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I agree with that mostly, except some consideration should be given to the critical feats that can do exceptional and awesome things.
do keep in mind here, that weapons with x3 crits actually lose more than spreadsheets say they do. When the crits are overkill the extra damage is wasted far more often than is done on a x2 with a larger critical range
These are both insightful comments and should be considered in addition to my "simplistic" summary of the issue. Another great observation (by Claxon I believe) was considering death by massive damage, which favors weapons with better critical stats.

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"Anti-Paladin" on its own isn't enough to tell us much about the character. Are they Int 13 or above? If so, a Halberd can make better use of the expanded options provided by the Combat Expertise and Improved Trip feat series. For that matter, so could a Horsechopper (which is basically "Halberd with the stats halberds should have had to begin with", so I always play it as one) due to their Reach property. In these cases I would advise halberd or horsechopper.
If not, then it really depends on the flavor you're going for. Brutal? Greataxe. Refined evil? Greatsword.

mcv |

Is it okay if I throw some math at this?
A falchion does on average 5 + x points of damage, where x is your damage bonus (whether that's from high Strength, Power Attack, magic weapon, sneak attack, favored enemy, or whatever). A greatsword does 7 + x damage.
For a falchion, 15% of its hits are crits. For a greatsword 10% of its hits are crits.
So a falchion is better than a greatsword when 1.15(5+x) > 1.1(7+x). Let's rewrite that:
1.15(x+5) > 1.1(7+x)
1.15x + 5.75 > 1.1x + 7.7
0.05x > 1.95
x > 39
So a falchion is better when you're damage bonus is larger than 39. That means that the fighter in my group, that does 23 + 2d4 damage, made the wrong choice. At least for now.
However, what happens when you get an improved crit range? That would mean 30% of the falchion's hits are crits, and 20% of the greatsword's.
1.3(x+5) > 1.2(x+7)
1.3x + 6.5 > 1.2x + 8.4
0.1x > 1.9
x > 19
So as soon as our fighter gets that increased crit range, his choice pays off. But you need a damage bonus of more than 19 and that improved crit range. If you don't go for that improved crit range, stick with your greatsword.

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Is it okay if I throw some math at this?
***math omitted***
Yes, math has been thrown at this (see previous posts), and similar conclusions were reached comparing greatsword and nodachi.
For a falchion, 15% of its hits are crits. For a greatsword 10% of its hits are crits.
This is not strictly true. Specifically, for enemies that require a high roll to hit, a die roll in the critical range of a weapon may not actually hit. This is especially important for enemies with high AC and for iterative attacks at lower bonuses on a full attack action.
However, this is not the final word on the issue, surely. You should read the thread. Others have made good points regarding massive damage and cool crit feats, all of which should be considered if choosing between these weapons purely based on mechanics.

Bigger Club |
This is from purely mechanical standpoint.
Greatsword: The reason to pick this is damage, that being said get rid of it after early levels, if you are a fighter or someone else who specializes in one particular weapon skip it. One noteworthy exception is that it is rather common item so if you play under a GM who doesn't like shops or custom orders from crafters etc. it might be a big deal. All that being said pick up either falcion, nodachi or falcata(nothing preventing from wielding it two handed), latter being best of the bunch and one of the few weapons that is mechanically worth it to have exotic weapon profiency if you can spare the feat.
Halberd: Only reason to pick this is for trip and since reach weapons are better for that nine times out of ten, pick up Hammer, lucerne or Horsechopper.
Great axe: Well sine we are talking antipaladin it can't be the reason of getting it as racial weapon. the x3 critical is nice for coup de grace but you can pick up X4 weapons for that if that were the reason for carrying a melee weapon.
So without further information on the character in question I would say falcata is mechanically the best choice, the fact that you -can- use it one handed is situational advantage but a very nice such, much better than brace for example. I can't be bothered to start making out the math when wielding it is superior to Greatsword/Nodachi/Falcion.
A note on the conversation in the thread is that Full BaB character will very rarely miss on roll of 15 with it's main attack apart from the last iteriative and even then it would probably need to be non-smiting non buffed attack against relativly high AC.(This is based on math in my head so no guarantees on it being correct. Also was assuming builds main thing would be about dealing melee damage.)

