word usage - How to pronounce fractions larger than a twentieth, where the last digit of the denominator is a 1 or a 2? i.e. one thirtieth is to 30 as _ is to 31

Disclaimer: I speak British English. I've noticed a lot of differences between the way Americans and Brits pronounce numbers.¹ Since the question concerns this, I thought it might be appropriate to draw attention to it case we inadvertently confuse each other. My question is not about these differences, I just wish to highlight them in case they cause confusion.

^{1. You seem to happily call a quarter "a fourth" sometimes,
we always call 131 "one hundred and thirty one",
we pronounce double digits in phone numbers like 12449 as "one two double four nine", and
I think we are much more likely to use expressions like "thirteen hundred" to mean 1300.}

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Question

When talking about fractions, I have frequently heard

1/2    a half
1/3    a third
1/4    a quarter
1/8    an eighth
1/64   a sixty fourth

1/56   a fifty sixth

etc.

Essentially the rule seems to be that, except for "a whole", "a half", and "a quarter", the word matches the ordinal number; that is to say:

    Numeral       Ordinal         Fractional
    one           first           whole
    two           second          half

    three         third           third
    four          fourth          quarter
    five          fifth           fifth
    six           sixth           sixth
    fifty-seven   fifty-seventh   fifty-seventh

Even though ¹/₄ is a quarter, ¹/₆₄ is a sixty-fourth.

So what’s ¹/₆₂? A sixty-twoth? A sixty-second? Surely not a sixty-half!

I know that simply saying one over sixty-two can usually work, but I'm asking specifically for the word itself, i.e. if I divide a huge pizza into 21 pieces, what are the pieces? Other than baker’s twentieths.

Summary: Can anyone point me to any sources (whether style guides or common usage studies or anything else) that discuss the pronunciation of fractions; specifically one that discusses this separately from ordinal numbers, rather than how to form ordinal numbers in the first place.

Answer

You asked for sources.

• Americans pronounce fractions with denominators ending with 1, 2, 3, as in twenty-firsts, twenty-seconds, twenty-thirds. For confirmation, here is a definition from
  Merriam-Webster, one of the canonical American dictionaries.

thirty-second 2 : the quotient of a unit divided by 32 : one of 32 equal parts of anything thirty-second of the total>

The word thirty-twoth does not appear in the Merriam-Webster dictionary, no matter how you spell it. While a few people may use thirty-twoth, it's definitely non-standard.

• There is a difference when the denominator is 4. Americans use both fourths and quarters for one piece of something divided into four equal parts (except for hours, which are always quarter hours) while in the U.K., these are usually quarters.
  Oxford Dictionaries Online has

fourth 2. chiefly North American A quarter:
'nearly three fourths of that money is now gone'

Merriam-Webster has both

fourth : one of four equal parts of something
quarter : one of four equal parts of something