Episode 6: The Hidden Mathematics of Music | Dormant Knowledge Sleep Podcast

In this episode of Dormant Knowledge, the educational sleep podcast for curious minds, Deb uncovers the hidden mathematical world that shapes every piece of music you've ever heard—the fascinating realm of tuning systems.

Episode 6: The Hidden Mathematics of Music | Dormant Knowledge Sleep Podcast

Host: Deb
Duration: ~66 minutes
Release Date: September 15, 2026
Episode Topics: Mathematical music theory, tuning systems, cultural variations in pitch organization

Episode Summary

Why does a piano sound different from a violin playing the same note? How do we decide what sounds "right" and what sounds "wrong"? In this episode of Dormant Knowledge, the educational sleep podcast for curious minds, Deb uncovers the hidden mathematical world that shapes every piece of music you've ever heard—the fascinating realm of tuning systems.

Journey back 2,500 years to ancient Greece, where Pythagoras discovered that simple mathematical ratios create beautiful musical intervals. But his "perfect" system contained a fundamental flaw: it was mathematically impossible to make every interval sound harmonious in every key. This discovery launched centuries of creative problem-solving, cultural innovation, and ultimately led to the "great compromise" that became our modern musical standard.

Explore how different cultures approached the same mathematical puzzle in radically different ways. Arabic music embraced quarter-tones that don't exist on Western pianos. Indian classical music developed twenty-two microtonal intervals per octave. Indonesian gamelan orchestras created tuning systems so unique that each ensemble became its own musical universe, with instruments that couldn't be interchanged between groups.

Discover the "keyboard crisis" that drove European instrument makers to desperate measures—organs with nineteen keys per octave, split black keys, and other musical nightmares designed to accommodate multiple tuning systems. Follow the story of how Johann Sebastian Bach's "Well-Tempered Clavier" helped convince the musical world to accept mathematical imperfection in exchange for unprecedented compositional freedom.

This episode reveals how colonial expansion and recording technology spread equal temperament globally, nearly erasing centuries of microtonal traditions, and how digital music technology is now creating a renaissance of alternative tuning systems that would have been impossible to implement before computers.

What You'll Learn

  • Discover how Pythagoras uncovered the mathematical basis of musical harmony and why his "perfect" system was ultimately unworkable
  • Learn about the "Pythagorean comma"—a tiny mathematical discrepancy that plagued musicians for over a thousand years
  • Explore how Arabic, Indian, Chinese, and Indonesian musical traditions developed completely different solutions to the same tuning problems
  • Understand why European keyboard instruments drove the adoption of equal temperament and how Bach's compositions demonstrated its possibilities
  • Learn how the human voice naturally gravitates toward pure mathematical ratios, regardless of cultural tuning systems
  • Discover how recording technology and mass production favored standardized tuning over traditional microtonal systems
  • Explore the modern "xenharmonic" movement and how digital technology enables impossible tuning experiments
  • Understand the psychological and physiological research on how different tuning systems affect listeners

