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I am reading the Guide to the buffer bard by Fiznab and I don't understand a detail of the buff.

Some raw numbers to begin: at level 9 you are going to buff your pals for +9 to attack (+4 inspire courage, +2 master performer and greater, +2 flagbearer and banner, +1 haste), +8 damage (+4 inspire courage, +2 master performer and greater, +2 flagbearer and banner), +30 ft movement speed, +1 attack at their max base attack bonus; this is only the BASIC buffs that came from banner, inspire courage and haste.

At level 9, you have

Inspire Courage (Su): [...] An affected ally receives a [...] +1 competence bonus on attack and weapon damage rolls. At 5th level, and every six bard levels thereafter, this bonus increases by +1, to a maximum of +4 at 17th level. [...]

How the +2 inspire courage became a +4 inspire courage?

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I think you read it a bit too fast. Just over, you can see this:

We are going to build a core bard and choose aasimar emberkin as a race; we will use our alternate favored bonus to enhance inspire courage till level 9, than inspire heroics until level 19.

The alternate favored bonus used is

Bard Choose one bardic performance; treat the bard as +1/2 level higher when determining the effects of that performance.

At level 9, you have +2 from here. Well it used to be. An errata was written and now it is +1/6. A post-errata correction would be:

+8 to attack (+2 inspire courage,+1 from alternate favored bonus, +2 master performer and greater, +2 flagbearer and banner, +1 haste), +7 damage (+2 inspire courage,+1 from alternate favored bonus, +2 master performer and greater, +2 flagbearer and banner)

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    \$\begingroup\$ The guide was presumably written using the pre-errata version of the aasimar bard FCB, which was +1/2 level. The specified numbers don't work out if it's +1/6. \$\endgroup\$ – topquark Jul 15 '18 at 16:27

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