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Stiff vs floppy- where do you feel it? How does it matter to you?

As is common in this forum, there is confusion between 'stiffness' and 'tension'. Also as is common, i am attempting to explain =) (being a qualified physicist).
There are actually 3 aspects here:

1. Tension
This is the actual scientifically-measurable 'tightness'. It is a force acting parallel to the string, measured in 'pound force', commonly abbreviated to 'pounds' and seen on tension charts and tension calculators. This is how tightly the string is being stretched by the tuner.
'Tension' is a fairly scientific word and should not really be used for anything else. The word 'tense' also means 'tightness caused by a stretching force'. So it is fairly clear what 'tension' means.

2. Stiffness
This is the actual scientifically-measurable 'rigidity', how much the string structure resists being bent into a curve. It is independent of tension.
For example, flatwounds are known to be stiffer than roundwounds.
Think of a metal spoon, it has zero tension because it is not being stretched, however it is very stiff.
When a string is off the bass, and therefore at zero tension, and you coil it up to fit it in the 6 inch envelope, the resistence you feel is the stiffness.

3. How it feels
This is not scientifically measurable and is rather subjective.
Players use many words for this, including 'tension' and 'stiffness'. Usually when players describe a string they are referring to 'how it feels'.
It is determined by a mixture of actual tension and actual stiffness. So 'how it feels' can be determined by varying mixtures of actual tension and actual stiffness, so you cannot really judge either of those from how it feels.

Also, FWIW gauge does not control tension in a lot of cases. Fatter strings are often more floppy IME.
At a particular scale length and pitch, scientific tension is determined only by string mass, which is mostly determined by gauge (with a little variation due to materials and construction). So it seems you are not referring to scientific tension by 'tension'.
At fixed scale and pitch, a fatter string will have more actual tension and more actual stiffness, so will be less floppy.

Fatter strings seeming more 'floppy' may be your observation that a larger mass string has more momentum so tends to be more prone to flopping around from side to side.
For example, a guitar top E .009 string at 15 pounds tension is well-tensioned, not at all floppy and has good tone. However a bass B .130 string at 15 pounds tension is a completely impractical undertensioned floppy mess with a bad tone.
Larger mass strings need more tension to keep the vibration under control, the minimum practical tension is higher.
 
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When I played in raucous punk or metal bands, i preferred a stiffer string because I'd get adrenalized while playing and dig/pluck hard. With a floppy string, my feel and tone got sloppy in those cases. So something like hex core GHS Boomers were ideal. I'm not a fan of heavy gauges, so 45-100 hex Boomers were great, because they were stiffer than other strings in the same gauge.

On the other hand, for more nuanced material where I use a lighter touch, I like a more flexible string, like a a round core roundwound akin to DR Pure Blues.

Finding that ideal "one size fits all" string feels like a lifelong quest, like our neverending "GAS" quests for "THE" tone, you know?
 
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I don't want them to feel like spaghetti noodles of course, but I love low tension thin strings and as low of an action as I can possibly get them without them choking out from touching the frets too much. A little occasional rattle is fine, but I draw the line at them getting choked out. I use Dunlop Super Bright Nickel rounds and Dunlop flats 40-100 (40-120 on my 5) on most of my basses, partly because they sound great, partly because they're low tension and thin, and if they made them thinner, I'd probably like them even more.
 
I generally prefer 45-105 flats, so they're pretty stiff. On the few - very few - basses I have that wear rounds? Same gauge strings as the flats, and again, pretty stiff. However, I will use lighter gauge/less stiff flats - If that's what it takes to get the sound I want from a particular bass. Yes, even those noodle-y TI Jazz Flats...;)
 
Used to be in the stiffer zone with 45-105 + 130 (even had switched to 135) for rounds and 45-100 + 132 for flats, regular hex core, regardless 34"/35" scale length.

Now, I prefer more flexible strings with thinner hex core 45-105 +125 for my 34" and 40-100 + 120 for my 35". More apparent ever since I've had my first tapewounds, which obviously are on the floppy zone. They made me learned that it is all about controlling my finger's attack.

I still use 32 + 45-105 + 130 regular hex core (stiffer) strings though, but they are on a multi-scale bass, so the higher strings aren't too stiff anymore.
 
It matters to me in terms of how different the feel is with, and without, stage Adrenaline and the antics of jumping and playing a higher energy rock show, old Beatle tunes, barnburners, bluegrass, and the like.

I don't mind the top end strings being a step lighter, but am finding these days that I really prefer what in most sets will be around a .105 E-string. Because the E-string tends to be the floppiest in any given set of standard strings (go balanced!) , and with my playing, I have a hard time getting not only the tone in terms of fundamentals, out of a lighter E-string, but the physics of how it vibrates in relation to how I pluck, results in some sloppy interaction which I find clears up with a heavier string.

I mostly play with two fingers, floating thumb, but in recent years I often switch to thumb and double thumb with the palm muting for really tubby walking lines. Which is great with the 40-100 type strings. And I slap and pop about 12 notes each per set list. Unless ..... :bassist:

And I do like the lighter top end for the slap/pop. But lately, cannot abide the tone on the lighter gauge E's and A's. And D's are usually way the hell too tight in standard sets, not sure why that is. Especially phosphour bronze, what the hell. The D string in those is like 15# heavier pull. One of these days I want to try the lower 4 of a 5 string set on ABG, nickels, because D'addario nickels are real softies. You can actually tune a nickel B-string up to E and it's still less tension than the stock phosphour bronze D-string would be. And doesn't ABG need more mass on the low string, not less??? Must try...