What's the Deal With Just Intonation?
The Exclusive Interview
CT: Hey, congrats on all the Super Bowl commercials. People are already calling it the "Just Intonation Bowl."
JI: Thanks. I'm not used to all this attention.
CT: So, what's the deal with you?
JI: The… "deal?"
CT: Your name is really on the tip of the collective tongue. And yet it seems like no one really understands what you're actually about.
JI: Yeah, it's been strange. I get misquoted a lot.
CT: Not just that… people keep tying themselves in knots trying to find a way to explain you. You're really hard to pin down. I thought I might give you the chance to explain yourself in your own words.
JI: This might be a short interview.
CT: Really?
JI: I don't have all that much specific to say about myself.
CT: I've noticed. No socials, no AMA, no influencer endorsement tie-ins… and your website looks like it hasn't been updated since 2003. You rarely do interviews, and when you do, it seems like you’re just making stuff up: do you really drive a tank around London and live in an abandoned bank? What's all this stuff about lucid dreaming?
JI: I think you might be confusing me with a student I once had. Brilliant guy. Really “got it.” Went a little nuts.
CT: Okay so who are you, then? This is your chance to tell the whole story. What are you about?
JI: I’m rational numbers in music.
😐
CT: That’s it?
JI: That’s it.
[ … ]
CT: I'm sorry but no, that can’t be "it." That's so broad it's basically meaningless. Rational numbers in music… is this entire thing some kind of elaborate troll? I've been to your Wikipedia page. It's got one of those warnings at the top: "this article may be too technical for most readers to understand," it says. "This article's factual accuracy is disputed," it says. It's six thousand words long. You're clearly some kind of esoteric theoretical construct. Uptown, the academics fight about you in those stuffy peer-reviewed journals… and downtown, the hippies repurposed an entire building in TriBeCa just to get you in the right lighting. No one can even agree on what you look like!
JI: Why are you so mad?
CT: Because you couldn't possibly be that simple. You don't get to just be all la-di-dah like this. This is serious. Do you even know how many rational numbers there are? There are infinity of them! Can't you at least, like choose a set of just the good ones? Pick 12? Or 128!
JI: They’re all good numbers, Chad.
CT: It's Chris. Okay, I guess if you really are just about "rational numbers," you should probably start by telling people what those are. Is there one you can recommend, for beginners?
JI: Yes, of course. The number 2 can be expressed as "2/1." It's rational.
CT: I know this one. It's an "octave." Can I call it that?
JI: Sure, if you like. I get a little confused, though. That word comes from Latin, implying "eight." We're not there yet. This number is "2."
CT: Give me one more.
JI: The number 1.5 is rational: it can be expressed as "3/2."
CT: Punching 3/2 into my calculator: 1.5 — it checks out.
JI: A rational is any number that can be expressed as a fraction of two integers. It’s one whole number divided by another.
CT: I was worried you'd describe them that way. I know that's straightforward to us, but you might consider dumbing it down for the general public, especially if there are musicians in the room. My music majors are all scared of fractions.
JI: I'm surprised to hear you say that — I sometimes sit in on their rehearsals and hear them speak fluently about "fifths," and "6/8," and "quarter" notes, and "Amin9/D." These fractions all seem to mean something to them. They even do some very strange math… they divide a "fourth" into five "half" steps, and a “sixth” into five “whole” steps. I heard one musician ask another whether "7/8" is equal to "2+2+3" or "3+2+2." I wasn't even aware there was a difference. Their calculations perplex me — and my simple arithmetic feels like kid’s stuff in comparison. Aren't you underestimating them?
CT: Those examples aren't the same though. Those are music theory terms. Rational numbers are "real" math. My music majors hated school and ignored math in particular. They didn't think they needed it. They assumed it was none of their business.
JI: And that's their own fault, I suppose?
CT: I heard you're a "tuning system." Isn't that true?
JI: Maybe… but in a broader sense than people think.