Episode Transcript

[Soft ambient music fades in]
Deb: Welcome to Dormant Knowledge. I'm your host, Deb, and this is the podcast where you'll learn something fascinating while gently drifting off to sleep. Our goal is simple: to share interesting stories and ideas in a way that's engaging enough to capture your attention, but delivered at a pace that helps your mind relax and unwind. Whether you make it to the end or drift away somewhere in the middle, you'll hopefully absorb some knowledge along the way.
We have really enjoyed working on this podcast so far, and we hope to keep it going with your support. You can find us at dormantknowledge.com or follow us on social media @dormantknowledge on Instagram and Facebook, or @drmnt_knowledge—that's d-r-m-n-t-underscore-knowledge—on X.
Tonight, we're exploring something that touches every piece of music you've ever heard, yet remains largely invisible to most of us. We're diving into the hidden mathematics of music—specifically, the fascinating world of tuning systems. How do we decide what sounds "right" and what sounds "wrong"? Why does a piano sound different from a violin playing the same note? And what happens when different cultures approach these questions in completely different ways?
[Music fades out]
So settle in, get comfortable, and let's begin our journey into the mathematical mysteries that shape every melody, every chord, and every song you've ever loved.
[Sound of papers shuffling]
You know, it's interesting... most of us take for granted that when we sit down at a piano, the notes just... work together. We don't think about the centuries of mathematical wrestling matches that went into making that possible. But actually, um, there's this whole hidden world of compromises and approximations happening every time you play a chord.
Tonight we're going to explore how humans have tried to solve what's essentially an impossible mathematical puzzle—and how different cultures came up with surprisingly different solutions.
Section 1: The Ancient Mystery
So let's start about 2,500 years ago, with a guy you've probably heard of but maybe not in this context. Pythagoras—yes, that Pythagoras, the one with the triangle theorem—was apparently the first person in Western history to really dig into the mathematics of music.
[Yawns softly]
Now, the story goes... well, actually, like a lot of ancient stories, we're not entirely sure how much of this is true versus how much is legend. But the traditional account is that Pythagoras was walking past a blacksmith's shop and noticed that different sized hammers made different pitched sounds when they hit the anvil. And being, you know, a mathematician, he got curious about whether there was a pattern.
So he goes home—or back to his school, I guess—and starts experimenting. He takes a string, like a simple musical string, and he figures out that if you press down exactly halfway along the string, you get a note that's an octave higher. The relationship is one to two. One part of the string to two parts of the string. And that octave... it sounds like the same note, just higher.
Then he discovers that if you press down at one-third of the string length, you get what we now call a perfect fifth. That's the interval between, say, C and G. And if you press at one-quarter, you get another octave. Three-quarters gives you a perfect fourth.
[Sound of chair creaking]
What fascinated Pythagoras was that these simple numerical ratios—one to two, two to three, three to four—produced what sounded like the most beautiful, most harmonious musical intervals. It was as if mathematics and beauty were the same thing. Which, for Pythagoras, they actually were. He believed that numbers were the fundamental reality underlying everything in the universe.
His followers, the Pythagoreans, developed this into a whole cosmology. They talked about the "music of the spheres"—the idea that the planets themselves created musical harmonies as they moved through space, based on their mathematical relationships to each other. It sounds mystical to us now, but you have to remember, they'd just discovered that mathematical ratios literally create musical beauty. It must have felt like they'd uncovered a secret of the universe.
So Pythagoras creates what we now call "Pythagorean tuning," based entirely on perfect mathematical ratios. And for a while, this works pretty well. You can build scales, you can make music, everything sounds... well, mostly good.
[Pauses, sound of drinking water]
But here's where things get interesting. And by interesting, I mean mathematically problematic in a way that's going to keep musicians and mathematicians busy for the next two thousand years.
See, when you start trying to build a complete musical system using only perfect mathematical ratios, you run into what's called the "Pythagorean comma." Now, comma in music doesn't mean punctuation—it refers to tiny intervals, tiny differences in pitch.
Here's the problem: if you start with a note, let's say C, and you build up a scale using only perfect fifths—the two-to-three ratio—you'd expect that after going around twelve fifths, you'd end up back at C, just higher up. Twelve perfect fifths should equal seven octaves, mathematically.
But they don't.
[Papers rustling]
You end up about a quarter of a semitone sharp. That tiny difference? That's the Pythagorean comma. And it means that if you tune everything to perfect mathematical ratios, some intervals are going to sound perfectly beautiful, and others are going to sound... well, pretty awful. Musicians called these bad intervals "wolf intervals" because they howled like wolves.
So you've got this beautiful mathematical system that works great in theory, but in practice, if you're a musician trying to play in different keys, some of your music sounds gorgeous and some sounds like... well, like wolves howling.
[Soft laugh]
And this is where our story really begins, because this problem—this tension between mathematical perfection and musical practicality—is going to drive innovation for centuries. Different cultures, different eras, different types of instruments, all grappling with the same fundamental question: how do you make the math work with the music?
Section 2: The Problem with Perfection
Now, you might be wondering, well, why not just... fix it? If you know where the problem is, can't you just adjust the system slightly?
And that's exactly what musicians tried to do, for about a thousand years. They came up with all sorts of ingenious solutions. Some of them would tune certain notes slightly differently depending on what key they were playing in. Others would build instruments with extra strings or extra keys to handle the problematic intervals.
[Sound of paper turning]
