abstract. Also, I find that primes can be much more approachable and easier to visually "see" as unique symbols, anyway.
You only need a handful. In this case I went with color2 emojis3:
The first three primes:
🟦 = 2
🟥 = 3
🟩 = 5
Products of these primes can now be used to represent any integer (says the Fundamental Theorem of Arithmetic)4
And integers can be used to represent any rational number (says some person on Wikipedia)5
And if multiplication and division are the primary mathematical operations I'll be performing on primes, representing them as emojis is even more efficient than numerals — I can just string them together to multiply them (without using the x operator).
🟦🟦 = 4
🟩🟥🟦 = 30
This makes it crystal clear that when we're seeing the number "30," we're seeing a collection of prime factors: 5 x 3 x 2. Because for musical applications, seeing the prime factors is often more useful than knowing what number they multiply out to.
And it looks pretty.
Later on, primes get really interesting: they start to look like different musical dimensions. For example, music constructed using only multiples of these three primes could be seen as three-dimensional music, and visualized as such.
How to use them and what to call them
Adjacent primes indicate they are multiplied (like variables in math):
🟥🟦 = 3 x 2 = 6.
A slash "/" indicates division, but is usually left in ratio form:
🟦🟦 / 🟥 = “4/3”
A ratio is a name — it represents one particular pulse or tone.
"4/3" is a name for the pulse four thirds as fast as 1/1.
”6/1,” or just “6” is the name of a pulse six times as fast as 1/1.
A colon indicates an interval — the distance between two or more ratios.
“3/2 : 4/3” is the interval between the ratios named 3/2 and 4/3.
(🟥 / 🟦) : (🟦🟦 / 🟥) = 3/2 : 4/3
The colon notation is especially useful when showing relationships between three or more integers or ratios, which comes up frequently in discussion of the harmonic series:
🟦 : 🟥 : 🟩 : 🟥🟦 = 2 : 3 : 5 : 6
Glossary/Layout
▶️ = Reference Rate for Pulses
aka "tempos" or "rhythms." (Larger symbols used)
⏩ = Reference Rate for Tones
aka "pitches." (Smaller symbols used)
The upper half of the screen represents the sensation of tone, and holds the scale tones as we collect them:
The lower left of the screen represents the sensation of pulse. The lower right contains the oscilloscope which allows us to see the individual pulses, and the waveforms of the tones:
"Bend Up"
Represents an increase in rate from pulse to tone.
A symbol moves from lower to upper half of the screen.
"Bend Down"
Represents a decrease in rate from tone to pulse.
A symbol moves from upper half of screen to lower.
⬛️ = 1 (non-prime)
Pulse: Quarter note aka the “tempo.”
Pitch: Unison — aka “fundamental” or “1st partial.”
🟦 = 2 (prime)
Rhythm: Eighth note
Pitch: One octave — aka “2nd partial.”
🟥 = 3 (prime)
Rhythm: Eighth note triplet
Pitch: One octave + a just perfect fifth — aka “3rd partial.”
🟩 = 5 (prime)
Rhythm: Quintuplet
Pitch: Two octaves plus a just major third — aka “5th partial.”
▶️ Push Play
⬛️ = 1
🟦 = 2
🟥 = 3
🟦🟦 = 4
🟩 = 5
🟥🟦 = 6
🟪 = …7?
Thanks so much for reading. You can subscribe for free:
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You can also buy me a coffee if you enjoyed this one.
Also worth mentioning: the experience of trying to make sense of all those abstract photos and color coded pie charts in the album art to Aphex Twin's Selected Ambient Works Vol II.
The Aphex fandom got on this one real quick: turns out that each image corresponded to a track on the record, serving as its wordless title. The cryptic attached diagrams gave the individual track lengths, without using any numbers.
My use of colors to represent numbers doesn’t imply any connection to color theory — this isn’t about mixing colors, or any correspondence to the visual frequency spectrum. That’s an interesting idea to explore, but not part of my inquiries, and I haven't personally seen any connection.
Before I settled on blocks of color for primes, I played with using other emojis. While the results are definitely more fun, they are a bit too visually confusing to be useful, and a little… cute. But I did like the idea, and may use it in the future.
🙂 = 2
🤠 = 3
😬 = 5
🫨 = 7
Every integer greater than 1 is either a prime or the product of some combination of primes.
- The Fundamental Theorem of Arithmetic.
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator. Every integer is a rational number.
- Some person from Wikipedia