What I Learned Writing an Album in Just Intonation | Hacker News Reader

What I Learned Writing an Album in Just Intonation

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159 pointspierrec93 comments

I'd love to get some feedback on this, because I'm not sure I've hit the original goal of this article.

For a long time, the topics of just intonation, harmonic lattices, and of their relationship with temperaments were difficult for me to understand, despite reading the theory on them. It's not until I applied the concepts that could say I understood them.

I tried writing the "missing article" that would have given me this understanding right away, banking on the added value of interactive figures. Right now I suspect that I failed in the same way as my early readings did: this article might be hard to understand unless you're already familiar with the topic, which is a bit of a chicken-and-egg problem. I think I glossed over some concepts, and it sometimes looks like I'm pulling numbers out of thin air in some examples. Nonetheless I hope it will help some people understand these topics. Maybe what's needed is a slower-paced version of this as a longer series.

The interactive harmonic lattice, with the commas clickable and audible, is first rate and is the best interactive illustration of JI problems I have yet seen.

It is not quite clear from the explanation, that since the "3" direction is also dividing by 2 implicitly, the note confusingly labeled "3" is really just 3/2 - the perfect fifth. Similarly, the note labeled "5" is actually what we hear as the 12TET major third - clarifying a couple of examples on this lattice (or playing a chord) would be good.

Then, the thing I think that might help is driving the point home a little more by creating a comma by hand - a "knight's move" to another tile 5 horizontal and 3 vertical, creates a ratio that almost but not quite undoes itself. The (unwieldy) math is 1*3*3*3*3*5*5/2/2/2/2/2/2/2/2/2/2/2 = 0.9887, and this is what creates the commas in JI. Hiding the octave transpositions, while making for a neat visualization of the lattice, confuses the underlying issue.

I'm now realizing this and I totally agree that hiding the octave transpositions might make things confusing.

Maybe this could be clarified by having a first lattice where this adjustment is not made, and everyone can observe how frequencies rapidly become extremely low or high, maybe to the point of being inaudible. Then we can ask "how can this problem be solved?" and introduce the concept of pitch class lattices where everything is transposed into one octave.

Regarding intervals, I've always seen the irony and should probably mention it. Yes, the classical concept of fifth is related to the 3-ratio, while thirds are related to the 5-ratio. I'd argue this falls under the "empty your cup" category, but yes, leaving it untold probably doesn't help with understanding.

To continue with an empty cup philosophy while still mentioning intervals, you can just posit "humans like how these 5- and 3-ratio intervals sound" and play chords using them to demonstrate. Then you don't need to handwave why this particular lattice was picked, and you can draw a comparison to 12TET's third and fifth.

What are your thoughts on the fringe 432 theory, but the actual roots of it in relation to the geometric significance towards whole number ratios and then what is never discussed: the tuning associated with it?

If you're talking about using 432 Hz vs 440 Hz it's nonsense. You could pick any frequency you want. The frequency you pick to tune an A to is arbitrary.

Yeah but you don't have the same knowledge apparently.

Same knowledge of what?

In other words, 3 * 3 * 3 * 3 * 5 * 5 is 2025, which falls well short of 11 octaves (2^11 = 2048). No series of 3's and 5's can ever multiply to a power of 2.

Oh, and one last point of feedback - it wasn't apparent to me that the "Aníron" progression was walking across the lattice. You should make the projected versions semitransparent to show the main chord progression walking as it does in the JI lattice.

Still, awesome diagrams.

I am very unfamiliar with the topic. Here's my takeaways from the article:

  - almost all music I've heard uses 12TET
  - JI is microtonal / an alternative tuning to 12TET
  - you're the author of the lovely ambient.garden, which uses JI
  - the audio widgets and visualizations were very helpful
  - JI has perfect harmonies, and 12TET approximates those enough to still sound good
  - JI opens up cool chord options
  - but there's extra pitfalls when making JI chords compared to 12TET 
  - still not sure what consonant, continuous, or misleading chords are. I only know what dissonance sounds like
  - still haven't looked at numbers; all I know is that there's ratios in there somewhere

> still not sure what consonant, continuous, or misleading chords are. I only know what dissonance sounds like

Consonance is the opposite of dissonance; so while a dissonant chord sounds jarring and unpleasant with beating, a consonant chord sounds harmonious, pleasant, smooth.

I'm not sure about the other two either, I don't think they're commonly used terms.

Maybe continuous refers to "continuo" in Baroque music, where the bassline and chords are separate but work together harmonically - in some tunings, perhaps this is harder.

I think misleading refers to chords where their function harmonically is ambiguous until they resolve "home" at a later time - for example, not using the root note of a chord, instead using an inversion or modulation that feels like the chords could lead in several directions, creating a kind of uncertainty and tension in the piece that feels satisfying to resolve when the later chord indicates which of the paths through the harmony were the "correct" one.

One definition of consonant intervals are those that share many overtones/partials in common [1]. This reduces the phenomenon of "beating" which is readily perceived as unpleasant dissonance. (Though small degrees of "beating" can be pleasant.)

[1] https://en.wikipedia.org/wiki/Consonance_and_dissonance#Acou...