But here's the thing—and this is where the mathematics gets really interesting—you can't actually fix it. Not completely. It's mathematically impossible to create a tuning system where every interval is both mathematically perfect and musically useful in every key.
This is because music exists in what mathematicians call "circular space." You expect that if you go up twelve semitones, you'll be back where you started, just an octave higher. But mathematical ratios exist in "linear space"—they just keep going up forever. The circle doesn't quite close.
It's like trying to fit a square peg in a round hole, except the peg and the hole are both made of pure mathematics, so you can't just sand down the edges.
[Yawns]
Medieval musicians dealt with this by basically avoiding the problem. They used modes—scales that didn't require you to play in every possible key. If you're only using, say, six or seven different notes in your entire musical system, you can tune those six or seven notes to sound really good together, and just... not worry about the rest.
But as music got more complex, as composers wanted to explore different keys and more sophisticated harmonies, this became less and less viable. You needed a system that could handle any key, any chord progression, any musical idea a composer might dream up.
And that's where things get really creative, because different cultures around the world approached this problem in radically different ways.
Actually, let me give you an example of just how different these approaches could be...
[Papers shuffling]
In Arabic music theory, they embraced the fact that not all intervals are the same. Instead of trying to force everything into twelve equal steps, they developed a system called "maqam"—m-a-q-a-m—that uses twenty-four different intervals per octave, including quarter-tones. Quarter-tones are intervals that fall exactly between the notes on a Western piano.
To Western ears, this can sound "out of tune," but that's just because we're culturally conditioned to expect equal temperament. In Arabic musical tradition, those quarter-tones aren't deviations from correct tuning—they're essential emotional colors that allow for much more subtle musical expression.
Indian classical music took yet another approach. They developed a system with twenty-two different intervals called "shrutis"—s-h-r-u-t-i-s. But here's the really interesting part: Indian musicians don't think of these as fixed pitches the way we think of piano keys. They're more like... guideposts, or landmarks. A skilled Indian classical musician can bend and shape these intervals depending on the emotional content they want to convey.
[Sound of settling into chair]
So while Western music was wrestling with how to make mathematics serve musical needs, other cultures were asking completely different questions. Some were asking: what if we embrace mathematical complexity instead of trying to simplify it? Others were asking: what if pitch is fluid rather than fixed?
These aren't just academic differences. They create completely different musical worlds, different emotional possibilities, different ways of thinking about what music can be and do.
[Soft ambient music begins to fade in]
Deb: I'm going to take a quick break here. When we come back, we'll dive into how Western instrument makers and mathematicians spent centuries trying to solve the tuning puzzle, and why the solution they eventually settled on was both brilliant and... well, a bit of a compromise.
[Music plays for transition]
Deb: Welcome back to Dormant Knowledge...
[Music fades out]
Section 3: The Instrument Revolution
So we left off with this mathematical puzzle that was driving musicians and instrument makers crazy for centuries. And the thing is, different instruments made the problem worse in different ways.
[Papers rustling]
Take keyboard instruments, for example. When you build a harpsichord or a pipe organ, you're essentially locking in your tuning choices. Each key corresponds to a specific string or pipe, and once it's built, that's it. You can't bend the pitch the way a violin player can adjust their intonation by moving their finger slightly.
This created what we might call the "keyboard crisis" in European music. See, if you're a lute player or a singer, you can adjust your tuning slightly for different pieces of music. You can make your perfect fifths truly perfect for one song, then adjust slightly for another song that uses different intervals.
But if you're an organ builder in the 1400s, you need to make a single decision that's going to affect every piece of music played on that instrument for the next couple hundred years.
[Yawns softly]
And organs were... well, they were huge investments. We're talking about instruments that took years to build, cost fortunes, and were expected to last centuries. Some of the organs built in the Renaissance are still being played today. So organ builders couldn't just experiment freely—they needed to pick a tuning system and stick with it.
This led to some fascinating regional variations. Different areas of Europe developed their own approaches to the tuning problem. Some organ builders would create keyboards with extra keys—black keys split into two different pitches, or extra notes squeezed in between the usual twelve.
I've seen pictures of these keyboards, and they look... well, they look like musical nightmares, honestly. [Soft laugh] Some of them have nineteen keys per octave instead of twelve. Imagine trying to play Bach on one of those.
But string instruments presented the opposite problem. A violin player, or a cellist, they're constantly making tiny pitch adjustments. It's not even conscious—their fingers naturally drift toward what sounds most in tune in the musical context. A good string player will tune their perfect fifths to be truly perfect, not equal-tempered approximations.
[Sound of chair shifting]
This is why, even today, orchestras sound different from pianos playing the same music. The string section is constantly micro-adjusting their intonation to create perfectly tuned intervals, while the piano is locked into its equal-tempered compromises.
And then there's the human voice, which is... well, it's the most flexible instrument of all. Singers naturally gravitate toward what musicians call "just intonation"—tuning that makes each chord sound as pure and resonant as possible. Which is beautiful, but creates yet another layer of complexity when you're trying to accompany singers with fixed-pitch instruments.
So by the Renaissance, European music had this whole ecosystem of different tuning approaches depending on what instruments you were using and what kind of music you were playing. It was sophisticated, but it was also... chaotic.
Something had to give.
Section 4: The Great Compromise
