by glitcher on 3/27/24, 5:28 PM with 84 comments
by FLT8 on 3/27/24, 6:29 PM
If you've ever tried building an additive synth, this probably seems obvious. If you play two tones with exact ratios together, the brain has a tendency to interpret them as a single tone with a different timbre. By making the ratios slightly off (don't have to be by much at all), or by making the ratios vary a little over time (eg. Subtle vibrato), the brain is able to pick out the "consonance" as a chord, and it sounds far more interesting.
by mrob on 3/27/24, 6:53 PM
William Sethares developed a model of consonance and dissonance that accounts for this:
https://sethares.engr.wisc.edu/paperspdf/consonance.pdf
Informal explanation:
by leblancfg on 3/27/24, 6:16 PM
by re on 3/27/24, 6:26 PM
The other article posted also has a audiovisual version of some of the study findings which helps clarify the results: https://phys.org/news/2024-02-pythagoras-wrong-universal-mus...
by ben7799 on 3/27/24, 6:36 PM
Whenever we find something new people like Music theory has to adapt.
I think it would be impossible to prove any unifying theory like this unless you can actually test all possible things someone could hear and western scales & such obviously don't encompass all that.
by Nav_Panel on 3/27/24, 8:25 PM
One tidbit I always think about:
> I've had interesting experiences playing just-intonation music for non-music-major students. Sometimes they will identify an equal-tempered chord as "happy, upbeat," and the same chord in just intonation as "sad, gloomy." Of course, this is the first time they've ever heard anything but equal temperament, and they're far more familiar with the first sound than the second. But I think they correctly hit on the point that equal temperament chords do have a kind of active buzz to them, a level of harmonic excitement and intensity. By contrast, just-intonation chords are much calmer, more passive; you literally have to slow down to listen to them. (As Terry Riley says, Western music is fast because it's not in tune.) It makes sense that American teenagers would identify tranquil, purely consonant harmony as moody and depressing. Listening from the other side, I've learned to hear equal temperament music as a kind of aural caffeine, overly busy and nervous-making. If you're used to getting that kind of buzz from music, you feel the lack of it as a deprivation when it's not there. But do we need it? Most cultures use music for meditation, and ours may be the only culture that doesn't. With our tuning, we can't.
> My teacher, Ben Johnston, was convinced that our tuning is responsible for much of our cultural psychology, the fact that we are so geared toward progress and action and violence and so little attuned to introspection, contentment, and acquiesence. Equal temperament could be described as the musical equivalent to eating a lot of red meat and processed sugars and watching violent action films. The music doesn't turn your attention inward, it makes you want to go out and work off your nervous energy on something.
by andoando on 3/27/24, 7:10 PM
Even considering deviations/slight variations from ratios:
1. You have to consider that these are deviations from the mathematical ratios. 2. The variations themselves can follow ratios. After all, not all deviations or variations are equally interesting.
For example if you were to play 1xxx2xxx3xxx4
you can vary the next verse 1xx2xxxx3xxx4 and the next 1xxxx2xx3xxx4
Here the second note is oscillating one before the "ideal" ratio and one after.
by mingus88 on 3/27/24, 8:26 PM
I don’t want to sound cynical but ethnomusicology is a field unto itself and the idea that western tonal harmonies are a very small piece of the pie is something we’ve talked about for at least a hundred years
Even the concept of just intonation for a western scale is well established
by odyssey7 on 3/27/24, 6:23 PM
Clearly a revival of the quadrivium is needed.
by dsign on 3/27/24, 6:59 PM
With that said, just as languages tend to have culturally accepted "proper form" --grammar and spelling--so does music. Music theory is about what spelling and grammar the locals accept and like in their music, and is worth studying for that reason alone.
Now, I wonder what would Pythagoras think of our current musical genres...
by dr_dshiv on 3/27/24, 10:22 PM
You’ll find some juicy details on Pythagoras (his Olympic days, for instance) as well as an overview of harmony in neuroscience, computer science, aesthetic theories, etc.
by zigzag312 on 3/28/24, 8:10 AM
> "these preferences for slight deviations disappeared upon elimination of the upper harmonics, presumably because this eliminates the slow beating effect"
It seems like the title is oversimplifying results for the sake of clickbait sensationalism.
by reilly3000 on 3/27/24, 7:15 PM
by HPsquared on 3/27/24, 8:00 PM
by igorbark on 3/27/24, 7:10 PM
some key words/links to get you started:
- "local consonance"
- "consonance/dissonance curves"
- a seminal paper: https://sethares.engr.wisc.edu/paperspdf/consonance.pdf
- a more recent re-implementation with a cool video at the end: https://www.sebastianjiroschlecht.com/post/ondissonance/
the basic idea being, different timbres lend themselves differently to different tuning systems. so we can parameterize our models of tuning systems based on timbre
an important thing to keep in mind: consonant/dissonant doesn't mean "good/bad" or "pleasant/unpleasant". they're the output values of a mathematical model which we have a complex intuitive relationship with. other ways of thinking about it might be "simple/complex", "resolved/unresolved", "release/tension", but all are inaccurate in their own way
some areas i'd love to see progress in: - the work i've seen focuses on computational models, i.e. take a simple mathematical model of timbre, and directly compute the consonance/dissonance curve from it. but real instruments' timbre varies across many dimensions, some prominent ones being pitch, time, and dynamics. can we instead burn some CPU cycles and generate curves from a waveform? - what does this look like for triads? tetrads? ...? - put this in the browser! would make it so much easier to play with and present the ideas to less technical audiences - how can we use this to generate new instruments? can a synth automatically adjust its tuning system based on its parameters? can we start from a set of desired consonant/dissonant intervals and generate an instrument with a matching curve?
