Why aren't slight imperfections in consonant intervals extremely dissonant?

Question

In theory classes, we're taught that an interval's acoustic consonance is a function of how "simple" it is as a ratio, where "simple" means being a ratio of small integers. So a perfect fifth (3:2) is consonant while a minor second (16:15) is dissonant.

But if you're slightly off a perfect fifth you might have a ratio of 3001:2000 or 2999:2000 which, by this "simplicity" criterion, should be extremely dissonant. Instead, we hear a smooth transition from consonance as we move away from a perfect fifth. Also, since there are always slight imperfections (however small) in performance, by this logic it seems like all music should sound acoustically dissonant. It clearly doesn't.

How is this explained?

Answer 1

If we look at a dissonance curve like this one, from William Sethares' site:

We can see that the statement

an interval's acoustic consonance is a function of how "simple" it is as a ratio

only holds at all when are already looking at particular ratios that are already themselves reasonably far apart. For example, it's fair to say that the fifth (3:2) is more consonant than the m3 (6:5). However, as you rightly say in your question, 3001:2000 would be more consonant than 6:5.

The reason for this is that this kind of dissonance curve (for complex tones) is derived from the dissonance curve for simple tones, i.e. single sine waves:

And as you can see, this shows that very small differences are still quite consonant; slightly larger differences are more dissonant, and then even larger differences show less dissonance.

So "an interval's acoustic consonance is a function of how "simple" it is as a ratio" isn't true if we take it too literally. It might still be a reasonable statement if restricted in scope to being a 'discrete' function only dealing with justly-intoned common musical intervals, which in some contexts would be the 'intuitive' way to interpret it.

Note also that these curves depend on the harmonic series used to calculate them - see Dissonance: why doesn't the roughness curve have a dip for complex intervals like 7/6?, and note Endolith's helpful comments on his answer.

Answer 2

Try writing out the actual ratio as a continued fraction. One often hears the actual sound as one of the early convergenets. (Examples later if desired.) Sometimes one hears the "nice" ratio close to the actual ratio.

Answer 3

Let's say you have two notes sounding with a frequency ratio of 3001:2000. Say, at frequencies of 600.2Hz and 400Hz. The lower note's 3rd harmonic is 1200Hz. The higher note's 2nd harmonic is 1200.4Hz, just different enough to be perceived as gentle beats (one beat every 2.5 sec if I've done my maths right). But if the higher note's frequency were 610 Hz, its 2nd harmonic would be 1220 Hz, giving a difference note of frequency 20 Hz, which is roughly on the border line between beats and being audible as a pitch. It's when the difference note is of roughly that frequency that a slightly off consonant interval sounds extremely dissonant. Note that whether this happens depends on the difference between the frequencies of upper harmonics, not on the frequency ratio.

Answer 4

Apparently, there is this notion of "critical bandwidth" within the ear as well .. the idea that frequencies that are very close together, trigger the same ciliar cluster on the basilar membrane of the cochlea.
Similar to how a person, when they have two pencil eraser ends placed on their back, might only perceive that one end is there when in fact it's two ends very close together, the perception of one tone from two (or, in your case, the perception of a perfect-fifth interval from a quite-imperfect 3001:2000 ratio) is driven by the sufficiently-close triggering of the very same ciliar clusters on the basilar membrane as the 3:2 perfect fifth at the same pitch level.

Effectively, the ciliar clusters on the basilar membrane act as "partial quantizers" for frequency, not allowing minor frequency differences to be heard as sharp or extremely rough dissonances, but instead quantizing them to frequencies of standard difference: the "critical bandwidth". Only when a difference in pitches exceeds the local critical bandwidth of the ciliar clusters on that area of the basilar membrane of the ear, does distinct perception as two different pitches occur.