CT: Broader than the idea of "notes being in tune?"
JI: Yeah. Rhythms and tempos can also be "in tune." I consider them to be an essential part of what I’m about.
CT: I never thought of it that way… I would have said that rhythms are "in time."
JI: Is there a difference?
CT: Tell me about your name.
JI: "Justin?" Pretty common.
CT: I thought you might make that joke. I meant the “just” part.
JI: Right — that comes from the Latin "iustus," which can mean various things but I think my parents had the definitions "precise" or "exact" in mind.
CT: Being named "precise," did you turn out to be some kind of perfectionist? There's this idea of "nominative determinism" … like when sound guys are named "Mike." I totally believe in that.
JI: Me too.
CT: So, are you Perfect?
JI: "Perfectionist" doesn't really describe where I'm at today. I'm more "aspirational." I get criticized for having my head in the clouds, but I do live in the real world — it's just that my value system is based in precision rather than chaos.
CT: “Lawful good?”
JI: “Lawful neutral.” Ferociously loyal to certain guiding principles, yes… but with no interest in value judgements.
CT: But with accuracy as a guiding principle, isn't it frustrating when humans aren’t willing or able live up to your exacting standard?
JI: I like to think there’s elegance in precision, and I want to show musicians something beautiful and elegant to aim for. Besides, I think most of them would find the irrational numbers far more difficult to be exact about than the rational ones.
CT: There are "irrational" numbers?
JI: There sure are. They have decimals that go on forever without any repeating pattern. They are the opposite of rationals: they can't be represented as ratios of whole numbers.
CT: This must be rare?
JI: Not at all, there are an infinite number of those as well. Some are pretty famous. I do a dinner once a year just to "reach across the aisle," if you will. Pi showed up to the last one! I have to admit I was star-struck.
CT: I've heard of Pi.
JI: A nice number. Really down-to-earth. I don't totally understand the obsession though? The media have been calculating Pi's digits since who even knows when, and they're still doing it today. It’s a little bit sad.
CT: And irrational numbers can also apply to music?
JI: Yeah, one in particular: remember before when we were talking about "2?" Musical theorists and instrument builders have been asking for a number you can multiply by itself to get 2 forever. You might have heard of a "square root?" That's what they want. A square root of 2. There have also been calls for a "cube root" or even a "twelfth root," of 2. Turns out, these numbers are Real! They’re just irrational.
CT: Where in music would these pop up?
JI: At this point, they are the foundation of all of western music.
[ … ]
CT: Are you calling all of western music irrational?
JI: Only in every possible combination of pitches except for the "octaves."
CT: And that's… bad.
JI: It's neutral… irrational numbers applied to music have strengths and weaknesses just like I do. They’re just another tool in the toolbox. You can focus on one tool or another, or mix them in various ways.
CT: Have we always mixed these tools?
JI: No, irrational numbers as applied to music are a pretty recent technology. We've only known how to harness them for that purpose in the past 500 years or so. You might have heard of "temperaments" — scales full of irrational numbers. Even though most musicians wouldn't know how to build one themselves, these little devices are really convenient — they fit in your pocket, and never require adapters. As you know, it’s the most convenient technology that usually wins out. I do think we've lost something, though.
CT: What have we lost?
JI: A sense of infinite possibility.
CT: What are your thoughts on Big Temperament?
JI: I certainly understand how it developed. There were more and more "mouths to feed," musically… and it just wasn't practical to build new instruments for every genre of music, or to retune them between each piece. Industrialization and mass production of the machinery was essential for bringing music to market at scale. "Big T" built orchestras and Guitar Centers full of standardized equipment and composers scrambled to create endless product for them. It's simple capitalism.
CT: Farms become factories… and shops become supermarkets.
JI: Right. Although today it's more like one giant Amazon. Even temperament used to be an open market — there were a couple dozen different vendors to choose from. But after 150 years of mergers and technological advances, it's become a complete monopoly. Through pure market saturation, a single company: "12 tone equal temperament" won out, managing to convince both the composer and the listener alike: it's the only brand that ever existed.
