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.
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.
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.
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'.Also, FWIW gauge does not control tension in a lot of cases. Fatter strings are often more floppy IME.
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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