Before I go all Muzak Theory on y’all, here is the basic problem statement:
Tuning is a logarithmic function which is started or seeded by a starting tone. However, standard tuning methods are linear. This means given a starting tone, say A 440, the note that you would call G4 in that context has a different Hertz than if you based your starting tone on C261.6. but in each context us less-hairy-apes call it a G4.
What tuners do is… To use a radical term… Ear fuck what should be in tune. A is not, and never has been, 440. It is a lie we all accept for expediency. And equal temperment can diaf.
i will try and give examples, and noone will care (except for perhaps @PatRx2 and @mzed), and i will also get these examples wrong. but such is the life of a blazingly good musician that gave it all up for family life 
Let’s take an imaginary note, A. A has a value (a hertz, but don’t get hung up on that) of 1.
There are notes we call A, B, C, D, and E. but how do we derive what they are? we do it from what are called overtones (and this is so fuckin’ simplified i should just go get a bagel and stop typing). The first natural overtone, given a perfection string or tube, is a doubling or octave. So the first note we can derive is actual A2, or the numeric value of 2. trust me, this will make sense in a moment.
the third note is derived similarly, we double A2. this gives us what we call a ‘fifth’. the note name is called E, and it has a value of 4. 1, 2, 4. Now the obvious problem is this new note isn’t in a neat sequence, so we need to drop its octave so it is in between 1 and 2. so we divide it’s wave form, and it’s numerical (fake) hertz if you drop octaves is now something like 1.5. (the math is completely wrong, i’m back of the enveloping this shit. don’t make me do the actual equations :D).
so now you have A=1, and E=1.5. You get the value of E by looking up the logarithmic function of overtones, and ‘octaving down’. but how are B, C, D found? well, the same way.
As you go up the overtone chain, and interesting thing happens. I will give a slightly obtuse example, but hopefully we can follow along.
Root note -> octave -> fifth -> fourth -> third -> minor third -> second -> (what we call a step) -> then all the fun microtones @mzed, @PatRx2 like to fuck around with. And by measuring them and changing their octaves, you build scales like ABCDEFG. and the tuning for each one, if you measure them from the starting point–A=1, is perfect and sounds harmounious.
(saving this draft, one second, the other shoe will drop in a mo)
okay, back.
here is where things get fun. the moment you start your tuning with a note that is say E=1.5, and do the same logarithmic calculations, all the notes don’t match A=1. They just don’t. Some are close (fifths and fourths are usually close, thirds, are usually off by what, 8 cents, and minor sevenths are off by a whopping 17 cents in general), so to play in tune, you must be acutely aware of the context of what key you are playing in. Do you bend a seventh, or do you play it equally tempered?
to backtrack a tiney bit and explain a term: equal temperament is the concept that you take an octave–A1 through A2, and you literally divide the hertz by an arbitrary number. In the US it is usually 12, as seen on pianos with their black and white keys. There are many, many others, but really noone that isn’t a theory nerd has ever seen or heard of them
So this is why I hate tuners. They recommend that you play out of tune, in a specific way, that millions of people have accepted as ‘good enough’. But it has a couple of effects:
- When you hear a ‘good enough’ tuned song, you say, “I like that”
- When you hear a properly tuned song, you say, “I really felt that”
So I encourage all of you to buy a snark, a korg, a tuning fork, etc. it is a great way to start training. but the mind blowing enlightenment is when a single oboe quietly sounds a perfect note, and 40+ people all come together in serious, perfect harmony.
dontcha love analog wave forms?