Enter a German organist and music theorist named Andreas Werckmeister. Now, Werckmeister—and his name is pronounced VERK-my-ster, with that German W sound—wasn't the first person to suggest equal temperament, but he was one of the most influential advocates for what he called "well-tempered" tuning.
[Sound of paper turning]
Here's what Werckmeister proposed, sometime in the 1680s: what if we just... divided the octave into twelve exactly equal parts? Instead of trying to make some intervals perfect and accepting that others would be terrible, what if we made every interval slightly imperfect, but in a consistent, predictable way?
It's a radical idea, when you think about it. You're giving up mathematical perfection—those beautiful simple ratios that Pythagoras discovered—in exchange for practical flexibility.
In equal temperament, a perfect fifth isn't actually perfect. It's about two cents sharp of a true perfect fifth. Now, a cent is a tiny interval—there are twelve hundred cents in an octave—but it's measurable. Musicians with really good ears can hear the difference.
But here's the genius of the system: because every interval is off by the same proportional amount, you can play in any key and the relationships between notes stay exactly the same. A C major chord has the same internal structure as an F-sharp major chord, even though F-sharp major would have been unplayable in earlier tuning systems.
[Yawns]
Werckmeister published several books about this system, and... well, they're quite mathematical. He was really trying to convince musicians that this compromise was worth making. But theory is one thing—somebody needed to demonstrate that this system could create beautiful music.
Enter Johann Sebastian Bach.
Now, we're not entirely sure that Bach was specifically trying to promote equal temperament when he wrote "The Well-Tempered Clavier." The title might refer to one of Werckmeister's alternative systems, or to any number of other "well-tempered" approaches that were floating around at the time.
But what Bach definitely did was write forty-eight pieces—two preludes and fugues in every major and minor key—that demonstrated you could create gorgeous, complex music in keys that had previously been considered unplayable.
[Sound of papers shuffling]
F-sharp major? No problem. A-flat minor? Beautiful. Bach was essentially saying to the musical world: look, if you accept this tuning compromise, look at all the musical possibilities that open up.
And it worked. Within about a hundred years, equal temperament had become the standard throughout most of Europe. Not because it was mathematically perfect, but because it was musically liberating.
Of course, this raises an interesting question: what did we give up? What kind of music became impossible when we embraced equal temperament?
Well, we gave up the pure, resonant sound of truly perfect intervals. If you've ever heard a really good choir singing in just intonation—where they're tuning each chord to be mathematically perfect—there's a quality of resonance, a kind of ringing beauty, that you just can't get from equal temperament.
We also gave up some of the subtle emotional differences between different keys. In earlier tuning systems, each key had its own character, its own emotional color. C major felt different from D major, not just because it was higher, but because the internal relationships between the notes were slightly different.
Equal temperament made all keys equivalent, which opened up huge compositional possibilities, but it also flattened some of the emotional landscape of music.
[Pauses]
But here's the thing... this whole discussion assumes that Western music's approach to the tuning problem was the only possible approach. And that's where things get really interesting, because other cultures had been solving these same mathematical puzzles in completely different ways for centuries.
Section 5: Cultural Wanderings
Let's travel, in our imaginations, to medieval Islamic Spain, around the 11th century. Islamic scholars had inherited Greek music theory, including Pythagoras's work, but they took it in directions that European music never explored.
[Sound of drinking water]
Islamic music theory embraced microtones—intervals smaller than the semitones we're used to in Western music. Instead of twelve equal divisions of the octave, Islamic theorists worked with systems that included quarter-tones, creating scales with twenty-four or even more intervals per octave.
Now, this wasn't just mathematical abstraction. These microtones served real musical purposes. They allowed for much more subtle emotional expression, more nuanced ways of approaching and leaving important notes in a melody.
If you've ever heard traditional Arabic music and thought it sounded "exotic" or "different," part of what you're hearing is this expanded pitch vocabulary. Those quarter-tones create intervals that simply don't exist in Western music—emotional colors that equal temperament can't produce.
[Yawns]
Meanwhile, in India, classical musicians developed what might be the most sophisticated tuning system in the world. Indian music theory recognizes twenty-two different intervals within an octave, called shrutis. But—and this is crucial—these aren't fixed like piano keys. They're more like... territories, or neighborhoods of pitch.
A skilled Indian classical musician learns to navigate between these shrutis fluidly, choosing exactly the right microtonal inflection to convey the emotional content of the music. It's a completely different conception of what a "note" is.
Indian music also developed the concept of ragas—r-a-g-a-s—which are much more than just scales. Each raga is associated with specific times of day, emotional states, even seasons of the year. And each raga has its own subtle tuning characteristics, its own way of approaching and ornamenting particular intervals.
[Sound of settling]
Now let's hop over to Indonesia, where gamelan orchestras developed something even more radical: tuning systems that don't use octaves the way we understand them.
In Javanese gamelan music, they have two main tuning systems: slendro—s-l-e-n-d-r-o—which divides the octave into five roughly equal parts, and pelog—p-e-l-o-g—which uses seven unequal intervals. But here's the really mind-bending part: these intervals aren't standardized. Each gamelan orchestra has its own tuning, developed over generations.
So if you took a musician trained on one gamelan and asked them to play on a different gamelan, they'd have to relearn the entire pitch system. It's like... imagine if every piano was tuned slightly differently, and piano players had to adjust their muscle memory for each instrument they played.
[Pauses]
