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Tapered ball end strings with round core wire, not hex.

Did you specify taper length? As far as I've read, inharmonicity in tapered strings is minimised if the full diameter starts right after the bridge saddle.

Like this...

KSCOM 125T 231017.jpg
 
Exposed core enthusiast here.

Try GHS Contact Core strings. They are like the LaBella Super Steps, in that all strings are tapered, but one last layer of winding is left intact instead of fully exposing the core. A nice fit in between no taper and raw cores.
I too, have found some basses and definitely the Chapman Stick benefit greatly from the exposed core in terms of bright clear tone. Sustain issues? I wholeheartedly disagree, if you set things up correctly...

My theory on why some people think exposed cores are awful due to odd overtones:
Your string seat in the saddle is basically a U shape. Picture the period at the end of this sentence resting at the bottom of the capital U. Now imagine your fingers flailing wildly left and right on that period, which represents the thin core of the string. Now look at it with a capital V. My theory is, that ultra thin core needs a firmer seat in the saddle, or it can easily slither a bit in that over-sized U, and indeed, cause some odd harmonics and overtones as a result. (Okay, I'll concede - it may also result in a loss of sustain...)
If you like the brightness and it brings your bass to life like I've had it do for some of my instruments, you may want to simply file a micro-v at the bottom of your saddle, giving a firm seat for the exposed core, thereby establishing a stabilized, clearer, focused tone and increasing the chances of preserving a solid amount of sustain. If you decide to go back to standard strings, the thicker, wound string will simply settle back into the U, effectively bridging the micro-V. Of course, you'll need to be fastidious in your execution of the filing and mindful of any burrs that should be abraded smooth before stringing back up.

To reiterate for the OP, if you haven't tried the GHS Contact Cores, I think you may be missing precisely what you're looking for.
 
Check into Fodera strings.

Historically Fodera and DR were the main companies to use round core, and now DR is making Fodera's strings. But DR generally does not have tapered strings while Fodera did when they were making strings themselves. So I don't know if the new Fodera/DR's are tapered or not. I'm also not positive they are round core but I would have to think so.
 
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Did you specify taper length? As far as I've read, inharmonicity in tapered strings is minimised if the full diameter starts right after the bridge saddle.

The Newtone site asks the string length from the nut to the ball end of the string to determine the taper at the bridge and how much to taper behind the nut so the string will fit in the tuner's string post.
I'm putting this set on a 32" scale Ibanez 5 string that I installed a Hipshot B Style quick release bridge on.
I determined it needs to be 33".
 
The Newtone site asks the string length from the nut to the ball end of the string to determine the taper at the bridge and how much to taper behind the nut so the string will fit in the tuner's string post.
I'm putting this set on a 32" scale Ibanez 5 string that I installed a Hipshot B Style quick release bridge on.
I determined it needs to be 33".

I think @HaphAsSard was asking about the length of the tapered portion at the ball end, which averages around 2".

D'A 125T 211011.jpeg
 
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Check into Fodera strings.

Historically Fodera and DR were the main companies to use round core, and now DR is making Fodera's strings. But DR generally does not have tapered strings while Fodera did when they were making strings themselves. So I don't know if the new Fodera/DR's are tapered or not. I'm also not positive they are round core but I would have to think so.

I used Fodera strings and loved them. Then they got hard to get, then they quit making them and moved everything over to DR, whose Hi Beams I currently use and like far better than any other widely available round wound string that I've ever played. If DR would just make Hi Beams with a taper, my request would be totally solved.

I play with a light touch and have found, for me, that round core strings offer lower tension, better sustain, more flexibility and I get much more nuanced response when doing bends, playing chords or using vibrato.
They tend to be more forgiving of dead spots on a certain Fender bass, that I own, too

In all of my years, I've played practically every string out there and kept notes of what I like and dislike about them. I owned a music store for several years and had wholesale access to practically every brand which afforded me the ability to experiment. We've all had string sets, with stiff, uneven tension, that we subconsciously fight and adjust to when playing them. All I know is when I went to round core and determined the gauges to achieve balanced tension across the neck, my playing opened up considerably because I wasn't fighting the bass or even giving my strings a 2nd thought. It allows me to concentrate on the music and be more aware and interact with the other players that I'm playing with. Oddly enough, after switching over to lower tension balanced strings, I noticed that I'm not as physically or mentally tired after a show or long gig.

