Was there a D to TH sound change in English?

Question

I looked up the etymology of "father" and see what Etymology Dictionary says:

Old English fæder "he who begets a child, nearest male ancestor;"

It clearly says "fæder" with a D.
Wikitionary also has the D version:

From Middle English fader, from Old English fæder, from Proto-West Germanic *fader, from Proto-Germanic *fadēr, from Proto-Indo-European *ph₂tḗr. Doublet of ayr, faeder, padre, pater and père.

The TH was also D in middle English along with old English.
Also the word "mother":

"female parent, a woman in relation to her child," Middle English moder, from Old English modor

But the word "brother":

Old English broþor, from Proto-Germanic *brothar (source also of Old Norse broðir, Danish broder, Old Frisian brother, Dutch broeder, Old High German bruodar, German Bruder, Gothic bróþar), from PIE root *bhrater-.

I don't know what that is, but it is not D. So the change only took place in some words.
Can anyone explain what kind of change that was and what words it applied to?

rajah9 · Answer

Grimm's law is at work here.

Grimm's law consists of three parts which form consecutive phases in
the sense of a chain shift. The phases are usually constructed as
follows:
Proto-Indo-European voiceless stops change into voiceless fricatives.
Proto-Indo-European voiced stops become voiceless stops.
Proto-Indo-European voiced aspirated stops become voiced stops or
fricatives (as allophones). This chain shift (in the order 3,2,1) can
be abstractly represented as:
bʰ → b → p → f
dʰ → d → t → θ
gʰ → g → k → x
gʷʰ → gʷ → kʷ → xʷ
Here each sound moves one position to the right to take on its new
sound value.

(Source: Grimm's Law at Wikipedia)
Consider the chain shift in bold. This phenomenon is at work in the Latin dental to English tooth and in the examples cited by the OP, from Latin pater to OE fæder to English father.

Was there a D to TH sound change in English?

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