The key to the article, and to all of Western music, is about a set of weird mathematical coincidences involving the twelfth root of two. In particular, that twelfth-root-of-two to the fourth power is almost exactly 4/3, and twelfth-root-of-two to the fifth power is is almost exactly 3/2. And, of course, twelfth-root-of-two to the 12th power is exactly 1.

For some reason, not entirely well understood, when you play frequencies that are nice multiples together, they make pleasant patterns ("consonance"). They interfere and reinforce at regular intervals, and it sounds nice. When you don't get nice ratios, they make irregular interference patterns, and that sounds unpleasant ("dissonance").

So... take a scale (one frequency to twice-that-frequency), divide it into 12 equal parts (geometrically), and you get of those nice patterns. Each successive note is the frequency of the previous note times the-twelfth-root-of-two. That's the "twelve tone equal temperament". It makes it easy to write pretty music just by following some mathematical rules.

There's a Wikipedia article on the-twelfth-root-of-two:

https://en.wikipedia.org/wiki/Twelfth_root_of_two

EXCEPT... that those coincidences aren't perfect. They're very close, but tiny differences matter. Some intervals that should work, don't. Different scales take on different feelings, even though in theory they're just the same thing nudged up a few hertz.

Anyway... that set of coincidences is the basis of all of the music you're familiar with. Other musical cultures also take advantage of it. But there's no fundamental reason why those particular coincidences should be the only interesting music, and so people look around (often, to yet more world musical cultures) for alternatives.

I think you're maybe missing the main motivation, which is that "correctly tuned" overtones are in nature, in physics, in all sounds -- they aren't a social construct, unlike everything else about music. I think the argument from JI advocates is that our body's perception knows very well what a justly-tuned chord sounds like, etc.

They aren't though. No acoustic instrument has perfectly linear overtones. And even something like a flute, which probably comes closest, has an overtone system that varies with force and input excitation.

The idea of the harmonic series is a useful approximation, not a physical truth. Fourier applies to completely static waveforms, and says nothing about how waveforms change over time. If you try use Fourier for dynamic waveforms - which is most of them - you run into all kinds of problems and tradeoffs.

So the concept of whole number consonant ratios is an attempt to impose a mathematical ideal of perfection where it very much doesn't exist.

Clearly music is related to the harmonic series, but it's related around it, not trapped inside it. The dissonances and deviations create colour, tension, and movement.

This doesn't mean xenharmonic experiments aren't worth doing, and we should all just use 12TET and not think about anything else.

But it does mean these are experiments away from real acoustics into computed perfection, not back towards a perfect ratio heaven that music was exiled from by 12TET.

this might help, primer on JI ET and pythagorean intervals for violinists in string quartet context. Good demos 30 seconds in

https://www.youtube.com/watch?v=QaYOwIIvgHg

I would say that this is way too technical for me, however, I also only have the slightest hint of music theory knowledge to begin with. It seems to me like that isn't a problem though, because this is a very technical topic by nature.

Unrelated to the writing, but I was kind of disappointed that when listening to the examples, I cannot for the life of me hear a difference between just intonation and equal temperament. I know it's there, but I can't hear it at all.

Do you hear the differences when clicking the 275 vs the 277 in the third player? If so, you can try to focus on those frequencies and see if they stand out to you.

I'm familiar with music theory and microtonal music and the first examples are just too close to really discern easily. The examples of "chord 1/2" in the section "The Far-Out Solution: Freestyle JI" are probably the first example that makes the difference clear, if you play the 12TET and the JI overlapping slightly. That's because subtle differences in tuning are much more perceptible in harmony (chords) than in melody (single notes), since notes playing together are perceived with noticeable effects introduced by our brains such as harmonic beating and feeling tone.

I thought this video was good at explaining and showing the difference audibly https://youtu.be/zJTBT5u9AwQ?si=M3csNlck4El_eluH

I have limited pitch perception, but I could hear the 12TET chords “beating” while the just ones fade out much more smoothly.

This beating occurs when two waves just a bit out of tune interfere, alternating between constructive and destructive interference.

It's worth noting that part of the point is showing that the difference can be very small.

The one at https://www.osar.fr/notes/justintonation/#compare is really tiny, the largest difference being 3hz, you can kind of compare side to side using https://www.szynalski.com/tone-generator/ and see for yourself.

The difference is more perceptible below that, in the chord comparisons.

I think you could make the piece more interesting to the casual reader by grounding the subject in its history. It's been suggested by the book below that the dominance of equal temperament in western music is a relatively recent innovation, something like 120 years old, and is perhaps a consequence of the industrial revolution and the demand for mass manufacture of musical instruments having uniform qualities it produced.

Further reading: How Equal Temperament Ruined Harmony (and Why You Should Care), by Ross W. Duffin (2008).

My layman's takeaway from the book: Equal Temperament is a compromise tuning that allows a piano to access all major and minor modes, at the cost of the keys on the outer ring of the circle of fifths to be somewhat out of tune. An ET "C Major" "sounds best", and the further you move away from it in either direction, the worse the key sounds. Also, the fact that Beethoven and Mozart were aiming for just intonation and/or meantone in productions of their works seems to be sort of an inside secret among music maestros, with rigid adherence to equal temperament slyly pushed on competing rookies to keep them trapped in the lower ranks by virtue of their resulting weaker performances. But the subject is highly contentious in western music for sure.

