Q: You wanted me to ask about your new album?
A: Alien Dreamscapes in Just Intonation is now on Bandcamp and all streaming services!
Q: So you're writing music in quarter tones then?
A: Noooo, when people say this it hurts a little bit, even though I know that's not the intention. Quarter tones would mean I'm using equal divisions of the octave. I'm not using equal divisions, I'm exploring the beautiful variety of unequal divisions based on whole-number ratios known as just intonation.
Q: What is just intonation?
A: Tuning in whole number ratios based on the harmonic series. There is a theoretically infinite variety of possible just intonation tunings, since the harmonic series is infinite! More on this below.
Q: Why are you doing this?! Is there any chance you'll stop?
A: One of the things that I love about working in just intonation is that each tuning creates its own sound world, and often the music often naturally flows from simply improvising in a new tuning. With these sound worlds I have to let go of many things that I've learned about harmony and discover new paths for creating music.
I am a classically-trained pianist who has played all sorts of Western classical music for most of my 50 years on this planet, and I'm especially fond of Shostakovich, Prokofiev, and Chopin. In the past five years I've been studying Carnatic rhythm, singing ragas, and listening to Indian classical music frequently, as well as Michael Harrison's gorgeous piano music, so what is coming out now is a result of all those things are floating around in my mind.
David B. Doty puts it like this: "[Just Intonation] is a gateway to a new and expanded palette of musical intervals and subtle distinctions hitherto unknown to Western composers. Ultimately, it is a method for understanding and navigating through the boundless reaches of the pitch continuum—a method that transcends the musical practices of any particular culture." https://soundamerican.org/issues/just-intonation/why-how-and-wherefore
Q: Ok, whatever, to each their own. I still don't understand what just intonation is. Your answer about ratios means nothing to me and is frankly a little bit annoying.
A: Sorry, I am required to ramble on about ratios at least a little bit. I use ratios to refer to specific intervals since note names aren't specific enough. In just intonation, you can have different types of (some wider, some narrower) Major 3rds or half steps, etc., all in the same tuning. It's beautifully irregular. For example, I need to write more information than just "minor 7th" or "B-flat" because that doesn't tell you if you should play the gorgeous harmonic 7/4 B-flat, the higher 16/9 one, or the even higher 9/5 B-flat. You could play any of those three B-flats and it would be sheer chaos!!
So that's one reason that in just intonation, I'm using whole-number ratios, like 5/4 is one type of Major 3rd, and 16/15 is one type of half step.
In order to explain just intonation more, it will help to explain equal temperament.
If you think the math for just intonation is confusing, check this out. :)
Every single half-step or semitone in equal temperament is the 12th root of 2. This is an irrational number, but if we round it to 19 decimals, it's 1.0594630943592952646. (Pi is the most famous example of an irrational number, infinitely non-repeating.)
Q: Ughhh omg I didn't ask for all that math, how does it sound?!
A: Hey look, someone on Musescore made this so that you can listen to side-by-side comparisons: https://musescore.com/user/49891829/scores/14472835
Just intonation can sound consonant, peaceful, and resonant. It doesn't have to, but it can! A 5/4 JI 3rd sounds sweet and consonant to me, where as the M3 in equal temperament is wobbly. The 5/4 JI third is perfectly in tune with the 5th note in the harmonic series, whereas the equal temperament major 3rd clashes against it. Equal temperament can't quite sound as consonant or peaceful, because those pure and resonant intervals just don't exist in that tuning.
Western keyboard instruments are currently tuned in equal temperament. Since equal temperament divides the octave into an equal number of parts, it works well for music that modulates to different key centers, since all the intervals stay the same in every key. It's perfectly consistent. But one big sacrifice is that none of the intervals in equal temperament ring pure except the octave, they are all adjusted, some quite a lot (like major thirds) and some not very much (like perfect 5ths.) The other sacrifice is that it's perfectly consistent.
According to Kyle Gann: "Equal temperament - the bland, equal spacing of the 12 pitches of the octave - is pretty much a 20th-century phenomenon. It was known about in Europe as early as the early 17th century, and in China much earlier. But it wasn't used, because the consensus was that it sounded awful: out of tune and characterless. During the 19th century (for reasons we'll discuss later), keyboard tuning drifted closer and closer to equal temperament over the protest of many of the more sensitive musicians. Not until 1917 was a method devised for tuning exact equal temperament." https://www.kylegann.com/histune.html
Gann's statement is a wee bit harsh and may be controversial since other sources say that equal temperament started becoming established in France and Germany around 1800, but my understanding is that he's saying the method of tuning this way didn't become like totally precise until 1917, much later than I would have guessed. One of the reasons I wanted to write about this is so that I can get clear on what I know and what I need to review in order to be able to explain all of this well, and although I have read a lot about the history of tuning, this is an area I must review in order to accurately explain it. Meanwhile, see my resources list below for books I recommend!
I do still love all sorts of music written in equal temperament, and I prefer to celebrate all the things we can do with just intonation instead of attacking what has become our standard tuning in the West. I do find reading other people's vicious attacks pretty fun, though, in all honesty, so I'm not above enjoying that. I think it's vitally important to know that there are other options, though.
