I think of it as scenes.
The recomended adventuring day has 2-4 scenes per long rest. Between each scene, you can get a short rest.
An average scene at "local hero" tier (level 1-4) is a (deadly) encounter, a (hard and an easy), (2 medium), (4 easy), or (1 medium 2 easy) encounters roughly. Scenes can be made harder by exceeding this budget. As you gain tiers the scene budget edges up a bit.
(You can give a scene a 4 point budget, and easy=1, medium=2, hard=3, deadly=4. Then you can ramp up or down scene budgets based on what your players can take, and how hard the day/scene should be.)
Also, the DMG math works best for single-class, featless PCs with not that many, nearly random magic items.
If your characters optimize well, then scene budget creeps up.
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If you want more a of a sandbox style world, you can break the world down into 4 tiers - T1, T2, T3, and T4.
A T1 scene budget is a CR sum of 1-6 (~PC level 1-4)
A T2 scene budget is a CR sum of 6-15 (~PC level 5-10)
A T3 scene budget is a CR sum of 15-22 (~PC level 11-16)
A T4 scene budget is a CR sum of 22-30 (~PC level 17-20)
A T5 scene budget is a CR sum of 30-60+
Note that for single monsters above CR 20, you have to rescale their CR to CR*2.5-30, as monster CR after 20 scales faster than it does before 20. (so a CR 30 is really 45).
Encounters where things are "broken up" are easier than if they are all together.
The PC level ranges is for a 4 person party. For other size parties, add up PC levels and divide by 4 is a half-decent way to do it.
(Note: "adding up CR" is imprecise, but it is really easy and gets you pretty close.)
So you could start the players off in a village, with a few problems.
You seed the area with mostly T1 scenes, and some non-aggressive (at least immediately) or hidden T2 scenes, with the T1 scenes leading to information on the T2 ones.
You can also level up using CR per player.
1: 2 CR/PC
2: 4 CR/PC
3: 8 CR/PC
4: 12 CR/PC
5-10: 5 + level*3 CR/PC
11-20: level*2.5 CR/PC
Time in each tier is:
T1: About 25 T1 scenes to reach T2
T2: About 66 T2 scenes to reach T3
T3: About 43 T3 scenes to reach T4
T4: About 20 T4 scenes to max level.
You can use this to help populate your world.
Now, over time, your scenes can move and merge. Two scenes can merge into a scene 1 or 2 Tiers higher.
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In my case, I'm building a world using roughly this mathematics. I'm using gritty rests, so each Scene is an adventuring day, and 2-4 scenes get a long rest (or a week) tacked on; so 10 days per adventuring day, or an average of 3 days per scene.
Based on that I can build a rough calendar of when PCs reach each tier.
Day 1: Tier 1
Day 75: Tier 2
Day 275: Tier 3
Day 400: Tier 4
Day 470: Level 20
This assumes "efficient adventuring". If we throw in a +33% factor for downtime not attributed to adventuring:
Day 1: Tier 1
Day 100: Tier 2
Day 350: Tier 3
Day 500: Tier 4
Day 600: Level 20
So I can populate the world in such a way that the Scenes "level up" over time; basically, unsolved problems get worse.
To do this I need about 150 scenes for my players to encounter over 20 levels. If I want a sandbox, I need 3x that number at least -- call it 500.
Now I can clump these down a bit. I can have something 10x bigger than a scene -- an arc? -- that I can assign a tier to. 10 scenes takes 10 days of adventuring plus 3 long rests, so 31 days, or a month of in-world time.
I don't have to detail these arcs yet. I just have to know how beefy they are at Day 1.
T1: 8 arcs prepped, 2.5 arcs to beat tier.
T2: 20 arcs, 6.5 arcs to beat tier.
T3: 12 arcs, 4.5 arcs to beat tier.
T4: 6 arcs, 2 arcs to max level
Arcs can be linear or not. If they branch, the length should be about 10 scenes (give or take).
An example of an Arc could be a Kobold cult. I want it to be a T1 arc.
10 T1 scenes, total CR from 1 to 6. So each "scene" has 8 to 42 Kobolds in it (!), and the Arc has 80 to 420 Kobolds.
That is a lot of Kobolds. Will get boring.
So I'll say 4 of the scenes in this arc are Kobold based. Kobold bandits. A Kobold patrol. A group of "rebels" in a cave. The temple itself with a shaman, Guard Drake and some Kobolds (low T2 scene, climax).
A random 1d4 dire wolf encounter (CR sum to 1-4) while exploring (or similar).
Maybe the Kobolds are raiding a trade route. And the city thinks it is the hill folk they are paying tribute to, but are cheating. So a the hill folk scene (CR 4ish) (which can be talked around!) can be added to the arc.
The Kobolds are in a Temple. The approach to the temple 1d2 gargoyles guarding it (CR 2-4). Ok 7 now. Getting close.
A stream they have to ford has a Water Weird in it (CR 3).
There will be 3 different paths to follow. 2 of them are wrong, and need creatures. 1d4 Harpies (CR 1-4) along one path, and the other has a Giant Spider and some Spiderlings (CR 1+).
Ok 10 scenes in this arc. If they exhaust it, they are half way out of the tier, and it should take them a month, give or take.
If they wait too long, the Kobolds "level up" and complete their ritual. The result is a T2 threat.