I disagree. I really liked that this post focused on the subject matter, rather than the history. History can be interesting in itself, but I find that when trying to actually learn something it is just a layer of distraction. Maybe “casual“ reader would find it interesting to read about the history. But more technically inclined readers might then just bounce off the piece because they would have to weigh through history in order to get to the part which they actually found interesting.

I haven't read this book, but that takeaway doesn't sound right. In ET, all keys are equally out of tune--that's a natural consequence of it being equal. In contrast, it's just intonation in which chords become progressively further out of tune the further they get from the center.

It's true that ET is a compromise in some ways, but it actually opens up the possibility for radical modulations into more distant keys without having to adjust intonation on the fly. In that sense, just intonation is also a compromise.

If you have not read it, you might find the last part of the blog post most interesting. It covers shortcomings of just intonation.

Better reading IMO: Temperament by Stuart Isacoff. It is a small-ish music history book that covers the whole subject pretty thoroughly, with various tuning options that have been tried through the centuries, while treating the tradeoffs between them in a fair manner.

Right now I suspect that I failed in the same way as my early readings did: this article might be hard to understand unless you're already familiar with the topic

I really enjoyed this article (although I already knew the core concepts, so I may be confirming your suspicion, sorry!)

The visualizations are terrific, especially the ones with the The Lord of the Rings chord progression. Despite knowing the concepts, I hadn't fully appreciated how TET forms a tiling; visualized that way, the "wrap around" the circle of fifths really hits home.

Excellent work, thank you.

I think it is a really great article but I am a chef too already familiar with the ingredients and style of cooking.

I think the main issue is getting people to try new dishes.

Of course, the dishes have to be good and not just the chef fooling around. Sometimes this style seems like "hey try this new exotic dish! It is made from mixing organic ice cream, grass fed beef, fair trade coffee and sriracha. Oh you think it sucks? Well that is because your taste buds are use to standard western food!"

I think of how no one needs an intellectual explanation as to why pad thai is good even if they only ever had pizza and burgers.

I think you need some info in the start of the article about why 12tet was invented, what problems it was solving and how it fails. Like - In Bachs day you'd have to retune the instrument if you changed key signatures and you could only modulate a few steps away before the ratios sounded really bad. So 12TET opens up the possibilities to modulate far away from your original and requires less retuning, at the cost of adding various amounts of dissonance to intervals. I always find it interesting that each key signature in 12TET has a unique sound (Like CMaj vs EMaj) due to inconsistent ratios in perfect fifths, etc, and that "offness" is not necessarily a bad thing. You could also mention what most people think "microtonal" is (adding extra notes in between half steps) and why just intonation is different.

IMO music is all about the balance of the harmonic/expected and dissonant/unexpected at various timescales. The experience of Just Intonation is interesting because at first it sounds wrong due to us expecting 12TET but once you recalibrate your expectations it sounds more pure and harmonic, like how choirs or string quartets slip into simpler ratios instead of sticking exactly to 12TET.

I think with some more context/succinct summary at the beginning (which is REALLY HARD to write about for a general audience) the rest of the article would be more clear for people not steeped in music theory. A way I simplify it for others is by framing it all as simple ratios vs more complicated ratios, and how harmony is a kind of alchemy in mixing those ratios together in the right amounts.

> each key signature in 12TET has a unique sound (Like CMaj vs EMaj) due to inconsistent ratios in perfect fifths

I thought this wasn't true in 12TET, because in that case a perfect fifth interval is always a ratio of the twelfth root of two to power seven, or almost 3/2, no matter the starting point, C or E.

Sorry, that was an incorrect thing I was taught and I never checked it... The ratio of a perfect fifth in 12TET is always 1.498307, not exactly 1.5, but it's close enough to 3/2. The deviation from "simple ratios" is more or less depending on the interval (P5(3/2): -0.001692923, M3(5/4): 0.00992105, m3(6/5): -0.010792885).

I really appreciate at the end how you explained some of why 12TET is so popular. There was a time when it was the new kid on the block, and there's a reason it took over. Most articles of this type are so busy taking a dump on it that you can't get a clear answer to that question.

I am a firm believer in the idea that if you want to know how to improve on something, you must clearly understand where you are coming from with current practice... and if your understanding of "current practice" is that it is entirely bad with no redeeming qualities, you don't understand current practice. We see that a lot in the programming space; there's a recurring set of ideas like "everything should be visually programmed everywhere!" that always come from people who spend a lot of time complaining about text without demonstrating any understand that it is there for a reason, indeed, many reasons, and if you casually throw all those good reasons away in your attempt to be new and fresh, you are starting so far to the negative that you stand no hope of producing anything new and useful.

Personally I really appreciate 12TET's ability to wrap the harmonies around as you show, and I've never thought about it that exact way, so I also appreciate the visualization. That doesn't mean it needs to be the only tuning used by anyone, though.