"Musical temperament was a response to the frustrating discovery that nature's proportions, in spite of man's best efforts to force them into a regimented, reliable scheme, follow their own inexorable paths." -Stuart Isacoff, Temperament, How Music Became a Battleground for the Great Minds of Western Civilization, p. 95 (This explains how my love for chaotic gardening aligns with my love for just intonation.)
The last thing I want to say before this turns into a textbook: All music software and electronic keyboards should allow people to choose other tunings, otherwise equal temperament is unfortunately imposed on cultures that historically have not used it, and insidiously eats away at the character and richness of their music. Thankfully, this is changing, with lots of help from Khyam Allami, a musician who created free software called Apotome and Leimma: https://khyamallami.com/Apotome-Khyam-Allami-x-Counterpoint
Q: Wait, I have more questions! What are limits, like why do you say something is 3-limit, 5-limit, 7-limit, or 11-limit in your titles?
The limits refer to the highest prime number included in the tuning. So 3-limit tuning means the ratios I'm using all contain multiples of 3, or the 3rd harmonic, which is approximately a perfect 5th. So in 3-limit JI, essentially, I'm stacking 5ths to create all the notes in that tuning. 5-limit JI means that now I've included the 5th harmonic and multiples of 5 as well as 3. 7-limit means now I can go in three directions to create notes, using multiples of 3, 5, and 7.
What are wolf tones?
A: If we stack 12 pure 3/2 fifths, we don't land exactly 7 octaves higher, but slightly higher. This difference between stacking 12 pure 5ths and the place we started is known as the Pythagorean comma. When Michael Harrison explained in a lecture on just intonation that the circle of 5ths is not actually a circle but a spiral, my brain exploded and has obviously never been the same. So if we try to tune all of the 5ths pure, we end up with one that howls like a wolf. This is what Stuart Isacoff is talking about with nature's proportions following their own inexorable paths. The Pythagorean comma is the discrepancy between 12 justly tuned perfect 5ths (3:2) and seven octaves (2:1).
Q: What are commas?
A: Very small intervals that result from the difference between different ways of tuning just intervals. For example, you can tune a Major 3rd 5:4 (a sweet 5-limit M3) or 81:63 (a wider, very wobbly 3-limit M3). The difference between these two 3rds is 81:80, known as the syntonic comma.
Q: What is a Lumatone?
A: A Lumatone is a MIDI-controller with 280 hexagonal keys that you can program with any tuning or light scheme you want. I've also programmed a traditional piano keyboard (with 88 keys and 12 per octave) in just intonation. I like both ways. The Lumatone offers so many possibilities that it can get a bit overwhelming sometimes, whereas being limited to 12 notes per octave can actually be somewhat freeing.
Q: What are some resources for learning more about this?
A: Jacob Adler's Microtonality Crash Course—next one starts October 19th, 2024! Email jacobadler@gmail.com to join. I can't recommend this highly enough.
Michael Harrison's raga classes on Zoom, ongoing, best way to simply experience just intonation without having to do any math: https://www.artslettersandnumbers.com/indian-ragas-with-michael-harrison
The Arithmetic of Listening: Tuning Theory and History for the Impractical Musician, by Kyle Gann
The Just Intonation Primer, by David B. Doty
Harmonic Experience, by W.A. Mathieu
Temperament: How Music Became a Battleground for the Great Minds of Western Civilization, by Stuart Isacoff
How Equal Temperament Ruined Harmony (and Why You Should Care) by Ross W. Duffin
If you're looking for more extreme xenharmonic nerdiness: Now and Xen podcast, interviews by Stephen Weigel
Q: What pieces can I listen to?
A: My album, Alien Dreamscapes in Just Intonation!
Revelation, Just Constellations, Seven Sacred Names, by Michael Harrison
The Well-Tuned Piano, LaMonte Young
Indian classical music: I need to get more specific here, but I'm already overwhelmed with trying to answer all these rarely asked questions.
If you actually made it through all that, thank you for reading! And if you didn't, that's ok, I'm trying to get better at explaining all of this and this was an early attempt.
Q: Is it possible to discuss JI without including some kind of chart?
A: No, this is not allowed. Here you go.
| JI ratio | cents in JI | cents in equal temp |
Unison | 1:1 | 0 | 0 |
Half step | 16:15 | 111.731 | 100 |
Minor tone | 10:9 | 182.404 | 200 |
Major tone | 9:8 | 203.910 | 200 |
Minor third | 6:5 | 315.641 | 300 |
Major third | 5:4 | 386.314 | 400 |
Perfect fourth | 4:3 | 498.045 | 500 |
Augmented fourth | 45:32 | 590.224 | 600 |
Diminished fifth | 64:45 | 609.777 | 600 |
Perfect fifth | 3:2 | 701.955 | 700 |
Minor sixth | 8:5 | 813.687 | 800 |
Major sixth | 5:3 | 884.359 | 900 |
Harmonic minor seventh | 7:4 | 968.826 | 1000 |
Grave minor seventh | 16:9 | 996.091 | 1000 |
Minor seventh | 9:5 | 1,017.597 | 1000 |
Major seventh | 15:8 | 1,088.269 | 1100 |
Octave | 2:1 | 1,200.000 | 1200 |
