What are these instruments? I guess their behavior has to be described by some interesting partial derivatives equations that give rise to non-integer multiplies harmonics.

So, what are they? I'd like to know.

Bells, gongs, drums, and other instruments that can be a modelled with a 2D surface that vibrates in an assortment of simultaneous modes are semi-harmonic.

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.

Also worthwhile to point out that brainwave coupling is based on "octaves" (i.e., roughly speaking -- gamma is double frequency of beta, which is double alpha, which is double theta, which is double delta....)

Just from experimenting with synthesizers, I’ve noticed that the perceived pitch of a tone is defined more by the relative frequencies of the harmonics, rather than their absolute frequencies. So, you can take a harmonic series like 100Hz, 200Hz, 300Hz... and add a constant to each of them and get something like 123Hz, 223Hz, 323Hz... and it will still sound more like 100Hz than 123Hz.

EDIT: Actually adding 23hz is probably way too much for the result not to just sound dissonant.

You are probably observing the missing fundamental phenomenon: https://en.wikipedia.org/wiki/Missing_fundamental

I think that is different – that’s simply the case where all the harmonics are integer multiples of the fundamental, but the fundamental itself is missing. Every cycle of the waveform will be identical, so the period will unambiguously be the frequency of the missing fundamental.

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.

There's a double bass player trick for when you don't have a low C extension to play the low note up an octave plus the note a fourth below. This creates an illusion of the low note being present. It works best with a bow since it's hard to get the simultaneous sounding of the notes perfect pizzicato.

A piano is ever so slightly inharmonic. This is due to the rigidity of the strings. Only a massless, 100% flexible string has perfect overtones.

You probably know this, but due to the 12 note tuning, most intervals on a piano are slightly off from perfect intervals. EG, on a string instrument or with voice it would be natural play notes a major third away such as C and E with a frequency ratio of exactly 5 to 4. There's no way to get every natural ratio exactly right and have 12 notes in an octave, so most of the intervals on a piano are tuned to approximations of an exact frequency ratio.

So pianos have at least 2 things working against having perfectly consonant chords.

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

Which is why I once sang in a choir for a couple of years. Nothing beats perfect harmony.

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.

When you're singing in perfect harmony, are you trying to be in tune with the people singing simultaneously to you or with the previous note sung? Presumably there are times when you can't do both at once.

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.

Oh yes, sometimes you actually notice this, often when approaching a certain note from above and below. Depending on the harmonic function, the major seconds are not always the same size and you end up singing 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.

If you try to keep in tune with previous notes, you will often start wandering away from the original key. This video has a good example:

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.

Really nice! Just got a chance to watch it. Such a nice explanation and he gives very nice examples of how intonation is a puzzle and a bit of an art, even for choirs.

Maybe not so much for a whole band, but for the classical guitar, people seem to tune the instrument slightly different for each piece (in a given key) and sometimes in the middle of a performance! Anecdotally, moving from something in e to d sounds terrible without adjustment.

But equal temperament maintains perfect octave structure. Everything else shifts.

And you could tune a piano that way. But it would only be in tune for the particular key that you tuned it to. So if it was tuned to C (as is normal for a piano) it would be (more noticeably) out of tune for any other key. Further, since other instruments are normally tuned for other temperaments, you'd be out of tune relative to them, even in your chose key.

That's one of the advantages of digital pianos. Many models have a setting where you can tune it for the key of your song. Then the basic chords on that key sound perfectly in tune, and the chords of nearby keys sound slightly dissonant, thus creating a beautiful tension.

This is what I mean, that the octave structure is preserved in equal temperament.

https://images.squarespace-cdn.com/content/v1/5230e9f8e4b06a...

I believe the human voice is the most popular instrument, world wide by a long shot followed by clapping. You are probably right that most drums are inharmonic, but I wonder what percentage of percussive instruments have a strongly identifiable fundamental anyway (or some tone if not the fundamental). At any rate, perfect intervals sound really consonant in singing, so I'd expect a bunch of cultures to independently stumble into that knowledge because singing is so popular. I didn't read the actual study, but I'd lean towards distrusting a single study or two, and in this case, wonder if this group of people is into polytonal singing.

> 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 am skeptical of this. I maintain that 1:2 is special. I would love to find more evidence or resources about this.

You can play with an online synthesizer to create timbres whose octaves are dissonant:

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...

[1] https://sethares.engr.wisc.edu/ttss.html

1:2 is special in that it gives the maximum possible consonance when using harmonic timbres, but there are other good scales built around different ratios. See:

https://en.wikipedia.org/wiki/Bohlen%E2%80%93Pierce_scale

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

You might want to be wary of your bias. From your statements, it seems as though you're somewhat dismissive of this study and are searching for studies/sources that support your point of view. If you find yourself dismissing the person you're replying to and other comments in this thread, even though they raise interesting points about instruments, you might want to think about whether you're being open-minded.

Maybe. If you knew me, I think you'd be hard pressed to label me as someone who isn't open minded. But these are not arbitrary positions. I study harmony intensely, integrating neuroscience, music theory and classical philosophy. Did you know that the Pythagorean test of small integer relationships in consonant bronze chimes (described by Plato and Aristoxenus) is considered to be the very first quantitative, hypothesis-driven experiment in western history? And yet, we still don't know the basis for consonance and dissonance!

I will only say that my "bias" got me what I wanted-- evidence about the phenomena! (And thank you for that)

I think that the OP is maybe making too much drama about it. Skepticism (even extreme skepticism) is always welcome in a scientific/technical discussion!

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.

> Then, playing two sounds an octave apart will match exactly all these frequencies and it will sound nice.

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.

Yes, people actually strongly prefer imperfect consonance over perfect consonance. I've just collected some great data on this.

Do you plan to publish this data? I'm very interested.

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.

> You can argue that the octave is a "basic phenomenon" inasmuch a vibrating string is basic.

Or...the human voice?

A drum is a good example, though we don't often think of it as a melodic instrument. The resonant modes of a drum don't look anything like someting as nice as those of strings, with their integer multiple frequencies. Makes you wonder, what music might have been like if our ear was more "drum-based" than "string-based" (however that might work).

Music already "is" something different! not all music is western. One of the most strikingly different kinds of music (besides synthesized stuff) is gamelan music, which is performed on a set of non-harmonic bells, which sound consonantly at non-integer intervals.

Thanks! I've been listening to gamelan music for the past couple of hours now, and have thoroughly enjoyed the experience. From my limited exposure, I get the impression that the music leans on repetition of sound patterns to give the listener something to hold onto musically, creating more of an atmosphere of sound than a melody you might hum along to. Is that fair to say? Perhaps I just need more exposure before things "click".

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...

So there's this thing called a wave, and waves with wavelengths that form a simple ratio interact to form periodic waveforms that are observably stable. Pulses too, actually.

> waves with wavelengths that form a simple ratio interact to form periodic waveforms

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.

Stable simple waveforms are observably more consonant than those with many transients. Nobody would have this dumb, stubborn discussion about rhythms.

You are right, this is not a discussion worth having when you can do the experience. If you have a digital synthesizer, just try the experiment that I describe and listen whether you can tell the periodic from the non-periodic waveform. You say that they are perfectly distinguishable, but I just did the experiment (even drawing the lissajous figures) and there's no real difference except the perceived pitch.

I didn't say they were perfectly distinguishable. You have reasons simple ratios don't result in physical waveform attributes relevant to music. I posit you should just really think about that for like five seconds. Otherwise I don't know what to tell you. Look up Pythagoras or maybe what a wave is or something.