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falcata... being best of the bunch and one of the few weapons that is mechanically worth it to have exotic weapon profiency if you can spare the feat.
The falcata does less damage than the greatsword, greataxe, and nodachi... until a character has +23 damage.
The keen falcata is a powerful thing indeed! It does more average damage than a keen greatsword or keen nodachi at just +11 damage or beyond. At mid-high levels, it seems a Keen falcata has a mechanical advantage in this sense.Is it worth a feat? Mechanically, I'm not sure. It depends on what feat you'd be giving up for it.
Full BaB character will very rarely miss on roll of 15 with it's main attack apart from the last iteriative and even then it would probably need to be non-smiting non buffed attack against relativly high AC.
If you look at the AC of monsters near and slightly above a character's level, you'll see that the second iterative attack can often miss on a 15 for the enemies with higher than average AC. Not every monster will be at average for it's CR. It's definitely not "rare" to miss on a 15 with an iterative attack, but the AC of the monster would have to be at least slightly above average, or buffed with Shield, Barkskin, etc. If you want to consider buffs, also consider that enemies can be buffed. There are many CR-appropriate monsters in the Bestiary where a full BAB character can miss on iterative attacks with a roll of 15.

Bigger Club |
Well buff advantage most of the time goes to characters instead of monsters, against the humanoid races we can assume equal.
But like I said it's not a full build or anything I did. But let me put some numbers down.
Let's take level 11 the earliest you have two iteriatives.
BAB 11/6/1
Weapon Focus +1
STR +6
Smite +4
Weapon +3
= +25/+20/+15
So with Roll of 15 Even the last one hits AC 30 that would hit most CR 11 monsters. Power attack would lower it, but buffs will rise it higher. So yeah 11th level with just smite when power attacking you are hitting AC 27 with a roll of 15. Now as I mentioned if it was unbuffed and not smite attack then it would be more common. But on a very brief look only CR 11 that had AC 28 or more was a dragon most had considerably lower than that. CR 12 started to have 27-28 as the average when with CR 11 it was more around 25. Those with higher than average will miss more often naturally but then again when you are met with high AC foe you will use those buffs. So imho as of now it seems that the statement holds water, vast majority of the time roll of 15 will hit.
And yeah I was assuming keen/improved critical when I was talking about falcata. One noteworthy thing is that Antipaladin can add keen to a weapon as early as 5th level via fiendish boon.

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Ok, so here's what I worked out. Assume a level 10 fighter. All feats are the same for both weapons. Assume we are fighting a CR10 mob with 24 AC. This is what you have:
Attack bonuses:
BAB +10
Weapon Training +2
Str +6 (assuming 22 Str)
Weapon Focus +1
Greater Weapon Focus +1
Weapon bonus +3
Power attack 0/-3
Total: +23/15
Damage bonuses"
Str +9 (again, assuming 22 Str)
Weapon Specialization +2
Weapon bonus +3
Power Attack +9
Total: +23
So the DPR formula is h(d+s) + tchd where:
h = chance to hit in decimal form
d = average damage per hit
s = precision damage that isn't multiplied on a hit
t = crit multiplier of weapon
c = chance to hit in decimal form
For the greatsword, assuming a +3 keen greatsword:
DPR = .95(30+0) + (2)(.2)(.95)(30) + .6(30+0) + (2)(.2)(.6)(30)
= 28.5 + 11.4 + 18 + 7.2
= 65.1
For the nodachi, again assuming a +3 keen nodachi
DPR = .95(28.5+0) + (2)(.3)(.95)(28.5) + .6(28.5+0) + (2)(.3)(.6)(28.5)
= 27.075 + 16.245 + 17.1 + 10.26
= 70.68
We'll do 1 more just to make things fair. Maybe you didn't get keen on your greatsword because you think an extra +1 would be better. That +1 doesn't change the chance to hit on the first attack, but it does on the iterative. It also raises the average damage of each attack by 1.
For the greatsword, assuming a +4 greatsword
DPR = .95(31+0) + (2)(.1)(.95)(31) + .65(31) + (2)(.1)(.65)(31)
= 29.45 + 5.89 + 20.15 + 4.03
= 59.52
Just for gits and shiggles let's compare this with a katana, which is 1d8/18-20/x2.
For the katana, assuming a +3 keen katana
DPR = .95(27.5) + (2)(.3)(.95)(27.5) + .6(27.5) + (2)(.3)(.6)(27.5)
= 26.125 + 15.675 + 16.5 + 9.9
= 68.2
Higher than the greatsword and only a couple of DPR below the nodachi. So that shows how important increasing the critical range on your weapon is. It's part of the reason why any min-maxing TWF fighter uses kukri (1d4/18-20/x2) instead of something with 1d6 and 19-20/x2.

