What is “warmth” and how do you get it?

According to Technical Director of Benchmark John Siau, DAC1 that I own, was specifically designed not to sound warm but rather natural without enhancement of any harmonics odd or even. Enhancing even harmonics does sound really nice with guitar or voice but not so with instruments that poses a little more complex than regular overtones harmonic structure. Piano is such instrument (as well as percussion instruments) and when it is reproduced on very warm equipment it sounds almost like out of tune.

On the other hand SS gear often designed to sound hi-fiish or for good spects on paper enhances (because of transient intermodulation introduced by deep negative feedback) odd harmonics. Our ears are very sensitive to odd harmonics since loudness cues reside there (like 7th or 9th harmonic).

Neutral sound (whatever it is) is the best but I understand that staying a little on warm side protects from brightly recorded material. Fortunately my Hyperion speakers sound slightly warm with zero sibilants on any CD - and it is in combination with Benchmark DAC1 and unforgiving class D amp.
My older speaker had aluminum dome tweeter and was unbearably bright in the same setup.

Your post should have read , " Warmth why would you want it ". Warmth is an added artifact of electronically produced music . A properly tuned acoustical instrument should not have the added warmth that you here on many systems . Having tuned musical instruments in the past , one thing we listened for was any unnatural warmth , if you could hear warmth something was wrong . Warmth was never considered natural or desirable . It would altar pace, pitch, flow and timing and make an instrument sound less neutral , most easily noticeable on sting instruments . Have A musical day

Having an always "warm" system is like a preference, choice , or flavor and may not be for everyone. I have a small body acoustic guitar and if played back through my system I wouldn't want or expect it to sound warm. I also have a big bodied mellow and bassy sounding guitar and wouldn't want that one to sound thin or bright. I can choose which guitar to play for different sound but I want my system to be neutral and have the capability of being true to the source. That's my preference.

'Warmth' for me is created by a tonal balance that makes reproduced large-orchestra music sound like real music played in a good hall. Of course, a 'good' hall is one that supports the lower frequencies of the orchestra so that the music doesn't sound 'cool' or 'thin'. 'Rich' is another word for warmth. I think the recording processes remove warmth from the music, and a music-reproduction system must recreate it. Warmth is defined, for me, as a tonal balance with a dB or 3 more energy in the lower MR/upper bass...mayb eup a dB by 300Hz, rising to maybe 3dB by 100Hz and maintaining that thru the bottom octave.

Of course, one can go too far on the warmth scale, which turns to 'thick', and that too doesn't sound real to me.

Systems that measure flat tonally are too thin and too bright for me.

Vacuumtubes in the system help retain natural musical warmth, but IMO the speaker system is most responsible for a system sounding warm and real...or not.
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here is a definition of warmth:

a slighy peak in the upper bass/lower midrange in conjunction with a dip in the upper frequencies, usually starting at about 3k.

it is definitely a coloration.

there is a difference between accuracy of timbre and warmth.

when attending live unamplified music, one usually does not use the term warm to describe a musical experience.

what distinguishes live music from listening to a stereo is accuracy of timbre .

an instrument soundws real when you hear it, but not real, maybe almpst real when you hear it on a recording.

very often "warm" is synonym for tube coloration of a euphonic nature.

it is an audiophile term which is not part of the definition of music.