Lord Sessadore
Explorer
Something you missed: Scale (+7 AC) plus a heavy shield (+2 AC and Ref) gives the sword-and-board and axe-and-board fighters an AC of 19, as well as a Reflex defense +2 higher than the others. That puts them ahead as a pure defender, both because their AC is ahead (albiet only by 1 point), but neither TWD or the Defensive weapon property grants a bonus to Reflex....
To find out lets examine five cases, a tempest fighter with two shortswords, a one handed fighter with a longsword and shield, a one handed fighter with a battleaxe and shield, a two handed fighter with a greatsword and a two handed fighter with a maul. Lets assume they're all wearing the best armor that takes advantage of their class features, and have the same abilitys scores (18 Str), lets further assume that they're all 1st level and are attacking the same first level soldier (AC 17). With these assumptions the defining elements are the proficiency bonus and base damage of their weapons, lets look at each case:
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Hmmm I guess you're right with PHB only weapons, the tempest fighter is balenced. I think other AV weapons might be a problem as well, not just the double weapons. Specifically anything with the defensive property since it allows the tempest fighter to match the AC of the sword and board fighter and the damage of the greatweapon fighter (this could be mitigated if the bonus from the defensive weapon property was rephrased as a shield bonus rather than an unnamed bonus).
Edit: As a point of reference here's what an Urgosh temest fighter would look like in the above comparison:
AC 18
Proficinecy bonus: +2
Total attack bonus: +7
Damage: Range 7-18, Average 12.5
Chance to hit the example soldier: 60%
Expected damage output: 7.5
However, to properly compare the expected damage of a double-weapon tempest fighter to the other kinds of fighters, remember that all double weapons require a feat. So instead of comparing the Double Axe** user to normal martial weapons, you need to compare it to superior weapons (and since we're using double weapons, I'm going to use other weapons from AV too). Giving you something more like this (each fighter spends 1 feat on a Weapon Proficiency):
Tempest fighter with two Katar*
AC: 17
Proficiency bonus: +3
Total attack bonus: +8
Damage: Range 7-12, Average 9.5
Chance to hit the example soldier: 60% hit + 5% crit
Expected Damage output: 5.7 + 0.775 = 6.475
Fighter with Shield and Bastard Sword
AC: 19
Proficiency bonus: +3
Total attack bonus: +8
Damage: Range 5-14, Average 9.5
Chance to hit the example soldier: 60% hit + 5% crit
Expected damage output: 5.7 + 0.7 = 6.4
Fighter with Waraxe and Shield
AC: 19
Proficiency bonus: +2
Total attack bonus: +7
Damage: Range 5-16 Average 10.5
Chance to hit the example soldier: 55% + 5% crit
Expected damage output: 5.775 + 0.8 = 6.575
Fighter with Fullblade
AC: 17
Proficiency bonus: +3
Total attack bonus: +8
Damage: Range 5-16, Average 10.5
Chance to hit the example soldier: 60% hit + 5% crit
Expected damage output: 6.3 + 1.125 = 7.425
Fighter with Mordenkrad (superior 2h hammer)
AC: 17
Proficiency bonus: +2
Total attack bonus: +7
Damage: Range 8-16, Average 12
Chance to hit the example soldier: 55% hit + 5% crit
Expected damage output: 6.6 + 0.8 = 7.4
Fighter with Execution Axe
AC: 17
Proficiency bonus: +2
Total attack bonus: +7
Damage: Range 7-16, Average 11.5
Chance to hit example soldier: 55% hit + 5% crit
Expected damage output: 6.325 + 1.175 = 7.5
Tempest fighter with Double Axe (off-hand side)**
AC 18
Proficiency bonus: +2
Total attack bonus: +7
Damage: Range 7-16, Average 11.5
Chance to hit the example soldier: 55% hit + 5% crit
Expected damage output: 6.325 + 0.8 = 7.125
*Katar was the best I could find as an upgrade to short swords. The kukri in AV is still 1d6 and brutal 1, but only +2 prof, so I felt katar was a better comparison. In any case, I don't think the tempest fighter has good options for superior, off-hand weapons - same expected damage as normal fighter with a bastard sword, just slightly higher crits.
**Read WotC_Logan's post just before I posted, so I switched the ugrosh to a double axe, using the off-hand side (since that is now the highest damage off-hand weapon).
So, with Wotc_Logan's recent clarification as to the intent of double weapons, we see that using a double weapon puts the tempest fighter using a double weapon somewhere between a normal sword-and-board fighter and a normal 2h fighter. I think that's fair, don't you?
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