When i play on E and A string on 12th and further frets i get a very dull sound and strings just don't ring, although i can make out the notes. The sound is similar to when you hit a metal barrel or pot with some water in it. And it's definitely not a problem with electronics, because the problem is still there when im not plugged in. On the recording i play on E string on 5th, 10th, 20th, again 20th and then an open E. So if you compare open E, 10th and 20th frets you'll hear that 20th fret doesn't ring at all. It's hard to make out, because my phone mic hardly catches ringing, but it's still enough. Sorry for any grammar mistakes, english isn't my first language.
Larger diameter strings will choke a bit at shorter lengths. You have two choices as I see it: 1. Use a lighter gauge string - they will have more flexibility and thus better "speak" at the shorter length (beyond the 12th fret) 2. Use a string that is known for its flexibility. Dunlop Super Brights come to mind, though I have had intonation problems with them. Also be sure you set the witness points on the strings at the bridge - that often helps.
Your English grammar is fine, better than many native writers. As @Turnaround states and @johnh89 implies, sound changes as a string is shortened. It would be possible to make a piano with shorter bass strings, but it wouldn't sound right. Another thing to check is height of pickup. If too close to the string, the magnet may exert some force on the string.
Yeah, i thought of changing strings that have brighter sound. I have standard strings that came with bass, so it's probable that they are the problem. Thank you for mentioning witness points, will be pretty helpful when I'll be installing new strings.
I actually noticed recently when i tried to lower the action on bass that i can't really do that, cause pickup is in the way, so I'll try lowering it. Btw do i need to keep bridge pickup closer to the string than neck pickup? I heard that it affects the tone (treble especially).