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I found a HiNative question where someone asked about the difference between
[痛]{いた}さ and [痛]{いた}み. A native Japanese speaker gave this answer:

> 「痛さ」は形容詞「痛い」が名詞化した名詞です。
>
> 「痛み」は動詞「痛む」の連用形が名詞化した名詞です。

I was taught that み was a nominalizing suffix that is placed after
[形]{けい}[容]{よう}[詞]{し}. However, this answer seems to suggest that it is just
the [連]{れん}[用]{よう}[形]{けい} of a verb ending in ～む.

This made sense to me at first;
[痛]{いた}む、[悲]{かな}しむ、[苦]{くる}しむ、[楽]{たの}しむ、[愛]{いと}しむ →
[痛]{いた}み、[悲]{かな}しみ、[苦]{くる}しみ、[楽]{たの}しみ、[愛]{いと}しみ.

However, you then have words like nidan [温]{あたた}む, which has a
[連]{れん}[用]{よう}[形]{けい} of [温]{あたた}め, but there is an [温]{あたた}かみ, and not to
mention the extra か in there from the [形]{けい}[容]{よう}[詞]{し} [温]{あたた}かい.

So what is the deal with ～み? Where exactly does it come from? Or is it that
[痛]{いた}み is just the [連]{れん}[用]{よう}[形]{けい} of 痛む, but み in [温]{あたた}かみ is
something grammatically separate?
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2021-01-15T05:19:36.393
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question
verbs etymology adjectives suffixes nominalization
What's the origin of ～み following a 形容詞?
_" All of the み!"_

More seriously, all of the ～み here are indeed just the 連用形【れんようけい】 of the
verb-forming suffix ～む, with a base underlying meaning of "seems like, looks
like, appears like: having that quality". This ～む is very likely cognate with
suppositional and volitional verbal suffix ～む, which, via regular sound
shifts, became the modern volitional verb ending ～おう (like in 行【い】こう) or ～よう
(like in 食【た】べよう). This is also likely cognate with similar suffix ～びる (as in
words like 大人【おとな】びる, "to be or seem like an adult"; this had older form ～ぶ,
exhibiting not-uncommon //m// ↔ //b// alternation). And, this is all likely
also cognate with 見【み】る, and with 目【ま・め】.

(Consider the English cluster around _see_ and _seem_ , _look_ and _look like_
, etc.)

Note that this is looking **derivationally** , at where things come from and
how things have changed through time. My impression from various Japanese-
language materials is that most folks may well view ～み as just _" a
nominalizing suffix that is placed after [形容詞]{けいようし}"_, and they don't think
too much about how it might be related to other things.

(To be fair, most native speakers of any language don't think too much about
the language itself -- it's up to us word nerds and language geeks to do that
kind of pondering. ✨)

Regarding your observation that some verbs that end in ～む have a 連用形 of ～め～,
that's actually a slightly different derivation: あたたむ is a verb, not an
adjective + derivational suffix. あたたむ is the older form of modern あたためる. The
related adjective + derivational suffix word is instead あたたかむ, which indeed
has a 連用形 of あたたかみ.

That said, the ～む that becomes ～める in あたたむ is in turn likely cognate again
with the same ～む in あたたかむ -- but that involves a discussion of verb
conjugation paradigms that gets quite deep and involved, which would go beyond
the scope of this question. :)
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> Is it that 痛み is just the 連用形 of 痛む, but み in 温かみ is something grammatically
> separate?

Yes, 痛み is the noun form (転成名詞) derived from the 連用形 of the verb 痛む, which
derived from the i-adjective 痛い.

痛い → 痛む → 痛み  
悲しい → 悲しむ → 悲しみ  
苦しい → 苦しむ → 苦しみ  
あわれな → あわれむ → あわれみ

The み in 温かみ is categorized as a suffix (接尾辞). According to some 国語辞典:

> み〘接尾〙  
>  ❶ 《形容詞・形容動詞語幹に付いて名詞を作る》  
>  ㋐ そのような性質や状態である、そのように感じられる意を表す。  
>  「赤みを帯びる」「温かみのある人柄」「甘み・新鮮み・ありがたみ」  
>  ㋑ そのような状態である場所の意を表す。  
>  「高みに立つ」「深みにはまる」「茂みに入る」  
>
> ◇表記㋐㋑とも、「味」と当てるが、近年はかな書きが一般的（ただし、漢語形容動詞語幹に付くときは、「味」も多い）。なお、これは和語の接尾語で、「甘味料」「人間［醍醐］味」など漢語の「味」とは別。  
>  （明鏡国語辞典）

> み［接尾］  
>  １ 形容詞・形容動詞の語幹に付いて名詞をつくる。  
>  ㋐そういう性質や状態、また、そういう感じを表す。「暖かみ」「ありがたみ」「新鮮み」  
>  ㋑そういう状態の場所を表す。「深みにはまる」「茂みに入る」  
>  （デジタル大辞泉）

あたたかい（「あたたかな」もあります）→ あたたかみ  
やわらかい（「やわらかな」もあります）→ やわらかみ  
ありがたい → ありがたみ  
おもしろい → おもしろみ  
青い → 青み  
新鮮な → 新鮮み  
真剣な → 真剣み  
嫌な → 嫌み
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