This creates a completely different relationship between musician and instrument. Instead of instruments being interchangeable tools that conform to a universal standard, each gamelan becomes a unique musical universe with its own internal logic.
Chinese classical music developed yet another approach. Traditional Chinese theory uses a five-note scale—pentatonic, meaning five tones—but the tuning of these five notes is based on a cycle of pure fifths, similar to Pythagorean tuning but without trying to force it into a twelve-note system.
What's fascinating is that this pentatonic approach actually avoids many of the mathematical problems that plagued Western music. By using fewer notes, you can keep more of them in perfect mathematical relationships without running into the comma problems.
[Sound of chair creaking]
Each of these cultural approaches reveals different assumptions about what music is supposed to do. Western music became obsessed with harmonic complexity—building elaborate chord progressions that work in every key. Islamic music prioritized melodic sophistication and emotional subtlety. Indian music emphasized spiritual and emotional transformation. Indonesian music created communal experiences where the uniqueness of each ensemble was part of the aesthetic.
None of these approaches is "right" or "wrong"—they're just different answers to the fundamental question of how to organize pitch into meaningful musical expression.
But they do sound different. Dramatically different. If you grew up with one system, music from another system might sound exotic, mysterious, beautiful, or... honestly, just strange.
[Yawns softly]
Which brings up a really profound question about music and human perception: how much of what we think sounds "natural" or "correct" is actually just cultural conditioning?
Section 6: The Science of Musical Perception
[Sound of papers shuffling]
So let's dig into what's actually happening in our brains when we hear these different tuning systems. Because it turns out, there's some fascinating research on whether musical preferences are innate or learned.
On one hand, there does seem to be some universal basis for why certain intervals sound consonant—meaning harmonious—and others sound dissonant, or tense. This goes back to the physics of sound waves.
When you play two notes together, their sound waves interact. If the frequencies are in simple mathematical ratios—like the two-to-three ratio of a perfect fifth—the waves reinforce each other in regular, predictable patterns. Your ear, and your brain, can process this regularity easily. It sounds "smooth."
But if the frequencies are in complex ratios, the waves create irregular interference patterns. Your auditory system has to work harder to process these sounds, and that extra effort registers as tension or dissonance.
[Drinking water]
So there is a physical, measurable basis for why Pythagorean ratios sound consonant. It's not just cultural—it's rooted in the mathematics of wave physics and the architecture of human hearing.
But—and this is a big but—cultural conditioning plays a huge role in what we consider "normal" or "acceptable." Western listeners who grow up with equal temperament learn to hear those slightly imperfect intervals as "correct." We stop noticing that the fifths are a bit sharp and the thirds are a bit flat.
Meanwhile, listeners who grow up with other systems develop different expectations. Indian classical musicians can hear microtonal distinctions that would be completely invisible to most Western musicians. Arabic music listeners are sensitive to quarter-tone relationships that sound like "wrong notes" to ears trained on equal temperament.
[Pauses]
There's actually been some really interesting research on this. Studies have shown that infants, before they've been exposed to much music, show preferences for simple mathematical ratios over complex ones. But by the time children are six or seven years old, they've already absorbed the tuning conventions of their musical culture.
So we're born with some universal tendencies toward consonant intervals, but we quickly learn to overlay cultural expectations on top of those tendencies.
This explains why you might hear traditional Chinese music and think it sounds "exotic," while someone who grew up with that tradition might hear Western orchestral music as harmonically cluttered or emotionally cold.
[Yawns]
Neither perception is wrong—they're just based on different learned expectations about how pitch relationships should work.
But here's where it gets even more interesting: some aspects of tuning seem to be tied to the physical characteristics of human vocal anatomy.
Section 7: The Voice as Universal Tuning Reference
[Sound of chair adjusting]
You know what's interesting? No matter what tuning system a culture develops, singing voices tend to do something similar across all traditions. They naturally gravitate toward just intonation—meaning intervals that are based on simple, pure mathematical ratios.
When a choir sings together without instrumental accompaniment, they instinctively adjust their pitch to create the most resonant, most mathematically pure harmonies possible. They're not thinking about mathematics—they're just listening and adjusting until the sound feels most "locked in," most stable.
This suggests that while cultural conditioning shapes a lot of our musical expectations, there might be something universal about the human voice as a tuning reference.
[Papers turning]
In fact, this is part of why many music theorists argue that just intonation is the most "natural" tuning system. When you sing, or when you play an instrument that allows for pitch flexibility like a violin or a trombone, you naturally drift toward those pure mathematical ratios.
The compromise systems—like equal temperament—only became necessary when we started building instruments with fixed pitches that needed to work in multiple keys.
But different cultures made different assumptions about the relationship between voice and instruments. In Western classical tradition, instruments increasingly became the primary reference point. We tune our voices to match pianos, not the other way around.
In Indian classical music, it's the opposite. The voice remains the ultimate reference, and instruments are expected to mimic the fluid, microtonal capabilities of the human voice. That's why Indian instruments like the sitar have sympathetic strings and moveable frets—they're designed to access the same pitch flexibility as a singer.
[Sound of drinking water]
Arabic music takes a middle approach. Instruments and voices both reference the same microtonal system, but the system itself is more fixed than Indian music while being more flexible than Western equal temperament.
What's fascinating is how these different approaches shaped the development of musical instruments in each culture.
