The paradigm I would have liked to see is where two tones are held constant and the participant moves a third tone to the location that sounds the best or the worst.
It is remarkable work. But I don't believe it. Octaves are just too basic a phenomena to be viewed as a cultural invention.
Octaves are consonant or dissonant depending on the timbre of the sound. For sounds produced by harmonic instruments, like a vibrating string or air vibrating inside a long tube, the sound is a superposition of waves whose frequencies are integer multiples of a fundamental frequency. Then, playing two sounds an octave apart will match exactly all these frequencies and it will sound nice. But there are other instruments (not privileged in the western music tradition) whose timbre is not composed of integer multiples of a fundamental frequency; and in these instruments octaves sound very dissonant.
You can argue that the octave is a "basic phenomenon" inasmuch a vibrating string is basic. Yet, from the point of view of a person who uses a synthesizer, the octave has nothing special with respect to other intervals.
I might compare this non-harmonic genre with the artist Sevish, whose work I've enjoyed a great deal. He too tries to step out of the realm of 12-tone equal temperament. Some of his work is quite reminiscent in structure to this gamelan music (e.g. Desert Island Rain). However, he also manages to build melodies and something you could probably call chord progressions (though I wouldn't know), in a new and very foreign musical world (his entire album Harmony Hacker is amazing). Besides the music being amazing, getting used to this new landscape is enjoyable in and of itself.
My point about the drums, though, was more aimed at what scales, chord progressions and melodies might be developed by a species who had as harmonic basis the drum's spectrum of resonant frequencies. We, by comparison, have the integer multiple of some base frequency, the canonical "harmonic sequence", whereas for them it would be quite different: https://en.wikipedia.org/wiki/Vibrations_of_a_circular_membr...
I don't think this level of consonance is necessarily something one has to consider as sounding "nice". It could just as well be thought of as sounding "hollow" or "uninteresting" or "weak", compared to more dissonant harmonies.
I have found that the "worst" audible beating has frequencies between 6hz and 20hz. Higher than 20hz you do not perceive it, and lower than about 6hz it becomes an agreeable "tremolo" instead of as annoying beating. Thus, it would seem that when the frequencies of two notes differ between 6 and 20hz you get the worst dissonance possible. When you are in the middle of the scale, this is more or less about a semitone.
So, what are they? I'd like to know.
Pipes and strings are better modelled by a 1D resonator, which is more likely to allow integer overtones - although they can still have inharmonic elements due to stiffness and - in the case of orchestral strings - rotation caused by the scraping bow.
None of which changes the fact that octaves are primary in any instrument which produces a range of pitches with a clear and reasonably sustained fundamental.
In fact researchers rely on the concept of pitch chroma/pitch class to distinguish between absolute frequency. Humans reliably hear the octave/not octave distinction, as do some animals. Obviously the animals aren't musically trained.
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5479468/
But it's not enough to say that "octave perceptions are learned." In this experiment it's more likely that octave use is learned in musical contexts.
There's a huge difference between saying that someone can't hear octave relationships at all, and that they can hear them perfectly clearly, they just don't find them culturally relevant - perhaps because their music theory is built on a grid of absolute pitches, and not on a repeating pattern of pitches.
EDIT: Actually adding 23hz is probably way too much for the result not to just sound dissonant.
In my case none of the harmonics are an integer multiple of the perceived overall pitch. And while the waveform will be aperiodic, it shows clear signs (on visual inspection) of similarity at the perceived frequency.
So pianos have at least 2 things working against having perfectly consonant chords.
Similarly, I was once convinced everyone in our jazz-rock band was out of tune after I had experimented with alternative tunings at home the whole day. Yeah, equal temperament is really quite a bit out of tune, but somehow we seem to manage just fine.
For example if you were singing a major second above someone else, and then you had to jump a perfect fifth and they had to jump a major sixth to end up singing in unison. At least one of those intervals is going to be out of tune.
So, one always tries to understand the harmonic function of the choir as a whole in relationship with the structure of the piece. For example, if tension is needed, one can sing even more towards a dissonant diminished fifth. Close harmony is all about that.
https://www.youtube.com/watch?v=TYhPAbsIqA8
So in a choral context, you would typically aim to be in tune with the people singing simultaneously. The root note of each chord is pitched according to equal temperament.
This has its downsides as well. Consider the chord progression Em -> A7: both chords contain the note G, which in the first chord acts as a minor third, and in the second chord as a minor seventh. The minor third should be pitched 16% of a semitone (cents) above equal temperament, and if we're doing really just intonation and pitching minor sevenths as harmonic minor sevenths, the minor seventh should be pitched 31 cents below equal temperament. So consider a voicing where one voice should hold a G across both chords: this means that even though it's singing the "same" note, the voice should drop 47 cents when the chord changes - almost a full quarter tone!
This video from the excellent Voces8 ensemble has an example of how this might sound, at 55:50 (between the first two notes):
https://www.youtube.com/watch?v=dDXbQ-2_sns
I highly recommend watching the whole thing, it goes into a lot of detail about the practical implications of singing in just intonation. For the problem described above, one solution is to avoid the issue by tuning minor sevenths using equal temperament in tricky cases like this one.
https://images.squarespace-cdn.com/content/v1/5230e9f8e4b06a...
I am skeptical of this. I maintain that 1:2 is special. I would love to find more evidence or resources about this.
https://raw.githack.com/CindyJS/ScaleLab/master/index.html
There's the book by Dave Benson [0] (available online), mostly about the mathematical modeling of instruments, that has a nice ethnomusicological compendium of instruments with weird timbres.
And then there's the infamous book by Sethares [1] that is all about the dependency of harmony on timbre.
[0] https://homepages.abdn.ac.uk/d.j.benson/pages/html/maths-mus...
I will only say that my "bias" got me what I wanted-- evidence about the phenomena! (And thank you for that)
EDIT: Regarding the basis for consonance/dissonance, the mathematical part of it is straightforward. When superposing pure waves of close frequencies you obtain beating (a slow frequency modulation of the amplitude of your sound), and beating does not appear when you superpose pure waves of very different frequencies, regardless of the interval, integer or not. Thus, the only dissonant intervals of pure sinusoidal waves are those that are very close to the unison. If you compound this with the fact that western instruments have harmonic spectra, you see why some intervals are consonant and dissonant: the dissonant intervals are those that have some partials that are close, but not exactly, unison.
Or...the human voice?
That is certainly not the case. The superposition of waves is linear, and waves of separate frequencies do not really "interact" besides being linearly combined. The fact that a waveform is periodic is not perceptible in our ears. We hear the frequencies separately and the ratio between them does not matter.
This is not a theoretical concept, but a physical observation that anybody can check. You can readily try it with a synthesizer. Create two waveforms f(x)=sin(x)+sin(2x) and g(x)=sin(x)+sin((2+e)x) where "e" is a small irrational number. The wave f is periodic and the wave g isn't. They sound just the same: you cannot tell which one is periodic just by hearing them.
The fact that those people don't rate consonant chords as more pleasant than dissonant ones is also interesting.
Also, cursorily reading a study that contradicts one's preconceptions and going like "nah, my gut feeling doesn't agree with it" is so quintessentially Hacker News ;-)