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