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So with Roll of 15 Even the last one hits AC 30 that would hit most CR 11 monsters.
Sure, if you assume a +4 smite bonus for an Antipaladin, who can only smite good... then he'll hit everything on a 15. Not exactly rocket science, and it's very optimistic. We're looking at this without large buffs like smite and favored enemy etc.

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Ok, so here's what I worked out...
***math omitted***
This is all good, and agrees with the conclusions reached in the rest of the thread (summary quoted below).
So, perhaps we can agree to a summary for the OP?
1) non-Keen Greatsword does more average damage than a non-Keen Nodachi, until high levels (+28 damage)
2) Keen Nodachi starts to outdamage the Keen Greatsword (on average) when you reach +13 damage, specifically when your needed die roll to hit is less than or equal to 15
3) When an enemy is very difficult to hit (requiring a roll of 16 or higher) the greatsword has more average damage, unless you have a ton of extra +damage
If you wanted to consider taking Exotic Weapon Profiency, it was pointed out that the Falcata (with Keen) is a solid choice...
The falcata does less damage than the greatsword, greataxe, and nodachi... until a character has +23 damage.
The keen falcata ... does more average damage than a keen greatsword or keen nodachi at just +11 damage or beyond.
I think everyone here has been in some agreement about these mechanics, with the exception of a small, silly debate about whether missing on 15 is just "uncommon" or actually "extremely rare"... So I think we've done a decent job of answering the OP's question on the mechanical front.

Major_Blackhart |
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I would say go with whatever flavor suits you.
I personally have a thing for half-orcs and other more monstrous races.
I've built several weapon masters, one that used a curved elven blade, something he took from an elf that he strangled early on. One that used a large falcata (as well as a bunch of other, normal sized ones.
I've built an unbreakable fighter that used a large dwarven waraxe, simply because of flavor purposes.
Go what you want to have fun with.

Bigger Club |
Bigger Club wrote:So with Roll of 15 Even the last one hits AC 30 that would hit most CR 11 monsters.Sure, if you assume a +4 smite bonus for an Antipaladin, who can only smite good... then he'll hit everything on a 15. Not exactly rocket science, and it's very optimistic. We're looking at this without large buffs like smite and favored enemy etc.
That was not what I stated. Just a note not accusing you of delibaratly missquoting me.(it is in my first post of the thread) But the original statement says that apart from last iteriative, unbuffed and not smiting missing on 15 is rare. So I never claimed anything other.
On the other matters. I am not sure if you were criticising my assumption of +4 on the smite, so just in case I was assuming CHA 16(15 and level increase) with +2 headband. Related to that I would dare assume that in a campaing where you are playing antipaladin good aligned opponents will be vastly more common than in more generic one. When you are fighting non-good foes that is when the buffing and fiendish boon are used. Also my fault for using Full BaB class term the way I did, I was referring to the fact that antipaladin is such a class not all of them.(While I am pretty sure it would be similar results with ranger being hardest to compare since favored enemy is dependant on the campaign.)
But let's see we need to be talking either over APL CR foes or equal CR(with a single opponent) that are not good, the anti paladin does not have much in the way of buffs and that the opponents had higher than average AC and you have to roll the 15 in that fight with the last iteriative. I would say those occurances are rare, perhaps it is just my experience here coloring my opinion.
That being said yes the miss on 15 is a factor when thinking about the wich one is best no matter how rare of an occurance it is in actual gameplay as long as it happens even once. Either way I think we are more or less in agreement, just a matter of degree.
And in relation to the OP, hard to give anything more spesific information than from the mentioned weapons greatsword is the best on generic build. Without more information can't really say anything more. That being said in most games the choice of a weapon apart from if you are wielding two handed, sword and board or TWF it does not really matter all that much.

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Lots of text
You say "rare". I say "uncommon". Yet the difference between these is so small... let's just agree to saying that this occurrence is "uncommon/rare" depending on the campaign.
Then everyone will be happy, we can have hugs and sing Kumbayah, while eating Hot Pockets. Because everyone loves Hot Pockets. Don't even try to deny it.