Page 2 of the Classes N1.odt says that multiclassing isn't allowed, with the one specific exception detailed under the bladesinger.

Under that subclass, we get this:

So, where are the rules for multiclassing? Online they can be read here:
https://5thsrd.org/rules/multiclassing/
But they are on page 163 of the PHB as well.
To multiclass, you have to meet the prerequisite stats for the class, and the rules direct you to a table. The prereqs are ALSO printed in the rules for each class, and this table repeats it- which is odd in my opinion. For those curious, ninja is Dexterity 13 and Dark Knight is Strength 13 and Charisma 13.
To multiclass bladesinger, you are doing a wizard (Intelligence 13) and fighter (Dexterity 13 or Strength 13) multiclass. While some multiclass combinations are annoyed by this restriction, bladesingers are not.
The rules then tell us to track different hit dice separately, so a Wizard 6 / Fighter 1 would have 6d6 and 1d10 as hit dice.
The rules then tell us how proficiency bonuses work:

This is interesting because it refers to "total character level", maybe the only place in the rules to define "character level", an otherwise familiar term from 3e. You can tell someone sat in a meeting pitching this, the other two options being adding it as a column to the multiclassing spell table, or providing the formula of:
(FLOOR((SUM(all class levels)-1)/4))+2.
Then we get to the big one:

This information, vital to multiclassing, is NOT under each class. It is nowhere but here. A wizard becoming a fighter gets "Light armor, medium armor, shields, simple weapons, martial weapons". For a bladesinger, this list isn't that interesting. Bladesingers are greatly limited in any armor but light, and can't benefit from their subclass features while wearing a shield. Nonetheless, it's perfectly reasonable to be interested in wearing a shield and stacking it with a legendary lustrate star if you have those coming out of your ears at 20th level- just be aware that it turns off bladesong completely (which means sacrificing every subclass power except Extra Elven Attack)- and takes an entire action to doff.

For those curious, Dark Knight gets the exact same thing as the Fighter (above) and the Paladin, and the ninja is an exceptional deal with "Light armor, proficiency with Stealth or one skill from the class's skill list if already proficient in Stealth, thieves' tools, longswords, shortswords". This list is longer than the average list because the ninja needs light armor, thieves' tools, and ninja weapons to have its core functionality. Note also that even though it appears you can't multiclass into heavy armor, there are ways to cheese that- the main one being to take a cleric domain that knows heavy armor. Regardless, the main way to get heavy armor is to start with a class that has it- artificer being the fast way for wizards. Stuff I don't really want marring anyone trying to conceive of a character, mostly as detailed in my hardcore rant thread.

Then we find the heart of multiclassing:

This elaborates on the main design for this process- you can get levels piece by piece. Since each level in 5e is clearly* designed to have the exact same "power weight", this is the main selling point, the entire reason you'd go through this. Sadly there are some exceptions- with the exception of Channel Divinity, they all make sense.

For the Bladesinger, one level of fighter grants Fighting Style and Second Wind. There's exactly four useful choices for a Bladesinger for Fighting Style.
1- Archery. This doesn't work with any of the features except Extra Elven Attack, but a +2 is a +2.
2- Defense. This is, design-wise, the worst choice. It is, however, a very strong bonus as +1 AC is a lot of mitigation because Bladesingers can get a really high AC, and each point is better than the last.
3- Dueling. A +2 to damage rolls any time you are in melee doing stuff. This also works with things like Toxic Cyclone Strike's initial hit. This is the first or second place pick, competing with Defense.
4- Superior Technique. This is generally not a great option, but your superiority dice can be used defensively or offensively and come back on a short rest.
Second Wind isn't nearly as exciting as it would be for a 1st level fighter. It also scales with fighter level, hurting it for the bladesinger. But it is some hitpoints every short rest.
The second level of Fighter is Action Surge, which gives you an entire action every short rest. This action has no restrictions in any way- cast a spell, attack twice, open a door. I know, I know- opening THREE doors a round! This is the main draw of two levels of fighter.

The Bladesinger feature I put in that allows multiclassing doesn't allow for a third class, where the fighter picks a subclass. I didn't allow that mostly because the third level subclass is a big important lore part, but for the Bladesinger, it is whatever works best with his job, and isn't nearly as cool. Bladesingers already have an identity, they don't need a second one that never grows.

As a note, the way spellcasting mostly works is that you add your full caster levels and half of your halfcaster levels (the dark knight counts as the paladin here, and the artificer is half rounded up as described in yet another fucking place), and then consult the multiclassing spellcasting chart. Which is the same as all the other spellcasting charts, really. Your highest level spells are still determined by your level in your class- a wizard 10 / cleric 5 would have the same spellcasting slots as a wizard 15 or cleric 15, but you can't go above 5th level wizard spells or 3rd level cleric spells. This defines an entire arc of spellcasting hybrids that still don't work great, as going more than 3 levels in another class pretty much destroys your high level spell casting, and you end up trying to find out which spells are badass when upcast (this is also why upcasting fireball to 9th level isn't anywhere close to meteor swarm).

Anyway fighter doesn't give you shit for caster levels, so you're basically just stuck at whatever your wizard casting is exactly. You get the same exact results staring at the multiclassing chart as the wizard one.

Ok, so that's it for the multiclassing rules. What didn't we see? Well, nothing talked about saving throws. In 3e, saving throws often got special boosts for multiclassing, but in 5.X there's no bonus at all. In a game where you can pick either fighter or wizard at 1st level and multiclass freely, fighters wins on this comparison, narrowly. The fighter knows strength, which is great for the bladesinger who has nothing for that, and constitution, which is incredibly good for any caster. Out of the casters, the only one starting with Con proficiency is the sorcerer- and it's definitely considered a perk for him. The wizard gives wisdom, likely the second best of the big three, and intelligence, which the wizard isn't bad at and is very rare.

So this all adds up to that earlier text: The bladesinger gets +4 hit points, as if he had started with a maximized d10 instead of a maximized d6, and then gets to pick between two "good" saves, Con or Wis. Then he gets to pick between two "bad" saves, Str or Int, and the results of these picks replace what the Wizard had at 1st level. This way I am sure to make up for the fact that the Bladesinger couldn't have actually started as a fighter due to my restrictive multiclassing rules.

There's actually other multiclass builds that seem neat, but I feel the need to whitelist them because I just don't have confidence in the design. So right now there's just the one. I floated an idea to Zem in email several months ago about allowing it based on power source, but he didn't seem to think that was cool, and that was enough to drop it from things I'm thinking about. Any thoughts on cool concepts that multiclassing should be serving? Such ideas could be added piecemeal as whitelists or entire other classes.




*redditors actually can't see that this is the design, either being unable to do math or too caught up on the fact that every level is not, actually as good as every other. Regardless, it was discussed when the thing was being designed, and you can reverse engineer the math pretty easily to see what they were going for, even if they didn't actually build it that way.