Western instruments became increasingly standardized and mechanized. We developed piano actions that could produce consistent dynamics across all keys, string instruments with precisely calculated fingerboard measurements, wind instruments with carefully calculated bore dimensions and tone hole placements.
The goal was consistency and interchangeability. Any piano should be able to play any piece of music. Any violin should be able to join any orchestra.
[Yawns]
But other cultures prioritized different values. Indonesian gamelan instruments are often made in matched sets, where each instrument is tuned specifically to resonate with the others in its ensemble. You can't just swap out one instrument for another—each gamelan is a complete, self-contained tuning universe.
Indian instruments often include design features that enable microtonal expression. Sitars have curved frets that allow for pitch bending. Vocals techniques include elaborate ornaments that explore the spaces between Western semitones.
These aren't just aesthetic choices—they represent fundamentally different philosophies about what music is supposed to do and how precision should be balanced with expression.
[Soft ambient music begins to fade in]
Deb: Let me pause here for another short break. When we return, we'll explore how equal temperament eventually conquered the world, and what that meant for global musical diversity.
[Music plays for transition]
Deb: And we're back...
[Music fades out]
Section 8: The Global Conquest of Equal Temperament
[Sound of papers shuffling]
So how did this European compromise solution—equal temperament—end up becoming the global standard for music? Well, it's a story that involves colonial expansion, industrialization, and the rise of recorded music. Not exactly romantic topics, but they're crucial to understanding why the world sounds the way it does today.
The spread of equal temperament wasn't really about musical superiority. It was about technological compatibility and economic efficiency.
[Yawns softly]
When European powers expanded globally in the 18th and 19th centuries, they brought their musical instruments and their tuning systems with them. But more importantly, they brought their approach to instrument manufacturing—mass production, standardization, interchangeability.
If you're going to manufacture pianos in factories and ship them around the world, you need a universal tuning standard. Equal temperament was perfect for this because it was mathematically precise, easily replicable, and worked with any musical style that could be adapted to keyboard instruments.
[Sound of settling]
The real turning point came with the invention of recording technology in the late 1800s. Suddenly, music wasn't just a live, local experience—it could be captured, reproduced, and distributed globally.
But recording technology had a bias toward equal temperament. Piano rolls, gramophone records, radio broadcasts—they all favored instruments with fixed, standardized tuning. The subtle microtonal adjustments that singers and string players made instinctively couldn't be easily captured or reproduced.
Plus, if you were a record company trying to sell music internationally, equal temperament was simply more practical. A recording made in New York could be played anywhere in the world without worrying about local tuning conventions.
[Papers rustling]
This created a feedback loop. As recorded music became more popular, musicians started training themselves to match the tuning of recorded performances. Music education became increasingly standardized around equal temperament. And as new generations grew up with this as their reference point, alternative tuning systems started to seem exotic or old-fashioned.
By the mid-20th century, equal temperament had become so dominant that many traditional tuning systems were endangered. Young musicians in cultures with ancient microtonal traditions were learning to play equal-tempered instruments instead of their traditional ones.
It wasn't usually an intentional cultural suppression—it was more like... technological drift. Equal temperament was simply more compatible with the emerging global music infrastructure.
[Yawns]
But something interesting has happened in the last few decades...
Section 9: The Digital Renaissance
The rise of digital music technology has actually made alternative tuning systems more accessible than they've been in centuries.
[Sound of chair adjusting]
When music is created digitally, tuning becomes purely mathematical. You can program any interval you want, any ratio you can imagine. You're not limited by the physical constraints of building instruments or the economic pressures of mass production.
This has led to what some people call the "xenharmonic" movement—x-e-n-h-a-r-m-o-n-i-c, meaning "foreign harmony." Musicians and composers who are exploring tuning systems that would have been impossible to implement before digital technology.
Some of them are recreating historical tuning systems with mathematical precision. You can now hear Bach played in actual well-temperament, or medieval music in authentic Pythagorean tuning, with all the wolf intervals intact.
Others are inventing completely new systems. There are composers working with scales that divide the octave into nineteen equal parts, or thirty-one, or fifty-three. There are pieces written in tuning systems based on the mathematical relationships between prime numbers, or the golden ratio, or pi.
[Papers turning]
Now, most of this music sounds... well, it sounds pretty weird to ears trained on equal temperament. But that's exactly the point. These composers are exploring emotional and aesthetic territories that simply don't exist in conventional tuning.
There's this composer named Harry Partch—P-a-r-t-c-h—who spent most of the 20th century developing what he called "just intonation" instruments. He built marimbas, guitars, and percussion instruments that could play pure mathematical ratios instead of equal-tempered approximations.
Partch's music sounds like... well, it's hard to describe. It's familiar enough that you recognize it as music, but strange enough that it feels like music from another planet. The harmonies have this crystalline purity that you just can't achieve with conventional instruments.
[Yawns softly]
And now, with computer software, any musician can experiment with Partch-style tuning without having to build custom instruments. You can download apps that let you play scales with seventeen notes per octave, or thirty-four, or any number you want.
The internet has also connected musicians interested in alternative tuning in ways that would have been impossible before. There are online communities dedicated to sharing microtonal compositions, forums where people debate the emotional characteristics of different mathematical ratios, YouTube channels devoted to exploring historical temperaments.