On to the music itself. One thing that I have found when I listen to a lot of "microtonal" music is that for music that is nominally more "consonant", it ends up with things I would not call "dissonant" per se (at least, in the traditional ways that 12TET has dissonant intervals), but a whole bunch of intervals I'd just call sour. Now, I won't deny that despite my general openness to alternate tuning pretty much everything I listen to is 12TET and I'm adjusted to that just like everyone else, but when I listen to Baroque-period music in pre-12TET tunings (as they were likely performed then), I hear the differences versus 12TET, but I don't hear that sourness in those pieces. It seems to me that if the goal is to seek greater consonance in tunings (and I acknowledge those who do not have that goal, this post is specifically about those who are seeking that) I would be hearing music that actually is generally more consonant. I would be more interested in alternate tunings if the resulting music was not so sour.

I really appreciate that your "ambient garden" is both different from what I'm used to hearing, but also I find it quite approachable. Sure, a bit of adjustment is necessary, or at least helpful, but it seems like one could actually argue that there is indeed greater consonances, and interesting contrast to conventional intervals, in that music, rather than just... sour intervals everywhere.

My basic problem is that I'm really struggling to hear the difference between many of your samples :-) And I've already got a bit of experience with microtonality (DSP background, playing and have tuned multiple instruments, reasonable pitch perception). One of the big problems for me is that it's not so easy to hear them side-by-side; you have to stop one of them, then start the other (unless you want them both at the same time, which you probably don't), and both the start and stop buttons are pretty laggy. FWIW, most audio comparison tools I've seen (typically centered around compression or restoration) include some way of switching seamlessly between the two, without restarts.

If I may provide some constructive criticism as someone who also has a background in tuning and tuning theory: I believe the article is pretty cool and you clearly have a good background on modern microtonal music. However, it is also clear that you do not have a good background on historical tunings (there are some factual errors in this regard).

Modern sources often implicitly assume that you understand tuning theory before teaching it, I agree. The understanding they often expect is this historical theory, though. I find that understanding historical tuning based on the idea of scales to be sort of tough to understand, while the circle of fifths is a lot more useful, and before 1900, pretty much nobody wrote about tuning in terms of scales since for all practical purposes it is very easy to hear frequency relationships in fifths and fourths rather than hearing them from steps.

Just Intonation is built on top of the harmonic series (not the circle of fifths), and I would suggest that you introduce the concept of the harmonic series before thinking about scales and fixed temperaments (your "boring" solution). As someone who does understand Just Intonation pretty well, I found the idea of going straight to a concept of scales very confusing, personally. The scale for true just intoned tunings is built on top of the harmonic series relationships you showed later, and those don't all have nice intervalic ratios.

Factually, the intervals were wrong for a typical Pythagorean scale: your fifth is 2^18/3^11 when the correct relationship for the typical notion of Pythagorean tuning is 3/2. Your scale stacks fourths based on 4/3 relationships, leaving the most important interval in the scale as the so-called "wolf" fifth. Most practical notions of Pythagorean tunings used fourths going one way and fifths going the other from the tonic, leaving the F#-C# relationship as the "wolf" (if the tonic is C), while all other fifths are pure. If you illustrate this with a circle of fifths rather than a graph showing scale degrees, it's a lot easier to understand. Pythagorean tuning also isn't really a type of just intonation since it doesn't use a harmonic series on a single note, but the exact definition of "just intonation" varies by source (some would say that even a temperament like Werckmeister is "just intonation").

All that said:

I would suggest you start with some ideas about tuning, introduce the harmonic series, introduce that "just intonation" is about preserving harmonic series relationships, and then go straight to your "freestyle" section. Eschew the notion that you need to understand the history of unequal temperaments to understand the present concept of the tension between the equal-tempered compromise scale and the just-intoned harmonic series.

Since you're more interested in modern theory, and your innovation and interests are there, I think you can skip the history and just get straight to the math and the lattices. Whether you use circles of fifths if you do this is up to you (I'm guessing you shouldn't - they are not so useful for modern tuning ideas).

Thanks for bringing in your understanding of historical tunings. Maybe a simpler correction would be to describe the example as "a Pythagorean tuning" rather than "THE Pythagorean tuning", which would be (according to some) technically correct despite not being historically representative. The goal of that example isn't to provide historical background, it's to provide one of the simplest possible derivations of 12 usable tones from just intervals. I believe that that the 3 being a multiplier or a divisor is a matter of perspective: if you start from the last tone in my example, the same scale is derived from reversed fractions. Starting from the middle might be closer to what you're describing, hopefully I'm getting that right.

Grounding the whole thing in the harmonic series on would certainly make sense.

You are correct about the ratios not mattering that way, but it is jarring for someone who knows about this stuff to see 4/3 (the fourth) instead of 3/2 and to see the positioning of the wolf being the I-V relationship in the scale. I would suggest that if you want to keep it, you should be explicit that the frequency relationship on this fifth is far off, and I would suggest drawing a circle of fifths (which will be a lot more clear after seeing a fifth in the harmonic series). Instinctively, that section feels very wrong despite being (mostly) right. I would personally suggest dropping it entirely - you don't need the history.

