On the Interpretation of 2n+1

The other day, regarding the number of positions that can be set on Shimano's electric rear derailleur, I expressed it as 2n+1 with plus/minus n stages plus the initial position.
The 9070 series is n=12 with 25 stages, but
the R9150 series is n=16 with 33 stages...
or rather 32 stages, as I wrote.
This is because Shimano defines it that way.
IMG_6976smsn5.jpg
↑Excerpt from the 6770 series manual
15 stages of adjustment inward from the initial position, 15 stages outward,
for a total of 31 stages of adjustment possible.
That's what it says.

IMG_6977smsn5.jpg
↑Excerpt from the 9070 series manual
12 stages of adjustment inward from the initial position, 12 stages outward,
for a total of 25 stages of adjustment possible.
That's what it says.
Why'd it get fewer than before

IMG_6978smsn5.jpg
↑Excerpt from the R9150 series manual
16 stages of adjustment inward from the initial position, 16 stages outward,
for a total of 32 stages of adjustment possible.
That's what it says.
For some reason, the initial position is not being counted as 1 stage.
I've copied the original text verbatim up to this point,
but even though this portion is just a copy-paste with only the numbers changed,
it doesn't match the 2n+1 interpretation, which is strange.

IMG_6980smsn5.jpg
From the R9250 series manual
18 stages of adjustment inward from the initial position, 18 stages outward,
for a total of 37 stages of adjustment possible.
That's what it says.

In an earlier post I wrote,
"If the number of these stages hasn't changed but
the sprocket count has increased, then
the adjustment range per stage would become coarser."
But to be precise,
"if the sprocket count increased but
the tooth spacing remained unchanged" then
this wouldn't necessarily apply.
For road bikes, there haven't been
examples of that since 9-speed onward.
In Shimano's case, 9-speed and 10-speed
were somehow crammed onto the original 8-speed freehub body,
and with 11-speed they extended the freehub spline,
but 12-speed is crammed in there somehow too.

Hypothetically, if 13-speed came out in the future, there's no way Shimano would have
the flexible judgment to hop on SRAM's XDR

they might switch road to micro-spline,
and looking at past trends, if that happened,
the current Shimano wheels with 12-speed-compatible freehub bodies
would likely remain incompatible,
just like the WH-7900 that didn't support 11-speed.
Though when they do that,
they'll probably simultaneously introduce something like "Road Boost"
and change the hub width.
And if Road Boost does come out,
I can already imagine crappy magazines and crappy curation sites saying
"Emergency test ride! The increased rigidity from the new hub width is palpable!"
and it makes me want to puke.
Just because the overall hub width gets wider,
the wheel structure from hub flange to rim
won't stop being a wimpy wheel
(Yo! Thanks for the rebuilds!)
and riding around.

The shift from quick-release to through-axle
is a bigger factor than the change in overall hub width, I think.
Road rear wheels went from 130mm-width quick-release
to 142mm-width through-axle,
but the flange width and dish amount of the rear hubs in these two standards
aren't that different, so
if you were to compare a "130mm-width through-axle"
with a "142mm-width quick-release,"
the former would probably have less wobble and higher rigidity.
Less wobble can be observed by
the likelihood of the phenomenon where
when you remove the rear wheel and immediately reattach it,
the disc brake, which wasn't rubbing on the rotor before removal,
now rubs against it.
This is especially noticeable in cyclocross,
where disc brakes began to spread
before the through-axle conversion.
Though riders who've been racing MTBs since way back
probably experienced this even earlier.
There's a product from DT where they take a DT rear hub,
convert both ends to their proprietary 10mm through-axle,
and instead of fixing it with the hub shaft under the frame's dropout,
they fix it with the 10mm through-axle,
essentially making a pseudo-through-axle quick-release hub.
Since the frame side is a cutout rather than a through hole,
I don't think you'd get the same satisfying click as a true through-axle,
but it's an interesting idea.

Addendum: I wrote "I don't think you'd get the same satisfying click as a true through-axle,"
but according to someone who's actually used it,
the feel is "rather equivalent to a through-axle,"
and the disc brake engagement is noticeably different, they said.
Thanks always for the comments.


Oops, I got off track.
Regarding examples where the sprocket count (speed) increased but
tooth spacing remained the same.
Shimano's CUES (Cues) comes in
four grades: U8000 series 11-speed, U6000 series 11-speed,
U6000 series 10-speed, and U4000 series 9-speed,
with unified tooth spacing across 9-, 10-, and 11-speed
and using only 11-speed chains, so
if CUES had an electric groupset,
even if you increased the speed count,
the coarseness per adjustment stage would remain unchanged.

For Shimano's electric groupset,
the rear derailleur adjustment reference positioning is,
across all 10-12 speeds, to set the low gear as 1,
shift from there four gears toward the top to gear 5,
enter adjustment mode, and adjust down toward the low side
until you feel the chain plate rubbing against gear 4,
then adjust toward the top side a few times.
The number of times, though, is:
6770 10-speed (n=15) 4 times
9070 11-speed (n=12) 4 times
R9150 11-speed (n=16) 4 times
R9250 12-speed (n=18) 5 times
The lateral movement per stage for each speed count
may or may not be the same,
but the fact that 9070 and R9150 are both 11-speed and both require 4 times means
these two n values have the same movement amount (※), and
the R9150 has a working range that's plus/minus 4 stages
wider.
In that case, if 9070 compared to 6770 is 10-speed→11-speed yet
2n+1 drops from 31→25 stages,
meaning you'd call it "coarser adjustment per stage,"
then R9150 would also be coarser compared to 6770.

※It might not be the same, just that the adjustment reference of 4 times
stays constant

Also, R9250 becoming 12-speed with
narrower tooth spacing yet
requiring 5 times for adjustment reference suggests
the adjustment range per stage is finer,
which would mean the full adjustment width
of R9250's 37 stages might actually be
slightly narrower than R9150's 33 stages (←my interpretation; Shimano says 32 stages).
Hmm, dunno.
Well, as long as you can adjust the shifting, no problem.

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