TL;DR (Actual Question:) I'm wildered; so please explain as though I were 10 years old.
What are the similarities and differences? This doesn`t compare all 4 nouns simultaneously.

A Student's Introduction to English Grammar (2005; by Huddleston and Pullum) uses all 4 terms (3 concurrently, in the penultimate para of page 53). But the book never defines each. OED's definitions (below) are uselessly circular. ♦ Because OED's definition 2 for the noun 'modal' just links to the definition of the ADJECTIVE 'modal', I quote the latter.

modal {adj} = 5. a. Of or relating to the mood of a verb. Of a verb or other element: expressing or used to express modality.

modality = 5. Grammar. The quality or fact of being modal (see modal adj. 5).

mode = 2. a. Grammar. = mood n.2 1.

[ODO] mood2 = 1. Grammar. a. A category or form which indicates whether a verb expresses fact (indicative mood), command (imperative mood), question (interrogative mood), wish (optative mood), or conditionality (subjunctive mood). (cp OED mood n.2).

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    I think this has been answered here: linguistics.stackexchange.com/questions/11648/… – fdb May 6 '15 at 20:01
  • @fdb Thanks. I did read that, but none of the answers explicitly compare all 4 together and simultaneously. So I remain bewildered. – AYX.CLDR May 6 '15 at 20:03
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    They are all just aspects of the same semantic phenomenon. Consider that imperative and interrogative moods are constructions in service of modality, usually but not exclusively deontic modality; the imperative mood means the same thing as deontic must, and the interrogative means either the same thing (as in a classroom) or a softened deontic should, would, could, etc, in demanding or requesting an answer. Optative means want or will, both deontic modal concepts. And Subjunctive and Conditional refer to variants of may, might, possible. It's a good mnemonic. – jlawler May 6 '15 at 21:44
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    Modality is the phenomenon, linguistic and logical; mood is the category (not tense, voice, person, aspect, etc -- btw, mode is just a variant spelling of mood); and modal is the adjective, labelling such things as modal auxiliary verbs, deontic modals, modal polarity items, etc. Just like fungi, fungus, fungal, fungicide. – jlawler May 6 '15 at 21:47

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