You're confusing orthography and phonology.
I know the difference. They always teach vowels and consonants as letters instead of as phonemes, and most people seem to use them that way, so I just have to talk about the phonemes corresponding to those letters. I also don't know IPA very well, and I can't assume anyone else does, so I tend to just say things like "y sound".
Also, there isn't a y sound in "mate" if you pronounce it at normal speed.
http://dictionary.reference.com/ has my as /meɪ/ and mate as /meɪt/. Vowels are a lot more vaguely defined than consonants, so I don't know how consistently dictionaries use the same letter, but it has to have something close to an ɪ in it, or it would just be "met".
But there's no reason for the r in "beard" to have to be a vowel since it's followed by a consonant, since that's never stopped most other consonants before.
You can have multiple consonants in a row like that, but there's always caveats. You can't follow an n with a b, for example. This is because consonants are difficult to pronounce consecutively, unless there's some reason that those two work particularly well. r is like a vowel, and can be placed next to any consonant.
I also don't know IPA very well, and I can't assume anyone else does, so I tend to just say things like "y sound".
That's the problem right there though, you're assuming that 'y sound' corresponds to the letter Y in English. The letter Y can represent either the phoneme /j/ (pronounced as the syllable-initial y), or the smallcaps i. The general rule is that syllable-initially Y represents /j/, elsewhere it represents the smallcaps i. Same goes for W, it's /w/ syllable-initially, /u/ (or smallcaps omega, or barred-u depending on your dialect) e...
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