For example, both 悲しさ and 悲しみ are glossed as “sadness” in JMdict. What's the
difference?

According to _A Dictionary of Basic Japanese Grammar_ , 〜み “is more emotive
and concrete characterization of some state”, while 〜さ “describes, in an
analytical manner, the degree of state represented by an adjective”.

The book uses a few examples to highlight the difference, but I can't make
sense of it.
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nuances adjectives suffixes nominalization
What is the difference between -さ and -み suffixes to make a noun out of an adjective?
`〜さ` seems to describe a "measurable" amount, while `〜み` seems to describe a
general concept of the adjective.

>   * 悲しみ - the general concept of sadness  
>
>   * 映画の悲しさ - the (amount of) sadness of that movie (possibly compared to
> other movies).
>

That's how I tend to compare them.

Also note that many of these types of adjective have corresponding verbs, such
as `悲しむ`、`楽しむ`、`痛む`. The `～み` is just the stem of the `～ます` form, which is
often used as a noun. So this might help distinguish them a little. For
example:

> 帰り (the return), 踊り (a dance), 痛み (a pain), 楽しみ (enjoyment, fun)
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They both mean the same thing but the nuance is as follows:

```

    〜さ (as in 悲しさ、楽しさ、痛さ) indicates a degree or an amount of 〜
    〜み (as in 悲しみ、楽しみ、痛み）indicates a state of being
    
```

I find the following contrasting examples as definitive:

> A:「痛さはどれくらいですか？」 = implies amount
>
> B:「痛み **の程** はどれくらいですか？」 = we add 程{ほど} to indicate an amount

However, to make things easier (or harder) **B** is starting to imply "an
amount" too (I'll see if I can find an example).

My supposition is that **A** was originally "the correct way", but so many
people used the incorrect way that _it_ is becoming correct; _just like in
English, meh!_

On OKWave I found the following useful examples:

```

    「痛み」：　痛むこと、痛んでいる状態にあること = Pain, in a state of pain 
    「痛さ」：　どれだけ痛んでいるかの状態・程度 = A state indicating how much pain, an amount
    
    ○「私達の心の痛みがわかりますか」＝「私達の心が痛んでいることがわかりますか」
       => Do you understand that our hearts are pained (by this)?
    
    ○「私達の心の痛さがわかりますか」＝「私達の心がどれだけ痛んでいるかがわかりますか」
       => Do you understand exactly how much our hearts are pained (by this)?
    
    △「かなりの痛みだ」(NG)
    ○「かなりの痛さだ」＝ very painful
    
    ○「かなりの痛みがある」＝ very painful
    △「かなりの痛さがある」 (NG)
    
    Note, preceding「がある」the most appropriate option is「痛み」.
    
    ○「軽い痛みがある」＝ a little painful
    △「軽い痛さがある」 (NG)
    
```

(Source: <http://okwave.jp/qa/q6754018.html>)
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the 悲しみ version could be re-written as 悲しく感じ, though not 100% accurate is a
way to easily remember what the meaning is. This would be the feeling of
sadness.

さ i like to liken to 差, which means difference or level. So 悲しさ would be level
of sadness
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