My main issue with all of these three is that they don't scale well, or at all in some cases. But one of the problems with with buffing these three is that if you buff then too much in gets into the other's territory, Find Familiar can't be stronger than Pact of the Chain, and Pact of the Chain can't be stronger than Beast Master.
Find Familiar: Having a Found Familiar can be pretty cool, it can scout, use the help action in combat, and make the Wizard no longer the squishiest member of the party. But the higher your level, the less likely you are to even bother spending the 10GP necessarily recast the spell when the Found Familiar inevitably dies. I don't want to buff the spell too much, but one of things I would be that it's proficiency bonus is equal to yours, another's thing I would add is that it's minimum hit dice become equal to half your proficiency bonus. (not sure about rounded up or down.) So than it's skills scale decently well and it becomes slightly more durable, one of my reasons for not increasing it's HP by more is cause 1, it's not as strong as Pact of the Chain, and 2, the Conjurer's level 14 ability will increase your Familiar's HP by 30 and I didn't want to reduce the value of that by too much.
Pact of the Chain: There is no point in picking anything except for the Imp, always pick the Imp, at least in the RAW version. The Chain Familiar is not as strong as the Beast, but it's definitely stronger than the Found Familiar, and also if you pick one of the unique creatures, your familiar is able speak and thus has the potential to be a psuedo party member in a way the Beast can't. It's proficiency would scale with it's level like the Found Familiar, but it's minimum hit dice would be equal to your proficiency bonus, that way your psuedo party member isn't made of glass and styrofoam, and there's actually a reason to pick something aside from the Imp, I'd also let them just use the max value for HP, so the Psuedodragon familiar of a 13th level Warlock would have 5d4+5 HP so 25 HP, still fairly weak but it can potentially survive a surprise AoE.
Beast Master: I actually don't have that many problems with it compared to most people, so long as the player picks the right creature, but I do think it's HP is a bit on the low side, 4HP per level is the same as Wizard with no Con bonus, so you're per bear will be the squishiest thing in the party, I don't want to buff it too much, so 5 times your Ranger level seems fairly reasonable, also have their hit die count increase per level, they can use hit dice during a short rest but their hit dice don't scale so it gets very infective at late levels.
Another thing I was considering was letting them get ASIs, not necessarily for the Found Familiar but probably for the Beast and Chain Familiar, but not necessarily at same rate as a character would.
What your thoughts on the original version of these and my proposed buffed versions of them?
Find Familiar: Your changes are not unbalancing, but ultimately it's a first level spell. Even without your changes it's one of the most powerful 1st level spells in the game, with a long duration, scouting, skill use, and help. You could make a higher level Find Greater Familiar spell to improve them, but you'd probably run up against what Pact of the Chain provides.
Beast Master: Like many people, I prefer the Revised Ranger Beast Master and just use that.
Pact of the Chain: Warlocks have a built in way to buff class features with invocations, so I made a custom one to do that. It's based on the revised ranger's companion bond.
Beastmaster - Prerequisite: 7th level Your familiar uses your proficiency bonus rather than its own. In addition to the areas where it normally uses its proficiency bonus, your familiar also adds its proficiency bonus to its AC and to its damage rolls. Your familiar gains proficiency in two skills (or tools) of your choice. It also becomes proficient with all saving throws. For each level you gain after 3rd, your familiar gains an additional hit die and increases its hit points accordingly. Whenever you gain the Ability Score Improvement class feature, your familiar’s abilities also improve. Your familiar can increase one ability score of your choice by 2, or it can increase two ability scores of your choice by 1. As normal, your familiar can’t increase an ability score above 20 using this feature unless its description specifies otherwise.
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My main issue with all of these three is that they don't scale well, or at all in some cases. But one of the problems with with buffing these three is that if you buff then too much in gets into the other's territory, Find Familiar can't be stronger than Pact of the Chain, and Pact of the Chain can't be stronger than Beast Master.
Find Familiar: Having a Found Familiar can be pretty cool, it can scout, use the help action in combat, and make the Wizard no longer the squishiest member of the party. But the higher your level, the less likely you are to even bother spending the 10GP necessarily recast the spell when the Found Familiar inevitably dies. I don't want to buff the spell too much, but one of things I would be that it's proficiency bonus is equal to yours, another's thing I would add is that it's minimum hit dice become equal to half your proficiency bonus. (not sure about rounded up or down.) So than it's skills scale decently well and it becomes slightly more durable, one of my reasons for not increasing it's HP by more is cause 1, it's not as strong as Pact of the Chain, and 2, the Conjurer's level 14 ability will increase your Familiar's HP by 30 and I didn't want to reduce the value of that by too much.
Pact of the Chain: There is no point in picking anything except for the Imp, always pick the Imp, at least in the RAW version. The Chain Familiar is not as strong as the Beast, but it's definitely stronger than the Found Familiar, and also if you pick one of the unique creatures, your familiar is able speak and thus has the potential to be a psuedo party member in a way the Beast can't. It's proficiency would scale with it's level like the Found Familiar, but it's minimum hit dice would be equal to your proficiency bonus, that way your psuedo party member isn't made of glass and styrofoam, and there's actually a reason to pick something aside from the Imp, I'd also let them just use the max value for HP, so the Psuedodragon familiar of a 13th level Warlock would have 5d4+5 HP so 25 HP, still fairly weak but it can potentially survive a surprise AoE.
Beast Master: I actually don't have that many problems with it compared to most people, so long as the player picks the right creature, but I do think it's HP is a bit on the low side, 4HP per level is the same as Wizard with no Con bonus, so you're per bear will be the squishiest thing in the party, I don't want to buff it too much, so 5 times your Ranger level seems fairly reasonable, also have their hit die count increase per level, they can use hit dice during a short rest but their hit dice don't scale so it gets very infective at late levels.
Another thing I was considering was letting them get ASIs, not necessarily for the Found Familiar but probably for the Beast and Chain Familiar, but not necessarily at same rate as a character would.
What your thoughts on the original version of these and my proposed buffed versions of them?
Find Familiar: Your changes are not unbalancing, but ultimately it's a first level spell. Even without your changes it's one of the most powerful 1st level spells in the game, with a long duration, scouting, skill use, and help. You could make a higher level Find Greater Familiar spell to improve them, but you'd probably run up against what Pact of the Chain provides.
Beast Master: Like many people, I prefer the Revised Ranger Beast Master and just use that.
Pact of the Chain: Warlocks have a built in way to buff class features with invocations, so I made a custom one to do that. It's based on the revised ranger's companion bond.
Beastmaster - Prerequisite: 7th level
Your familiar uses your proficiency bonus rather than its own. In addition to the areas where it normally uses its proficiency bonus, your familiar also adds its proficiency bonus to its AC and to its damage rolls.
Your familiar gains proficiency in two skills (or tools) of your choice. It also becomes proficient with all saving throws.
For each level you gain after 3rd, your familiar gains an additional hit die and increases its hit points accordingly. Whenever you gain the Ability Score Improvement class feature, your familiar’s abilities also improve. Your familiar can increase one ability score of your choice by 2, or it can increase two ability scores of your choice by 1. As normal, your familiar can’t increase an ability score above 20 using this feature unless its description specifies otherwise.