Etymology
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Words related to grandame

grand- 
a special use of grand (adj.) in genealogical compounds, originally with the sense of "a generation older than," first attested c. 1200, in Anglo-French graund dame "grandmother," also grandsire (late 13c.), from such use of Old French grand-, which perhaps is modeled on Latin avunculus magnus "great uncle." The partly-Englished grandmother, grandfather are from 15c. Other such words in European languages are formed with the adjectives for "old" or "best" (Danish bedstefar) or as diminutives or pet names (Greek pappos, Welsh taid). The French formation also is the model for such words in German and Dutch. Spanish abuelo is from Latin avus "grandfather" (from PIE *awo- "adult male relative other than the father;" see uncle), via Vulgar Latin *aviolus, a diminutive or adjective substitution for the noun.

The extension of the sense to corresponding relationships of descent, "a generation younger than" (grandson, granddaughter) is from Elizabethan times. The inherited PIE root, *nepot- "grandchild" (see nephew) has shifted to "nephew; niece" in English and other languages (Spanish nieto, nieta). Old English used suna sunu ("son's son"), dohtor sunu ("son's daughter").
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dame (n.)

c. 1200, "a mother," also "a woman of rank or high social position; superior of a convent," and an address for a woman of rank or position, used respectfully to other ladies, from Old French dame "lady, mistress, wife," from Late Latin domna, from Latin domina "lady, mistress of the house," from Latin domus "house" (from PIE root *dem- "house, household").

From early 14c. as "a woman" in general, particularly a mature or married woman or the mistress of a household. Used in Middle English with personifications (Study, Avarice, Fortune, Richesse, Nature, Misericordie). In later use the legal title for the wife of a knight or baronet.

Slang sense of "woman" in the broadest sense, without regard to rank or anything else, is attested by 1902 in American English.

We got sunlight on the sand
We got moonlight on the sea
We got mangoes and bananas
You can pick right off the tree
We got volleyball and ping-pong
And a lot of dandy games!
What ain't we got?
We ain't got dames! 
[Richard Rodgers, "There Is Nothin' Like a Dame," 1949]
grandmother (n.)
early 15c., from grand- + mother (n.1), probably on analogy of French grand-mère. Replaced earlier grandame (c. 1200) and Old English ealdemodor.
granny (n.)
also grannie, 1660s, according to OED, most likely a diminutive and contraction of grannam, shortened form of grandame, rather than from grandmother. The sailor's granny knot (by 1803), originally granny's knot, readily jammed and insecure, is a reef or square knot with the second part crossed the wrong way, so called in contempt because "it is the natural knot tied by women or landsmen" [Smyth, "Sailor's Word-Book," 1867]. Granny Smith apples (1895) are said to have been named for Maria Ann Smith (d. 1870) of Australia, who originated them. Granny glasses attested from 1966.