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At which point does being out of tune become a problem?

Notwithstanding that on a piano Ab and G# are the same note, the physics of it says that they actually are not exactly the same pitch. Some early music stringed instruments have a tiny "semi-fret" to compensate for that (the name of which escapes me at the moment).
Viola da Gamba, a wonderful instrument
 
I'm not asking because I have trouble tuning my instruments, I'm asking because I've never thought about how much out of tune an instrument has to be, to be noticably bad sounding.
Of course this depends on the instruments range, and the purpose of the instrument in the song, and so on. So let's just assume were talking about something like a Jamerson bass line.
I know from modular synths and orchestras, that the out of tune sound between different voices can actually be a good thing and pleasant sounding. But of course there is a line where it becomes noticably bad sounding.

To be even more specific, when I pick up my p bass to noodle around on it, it's usually a bit out of tune, but not so much that it would bother me. I still usually tune, but I noticed that I can even play along with songs, basically doubling the bassline, and it still sounds perfectly fine. I feel like I can be out by something like 20 cent and it's still just fine.

What are your experiences with this? Have you used controller out of tuneness for a specific sound before?
IMO, it's fine until it isn't. For me being out of tune as a bassist makes everything above it out of tune. The more intricate the arrangement is, the more problematic it becomes. In my experience I have not found that being out of tune of any amount as an improvement.
 
I spend a lot of time under headphones so even small increments annoy me. In the open air plus or minus a few cents I can tolerate but after that it becomes distracting.
100% this. If listening thru IEM or studio, absolutely tuning is crucial. Open air however, up to maybe 5 cents but I'm willing to bet shoegaze became a genre because of this lol
 
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As someone with very high sensitivity to intonation, ±20 cents all the time would drive me up the wall. Anything more than ±5 and my ears start to twitch. With that said, there is a big difference between 'sweet intonation' and 'being out of tune', especially when lots of continuous-pitch instruments (or voices) are playing together.
I can tell when it is out of tune, but to me in a busy mix, especially when I'm doubling an existing bass line, it just sounds thicker. Though with sustained notes it probably wouldn't work well.
 
Def when someone else hears it...I've heard a few live bass tones where I couldn't hear the difference half step up or down, not a tuning thing. The Pretenders were on TV and Farndon had a string at least a half step down(probably bumped the key). Couldn't miss it but they got through it. Once I've tuned up I've never has tuning change noticeably/audibly except outdoors.
 
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One of the best guitarists I've ever worked with is also kind of a slob, even with tuning. He doesn't use an electronic tuner (I gave him once and he quickly lost it) and sometimes his guitars keep going out like crazy.
He somehow gets away with it. Something about the way he plays is so rhythmic, it gets people's toes tapping and they don't notice the tuning. Maybe it's the way he plays bass lines & just "stabs" at the upper chords without letting them ring out long enough to soak in. (If it's really out, he will adjust on the fly.)
Compare that to a rookie who dutifully uses the tuner, then strums full basic chords and they sound worse than death.
 
I have tried to test my hearing by tuning my bass open A string with a clip-on tuner, and then tuning the other strings by ear, and then checking the result of my tuning by ear with what the clip-on says. When I do this, I find that I am (usually) pretty close to what the Snark says.

Without creating ones own way to test the ability to discern when, (for the sake of simplicity) two simple sine wave tones, (not complex sounds with overtone content like a musical instrument produces) are either the same or not by varying degrees.

Does anyone know of some sort of app designed to test just how far "out of tune" a sound must get for me to say I am able to recognize it is so? I envision something like two tones are produced, which could also be octaves apart, and you then say if they are, or are not in tune.
There are several.
If you have iphone, try "Tunerval" for single tones. It plays a tone then an interval, and you have to identify if it is #,b, or in tune.

For harmony practice, "Harmonize" is weirdly fun. It plays a drone and you have to match an interval with a slide.
 
When someone notices.

I once did a gig in which the manager complained to me before that 'everyone played out of tune'. So I took the most accurate singer I know and played tuning between every song. We got a complaint that we were 'out of tune'.

My suspicion is he had undiagnosed perfect pitch. Nothing is in Tune in pop music.

Other times I've noticed that I've drifted out nearly a semitone. No one noticed.

It depends.