Ampex 357 appears to be very close to Ampex 457, but not exactly. Coercivity is very close on 357, 407 and 457 (305, 330, and 320 Oe respectively.) 357 Retentivity is 100 points down from 457, but 100 points higher than 407. Frequency sensitivity @ 1 kHz and 10 kHz are a bit less for 357 than 457, but not much. I've always thought of 357 as on par with 407, but maybe it's technically in between 407 and 457 (making it completely unnecessary… probably why they don’t make it anymore).
I would characterize modern 457 as a new and improved 357, or just a better cut (closer to the center) in the manufacturing process.
The most important thing to know is that all these tapes are bias compatible as demonstrated by the coercivity figures above. Ampex/Quantegy 406, 407, 456 and 457 all require the same bias, as do BASF 911 and 3M 206, 207, 226 and 227… you get the idea. BASF 911 has a coercivity of 320 Oe, same as 456/457. BASF LPR-35 is 320 as well and fits into the same niche as 407/457 as a 1-mil tape for use on machines like the R8, 388 and 22-2.
But in the real world few care about this stuff -- all you need to know is that the tapes are “bias compatible” which is plain English for having a similar coercivity. 406/407 will have a smoother compression than 456/457, but the latter is brighter, though “glassy” if pushed too hard.
So if you have a typical machine set for 250 nW/m flux level, 456 will be bright and clean at 0 VU. On the other hand 406/407 at the same flux level will be closer to saturation and thus produce that pleasing distortion that everyone is after. But it has slightly less sparkle simply because the harder you push tape the lower the top end frequency response. It’s not much change though -- maybe 16 to 18 kHz as opposed to 20 kHz.
In answer to another question, yes I’ve had older (early 80’s) 3M/Scotch 226 fall apart all over my TSR-8 with sticky shed, but newer stuff (in the black box) was fine. 3M hasn’t made tape since about 1994/95, and everyone else is defunct except Quantegy and (God willing) RMGI bringing back the old BASF/EMTEC formulations.
Knowing this stuff is interesting in an academic sense, but if you’re really smart you’ll buy new Quantegy tape for your machine and forget about trying to bend Murphy’s Law in your favor when it comes to the canvas you put you precious work on.
There are really good tapes out there that fewer people are familiar with like BASF/EMTEC SM 468 and AGFA/BASF LM 526. 526 is bias compatible, but most people that still have it are jealously guarding this secret weapon. It’s something to look for though. It was designed for thousands of passes in mastering houses and is virtually indestructible. If you are going to put a tape in a time capsule, put it on 526, or 468 -- another of my favorites, which just happens to be bias compatible with GP9… imagine the possibilities there. All you would have to do is install a level switch and use two of the best tapes ever to come down the pike without having to re-bias. But I digress and now I'm rambling.
Hope that helps.