Example: Mae arnat ti ddwy bunt I fi. Why is ‘dwy’ mutated? Is it because there is a soft mutation after ‘ar’?
yup, ‘ar’ causes a soft mutation 
I thought it was due to the “logical subject” of it being ‘you’ - or maybe just that a form like arnat ti feels so similar to a short verb form like dylset ti, where ‘ti’ would cause softening because of being the subject…?
ETA - would Mae gen i ddwy bunt work the same?
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All forms of “ar” cause a soft mutation. Whether followed by the pronoun or not.
Same is true of:
at
am
gan
i
o
And probably some others I can’t remember
It’s not the ‘ti’ that’s causing the mutation, it’s the short form verb itself.
gen is a form of gan, which causes a soft mutation, so the dwy mutates to ddwy because of gan.
Hmm… I’ve been trying to work out how to politely disagree – both because Aran expects us to play nicely, and because I know full well that you speak more and better Welsh than I do. And in a way it doesn’t matter, since the original question was why does this happen: in the end, any linguistic explanation why is just a matter of trying to organise the observed data into patterns, so that we can say “see – x is like y, so that’s why”. It doesn’t really matter if we agree on the patterns, so long as we agree on the data – on what people do (and/or should) actually say.
But Gareth organises the causes for mutations into two basic categories: contact (this word is mutated because it comes directly after this other one), and grammar (e.g. finite verbs tending to have SM even when they don’t have a mi or a fe before them, just because they’re finite verbs).
He doesn’t always make clear which is which, mind you, but I’ve always taken the list of prepositions that cause softening to be about causing contact mutation – cancelled if some other word comes in between. So in the pdf that Sionned posted he gives the example bwrdd – ar fwrdd (table, on a table); but we could add to that ar y bwrdd, where the y comes in-between, and the SM is blocked.
On the other hand, the ‘SM after the notional (even if not grammatical) subject of a sentence’ is a grammatical one, that he illustrates there with rhaid i Dafydd fynd, where mynd gets softened because it’s Dafydd who has to go.
So I was looking for an example that might combine these points, and I found something in the proceedings of the Assembly, in a debate on the M4 relief road, where the phrase “if everyone had two cars” is given as pe bai gan bawb ddau gar. Now, I hope that’s natural, or at least professionally-translated Welsh; and I’d understand it as gan softens pawb by contact; bawb softens dau because pawb is the notional subject – the people who may or may not own the cars; and ddau softens car because, again, contact.
So in the original examples I don’t see how ar or gan can soften something that comes after an ending and a pronoun, like arnat ti – I’d have assumed the ending and the pronoun would block contact, and so it must be something more like the notional subject rule, because of situations where e.g. mae arna’ i ofn means the same as dw i’n ofni.
Does that make sense?
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And this, folks, is why we don’t worry about mutations: Imagine holding this kind of internal debate every time, on the spur of the moment, you need to know whether to say dyw or ddwy! 
That’s not to say that the discussion isn’t interesting, and ultimately it’s always satisfying to know the ‘right’ answer and why it’s ‘correct’.
But in the heat of a casual conversation, down the pub, or at work, or in a café? Not worth worrying about. At all.
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The preposition endings don’t change the fact that they are still prepositions (i.e. arnat is still ar, just applied to the second person, etc), and even though the pronoun ti is in there, I believe it can be considered part of the preposition rather than a separate blocking word because here it is an optional inclusion (mae arnat ddwy bunt = mae arnat ti ddwy bunt).
To be honest though, I’m not sure a) what the grammatical term for that would be, b) if that’s the real reason anyway - it could be c) that’s how the ‘reasoning’ has settled in my brain but I can’t remember if it was my supposition in the first place or something I learnt along the way! 
And I heartily agree with Rob - sometimes I can explain things with my “tip-of-the-iceberg” fluency in grammar, sometimes I can’t because I soon get out of my depth and end up confusing myself, but generally I’m happy to stick with the “that’s-just-how-it-is” method 
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In general, I’m perfectly happy with this – you (one) can speak fluently if you (one) can fly by the seat of your pants, whereas you can’t if you’re having to think through all the rules before you can say anything. And the ability to fly by the seat of one’s pants doesn’t necessarily lend itself to giving a clear answer to ‘why’, but sometimes there isn’t even an easy or clear ‘why’ to give, anyway.
