7 ms·

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, harmoni

• thrtythreeforty 2y ago
  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.
  • pierrec 2y ago
    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.
    • thrtythreeforty 2y ago
      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.
  • pyinstallwoes 2y ago
    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?
    • bongodongobob 2y ago
      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.
      • pyinstallwoes 2y ago
        Yeah but you don't have the same knowledge apparently.
        • bongodongobob 2y ago
          Same knowledge of what?
  • munchler 2y ago
    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.
• thrtythreeforty 2y ago
  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.
• wonger_ 2y ago
  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
  • lewispollard 2y ago
    > 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.
  • colanderman 2y ago
    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#Acoustics_and_psychoacoustics https://en.wikipedia.org/wiki/Consonance_and_dissonance#Acou...
  • jfengel 2y ago
    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 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.
• bigstrat2003 2y ago
  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.
  • JohnKemeny 2y ago
    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.
  • lewispollard 2y ago
    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.
  • javier123454321 2y ago
    I thought this video was good at explaining and showing the difference audibly https://youtu.be/zJTBT5u9AwQ?si=M3csNlck4El_eluH https://youtu.be/zJTBT5u9AwQ?si=M3csNlck4El_eluH
  • black_knight 2y ago
    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.
  • Levitz 2y ago
• sonorous_sub 2y ago
  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.
  • black_knight 2y ago
    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.
  • smrq 2y ago
    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.
• iainmerrick 2y ago
  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.
• eltoxo 2y ago
  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.
• klik99 2y ago
  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.
  • banana-lambda 2y ago
    > 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.
    • klik99 2y ago
      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).
• jerf 2y ago
  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.
• Sesse__ 2y ago
  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.
• pclmulqdq 2y ago
  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).
• riemannzeta 2y ago
  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!)