Judgment Definitions

This document defines the ordinary judgment model consumed by Scoring V3 and Accuracy V2.

Notation Summary

Symbol	Name	Meaning
$\delta$	Hit timing bias	Time difference between a player action and its ordinary judgment point.
$\mathcal{J}$	Ordinary judgment set	Set of ordinary judgment grades, including the default Miss element $m = \varnothing$.
$J$	Judgment mapping	Function that maps a hit timing bias to an ordinary judgment grade.
$\Sigma$	Ordinary judgment point sequence	Ordered sequence of ordinary judgment points in a chart.
$N$	Ordinary judgment count	Length of $\Sigma$ in this specification. Lightning notes are excluded.
$\Delta$	Hit offset sequence	Ordered sequence of hit timing biases.
$\Xi$	Ordinary judgment sequence	Ordered sequence of ordinary judgment results.
$\mathcal{J}_f$	Lightning judgment set	Dedicated judgment set for Lightning notes.
$\Phi$	Lightning judgment sequence	Ordered sequence of Lightning judgment results.
$F_s$	Lightning Exact count	Number of Lightning Exact results.
$F_m$	Lightning Miss count	Number of Lightning Miss results.
$H_f$	Lightning HP	Lightning-specific reserve consumed by Lightning Miss results.
$O_f$	Lightning overflow	Lightning Miss damage that exceeds remaining Lightning HP.

Terms

An ordinary note is a Dew, Drizzle, or Raindash Note. Ordinary notes produce ordinary judgments.

An ordinary judgment is a judgment result produced by an ordinary note. Ordinary judgments form the sequence $\Xi$ and are used by ScoreV3, AccV2, ordinary combo, All Perfect, and All Combo conditions.

Lightning Notes produce Lightning judgments. Lightning judgments form the separate sequence $\Phi$ and are excluded from $\Xi$ and the ordinary judgment count $N$.

An ordinary judgment point is a reference time used to judge a specific ordinary player action.

A hit timing bias is the time difference, in milliseconds, between the player action and the ordinary judgment point. Negative values indicate early hits; positive values indicate late hits.

A judgment grade is a union of continuous time intervals on the real line. For a judgment grade $l$:

$$l = \bigcup_{i = 1}^{n} I_i$$

where each $I_i$ is a continuous interval on $\mathbb{R}$.

The ordinary judgment set $\mathcal{J}$ contains all ordinary judgment grades and the default element $m = \varnothing$, which represents no produced judgment. Distinct judgment grades do not overlap, and the union of ordinary hit windows is continuous.

Judgment Windows

Milthm uses the following ordinary judgment windows.

Grade	Name	Hit timing bias interval	Window width
$s$	Exact	$[-35, 35]$ ms	$70$ ms
$p$	Perfect	$[-70, -35) \cup (35, 70]$ ms	$70$ ms
$g$	Great	$[-105, -70) \cup (70, 105]$ ms	$70$ ms
$n$	Good	$[-140, -105) \cup (105, 140]$ ms	$70$ ms
$b$	Bad	$[-155, -140) \cup (140, 155]$ ms	$30$ ms
$m$	Miss	$\varnothing$	Not applicable

The judgment mapping $J: \mathbb{R} \to \mathcal{J}$ is:

$$J(\delta) = \begin{cases} j, & \exists j \in \mathcal{J} \left(\delta \in j\right), \\ m, & \text{otherwise.} \end{cases}$$

Judgment Sequence

Let the sequence of ordinary judgment points be:

$$\Sigma = \{\sigma_1, \sigma_2, \cdots, \sigma_N\}$$

where $\Sigma$ is finite and non-decreasing. In this document, the ordinary judgment count $N$ is the length of this sequence. Lightning notes are excluded from $N$.

The hit offset sequence is:

$$\Delta = \{\delta_1, \delta_2, \cdots, \delta_N\}$$

The ordinary judgment sequence is:

$$\Xi = \{\xi_1, \xi_2, \cdots, \xi_N\}, \qquad \xi_i = J(\delta_i)$$

Lightning Statistics

Lightning notes use a dedicated statistic judgment set:

$$\mathcal{J}_f = \{f_s, f_m\}$$

where $f_s$ is Lightning Exact and $f_m$ is Lightning Miss.

Let the Lightning judgment sequence be:

$$\Phi = \{\phi_1, \phi_2, \cdots, \phi_M\}, \qquad \phi_k \in \mathcal{J}_f$$

Lightning statistics use the following state variables.

Symbol	Name	Meaning	Initial value
$F_s$	Lightning Exact count	Cumulative number of Lightning Exact results.	$0$
$F_m$	Lightning Miss count	Cumulative number of Lightning Miss results.	$0$
$H_f$	Lightning HP	Reserve value that absorbs Lightning Miss damage.	$256$
$O_f$	Lightning overflow	Overflow damage from the current Lightning Miss after Lightning HP is exhausted.	$0$

At the beginning of a chart:

$$F_s = 0, \qquad F_m = 0, \qquad H_f = 256$$

Lightning HP is capped at $256$. Ordinary Exact and Perfect judgments recover Lightning HP:

$$H_f \leftarrow \begin{cases} \min(H_f + 2, 256), & \xi_i = s, \\ \min(H_f + 1, 256), & \xi_i = p, \\ H_f, & \text{otherwise.} \end{cases}$$

Lightning Exact does not recover Lightning HP. For each Lightning judgment:

$$\begin{aligned} f_s &: F_s \leftarrow F_s + 1, \\ f_m &: F_m \leftarrow F_m + 1. \end{aligned}$$

Each Lightning Miss deals $64$ Lightning HP damage. Let $H_f^-$ be the Lightning HP before the damage is applied. The overflow damage is:

$$O_f = \max(64 - H_f^-, 0)$$

and the remaining Lightning HP is:

$$H_f \leftarrow \max(H_f^- - 64, 0)$$

If $O_f > 0$, the overflow is applied to the ScoreV3 combo score state and can reduce the final score. See Scoring V3.

Lightning judgments are counted only in Lightning-specific counters and are not added to the ordinary judgment sequence $\Xi$.

Lightning judgments, Lightning HP, and Lightning overflow do not affect All Perfect or All Combo conditions. These conditions are evaluated only from ordinary judgment counters.

When a Lightning Note and ordinary notes are judged at the same time, the Lightning Note is judged after the ordinary notes. If a Drizzle Note is already active, the Lightning judgment does not break the active Drizzle.