Pitch Recognition - Problems and Solutions?

Both Benedetti's Puzzle and the railsback curve are terms I had never heard of before.  Both of these are interesting topics, and just knowing what terms to search for now can open a flood of information.

In the Benedetti's Puzzle video (played at .75x speed to make it easier to follow) he talks about how a capella choirs can end up in a different key from where they started,  "through no fault of their own" and "it's built into the mathematics".

Normally I would assume that I'm simply not well trained for pitch recognition, but fact that I always almost always drift upward in pitch on the theremin, never down, makes me wonder if this effect isn't at least part of my problem.

Thanks for both of these suggestions.  Maybe it isn't just me . 

"In the Benedetti's Puzzle video (played at .75x speed to make it easier to follow) he talks about how a capella choirs can end up in a different key from where they started,  "through no fault of their own" and "it's built into the mathematics"."  - pitts8rh

I wonder how much is going a tad sharp is an OK sounding accentuation, so singers do that?  When I tune my guitar by ear, any strings that are off tend to be sharp compared to the others.  In a way it's an over correction, in a direction that sounds less worse than flat, and perhaps slightly sharp sounds slightly better than in-tune (until you play a chord)?  This has been tripping me up my whole life, so I always use a tuner, now that I have a good one handy.

The video blames it all on very subtle math, which strikes me a little like saying toilets flush in the opposite direction down under due to the Coriolis effect (which in fact is so exceedingly weak it's very difficult to come up with smaller scale experiments that demonstrate it at work).

I'd never heard of the Railsback curve either, and it's nice to give it a name and be able to do web searches on it.  This was an interesting paper:  https://asa.scitation.org/doi/10.1121/1.4931439.  I've always wondered how they design stretching piano tuning software, and indeed the mechanics of how they even sense pitch in electronic piano tuning software.

A little OT, but IMO 12-TET equal temperament doesn't get enough praise.  Sure, it's a compromise, but an amazing one.  The perfect fourth (4/3) and perfect fifth (3/2) are off by less than 2 cents, and the simpler ratios are the most important to hit because they're the most audible from a beating standpoint (I encountered this when developing the cubic plateau scaling in the D-Lev triple oscillator).

I love geeky discussions like this Tuning is a complicated mess. While we use mathematics in an attempt to explain and describe tuning practices, our ears fight against that, which is what causes the problems. Another significant problem has to do with equal temperament, which was developed, I believe, to solve the problems with tuning keyboard instruments and fretted string instruments in which each pitch must be fixed. Such a tuning scheme is simply a compromise that essentially allows such instruments to play relatively well in all 12 major and 12 minor keys in Western music.

HOWEVER (don’t you just love “howevers?” - LOL), instruments that do not have very rigidly fixed pitches - essentially, orchestral string instruments, woodwinds, and brass. Yes, even woodwinds have a fair amount of room to bend pitches up and down - although the oboe is used tune the orchestra because its pitch has the least amount of room for variance - pitch on the oboe is mostly locked in by the reed. Anyway, orchestras and other groups do NOT use equal temperament (GASP! Oh the horror of it all!!!)! LOL! For example, our ears tend to like thirds, sixths, and leading tones to be slightly high when playing in Major keys - because they sound “better” to our ears. What this means is that the “same” note will have a slightly different tuning that is dependent on the key! GASP! LOL. As an example, in C Major, A is the sixth and will be played just a little high and will be slightly out of tune with the piano (or other fixed-pitched instruments). However, in F Major, A is the third and will be played in tune with the piano (or other fixed-pitched instruments). So, does this mean that orchestras play out of tune when performing, let’s say a piano concerto? YES! GASP! Why? Again, the orchestra sounds better and a very slightly out of tune keyboard (on a few notes) isn’t noticeable to most people. It’s all a compromise.

BTW, a similar situation occurs with music using minor scales although non-fixed-pitched instruments will play the third SLIGHTLY flat - it helps emphasize the feel of minor keys.

One interesting (informal) discussion of this may be found here:Tuning

Finally, I think that the guy on the first video wasn’t necessarily saying that mathematics was driving pitch up, per se, but that our ears hear certain intervals, a certain way in relation to other intervals and that this eventually leads to a change in key, because of how we hear them. (Does that make sense? I’m thinking out loud now.) 

Very interesting 'geeky' discussion. For one thig, the theremin has made me look and listen from new angles at other musical instruments and singing as well, and also has urged me to study things like the history of scales, tunings and temperaments, and vocal/melodic concepts and techniques from Indian music. I'm mostly self-taught in music, and that means I kind of have theoretical and practical lagunes, and I'm going at it in my own autodidactic ways, learning as I go along and the need for some piece of theory and/or information arises. And always happy about some new pointers / keywords to further my knowledge.