Edit: I was thinking about your existing diagram a bit more, and here is the rework I might suggest without doing anything too major:

Do an up-a-fifth-down-a-fourth path through the chart, and that is easier to rationalize and understand than "mostly 4/3 with 2/3 ratios interspersed when we need to drop an octave":

C * 3/2 -> G

G * 2/3 -> D

D * 3/2 -> A

A * 2/3 -> E

E * 3/2 -> B

B * 2/3 -> F#

There is only one break from the pattern right here with two descending perfect fourths in a row:

F# * 2/3 -> C#

C# * 3/2 -> G#

...

You can also restart from C going down to F and Bb once you reach F# and that gives you the typical Pythagorean tuning, but then you will have to explain why that happens to readers who are unfamiliar with tuning.

I will admit that I already had some familiarity with the topic, but I feel you did an amazing job at explaining specifically the constraints you're under as a composer, and in elucidating how having digital tools doesn't magically solve the problems you face in harmonizing in just intonation over the course of a piece. I had not really understood that before reading your article, and at least suspected that we might transition to hearing mostly Just Intonation eventually as digital tools became more widely used.

Of course you could make it more accessible by starting from some even more basic principles and elaborating further on those foundations, but then you risk losing some of our audience that's already familiar with those. For me at least you struck a good balance. (No pun intended!)

For practical performance of alternate intonations, you can tune a guitar so that a particular chord shape presents perfect roots and 5ths, while arranging for a single string to be a perfect third (e.g, a few cents "flat" compared to equal temperament.)

When playing this same chord shape (or an appropriate derivative) up the neck, the relative intervals is maintained for other chords, allowing perfect intonation not just of the root chord but of other chords in the key family.

For chord shapes where a flatted string cannot be used as a 2nd or 3rd, it will sound objectionably flat, but for many chord shapes it's possible to fret the flat string with additional pressure to bring it back into tolerance.

Mastering this technique makes the guitar sound much more in tune with itself which is a wonderful sensation as a player.

(In case you didn’t know, it’s fun that) Jacob Collier plays a 5-string guitar in a tuning that takes advantage of this trick exactly. IIRC, from lowest to highest he tunes them a fifth, a fifth, a fourth and a fourth apart. By flatting the middle string a dozen cents or so, you can get a root-fifth-perfect tenth voicing of a major chord. Neat trick.

Speaking of Jacob Collier, this is a good time to mention the song where he “hacks” equal temperament to modulate to G-half-sharp without anyone noticing.

https://www.youtube.com/watch?v=Xd54l8gfi7M&t=3m17s

One more Jacob Collier mention: also generally releases albums using just intonation, I think, and especially does hand-tuning of thirds etc. I like his description of having to "monkey swing" when an e.g. flattened third needs to become a new root note.

Aren't you still fighting the fact that the frets themselves are tuned as 12TET?

The fight increases the faster/further you jump around the circle of 5ths. The 12TET fret spacing aligns octaves and 5ths with JI, so if you tune for a JI `I` chord, your other chords will be internally JI but only your `V` chord will be spaced perfectly from the `I` chord. The other chords will be closer to perfect JI the closer they are to `I` on the circle of fifths.

This microtuning isn't perfect in theory, but it's a favorable compromise for a lot of western music, especially for legato strumming where the inter-string intervals are heard more often than the inter-fret intervals.

Most guitars are not properly in equal temperament intonation, because that requires compensation at the nut.

In particular, an unwound G string can be off. I got sick of three decades of listening to that out of my main guitar, and earlier this year finally did something out of it with satisfactory results:

https://mstdn.ca/@Kazinator/112332540341043265

Without this string, I cannot get these G and D power chords to be in tune simultaneously:

  E   |---|---|-*-|
  B   |---|---|-*-|
  G  o|---|---|---|
  D  o|---|---|---|

  E  x|---|---|---|
  B   |---|---|-*-|
  G   |---|-*-|---|
  D  o|---|---|---|

If you tune the latter D, then the open G is too flat in the former G. If you sharpen the open G, the latter D is out of tune.

The problem is that when we fret the A note on the G string near the nut, it stretches; the string goes sharp. So the interval between the nut and that fret is wider than a tone, without the compensation that shrinks it.

The other strings have the problem, but the unwound G has it worse due to lower tension relative to mass. I'm so happy with the G compensation on that guitar, that it needs nothing else.

Anyways, there are videos out there which do A/B comparisons between ordinary guitars and just intonation. They feature uncompensated guitars, so they perpetrate a fallacy.

The guitarist must experience a properly intonated guitar with nut compensation first, then decide whether going to exotic intonations is worth it.

Here are three additional references on applied Just Intonation.

The first book is abstracted away from composition and provides a more general description of tuning systems.

The next two are composition focused, explaining how tuning system is an option for musical works.

1) The book "How Equal Temperament Ruined Harmony" provides excellent historical context on why ET was a great solution and how we kind of got stuck with it.

While it could be read by anybody, having a bit of musical knowledge OR physics of acoustics knowledge may benefit you.

https://www.amazon.com/Equal-Temperament-Ruined-Harmony-Shou...

There are also two composers who recognize ET as a constraint (as opposed to the "the only option").