[Sound of papers shuffling]
It's created this fascinating parallel universe of music that exists alongside mainstream equal-tempered music, but explores completely different aesthetic possibilities.
And this brings us to some really interesting questions about the future of music...
Section 10: Modern Questions and Ancient Echoes
[Sound of chair creaking]
So where does all this leave us? We've got this global musical culture based on equal temperament, but we also have digital technology that makes any tuning system possible. We understand more about the mathematics of music than ever before, but we also recognize that mathematical perfection isn't the only goal music can pursue.
There's been some interesting research recently on how different tuning systems affect listeners psychologically and physiologically. Some studies suggest that just intonation—those pure mathematical ratios—might have measurable effects on stress levels and emotional states.
[Yawns]
Other research has explored whether certain intervals have universal emotional associations, or whether everything we think we know about musical emotion is culturally learned.
These questions matter because they get to the heart of what music is for. Is music primarily about emotional expression? Spiritual experience? Mathematical beauty? Social bonding? Cultural identity?
Different tuning systems prioritize different answers to these questions.
[Papers turning]
There's also been renewed interest in historical tuning systems for performance of early music. More and more classical musicians are learning to play baroque music in the temperaments it was actually written for, rather than translating everything into modern equal temperament.
And the results are... well, they're revelatory, honestly. Music that can sound overly familiar in modern tuning becomes fresh and surprising when played in its original temperament. You hear harmonic relationships that equal temperament obscures.
[Sound of drinking water]
But perhaps the most interesting development is how electronic music has started to explore tuning as a creative parameter. Electronic musicians can change tuning systems in real time, morphing from equal temperament to just intonation to completely invented scales within a single piece.
This opens up compositional possibilities that would have been inconceivable to earlier generations of musicians. You can have melodies that gradually shift their internal tuning relationships, harmonies that exist in multiple temperaments simultaneously, rhythmic patterns that are based on mathematical relationships rather than time.
Some composers are even exploring "adaptive tuning" systems—computer programs that analyze the harmonic context of music in real time and automatically adjust each note to create the most consonant possible tuning for that specific moment.
It's like... imagine if every chord could be tuned perfectly for that chord, but then instantly retuned for the next chord. You'd get the best of both worlds—the pure resonance of just intonation with the harmonic flexibility of equal temperament.
[Yawns softly]
Of course, most people listening to music today don't consciously think about tuning at all. Equal temperament has become so normal, so transparent, that we hear right through it to the musical content.
But for those who do pay attention to these subtle mathematical relationships, we're living in a kind of golden age of tuning exploration. We have access to more musical possibilities than any generation in human history.
[Sound of papers shuffling]
And yet, in a way, we've come full circle back to Pythagoras's original insight—that mathematics and musical beauty are intimately connected. We just understand now that there are many different ways to make that connection, many different mathematical approaches to organizing sound into meaningful expression.
Closing Reflections
[Soft ambient music begins to fade in]
You know, there's something poetic about the fact that this whole journey—from Pythagoras to digital xenharmonic music—started with someone listening carefully to the sounds around them and wondering about the patterns they heard.
That curiosity, that willingness to question why things sound the way they do, led to centuries of mathematical investigation, cultural exchange, technological innovation, and artistic exploration. All because someone noticed that different sized hammers made different pitched sounds and thought, "Hmm, I wonder if there's a pattern there..."
[Yawns softly]
And in the end, what we've learned is that there isn't one "correct" way to organize pitch into music. Equal temperament isn't right and other systems aren't wrong—they're just different solutions to different musical problems, different ways of balancing mathematical elegance with practical flexibility, different approaches to the fundamental question of how sound can become meaningful.
The mathematics of music reveals something beautiful about both mathematics and music: they're not fixed, absolute truths, but rather human ways of finding pattern and meaning in the world around us. The ratios that Pythagoras discovered are real—they're built into the physics of sound waves. But what we do with those ratios, how we organize them, what emotional significance we give them, that's entirely up to us.
[Sound of chair settling]
Every time you hear a piece of music—whether it's a Bach fugue in well-tempered tuning, a Javanese gamelan with its unique pitch universe, an Arabic maqam exploring quarter-tone relationships, or an experimental electronic piece morphing between different temperaments in real time—you're hearing centuries of human ingenuity wrestling with the beautiful impossibility of making mathematics and music work together perfectly.
And maybe that's the most important lesson from our journey through tuning systems: perfection isn't always the goal. Sometimes the most beautiful solutions are the compromises, the creative workarounds, the moments when we accept that we can't have everything we want and figure out how to make something wonderful with what we can have.
[Music continues softly]
Thank you for listening to Dormant Knowledge. If you're still awake and hearing my voice, I appreciate your attention. But if you've drifted off to sleep somewhere along the way—which was partly the goal—then you won't hear me say this anyway. Either way, I hope some knowledge about the hidden mathematics of music has made its way into your consciousness, or perhaps your dreams.
Until next time, this is Deb wishing you restful nights and curious days.
[Music fades out]
END OF EPISODE

