7 ms·

What I Learned Writing an Album in Just Intonation

159 points by pierrec 2y ago

vunderba 2y ago
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.
- eternauta3k 2y ago
  I have no clue about this topic, but isn't Just Intonation what e.g. people sing when left to their own devices?
  - handity 2y ago
    Yes
- andybak 2y ago
  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.
pierrec 2y ago
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.
- 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.
  - 2y ago
benwen 2y ago
“ This also has the reputation of being terribly impractical, which is always a selling point for me.” That was truly LOL. Thank you.
- an_aparallel 2y ago
  posting related vinyl record collection comic: https://www.ratherrarerecords.com/wp-content/uploads/AlexGregory_cartoon_vinyl_1000.jpg https://www.ratherrarerecords.com/wp-content/uploads/AlexGre...
rob7cc 2y ago
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.
- keybits 2y ago
  To save people a few clicks, there's a wonderful visualisation of the generated music at https://ambient.garden/ https://ambient.garden/
thomasahle 2y ago
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-temperament-in-the-most-general-sense/66635 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.
- the_other 2y ago
  Why is it a sensitive subject?
  - pyinstallwoes 2y ago
    Wait until you hear about A=432Hz
    - the_other 2y ago
      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.
    - jameshart 2y ago
      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.
  - thomasahle 2y ago
    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.
muhrizqiardi 2y ago
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!
lovegrenoble 2y ago
Avez-vous la version française, monsieur ?
Sharlin 2y ago
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.
- kaapipo 2y ago
  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!
- cchi_co 2y ago
  [dead]
- naltroc 2y ago
  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.
- Levitz 2y ago
  I never thought about this, does a choir left to its own devices naturally gravitate to JI rather than ET?
  - PaulDavisThe1st 2y ago
    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).
kemmishtree 2y ago
[flagged]
nonrandomstring 2y ago
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 http://www.harmonics.com/lucy/lsd/colors.html [1] https://github.com/jacobjoaquin https://github.com/jacobjoaquin
- colanderman 2y ago
  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.
carterschonwald 2y ago
This is interesting, but the harpsichord based samples really made me not wanna listen. But that’s probably just me disliking harpsichord
- cchi_co 2y ago
  [dead]
- naltroc 2y ago
  Would it be more appealing if we relabeled "harpsichord" as "pluck triangle synth"? :P
  - carterschonwald 2y ago
    That would explain why it sounds so scratchy to me.
- pierrec 2y ago
  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 https://en.wikipedia.org/wiki/Inharmonicity
cchi_co 2y ago
[dead]
naltroc 2y ago
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-Should/dp/0393334201 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-Fulfillments/dp/030680106X 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-Part/dp/0901938300 https://www.amazon.com/Craft-Musical-Composition-Theoretical...
elevation 2y ago
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.
- cscheid 2y ago
  (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.
  - wrs 2y ago
    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 https://www.youtube.com/watch?v=Xd54l8gfi7M&t=3m17s
    - cjbprime 2y ago
      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.
- bwanab 2y ago
  Aren't you still fighting the fact that the frets themselves are tuned as 12TET?
sporkl 2y ago
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 https://www.youtube.com/@new_tonality [1]: https://www.dmitrivolkov.com/projects/pivotuner/ https://www.dmitrivolkov.com/projects/pivotuner/
- pierrec 2y ago
  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.
  - sporkl 2y ago
    yeah, in general I do want to open source it, but for the time being I’m a student and every little bit counts
flobosg 2y ago
For more pieces in just intonation, check out https://en.wikipedia.org/wiki/List_of_compositions_in_just_intonation https://en.wikipedia.org/wiki/List_of_compositions_in_just_i...
hypertexthero 2y ago
One of the linked music examples, “occultation” by Pure Code, is beautiful ambient music.
crtasm 2y ago
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
- pierrec 2y ago
  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.
  - pbronez 2y ago
    I’m on iOS with Safari - no audio either. Album plays fine on Bandcamp though!
  - crtasm 2y ago
    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
riemannzeta 2y ago
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?
- naltroc 2y ago
  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.
  - riemannzeta 2y ago
    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.
    - pbronez 2y ago
      How do you feel about Ring Modulation? It’s another step into weird. https://youtu.be/9PmHWap1T-E?si=9lbiTQSU1ARFBKeD https://youtu.be/9PmHWap1T-E?si=9lbiTQSU1ARFBKeD
      - riemannzeta 2y ago
        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.
riemannzeta 2y ago
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/10.1088/0034-4885/72/7/076001 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!
graycat 2y ago
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".

