I was always taught, as I'm sure many of you were, that at a low pitch the intervals are always best left wide - 5ths and octaves. I've been talking about this with a coujple of pupils, and I wonder if the reason I'm giving tallies with anyone else's (I was never taught why)

I start by showing them how an octave is a doubling of the frequency, so two octaves is x4 etc (surprising how many kids don't immediately twig that it's not 3!)

Then I say they must have been wondering what happened to three times, and with the little caveat that actually it's a few vibrations short (or is it a few vibrations over? I can never remember) but not to worry about that now, I show them how you find x6, and say that from C, the E just over 2 octaves higher is (roughly) x5 (x here being "times") thus the major triad sounds comfortable to us because it's maths/physics and not just because we chose those notes and have got used to them.

So the C triad in the middle of the piano is related to a fundamental a couple of leger lines below the bass stave. The triad an octave lower is still tolerable to our hearing, though its fundamental is pretty low (the lowest C on my piano) but the triad an octave lower than that (around the bottom of the bass stave) sounds very muddy in close position, and I surmised that it was because its fundamental is almost out of our normal range of hearing, and the one below that is definitely related to a pitch we can't detect except as a sort of motor-boat sound (I can get one of those on my little Yamaha keyboard) and almost impossible to hear the three pitches clearly

Is this something any of you have been taught? Does it make sense? (I have James Jeans' Science and Music somewhere, a wonderful read, and he may have dealt with it, but it's been a while and I can't remember)

"Roughly" only applies to instruments, like piano and harp, that are tuned in equal temperament and cannot be adjusted during performance. The "few vibrations over" are specific to the piano, and compensate for the stiffness of the strings. For the implications of this stiffness, see below.

Since Jeans wrote that, perceptual psychologists have done a lot of work on the innate response of the auditory system (ear and brain) to pairs of pure sounds ("sine waves"). They call this "acoustic consonance" (and "dissonance"), to distinguish it from consonance and dissonance as codified in conventional harmony, and have applied it to a suggestion by Helmholtz that dissonance results from partials of two tones being close but not identical.

When I was studying for my degree, I wrote an essay, which is on line here, about the writings of a 20th C musical analyst called Heinrich Schenker. Sections 2 to 6 discuss various aspects of what I think of as the "neo-Helholtz" theory of dissonance. My particular argument is that the harmonic series gives consonant intervals only in timbres with harmonic partials, and that not all instruments produce harmonic partials.

If you read these chapters, you will get a better explanation of dissonance than I can provide here, and you will also find references to the publications documenting this work However, two quotes below are relevant to the questions you raise.

From Section 4.2:

"The piano sound is produced by hammers striking long thin rods, some of which are loaded with a uniform wire winding. The modes of vibration of these long rods approximate fairly closely to those of a stretched flexible string, but the stiffness associated with their finite diameter is detectable and is taken into account by the piano tuner. Whereas the ideal, infinitely flexible stretched string would have free vibrations whose frequencies coincided with the harmonic series based on the lowest, whole string frequency, stiffness increases the frequency of the partials relative to the harmonic frequency. Once a piano tuner has established an equal tempered chromatic octave, his/her normal procedure (from which special purposes, such as to prepare a piano for playing with an electronically tuned electric bass guitar, may require departure) is to tune octaves above and below these notes by eliminating beats between the first partial (fundamental) of the upper note and the second partial of the lower one." Thus, because piano string partials are not harmonic, the frequency ratio of well-tuned piano octaves is greater than 2.

and from Section 3:

"This means that, in musical terms, from about the top of the treble stave upward the most dissonant interval [between pure tones] remains constant at somewhat less than a semitone, while at middle C it is 1.5 semitones, roughly doubling for each octave downwards from there."

IMO, this is the fundamental of acoustic perception that makes close intervals at low pitch sound rough. Of course, it is not a complete explanation, because it merely takes the problem back to the functioning of the human auditory system without explaining why it behaves as it does.