CT: And yet it never stops lobbying for power and influence.
JI: It's all they teach in the schools.
CT: You can't deny… it's a fabulous company with beautifully designed products.
JI: Oh for sure. It's a kind of synthetic utopia. Daphne Oram put it really nicely:
“The equal tempered scale — a brilliant compromise between nature’s ‘scale’ and the most satisfying scale that mathematicians can envisage.”
It's the most optimized twelve-note scale possible, dreamed of since the 16th century, and now it comes pre-installed on every piano, vibraphone, bassoon, organ, and electric bass in the western world. Only the churches still have the vintage instruments from before it became the universal standard. Today it’s so taken for granted that most people don’t even know it has a name.
CT: Like asking a fish “how’s the water.”
JI: Exactly… “what’s water?”
CT: But is there a problem?
JI: Utopia is a double-edged sword: it's a closed ecosystem — it rejects less expensive third-party peripherals, and needs to be continually updated every decade or so at a high cost to the programmers and the end users alike. Every update seems to break the last version, the company founders are long-gone, and the current leadership has thrown out the original charter. In some industries the feature-bloat gets worse and worse, making it bruisingly unintuitive and inaccessible for new users. In others, scientists have hyper-engineered the product down to a tiny palate of highly addictive flavors, marketed to the masses. Even our singing voices now qualify as ultra-processed junk food: they’re taken straight out of the air, genetically modified, and stripped of all nutritional value. We sound like robots.
A technology that once seemed full of possibilities now looks like the architect of its own destruction. As a child I used to be free — I was outside in nature all day. I would run through the orchards and eat the fruits straight off the trees. The juice would run down my arms.
CT: Are we still talking about music?
JI: Ultimately yes, although our ancestors might not have recognized it as such.
CT: I can see now why people often overlook what you are and focus on what you aren't. Rage-bait gets "engagement."
JI: Don't I know it. What I really am is super simple. Boring, even. I'm terrible at cocktail parties.
CT: I recently went to a concert and heard a piece that was "in 11-limit just intonation." Is that like saying a piece is "in E Major?" Are you a scale?
JI: Thanks for bringing this up: it's a common and completely understandable misconception. I'm no one scale in particular. Rather, I'm a principle that can be used to build scales: just choose pitches with frequencies in whole-number ratios (like 3/2 or 5/4). The number of scales is limited only by your imagination. You know Minecraft? If ratios are like "blocks," then I'm the "creative mode" where the player can build custom worlds. I'm like a "level-creator" for music.1
CT: But if you're "limited only by my imagination," what's the deal with the "11-limit" part of that piece's title?
JI: It can be useful to work within self-imposed limits. It can help a composer free up raw creativity by committing to a smaller palette of options.
CT: I've read that! Have you heard this quote:
“To chain oneself as a creator is a legitimate exercise of personal freedom. To be forced to endure such limitations because of a conspiracy of factors quite beyond the ordinary creator’s control is a musical horse of another color.” - Harry Partch
JI: Hahahaha I love Harry. He was so cantankerous. Did you know he dated Ramon literal Novarro?
CT: Wow, hot. Was Partch another one of your students?
JI: No, we just hung out a lot. But he was incredibly insightful — by the end I'd say he knew more about me than I knew about myself. He could see other points of view with crystal clarity… he just had a really hard time giving them license to exist when he disagreed. He got himself pretty angry sometimes.
One time I tried to tell him, you know, I'm a lot for people to deal with, why not allow them something quick and easy from time-to-time? Every meal doesn't have to be a tasting menu of the freshest farm-to-table vegetables. People's ears get tired, you know?
CT: I bet he didn't like that.
JI: He looked me straight in the eye and I swear he actually growled: "we may well pause to ponder the inscrutability of a man and his apparently lethargic cochlea.”