For me, when my gear is not right, I'm not right and it's only taken me 45 years to figure out how to improve it. I realize that every player is different, this just works for me.
 
I think @HaphAsSard was asking about the length of the tapered portion at the ball end, which averages around 2".

View attachment 5239248
Since this is my first set from Newtone, I'll measure the taper and see if they adjust them to compensate for the bridge type, distance and scale length. I do know some strings companies offer a longer taper to compensate for thru the body stringing like my old Lakland strings had.
 
Nickel or stainless?
Keep us in the loop, looking forward to comparisons...

To start, I'm going with Newtone's platinum series in stainless.

This from their website regarding their platinum strings:
"A finer outer wrap cuts down squeal, whilst the core-to-wrap ratio is set to give a string with a true fundamental note, meaning less compression is required for recording"

I guess they are saying the string velocity is more controlled requiring less need for using a compressor to even things out when recording.... we'll see.
 
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Since this is my first set from Newtone, I'll measure the taper and see if they adjust them to compensate for the bridge type, distance and scale length. I do know some strings companies offer a longer taper to compensate for thru the body stringing like my old Lakland strings had.

I once tried a tapered B on my son's Ibanez SR1205E. But due to the bridge design, where the distance between the ball end anchor and the saddle is shorter than most, the taper turned out to be too long by having too much of it being in front of the saddle, the unpleasant effect of which was weird overtones higher up the neck (aka inharmonicity).

Sadowsky on 1205 200406.jpg
 
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I once tried a tapered B on my son's Ibanez SR1205E. But due to the bridge design, where the distance between the ball end anchor and the saddle is shorter than most, the taper turned out to be too long by having too much of it being in front of the saddle, the unpleasant effect of which was weird overtones higher up the neck (aka inharmonicity).

View attachment 5239312

With large strings, tapered or not, due to their physical size and mass, the magnetic pull from the pickups will definitely make a string act exactly like you are describing.
And, by using a tapered string, it is sitting closer to the pickup if you didn't raise the action after you installed it.
I purposely set my pickup height by playing the last fret and lowering the pickup until the string weirdness disappears. You just have to compensate for the lower output with your volume and amp level controls.
You will be amazed at how much better strings sound without magnetic pull affecting their vibration.
 
With large strings, tapered or not, due to their physical size and mass, the magnetic pull from the pickups will definitely make a string act exactly like you are describing.
And, by using a tapered string, it is sitting closer to the pickup if you didn't raise the action after you installed it.
I purposely set my pickup height by playing the last fret and lowering the pickup until the string weirdness disappears. You just have to compensate for the lower output with your volume and amp level controls.
You will be amazed at how much better strings sound without magnetic pull affecting their vibration.
That may be well and good but isn't necessarily an explanation for what Michael described, nor does it disprove it in principle. Imagine the tapered part being one-foot long: the two differently-gauged segments of vibrating string would each divide into fuses producing dissimilar harmonics. This is, to a lesser degree, what happens when the normal diameter starts too far from the saddle.
 
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My experiments yielded best results when the taper is a hair from the saddle crest. I use spacers at the ball end to make that happen. It becomes a bit of a game because, as you change the distance between the taper and saddle, you also have to adjust the intonation.
 
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I got my Newtone set, just as I custom ordered them, and they are fantastic!
I used my personal gauge preferences with round core wire and taper on the B, E & A strings and could not be more pleased with their feel, sound, and intonation. They are like Hi Beams on steroids.
Thanks for the tip, I'm sold.
And that right there is what matters. Happy ears means happy playing.
 
I got my Newtone set, just as I custom ordered them, and they are fantastic!
I used my personal gauge preferences with round core wire and taper on the B, E & A strings and could not be more pleased with their feel, sound, and intonation. They are like Hi Beams on steroids.
Thanks for the tip, I'm sold.
How do they compare to Hi Beams(or Fatbeams)for stiffness?