2) Harry Partch built many of his own instruments to circumvent issues from 12ET and composed music just for them. He describes his conceptualizations of JI in "Genesis Of A Music":

https://www.amazon.com/Genesis-Music-Account-Creative-Fulfil...

3) Paul Hindemtith also has some opinions on the matter, though both Partch and I disagree with his points of view. Nonetheless he has some at-the-time innovating ideas, some of which are valid and others of which may not have aged well. You can read about them in "The Craft of Musical Composition"

https://www.amazon.com/Craft-Musical-Composition-Theoretical...

Sometimes I like writing stuff in odd scales and intonations. The "Lucy" tuning is also very pleasant [0]. I once met a producer who had guitars fretted to this tuning. It doesn't lend itself well to DAWs that have an underlying MIDI note logic. Practically it's better to work in Pd/Max with raw oscillators and arithmetic than to try shifting MIDI note values around with bend. The ZynAddSub engine (also used in Yoshimi synth) has a nice scales filter.

Probably worth mentioning Jacob Joaquin [1] who did this stuff in Csound back in the 90s. On one of his patches I based a generative music system for a game, it was the "Harmonic tree" - very similar to the system in the article - repeated multiplication by integer fractions at each branch.

Using simple timbres (like flute/organ/sine) you can make big but crystal clear sounding chords, and then slowly changing one "note" at a time you get an ever evolving soundscape that blows anything "Eno" off the map. Its in a sweet creative area between composition and additive synthesis.

[0] http://www.harmonics.com/lucy/lsd/colors.html

[1] https://github.com/jacobjoaquin

You can use the same construction as the Lucy scale with the periods of planets to produce true "music of the spheres"! Though the pitches tend to be clumpy, it does sound quite otherworldly.

Haven't read it fully (to be honest this topic is new to me), but it's funny that as I read this article, I also happen to listen to a microtonal music: Flying Microtonal Banana by King Gizzard & The Lizard Wizard.

Both the name of the album and the name of the band sounds funny, trust me when I say this: It's one of the best microtonal album out there. They really mixed the sound element of turkish(?) music into the album. Highly recommended!

Should be noted that any ensemble that consists only of continuous-pitch instruments (eg. strings, trombone, human voice, fretless guitar) is free from the shackles of 12TET and its harmonies naturally tune to as close to "pure" intervals as the performers’ skills allow.

The guitar doesn't event need to be fretless; there is a lot of leeway (even towards flat) even with frets if you know the technique!

Yes, this should be noted.

Naturally JI instruments include any instrument which has a continuous frequency domain, like trombone, fretless stringed instruments, and human voice.

Instruments with keys or frets may be "bendable". However, the direction and quantity of bend varies with with each tone it is paired with, making this incredibly impractical.

I never thought about this, does a choir left to its own devices naturally gravitate to JI rather than ET?

It seems to depend a lot on their age and experience. Years of only single 12TET will tend to pull them away from the JI "sound". However, I know a few choral singers who've tried (in midlife) to single JI and once they got over the initial "hump" they found it quite natural.

The "beating" that occurs in 12TET is inaudible to most western listeners because we've become so acculturated to it. You have to get some exposure to music where it doesn't happen before you even notice it (and thus seek out more consonant harmony).

Overall a good intro on the subject. I feel like at the start it might benefit from being a bit more explicit about pitches being frequencies and ratios being intervals; probably just worth a reminder for people who aren’t as familiar

Might be interesting to talk about how the usual ratios come from the harmonic series. For sounds that don’t produce a harmonic series, other potentially non-integer ratios can actually sound more consonant. The youtube channel New Tonality[0] has a bunch of great videos about this

Also wanted to mention that I’ve been working on a piece of commercial software[1] for working with freestyle/adaptive just intonation, if anyone’s interested

[0]: https://www.youtube.com/@new_tonality [1]: https://www.dmitrivolkov.com/projects/pivotuner/

Thanks for posting this. The approach of keeping the traditional 12 notes while changing the tuning on-the-fly is really interesting and maybe the right tradeoff. This is so spot-on because adaptive JI is mentioned as a possible solution at the end of the article. It's the classic HN move where a weird topic is being discussed and someone says "why yes, I've made tools that happen to exactly match the niche topic being discussed!"

In fact, it's probably not my business, but this topic is so profoundly niche (and likely to remain so for a while) that I personally wouldn't make this commercial. My reasoning (and the reason I open-sourced some projects) is that it will limit adoption while changing almost nothing about my income. Obviously I hope I'm wrong, maybe these techniques will pick up in popularity. It's also possible that I think this way simply because I'm bad at marketing.

yeah, in general I do want to open source it, but for the time being I’m a student and every little bit counts

This reminds me of when I stumpled into a music theory flame war by proposing the "scale": 1, 15/14, 15/13, ..., 15/8, 2. https://music.stackexchange.com/questions/66620/what-is-a-te...

This set of 8 nodes (can be transposed to whichever base frequency) has the lowest common denominator between any pair of nodes among all 8-node scales.

I thought that might make it a good choice for a just intonation scale (which can really be any subset of frequencies you build you music from), but it turned out a sensitive subject.

Why is it a sensitive subject?

