ImproviseImprovise is a static ability that functions while the spell with improvise is on the stack. “Improvise” means “For each generic mana in this spell’s total cost, you may tap an untapped artifact you control rather than pay that mana.”
The improvise ability isn’t an additional or alternative cost and applies only after the total cost of the spell with improvise is determined.
Multiple instances of improvise on the same spell are redundant. (CR 702.126) (Your artifacts can help
castTo cast a spell is to take it from the zone it’s in (usually the hand), put it on the stack, and pay its costs, so that it will eventually resolve and have its effect. A player may cast a spell if they have priority. See rule 601, “Casting Spells.”
To cast a card is to cast it as a spell. (CR 701.5) this spell. Each artifact you
tapTo tap a permanent, turn it sideways from an upright position. Only untapped permanents can be tapped. (CR 701.26) after you're done activating mana abilities pays for
.)
CounterTo counter a spell or ability is to cancel it: remove it from the stack without resolving it. A counter on an object or player is instead a marker that changes characteristics or interacts with rules or abilities; for example, a +1/+1 counter modifies a creature’s power and toughness. (CR 701.6; 122.1) target spell unless its controller pays
.