Abstract: Bemuse introduces an easier set of timegates, determined from statistical data of past 60000 games played on Bemuse, which will be applied to charts with level 1~5. Charts at level 6+ are not affected by this change.
Introduction
Bemuse has very strict timegates. More strict than LR2’s EASY judge (which most BMS uses):
Judgment  Bemuse  LR2#RANK 3


Meticulous / PGreat  20ms  21ms 
Precise / Great  50ms  60ms 
Good / Good  100ms  120ms 
Offbeat / Bad  200ms  200ms 
However, this makes this game very unfriendly for beginners.
Analyzing the impact of the problem
I analyzed the data of about 60,000 games played on Bemuse.
Assuming that beginners play level1 songs (7,000 games), here is the distribution of the given grades:
Grade  Minimum Score  Percentage 

F  0  35% 
D  300000  15% 
C  350000  18% 
B  400000  19% 
A  450000  10% 
S  500000  3% 
68% of all the level1 games ended with bad grades (F, D, C). Getting bad grades all the time is quite demotivating for new players.
First proposal
First, I put a proposal to add beginners timegates with the following thresholds:
Judgment  Normal  Beginner 

Meticulous  20ms  50ms 
Precise  50ms  100ms 
Good  100ms  160ms 
Offbeat  200ms  200ms 
The beginner timegates would be applied on level1 and level2 songs.
I chose to adjust the timegates over other options because:
I don’t want to introduce options (e.g. adding a beginner mode) as that would complicate the game and the scoreboard.
I don’t want to adjust the grading threshold as that would make the grading scheme inconsistent. I prefer to be able to say “If you get more than 450,000 you get an A” regardless of levels or settings.
Therefore, I decided to inflate the score for beginner charts, so that they get good grades easier, while retaining the rest of the game mechanics.
Feedback
I received several initial feedback from people in the BMS Chat Discord group:
People would have hard time adapting to normal timegates when it is suddenly 2.5x smaller.
Changing the timegates at level3 seems too sudden. A more gradual transition may be better.
Some people are not sure whether forcing an easier timegates on easier charts is a good idea or not.
Also:
 It will be very easy to get a full perfect score (555555). The scoreboard will be filled with the same scores if the meticulous timegate is too wide.
Several ideas are suggested:
Gradual increase in difficulty. As level increases, make the timegates stricter.
Separate beginner mode.
Binary judging (e.g. only Meticulous and Offbeat/Missed).
The plan forward
I decided to implement gradual increase in difficulty up to level 5, so that it does not affect any players playing level 6 and above.
I decided not to implement a separate mode due to the reasons listed in the previous section. Also, I decided not to do binary judging as that would result in many people ending up getting the same score (I try to avoid that).
So, I decided to implement 5 sets of timegates:
 Beginner timegates for songs at level 1 and 2.
 Level3 timegates for songs at level 3.
 Level4 timegates for songs at level 4.
 Level5 timegates for songs at level 5.
 Normal timegates for songs at level 6+.
But what numbers should I use for these sets of timegates?
Let’s understand the behavior of players more.
Segmenting the users
I obtained the table of scores at the 1st, 2nd and 3rd quartile for each level.
Note: Level 0 is the tutorial.
Level  Q1  Q2  Q3  Average  n(Games) 