Wait until you hear about A=432Hz

I've heard a little about this one. It's not that interesting, IMO. Some people think tuning even temperament down such that middle A becomes 432Hz (rather than 440Hz) is better for your health as well as the tone. I can get a little "woo" from time to time, but this one left me cold.

Obviously C4 should be 256Hz. This has the important property that it makes C-3 2Hz, which is another name for 120bpm.

In equal temperament, this makes A4 430.5Hz.

I think some people start from equal temperament. Then define a scale as the subset of those 12 nodes.

Any other subset of frequencies is just considered a "weird tuning" rather than a scale in its own right.

That's my best steel man. As you can see they deleted the question after a lot of discussion.

Oh hey, I've experimented a bit recently with the similar scale 1, 24:23, 12:11, ... 24:13, 2. Essentially a 12-tone version of yours. I've been calling it the "utonal" scale as it is a utonal [1] series. Complements what I call the "otonal" scale of 1, 13:12, 7:6, etc.

The trouble is, while intervals with the root are very consonant, arbitrary intervals don't sound great.

I've noticed though, that a utonal scale is very similar to the scale of harmonics [2], which is traditionally used relative to a moveable root note. So I intend to experiment in that direction.

[1] https://en.wikipedia.org/wiki/Otonality_and_utonality

[2] https://en.wikipedia.org/wiki/Scale_of_harmonics

What a marvelous article about intonation and temperament. It is one of those expositions that approach perfection in conveying the subject matter. I am no music theorist but rather an engineer with an appreciation for all things wave- or signal-oriented. I will want to read it multiple times to savor the explanations. The author's album based on just intonation is linked in the article ("A Walk Through The Ambient Garden") and I highly recommend it: it is a mesmerizing composition.

To save people a few clicks, there's a wonderful visualisation of the generated music at https://ambient.garden/

Readers who enjoyed this essay might also enjoy this one, which goes into further depth on the mathematics of musical intervals, the connection between the "blue note" in Equal Tempered jazz music, and more:

https://iopscience.iop.org/article/10.1088/0034-4885/72/7/07...

As an added bonus, the author draws an extended analogy between musical intervals and energy levels in quantum mechanics toward the end of the article, and indeed the study of the patterns in intervals between and among energy levels in the simple harmonic osciallator remains an active area of reasearch!

For more pieces in just intonation, check out https://en.wikipedia.org/wiki/List_of_compositions_in_just_i...

When I compose, I usually begin by sitting quietly and humming or singing out loud until I have a melody that I like. Because I was born and raised in the west, acoustically I'm naturally accustomed to a 12 tone equal temperament.

Atonal music is more difficult for me to grok, and I kind of imagine that for you to be fluent and compose in these sort of microtonal scales you would need to be introduced to it at a young age - in much the same way that it's significantly easier to learn a second language if you grow up in a bilingual environment.

I have no clue about this topic, but isn't Just Intonation what e.g. people sing when left to their own devices?

Yes

I'm no expert but your use of "atonal" and "microtonal" interchangeably here seems way off the mark.

"Atonal" usually refers to music composed without a fixed sense of key - and was originally used to describe music written using standard western 12TET.

It's a property of the music's relationship to key signatures - not it's tuning system.

This is interesting, but the harpsichord based samples really made me not wanna listen. But that’s probably just me disliking harpsichord

Would it be more appealing if we relabeled "harpsichord" as "pluck triangle synth"? :P

That would explain why it sounds so scratchy to me.

The choice of instrument is not entirely innocent. Harpsichords are rich in harmonics, and those harmonics are almost exactly multiples of the fundamental. This is perfect for revealing natural ratios in chords. Pianos have "bent" harmonics which would slightly obscure this. This is inharmonicity: https://en.wikipedia.org/wiki/Inharmonicity

If the author is here, the audio clips don't play for me. Firefox console has: Uncaught (in promise) TypeError: document.querySelector(...).contentDocument is null

Sure, I also use Firefox but I don't have this error, so that's not really enough detail for me to look into it.

I’m on iOS with Safari - no audio either.

Album plays fine on Bandcamp though!

Late reply but here's the full error in my console

   Uncaught (in promise) TypeError: document.querySelector(...).contentDocument is null
   innerElements https://www.osar.fr/notes/justintonation/ji.js?version=1723776197:14
   <anonymous> https://www.osar.fr/notes/justintonation/:70
   promise callback * https://www.osar.fr/notes/justintonation/:69

I feel like I can hear beat frequencies in the 12TET chords that I don't hear in the JI chord examples.

Anybody else hearing that? Or is that my imagination?

Yes!

We don't even have to listen to the audio examples to know that yes, ET chords produce beat frequencies. This is an artifact of the non-integer relationships created in ET chords that don't exist in the JI examples.

A piano will produce similar beat frequencies when playing a perfect fifth. Some ears are more sensitive to hearing it than others.

Thank you! I'm so glad to have that confirmed. I feel like I do hear them, and actually have come around to liking them. They feel quite different from what you get with a pure harmony in Just Intonation -- maybe the ephemerality of the flow of energy back and forth between different subharmonics adds to the excitement somehow.

