The special hardware was actually just a DSP at the ISP end. The big difference was before 56k modems, we had multiple analog lines coming into the ISP. We had to upgrade to digital service (DS1 or ISDN PRI) and break out the 64k digital channels to separate DSPs.

The economical way to do that was integrated RAS systems like the Livingston Portmaster, Cisco 5x00 seriers, or Ascend Max. Those would take the aggregated digital line, break out the channels, hold multiple DSPs on multiple boards, and have an Ethernet (or sometimes another DS1 or DS3 for more direct uplink) with all those parts communicating inside the same chassis. In theory, though, you could break out the line in one piece of hardware and then have a bunch of firmware modems.

The asymmetry of 56k standards was 2:1, so if you got a 56k6 link (the best you could get in theory IIRC) your upload rate would be ~28k3. In my expereience the best you would get in real world use was ~48k (so 48kbpd down, 24kbps up), and 42k (so 21k up) was the most I could guarantee would be stable (baring in mind “unstable” meant the link might completely drop randomly, not that there would be a blip here-or-there and all would be well again PDQ afterwards) for a significant length of time.

To get 33k6 up (or even just 28k8 - some ISPs had banks of modems that supported one the 56k6 standards but would not support more than 28k8 symmetric) you needed to force your modem to connect using the older symmetric standards.

The lower case "k" vs upper case "K" is an abomination. The official notation is lower case "k" for 1000 and lower case "ki" for 1024. It's an abomination too, but it's the correct abomination.

Line speed is always base 10. I think everything except RAM (memory, caches etc.) is base 10 really.

Yep. Though I found 42k reliable and a useful boost over 36k6 (14%) if I was planning on downloading something big¹. If you had a 56k capable modem and had a less than ideal line, it was important to force it to 36k6 because failure to connect using the enhanced protocol would usually result in fallback all the way to 28k8 (assuming, of course, that your line wasn't too noisy for even 36k6 to be stable).

I always avoided WinModems, in part because I used Linux a lot, and recommended friends/family do the same. “but it was cheaper!” was a regular refrain when one didn't work well, and I pulled out the good ol' “I told you so”.

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[1] Big by the standards of the day, not today!

Not sure about your last point but in serial comms there are start and stop bits and sometimes parity. We generally used 8 data bits with no parity so in effect there are 10 bits per character including the stop and start bits. That pretty much matched up with file transfer speeds achieved using one of the good protocols that used sliding windows to remove latency. To calculate expected speed just divide baud by 10 to covert from bits per second to characters per second then there is a little efficiency loss due to protocol overhead. This is direct without modems once you introduce those the speed could be variable.

Yes, except that in modern infra i.e. WiFi 6 is 1024-QAM, which is to say there are 1024 states per symbol, so you can transfer up to 10bits per symbol.

An ISDN BRI (basic copper) actually had 2 64kbps b channels, for pots dialup as an ISP you typically had a PRI with 23 b, and 1 d channel.

56k only allowed one ad/da from provider to customer.

When I was troubleshooting clients, the problem was almost always on the customer side of the demarc with old two line or insane star junctions being the primary source.

You didn’t even get 33k on analog switches, but at least US West and GTE had isdn capable switches backed by at least DS# by the time the commercial internet took off. Lata tariffs in the US killed BRIs for the most part.

T1 CAS was still around but in channel CID etc… didn’t really work for their needs.

33.6k still depended on DS# backhaul, but you could be pots on both sides, 56k depended on only one analog conversion.

56k relied on the TX modem to be digitally wired to the DAC that fed the analog segment of the line.

As someone that started with 300/300 and went via 1200/75 to 9600 etc - I don't believe conflating signalling changes with bps is an indication of physical or temporal proximity.