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All three nouns are derived from aberrant, the latter two are not used often I suppose considering that spell check considered them misspelt.

What are the difference between the three? Are they interchangeable?

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Yes, you could probably get away with using them interchangeably. However, as always, there are shades of meaning at play.

An "aberration" is something aberrant, like one penny in a jar of nickels.

"Aberrance" is the degree to which something is aberrant. There would be more aberrance associated with a snail in a jar of nickels.

"Aberrancy" names the concept of being aberrant. There's so much aberrancy in the world--a jar of nickels with a penny and a snail in it isn't really all that strange, considering.

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    I think OP is considering them all as nouns with the first definition, with the latter two taking similar forms to the noun "occurence" and "vacancy". Commented Jul 31, 2015 at 14:45
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This question verges on being off-topic as general reference, given that you can find the answer in a standard dictionary. But making sense of the dictionary data does require some understanding of how dictionaries convey information about (formally) undefined terms, so I'll walk through the relevant material on aberration, aberrance, and aberrancy to illustrate how this works.

In Merriam-Webster's Eleventh Collegiate Dictionary (2003), you'll find aberrance and aberrancy listed in boldface type near the end of the entry for aberrant; aberration has its own entry. Here are the relevant entries (with the dictionary's pronunciations of the term omitted because I can't replicate some of the type characters used for them):

aberrant adj {L aberrant-, aberrans, prp. of aberrare to go astray, fr. ab- + errare to wander, err} (ca. 1780) 1 : straying from the right or normal way 2 : deviating from the usual or natural type : ATYPICAL — aberrance naberrancy naberrantly adv

aberration n {L aberrare} (1594) 1 : the fact or an instance of being aberrant esp. from a moral standard or normal state 2 : failure of a mirror, reflecting surface, or lens to produce exact point-to-point correspondence between an object and its image 3 : unsoundness or disorder of the mind 4 : a small periodic change of apparent position in celestial bodies due to the combined effect of the motion of light and the motion of the observer 5 : an aberrant individual

If we accept that the meanings of aberrance and aberrancy are somehow limited by the definitions of aberrant (within whose dictionary entry they appear), we can see that aberration has a number of meanings that they do not share. It follows that aberration is not synonymous with aberrance or aberrancy. But in what respect are the latter two terms limited by the entry for aberrant?

When a dictionary parks a related form of a word at the end of an entry for a word (as the Eleventh Collegiate does here with the nouns aberrance and aberrancy and the adverb aberrantly at the end of the entry for the adjective aberrant), it is indicating that the words thus treated have the meanings given for the main entry but converted to the part of speech that the appended word is classified under, mediated by the meaning or meanings of the relevant suffix.

The Eleventh Collegiate offers these entries for the suffix -ance and the suffix -ancy:

-ance n suffix {ME, fr. AF, fr. L -antia, fr. -ant-, -ans -ant + -ia -Y} 1 : action or process {furtherance} : instance of a quality or process {performance} 2 : quality or state : instance of a quality or state {protuberance} 3 : amount or degree {conductance}

-ancy n suffix {L -antia, more at -ANCE} : quality or state {piquancy}

On the face of it, the -ance suffix has more possible meanings than the suffix -ancy, but the range of meanings that a suffix has in a particular instance depends on the root word it is attached to. Theoretically (according to Merriam-Webster), aberrance might refer to an action or process, a quality or state, or an amount or degree of its root word, while aberrancy can refer only to the quality or state of its root word; but in actual usage, the two words are virtually interchangeable.

In any case, whether you reserve the right to use aberrance in certain situations where the Eleventh Collegiate doesn't endorse using aberrancy (such as to connote the process of doing something aberrant), the two nouns share these meanings: "1 : the quality or state of straying from the right or normal way 2 : the quality or state of deviating from the usual or natural type : being atypical."

I doubt that many people consistently maintain a distinction between aberrance and aberrancy; most people either use them interchangeably or use one to the exclusion of the other. In contrast, though the first definition of aberration is quite similar to the sense of those two words, aberration has other meanings that they do not.

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