0  104686  154259  268021  198136  2765 
1  252622  346444  414653  327137  7336 
2  322991  381918  426359  366219  2560 
3  324557  381131  427655  368571  4374 
4  320880  385333  438184  371018  4677 
5  329700  388028  436259  375340  5827 
6  341833  397973  443670  385106  5423 
7  354577  411894  452297  392514  5189 
8  377016  421752  456300  408947  4247 
9  377461  429192  466141  415859  5267 
10  379397  431703  469782  416676  5373 
11  400372  443042  474009  428015  2611 
12  399461  437981  469937  422399  3610 
Then I decided 5 personas:
 Beginners: Level1 games that scored between Q1~Q2 (252622~346444).
 Level3: Level3 games that scored between Q1~Q2 (324557~371131).
 Level4: Level4 games that scored between Q1~Q2 (320880~385333).
 Level5: Level5 games that scored between Q1~Q2 (329700~388028).
 Normal: Level8 games that scored between Q1~Q3 (377016~456300).
This allows me to take a closer look at each persona.
The accuracy data
Along with each game’s score, Bemuse also collects the accuracy data, which is a histogram of note offsets.
Note offsets generally follows a normal distribution curve. So, by looking at the S.D. (standard deviation) number we could approximate the accuracy. We assume that the mean is 0ms.
First, start with the S.D. value itself:
$$ \sigma = 17.1\text{ms} $$
Calculate the probability that a note would receive a certain judgment, according to the Normal judge:
$$
\begin{align*}
P[Meticulous] & = 2(\Phi(0)  \Phi(\frac{20\text{ms}}{\sigma})) = 75.8\% \\
P[Precise] & = 2(\Phi(\frac{20\text{ms}}{\sigma})  \Phi(\frac{50\text{ms}}{\sigma})) = 23.9\% \\
P[Good] & = 2(\Phi(\frac{50\text{ms}}{\sigma})  \Phi(\frac{100\text{ms}}{\sigma})) = 0.3\%
\end{align*}
$$
Combine those probabilities to obtain the expected accuracy:
$$
\begin{align*}
E[Accuracy] & = P[Meticulous] + (0.8) P[Precise] + (0.5) P[Good] \\
& = 95.1\%
\end{align*}
$$
Which is quite close to the actual accuracy score reported by the game, 95.07%.
Therefore, the S.D. represents how precise the player is able to hit those notes, independent of the timegates used. The lower the S.D., the more precise.
Data of each persona
Next, we look at each persona, and find out:
 The average S.D. (used to approximate the timegatesindependent accuracy).
 The average combo bonus score (used to determine expected final score).
Here is the resulting data:
Persona:  Normal  Level5  Level4  Level3  Beginner 

E[S.D.]  38ms  50ms  52ms  52ms  64ms 
E[Combo bonus]  21340  14752  15009  14800  19912 
Then we calculate the grade a persona would get under the Normal judge:
Persona:  Normal  Level5  Level4  Level3  Beginner 

P[Meticulous]  40.1%  31.1%  29.9%  29.9%  24.5% 
P[Precise]  41.0%  37.2%  36.4%  36.4%  32.0% 
P[Good]  18.0%  27.2%  28.2%  28.2%  31.6% 
E[Accuracy]  82.0%  74.4%  73.2%  73.2%  66.0% 
E[Score]  431112  386685  380897  380768  349705 
E[Grade]  B  C  C  C  D 
Devising a new judgment scheme
Next, I devised a judgment scheme with the following constraints:
For each persona, the expected grade should be B. Expected final score should be 400000~450000.
It should still be impractical for experts to get a perfect score on beginner songs. The probably of obtaining a perfect score by an expert should stay below 1%.
For this, we need another persona: the Expert persona, which represents the games that scored over 540000. About 100 games were found, so this persona represents the top 0.16% of all the games. This persona have a S.D. value of 12 ms average.
For beginner songs, we calculate the chance of getting a perfect score by calculating the probability in which this persona would get 100 consecutive Meticulous judgments, $P[Meticulous]^{100}$.
For $P[Meticulous]^{100}$ to stay below 1%, the Meticulous timegate should be no more than 24ms.
With this constraints, it’s time to tweak the knobs!
Result
I played around with different numbers until I came up with this judgment scheme:
Judgment  Normal  Level5  Level4  Level3  Beginner 

Meticulous  20ms  21ms  22ms  23ms  24ms 
Precise  50ms  60ms  70ms  80ms  100ms 
Good  100ms  120ms  140ms  160ms  180ms 
Offbeat  200ms  200ms  200ms  200ms  200ms 
E[Score]  431112  408684  419273  429940  430191 
E[Grade]  B  B  B  B  B 
These sets of timegates will hopefully make Bemuse easier for beginners, while retaining the difficulty in the most part.