Show Notes & Resources

Key Historical Figures Mentioned

Pythagoras (c. 570-495 BCE) Ancient Greek mathematician and philosopher who discovered the mathematical ratios underlying musical consonance. His work established the foundation for Western music theory, though his "perfect" mathematical system created practical problems that influenced musical development for centuries.

Andreas Werckmeister (1645-1706) German organist and music theorist who advocated for "well-tempered" tuning systems. His mathematical approach to solving the tuning problems of keyboard instruments laid the groundwork for equal temperament's eventual adoption throughout Europe.

Johann Sebastian Bach (1685-1750) German composer whose "Well-Tempered Clavier" demonstrated the musical possibilities of accepting mathematical compromise in tuning. His forty-eight preludes and fugues in every major and minor key showed that equal temperament could produce beautiful music in previously unplayable keys.

Harry Partch (1901-1974) American composer who spent his career developing custom instruments capable of playing pure mathematical ratios instead of equal-tempered approximations. His work pioneered modern explorations of alternative tuning systems and just intonation.

Important Musical Concepts

Pythagorean Comma A tiny interval (about a quarter of a semitone) representing the mathematical discrepancy that occurs when building scales using only perfect fifths. This comma made it impossible to create a tuning system using pure mathematical ratios that worked in all keys.

Just Intonation A tuning system based on pure mathematical ratios that creates perfectly consonant intervals. While mathematically beautiful, just intonation creates practical problems when trying to modulate between different keys or play complex harmonic progressions.

Equal Temperament The modern Western tuning system that divides the octave into twelve exactly equal parts. While no interval is mathematically perfect, this system allows music to be played in any key with identical harmonic relationships.

Maqam (Arabic Music) A system of musical scales and modes that incorporates quarter-tones and microtonal intervals. Maqam theory recognizes twenty-four intervals per octave instead of the twelve used in Western music, allowing for more subtle emotional expression.

Shrutis (Indian Classical Music) Twenty-two microtonal intervals within an octave recognized by Indian music theory. Unlike fixed pitches, shrutis function more like "territories" of pitch that skilled musicians navigate fluidly for emotional expression.

Slendro and Pelog (Indonesian Gamelan) Two tuning systems used in Javanese gamelan music. Slendro divides the octave into five roughly equal parts, while pelog uses seven unequal intervals. Each gamelan ensemble has its own unique tuning developed over generations.

Modern Applications and Connections

Digital Music Technology Computer software now allows musicians to experiment with any conceivable tuning system without building custom instruments. This has led to renewed interest in historical temperaments and completely invented scales.

Xenharmonic Movement Contemporary composers exploring tuning systems that divide the octave into numbers other than twelve—such as nineteen, thirty-one, or fifty-three equal parts. These experiments create entirely new harmonic possibilities impossible in traditional Western music.

Adaptive Tuning Systems Computer programs that analyze musical context in real-time and automatically adjust each note to create optimal consonance for specific harmonic situations. This technology promises to combine the pure intervals of just intonation with the flexibility of equal temperament.

Historical Performance Practice Classical musicians increasingly perform baroque and earlier music using the original tuning systems, revealing harmonic relationships that equal temperament obscures and bringing fresh perspective to familiar repertoire.

Further Learning

Books:

  • "How Music Works: The Science and Psychology of Beautiful Sounds" by John Powell - Accessible exploration of musical acoustics and perception
    https://amzn.to/3VbhmMn (paid link)
  • "Tuning and Temperament: A Historical Survey" by J. Murray Barbour - Comprehensive academic treatment of Western tuning systems
    https://amzn.to/4nsPAXW (paid link)
  • "Genesis of a Music" by Harry Partch - The composer's own account of developing just intonation instruments and theory
    https://amzn.to/4nvXm3n (paid link)

Documentaries:

  • "The Music Instinct: Science and Song" - PBS documentary exploring the relationship between music and mathematics
    https://www.pbs.org/wnet/musicinstinct/
  • "Bali: Beats of Paradise" - Film exploring a couple who share their love of Indonesian dance and traditional music
    http://balibeatsofparadise.com/
  • "Raga: A Film Journey into the Soul of India" (1971/2010) - The classic documentary about Ravi Shankar that introduced Indian classical music to Western audiences, remastered in 2010

Online Resources:

Academic Sources:

Episode Tags

#Mathematics #MusicTheory #Pythagoras #Bach #TuningSystems #EqualTemperament #JustIntonation #CulturalMusic #ArabicMusic #IndianClassicalMusic #GamelanMusic #Xenharmonic #MicrotonalMusic #SleepPodcast #EducationalContent #MusicHistory #ScienceHistory #Acoustics #MusicPerception #DigitalMusic #HistoricalPerformance #SoundPhysics #MathematicalMusic