Corda + RS-1
I gave this combo a try. I also had the HP-1 handy, and several sets of pads (no genuine flat pads, alas).
RS-1 with bowl pads sounds like the RS-1. Peaky with an emphasized high end region. Vocals sounded tipped toward the highs and unnatural. Smooth sound, but nothing close to real.
RS-1 with bowl pads, rubber cement about 50% of the surround: better, but not great. Same kind of sound as above.
RS-1 with modded comfy pads (Grado SR-60 pad with hole cut in center). Much closer perspective, and the high end edginess is greatly reduced. Still a bit peaky
RS-1 with modded HD-414 pad. Best pad for the RS-1/Corda combo, and so far the best pad short of an original flat pad for the RS-1. Finally a pad that takes off the high end edginess. Voices sound intimate and personal. Real. This variant of the RS-1 is a good combo with the Corda.
HP-1 with bowl pad. Much better than RS-1, but midrange isn't full. Perspective is distant, as with RS-1, but doesn't have the same high end sizzle. Voices are more natural, but far away. Loads of detail.
HP-1 with modded bowl pad: Fills in the midrange from the standard bowl pad. Lots of detail, and a very good option for the HP-1
HP-1 with modded comfy pad. Close in perspective makes the music more intimate. Still well detailed, but may be some reduction in detail over the bowl pads.
HP-1 with modded Sennheiser pads. Very clear reduction in detailing over the modded comfy pads. Same intimate perspective, but there's a clear loss of information.
After this exercise, I'd say the modded Senn pads were the best pads in current production for the RS-1. However, I think the HP-1 does better with either the modded bowl pads, for a distant highly detailed perspective, or the modded comfy pads for a more intimate sound with more emphasis on the midrange.
All of this was done with the Corda, which is an excellent amp for either headphone, IMO. You need to go up the price scale or to DIY to do any better.