It's hard for me not to think of the electric guitar in this context. I love the purity of an acoustic played in a plangent tone like Django. But the incredibly rich spectrum introduced when an amplifier saturates and tons of additional harmonics and subharmonics that aren't supposed to be there gets introduced -- that's it own kind of excitement. Like falling into an abyss aurally.

How do you feel about Ring Modulation? It’s another step into weird.

https://youtu.be/9PmHWap1T-E?si=9lbiTQSU1ARFBKeD

I don't know that much about ring modulation, but having read a bit about it, I think I love at least most of the uses of it I know. The guitar solo in Paranoid Android is one of my favorites in any record. But I love pretty much everything Jonny Greenwood has ever done.

Ring Modulation adds a constant frequency above and below the source frequency, producing non-integer relationships between the input signal and the new signal.

That's why it sounds weird, because both of the new tones will almost never be harmonically related to the input signal.

that delightful "abyss" is really just noise. as in white/pink/blue noise.

Signals tend to sound "better" (richer harmonic content) when they have fuzz added. Convolution reverb is a great example of how mixing a noise sample into any source signal increases its apparent depth or richness.

Some saturators do the same thing.

Others distort the signal by boosting selected harmonics (making the waveform more square-like or more sawtooth-like for example, which is not the same as noise and has a different sound).

Clipping the signal by requesting amplitudes beyond the available range is another way to add "cruch" (noise) to a signal.

The actual implementation will vary per device but overall we like our sounds cut with a bit of noise.

One of the linked music examples, “occultation” by Pure Code, is beautiful ambient music.

“ This also has the reputation of being terribly impractical, which is always a selling point for me.”

That was truly LOL. Thank you.

posting related vinyl record collection comic: https://www.ratherrarerecords.com/wp-content/uploads/AlexGre...

Avez-vous la version française, monsieur ?

When I started with violin, since my academic major was math, looked into the 12th root of 2 and ratios of small whole numbers, dissonance from beats, etc. stuff.

Then I learned (partly from a really good violinist):

(1) On violin, can use the ratios of small whole numbers to check intonation. So, get to check the intervals of octave, fifth, fourth, major third. That is, play two notes, each on its own string, together, listen for the beats, and adjust pitch (move a finger) until the beats go away.

That approach starts with the tuning of the violin, notes G, D, A, E, each on its own string, starting with the first G below middle C and going up by fifths (frequency ratio 3/2). The beats are easy to hear, and making the beats go away is what essentially all violin players do, whatever slightly different frequencies might come from other approaches to tuning.

(2) For the rest, there is a single, simple overriding principle: Make it sound good. Uh, there is no practical way to play a violin and get all the frequencies anywhere except close, varying with vibrato, and sounding good!

(3) The math of the 12th root of 2, etc. is too precise for violin -- can't expect to play frequencies to such precision and with vibrato, on that wooden box, with flesh fingers, real violin strings, bow rosin, etc. and really don't need to.

E.g., for the math, there is no good guarantee that a note played on violin is really accurately what the math and physics (a solution of a certain differential equation) define as a periodic function -- thus, lots of considerations of overtones start to become imprecise and, thus, moot.

Again, make it sound good! So, sure, for a major third, ratios of small whole numbers can be relevant, but for a minor third, mostly just guess at the semi-tone and make the interval sound good!

(4) For semi-tone guessing: The famous Bach E-major Partita starts with E (one octave above the open E string, half way along the string), D# (down one semi-tone), and then the E again. So, it's the three notes, played quickly, with the D# inserted and left quickly, with the musical effect an introductory bright heralding -- the relevance of the semi-tone as the 12th root of 2 is lost and instead just play the E with the little finger and for the D# squeeze in the next finger until it passes for a semi-tone and gets the heralding musical effect.

(5) For much more, can play the Bach Chaconne with its many chords, sometimes on all 4 strings -- no way can a violinist try to honor some precise tuning arithmetic for all the intervals played. E.g., the piece starts in D minor and with the "D minor" chord, that is, D, F, A. On a piano that would be the first D above middle C, the first F, that is, 3 semi-tones, above the D, and the A, a fifth above the D. But on violin that would have both the D and the F on the same string, so Bach put the F an octave higher, on the E string, a semi-tone above the E string. Sooooo, to play that opening chord, play the D and the A together, and, continuing right away all with one down bow, with the A and the F together. Then the F is from the index finger at a guess of a semi-tone above the E string. So, the bow starts on the D and A strings, both open (no fingers on the strings) and, thus, brilliant, and then both the open A string and the fingered F. So, the A and F 'would' be just an okay major third but the F is up an octave so that the A and F are an interval of 8 semi-tones and, actually, dissonant. Since the F is on the E string which is still brilliant even though fingered and not open, all three strings are brilliant, and the opening is dramatic and in a minor key. Some of the piano and orchestra arrangements make this first chord dramatic!

In all that for that opening chord, on violin, the 12th root of 2 and ratios of small whole numbers don't play a central role, and the same for the rest of the piece, nearly all chords.

Sure Bach knew very well what he was doing, indeed, in that music, SHOWING that could make music in all the keys, and had all those pages and pages, for violin, in another book, for cello, in another book, piano with the intervals and chords "sounding good".

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