CT: Hahaha so bitchy!
JI: Anyway, I have no problem with limitations, and a particular fondness for setting up limits that exist within my world of rational numbers.
CT: Can you suggest a starting point for this strategy?
JI: I usually recommend focusing on my primes.
CT: You're about to talk math again, aren't you. You realize you're going to lose people?
JI: Chris!! This is fourth grade math.
CT: Just saying.
JI: Okay, I won't judge if you need to ask the internet for a refresher on what a prime number is. By using only multiples of certain prime numbers, you create music in a "prime limit."
CT: For example?
JI: The entire history of western music could be seen broadly as "5-limit." It is based only on numbers that are multiples of the primes 2, 3, and 5.
CT: Are you calling western music "limited?"
JI: Only all of it.
CT: And that's… bad.
JI: Not remotely. The music is wonderful. I'm a huge, huge fan. I have the complete box set on vinyl… could have bought a house with the shipping costs alone.
CT: And now people are giving higher limits a try?
JI: They are. The next three are 7, 11, and 13… Each successive prime is like adding another dimension to reality. That composer who wrote the 11-limit piece was doing a bit of showing off by putting that in the title… by using the first five primes, she sees the piece as 5-dimensional.
CT: Isn't it a little… intimate to reveal your primes right there in the title of the piece? Feels like oversharing.
JI: Yeah, I'm always telling composers not to talk about it. They can't help themselves. It comes from a good place — they want to give context to the listener. But for now it often just comes off as bragging. A prime limit is no value judgement. "Early" music didn't even use 5. It was all just small number multiples of 2 and 3. And that stuff is breathtakingly powerful. Transcendent. People think it's God speaking directly to them.
CT: That reminds me… a lot of people do associate you with "early" music, as well as stuff like "ambient" or "new age." But I expect you'll tell me you're not a "genre."
JI: You'd be right. I am not a genre.
CT: But wouldn't you at least admit to being "consonant?" People meditate to you.
JI: Consonance can be subjective, but in general: the smaller the rational numbers you choose to work with, the more consonant I am. I’ve heard definitions of myself with that part tacked onto the end: “… with a preference for smaller numbers.” But they’re projecting… I have no small-number agenda.
I suspect people associate me with consonance because they are used to hearing temperament, which is inherently dissonant. It has made them forget (or never learn) what true consonance really sounds like. It can be shocking to hear my small-number ratios when you've only ever heard irrational approximations of them.
CT: But I suppose you can be dissonant as well.
JI: Babe, I can be so dissonant you'll be begging for your mama.
CT: Thanks for taking the time to talk to me. You are as enigmatic as ever. People are hopefully entertained, if still maybe confused. I may have to have you on as a regular guest and get into some specifics?
JI: It would be a pleasure.
CT: Just one more thing: as I mentioned before, there's an ongoing argument about what you really look like. Many claim to have seen you, but the descriptions are inconsistent. People want to see how you're notated.
JI: I don't really notate myself. I work with various designers. I have closets full of notational systems and extended accidentals.
CT: And who are you wearing tonight?
JI: The gown is Ben Johnston, and the fragrance is "Alpha" by Wendy Carlos.
CT: Gorgeous. And the earrings?
JI: Thanks, they're from last season's Harry Partch “Utonality Diamonds” collection.
CT: Do you ever just throw on a grand staff, a couple clefs, and call it an outfit?
JI: For sure. I enjoy comfort as much as anyone. You can do a lot with basics.
CT: Next time give me a tour of your closet?
JI: That would be super fun.
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1 The author would like to admit that, being far too old to ever have actually played Minecraft, he only read about it on the internet to create this metaphor, and probably mis-represented it in some way. But relating Just Intonation to SimCity or the Lode Runner level editor wasn't working quite as well, and seemed tragically dated.
I enjoyed this a lot. I still don't really 'get' just intonation, but I think I understood at bit more about what it isn't.