But… (sorry!) I was still trying to find a good clear example, because now I’m trying to get my own understanding of it straight in my head, and I managed to come up with this:
“Mae ar y bwrdd iechyd ddyletswydd i ystyried y cais hwn” (The Board of Health has a duty to consider this request/application…)
How does this help? Well… a bit. Maybe. The thing is, you’re going to have a softening after arnat ti, whether we blame the ar or the ti, because you just do: that’s what people say, and they won’t change what they say just because we can or can’t rationalise it away. But when you’ve got a noun instead, I think it’s clearer that it can’t really be down to the ar: you’ve got y in the way, so it’s bwrdd not fwrdd, and then you’ve got bwrdd and iechyd in the way, but then you still soften dyletswydd because it comes straight after the people who’ve got the duty (as in the English version, “The Board has a duty…”).
So, yeah, I still think the original dwy bunt got softened because you’ve got 'em
I can see your reasoning, but let me just play devil’s advocate and throw this in -
if it’s the you causing the mutation as you suspect, and not the owe (‘ar’ in whatever person)…
mae arnaf ddwy bunt iddo fo
mae arno ddwy bunt iddi
mae arnynt ddwy bunt iddynt
mae ar y gwerthwr ddwy bunt i ti
mae ar y brynwraig ddwy bunt i fi
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Y bwrdd iechyd doesn’t cause the soft mutation here…so surely that reinforces the point that it’s the presence of “ar”?
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On a tangent - when I looked up “ar” in GPC it mentioned occasionally a hard mutation “caledir” l to ll after ar. It gives a reference to a grammar book from 1952, which i haven’t got. I suspect if true it’s probably highly unusual or maybe refering to middle Welsh, but would love to see an example.
No, no, that’s not devil’s advocate, that’s really helpful – because it reveals that we’re actually talking about the same thing at slight cross-purposes, possibly due to me not explaining myself clearly (or “differing to agree” as it’s called in our house…) So I’ll try to explain better what I meant, hopefully without sounding like I’m completely ignoring what you said – because, at the end of the day, it doesn’t really matter how one categorises it in one’s head, so long as it leads you to say the right thing without undue confusion or hesitation.
So I previously thought you meant that the ar caused the mutation in the same way as if I said Rŵan hyn, rhywle yn Rhydychen, mae na gath ar fwrdd – which is probably true: right now, there probably is, somewhere in Oxford, a cat on a table. And I was fixing on that being different from Mae’r gath ar y bwrdd – but that wasn’t really what you meant, because neither the cat nor the table owe anybody anything: it’s just ar meaning ‘on’ mutating the very next word (only) by contact.
On the other hand, when I said that the dwy bunt got softened because of you, I didn’t mean that it was because of it being ‘you’ as opposed to me, or them, or the customer: I meant that whatever comes after the person who owes the money – the person who’d be the subject of the sentence in English – gets softened, which matches your examples.
And I know thinking about it via English seems a bit back-to-front, but I think it’s basically what Gareth means by the ‘notional (not necessarily grammatical) subject’ of the sentence: grammatically I suppose it’s the wretched two pounds that’s the subject of all these sentences, but if I’m talking about the seller short-changing someone, it’s something that the seller does, not the money itself. It’s like when I’ve taught English to native-speaking adults, and I ask them for the ‘subject’ of a sentence like “It was your fault” (because we’re looking at subject-verb agreement, and I don’t want them to come over all Yorkshire and write “It were…”) – and they’ll say “you”, or possibly even “fault”, rather than “it”, because they’re thinking about meaning, rather than grammar.
So yes, we could just go with ar makes things soften when it’s being used in the sense of “owe” – but then we’re going to get things like:
mae gennyf ddwy bunt
mae ganddo fo ddwy bunt
mae gan y brynwraig ddwy bunt
…and we’ll say, OK, OK, gan does it, too. But then we’re going to wind up with a whole list to remember – like @AnthonyCusack’s – which is fine, but I think I’m going to find it harder to remember a list than just to get the feel for it as a pattern that works the same way as Rhaid i Dafydd fynd.
(Phew!)
Um… does that make slightly better sense?
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Yes, that makes more sense. And it also illustrates why it’s so easy to get bogged down with grammatical rules!
and that’s the key . There is indeed a list of prepositions that cause soft mutations, and it is indeed harder to remember a list (despite the mnemonic rhyme “Am, ar, at, dan, dros, drwy ,heb, i, o, gan”) than to ‘go-with-the-flow’
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