159 points by pierrec 2y ago

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.

eternauta3k 2y ago
I have no clue about this topic, but isn't Just Intonation what e.g. people sing when left to their own devices?
- handity 2y ago
  Yes
andybak 2y ago
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.

pierrec 2y ago

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.

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.
- 2y ago

benwen 2y ago

“ This also has the reputation of being terribly impractical, which is always a selling point for me.” That was truly LOL. Thank you.

an_aparallel 2y ago
posting related vinyl record collection comic: https://www.ratherrarerecords.com/wp-content/uploads/AlexGregory_cartoon_vinyl_1000.jpg https://www.ratherrarerecords.com/wp-content/uploads/AlexGre...

rob7cc 2y ago

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.

keybits 2y ago
To save people a few clicks, there's a wonderful visualisation of the generated music at https://ambient.garden/ https://ambient.garden/

thomasahle 2y ago

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-temperament-in-the-most-general-sense/66635 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.

the_other 2y ago
Why is it a sensitive subject?
- pyinstallwoes 2y ago
  Wait until you hear about A=432Hz
  - the_other 2y ago
    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.
  - jameshart 2y ago
    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.
- thomasahle 2y ago
  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.

muhrizqiardi 2y ago

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!

lovegrenoble 2y ago

Avez-vous la version française, monsieur ?

Sharlin 2y ago

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.

kaapipo 2y ago
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!
cchi_co 2y ago
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naltroc 2y ago
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.
Levitz 2y ago
I never thought about this, does a choir left to its own devices naturally gravitate to JI rather than ET?
- PaulDavisThe1st 2y ago
  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).

kemmishtree 2y ago

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nonrandomstring 2y ago

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 http://www.harmonics.com/lucy/lsd/colors.html [1] https://github.com/jacobjoaquin https://github.com/jacobjoaquin

colanderman 2y ago
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.

carterschonwald 2y ago

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

cchi_co 2y ago
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naltroc 2y ago
Would it be more appealing if we relabeled "harpsichord" as "pluck triangle synth"? :P
- carterschonwald 2y ago
  That would explain why it sounds so scratchy to me.
pierrec 2y ago
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 https://en.wikipedia.org/wiki/Inharmonicity

cchi_co 2y ago

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naltroc 2y ago

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-Should/dp/0393334201 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-Fulfillments/dp/030680106X 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-Part/dp/0901938300 https://www.amazon.com/Craft-Musical-Composition-Theoretical...

elevation 2y ago

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.

cscheid 2y ago
(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.
- wrs 2y ago
  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 https://www.youtube.com/watch?v=Xd54l8gfi7M&t=3m17s
  - cjbprime 2y ago
    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.
bwanab 2y ago
Aren't you still fighting the fact that the frets themselves are tuned as 12TET?

sporkl 2y ago

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

pierrec 2y ago
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.
- sporkl 2y ago
  yeah, in general I do want to open source it, but for the time being I’m a student and every little bit counts

flobosg 2y ago

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

hypertexthero 2y ago

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

crtasm 2y ago

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

pierrec 2y ago
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.
- pbronez 2y ago
  I’m on iOS with Safari - no audio either. Album plays fine on Bandcamp though!
- crtasm 2y ago
  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

riemannzeta 2y ago

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?

naltroc 2y ago
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.
- riemannzeta 2y ago
  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.
  - pbronez 2y ago
    How do you feel about Ring Modulation? It’s another step into weird. https://youtu.be/9PmHWap1T-E?si=9lbiTQSU1ARFBKeD https://youtu.be/9PmHWap1T-E?si=9lbiTQSU1ARFBKeD
    - riemannzeta 2y ago
      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.

riemannzeta 2y ago

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/10.1088/0034-4885/72/7/076001 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!

graycat 2y ago

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".