by riffic on 3/27/24, 6:56 PM
by graycat on 3/27/24, 8:24 PM
Gee, I'm plenty good with the math and applications of Fourier theory. Sure, some physics course solved the differential equation of vibrating strings. Early on my career did well with Fourier theory, the FFT (fast Fourier transform), and Navy ocean data. And with violin, I got a start in the music school of Indiana University and eventually made it through several pages of the Bach E-major preludio, the Bach Chaconne, and a transcription, as I recall, up an octave and a fifth, of that famous Bach solo cello piece. Been known to take the positive whole number powers of 2^(1/12). Dissonance? Sure, I really like the S. Barber Adagio for strings!!! R. Wagner's chords. The Franz Schmidt Intermezzo to his Notre Dame. The Sibelius violin concerto.
But, stilllll, a lot of your terminology I can't follow.
For an opinion: Mostly music is art, that is communication, interpretation of human experience, emotion. Given the roles of, say, vibrato, Fourier theory and 2^(1/12) do not directly explain all of the art.
For a perfect fifth, sure, as is standard, when take the violin out of its case, first tune the four strings G, D, A, E in perfect fifths and do this by listening for beats in the overtones. Then with the strings so tuned, often when the music calls for one of those 4 notes, just go ahead and play it on the open (look, Ma, no use of fingers on the left hand) string.
In particular, for
> perfectly in mathematical ratios
in real music from before Bach to after Barber, I doubt that "perfectly" happens very often or is even attempted. E.g., with violin, can't play much music if play only on the open strings, and as soon as put a finger of the left hand on a string, with vibrato or even without, there won't be much "perfectly mathematical".
by uwagar on 3/27/24, 6:40 PM
by jimbob45 on 3/27/24, 6:15 PM
Sounds like the author doesn't realize you can tune a piano. Actually, it doesn't sound like they know anything about music because the octave is surely universally harmonic.
by blitzar on 3/27/24, 7:09 PM
by srcreigh on 3/27/24, 7:29 PM
Study link: https://www.nature.com/articles/s41467-024-45812-z
Start with figure 3. Non-musicians generally favour pythagorian style harmonic ratios, while trained musicians tend to prefer equal temperament.
Each of the following intervals in the data confirm this:
- The pythagorian major 3rd ratio 5/4 is slightly flat compared to 5 semitones. Non-musicians nailed it, musicians prefer the sharp ET version.
- The range from 10-11 semitones is rated relatively high by non-musicians. This range includes the 7th harmonic, 7/4
- Even the 11th harmonic, taking place between 6(tritone) and 7(perfect fifth) is rated highly by non-musicians relatively to musicians.
- The "9" major 2nd octave, 9/4 JI, aka 14 semitones, is tuned more perfectly by non musicians (9/4) compared to musicians
Later the authors experiment with a artificial gong style instrument (fig 5). However its harmonics are remarkably close to several standard pythagorean ratios. You see 1.52f0 (slightly sharp perfect 5th 3/2); 3.46f0 (slightly flat harmonic 7th 7/4); 3.92f0 (slightly flat double octave 4/1). These harmonics are so close to a guitar or piano or human voice, it can't really even be considered inharmonic at all.
With this instrument, you get, yet again, more pythagorean intervals:
- sharp major 2nd, 9/8 ratio (makes sense; started with a sharp 3/2 and 9/8 is just (3/2)^2 modulo octaves).
- People still correctly tune the major 3rd 5/4 ratio (slightly less than 5 semitones) even without a 5th harmonic. bonkers
- Sharp 5th, which makes sense as the harmonic is 1.52f0 instead of 1.5f0
- Heavily emphasized harmonic 7th interval between 9 and 10 semitones
Later, the authors experiment with various mis-tunings of the pythagorean intervals (Fig 8). The most highly rated intervals are slightly sharp or flat from JI. This is evidence for JI intervals being pleasing--slightly sharp or flat creates a pleasing rhythm of Just harmony. It's like just harmony but better.
In conclusion, essentially every plot in the study points towards pythagorean, mathematically tuned harmony being more pleasing. Even in surprising ways. I could not find a single data point indicating otherwise, except for indoctrination of musicians towards ET leading to things like preferring sharp major 3rds and not recognizing the harmonic 7th interval.
by pfgallagher on 3/27/24, 6:29 PM
by henjodottech on 3/27/24, 6:40 PM