Screeching #2*

Right. So, waves are perceived to sorta be the same when the ratio of their frequencies is a power of 2, right? Actually, you could say a power of 2 or a ratio that is really close to a power of two, I guess? I mean, waves with a frequency ratio of 1.97:1 will be perceived to be really similar too, just not exactly the same; 1.97 is pretty close to 2, but 2 is pure and special and whole and strong, and 1.97 is impure, mundane, partial and weak, even if those two are pretty much the same thing for all practical purposes. I mean, at some point there's actually no perceivable difference - 1.999999 will be indistinguishable from 2. But 0.03 is not negligible. Also, we should keep in mind that the waves are usually generated in multiples, and even though 400Hz will be perceived to be similar to both 800Hz and 788Hz, some of their multiples might be too close together and clash (remember that thing where waves with frequencies that are close together feel eerie and stuff?). Like, let's say we have the following set of waves generated from a source:

440Hz, 880Hz, 1320Hz, 1760Hz, 2200Hz, 2640Hz, etc.

Now, if we generate a similar series from 880Hz, we'll get:

880Hz, 1760Hz, 2640Hz, 3520Hz, 4400Hz, 5280hz, etc.

All frequencies in this series are also in the previous series, so there's no clashing going on. Now, if we generate a series from 867Hz, we'll get:

867Hz, 1734Hz, 2601Hz, 3468Hz, 4335Hz, etc.

867Hz itself will clash with 880Hz, 1734Hz will clash with 1760Hz, 2601Hz will clash with 2640Hz. It's not going to be pretty. So it's best to stick to near-perfect 2:1 ratios to establish equivalence . Or maybe not; sometimes it might be necessary to use another value.

We should keep in mind that this entire thing about 2:1 and similar ratios signifying the sameness in the perception of two frequencies might be cultural (yes, your imagined wave-perceiving folks have cultures). Some cultures might have other ratios signify sameness, and other might have no such concept at all. Who knows. Also, let's agree that even though two waves with such ratios might seem to be the same in some weird sense, it's perfectly clear to the perceiver that the wave with the higher frequency is very distant from the one with the lower frequency (2:1 is a big ratio), even if they seem to be the same thing essentially, somehow (as we'll find out in 5 minutes, they're the same thing in terms of what sort of feelings they would create if they were accompanied by another wave that is not their equivalent). 


So, now that the entire thing about the equivalence property is cleared up, what about all the other ratios between a frequency and its double? We've already implied that each different ratio corresponds to a perception of a different "distance" between the waves. Well, here comes the interesting part: different distances are directly associated with different feelings and emotions in the perceiver's mind. Let's define a ratio category as "a ratio of integers and all the ratios that are close enough to it to be perceived to have the same feeling". A ratio category around 5:4 will correspond to "happy"/"cheerful"/"elementary positivity particle". A ratio category around 6:5 will correspond to "sad"/"gloomy"/"elementary negativity particle". A ratio category around 4:3 will correspond to"badass"/"clean"/"elementary neutrality particle". In addition, due to the equivalence thing from above, each ratio category around a given ration x/y evokes the same feeling as a ratio category around 2y/x. So, how many such categories do we have? And how similar do we mean exactly when we say "and similar"? Is 5:4 similar to 51/40? Is it similar to 6:4? Well, those things might vary with culture too. So, for now, let's stick with an arbitrarily chosen number, let's say 12. Right. So, we have 12 different "feeling-distance-ratio-categories". That's a stupid name, so let's call them intervals instead.

So, if we're given a wave with a frequency f, and another wave with a frequency g, we know that:

• If the ratio between f and g is in an interval around a ratio of a number that is a power of two and 1, then they will be perceived to be equivalent. 
• If the ratio between f and g is smaller than two, then it falls into one of the twelve intervals, and which interval it falls into decides what sort of a feeling will be perceived.
• If the ratio between f and g is larger than two, then the feeling that will be perceived will be what would be perceived between f and g*2ⁿ, where n is the smallest natural number such that the ratio between f and g*2ⁿ is smaller than two (this is just a fancy way of reiterating the equivalence thing from above).

Okay, so let's list the intervals we've got (with specific ratios chosen arbitrarily; it doesn't really matter whether we're defining a ratio category around 16/15 or 17/16, because "interval" is already an arbitrary measure of range that we've deliberately avoided constraining, and those kinds of ratios are close enough to definitely be included in intervals built around each other):

• Around 1:1; will be perceived as the same thing. Still, two frequencies might be close together to be in this interval but not close enough to avoid the clashing effect we've discussed. Has an equivalent interval around 2:1.
• Around 16:15; will be perceived as a clashing, irritating abberation. Has an equivalent interval around 15:8.
• Around 9:8; will be perceived as an unsettling clutter - as two things that are far apart enough to be easily separable, but still too close together to fit together in a pretty way. Has an equivalent interval around 16:9.
• Around 6:5; sad. Has an equivalent interval around 5:3.
• Around 5:4; happy. Has an equivalent interval around 8:5.
• Around 4:3; clean. Has an equivalent interval around 3:2.
• Around √2:1; wrong. Yes, I know, this one's exceptional in that it's not an integer ratio. Since it's exactly in the middle of our intervals, we want two of these ratios stacked together to be equivalent to one equivalence upward. Also, this one doesn't have a separate equivalent, since √2:1=2:√2. 

Actually, enough with the ratios. Let's convert those 12 to decimals:

• 1, equivalent to 2. 
• 1.06666..., equivalent to 1.875.
• 1.125, equivalent to 1.7777...
• 1.2, equivalent to 1.6666....
• 1.25, equivalent to 1.6.
• 1.333..., equivalent to 1.5.
• 1.41421...
• 1.5, equivalent to 1.333....
• 1.6, equivalent to 1.25.
• 1.666..., equivalent to 1.2.
• 1.777..., equivalent to 1.125.
• 1.875, equivalent to 1.06666...
• 2, equivalent to 1.

(There's 13 items in the list because the final item, 2, is not actually supposed to be there, and is just floating around there for demonstrative purposes).

A ratio will be said to belong to interval of whichever ratio it is the closest to (in terms of the difference of absolute values). We should note that equivalent intervals are not literally the same intervals; a ratio that belongs to the interval around 1.125 doesn't belong to the interval around 1.7777, it just evokes the same feeling as a ratio that would belong to the interval around 1.7777.

So, we now have a system where the perceiver can perceive two waves and feel a certain kind of emotion. Hell, it's not like we are constraining the perceiver guy to feel just one emotion at a time, right? So if there's more than two waves around at the same time, the perceiver can feel a combination of emotions. Example: we have waves with frequencies of 400Hz, 480Hz, and 600Hz, we have ratios of 1.2, 1.25, and 1.5 at the same time. 400 to 480 forms a "sad", 480 to 600 forms a "happy", and 400 to 600 forms a "clean". So, how does this make the perceiver feel? The answer is "sad" - the first interval from below has a little more weight than the second interval, and the big, surrounding interval is neutral, so the overall feeling is sad. Similarly, 400Hz, 500Hz, and 600Hz would produce "happy", "sad", and "clean", and the overall feeling would be "happy". Different numbers of waves forming different kinds of intervals can produce all sorts of combinations of emotions this way. 

* I lied.