Short answer

The PRICE vowel that we hear in the word wise, /waɪz/, has a systematic relationship with the KIT vowel which we hear in the word wizard, /'wɪzəd/. As we add syllables to the base of a word in English, we tend to reduce the length of the vowel in the base. This is so that we can accommodate the new syllables and still preserve the perceived stress timed rhythm of English. When we add syllables to a base word containing the PRICE vowel, the vowel is very likely to change to a KIT. Other pairs of long and short vowels have this same type of relationship in English.

Full answer

What the Original Poster is observing here is a side-effect of the way English organises stressed syllables in connected speech.

English is a 'stress-timed' language. What this means is that the syllables in English utterances do not come at regular intervals in the way that they do in Japanese for example, or in Spanish. Instead English utterances give the impression that the stressed syllables come at regular intervals. In actual fact this is not strictly what's happening, in the sense that although they give this impression, the stressed syllables do not occur at strictly regular intervals at all.

The upshot of this is that the following utterances, for example, will take roughly the same amount of time to pronounce in English:

fish, peas, beans, rice.

fish and then some peas and a banana and a casserole.

The first utterance there has only four syllables, but the second has fifteen. However, in normal relaxed speech, these sentences will have approximately the same duration. This is because they each have four stressed syllables:

fish , peas , beans , rice .

, , , . fish and then some peas and a banana and a casserole.

Now, if you're a native speaker, if you tap out the stressed syllables as you say the two utterances, you'll realise that the stressed syllables seem to come at regular intervals, regardless of what other material comes in between them.

English has a system for enabling this to happen. One part of this system is that many grammatical words have two forms. They have a strong form which we use if these words are stressed (or stranded). They have a weak form which we use if they are not stressed. So for example, the strong form of can is /kæn/ and the weak form is /kn/ or /kən/. You will be able to hear the difference in the following examples:

I can dance.

I can dance.

What you will notice is that these weak forms have a reduced vowel, most usually a schwa, /ə/. This vowel is easy to pronounce very quickly because it does not require any large movements of the articulators.

Another effect that we find is that a given stressed syllable will become shorter when more unstressed syllables are added. So the if we compare the /mæn/ ('man') in the word man and the /mæn/ in the word manager, we will find that /mæn/ is much longer in the first word than in the second. Compare:

He's a man .

. He's a manager.

The reason for this is that with the word manager, we need to be able to squash the rest of the word in before the next stressed syllable arrives. Reducing the length of the vowel in /mæn/ enables us to squish in the next two syllables without radically increasing the length of time it takes to say the word. This is known as rhythmic clipping.

Within words themselves we can see processes similar to the ones above at work. English has families of vowels. In British English we can notionally divide these families according to the perceived length of the vowel. So in one group we have the long vowels, /i:, u:, ɑ:, ɔ:, ɜ:/ (as in bead, booed, bard, board, bird), and the diphthongs. In the other we have the 'short' vowels, which include, amongst others /ɪ, e, æ, ɒ, ʌ/ (as in bid, bed, bad, bod, butt).

Now, there seems to be a systematic relationship between many long vowels in English and other specific short vowels. If you speak English, these qualities will seem to be logical. In fact, they aren't. In terms of the actual sound there is almost no phonetic relationship between the long vowels and their short vowel counterparts in modern English. For example, the vowels in the words weight and bad are not very similar. We represent these sounds by the symbols: /eɪ/ and /æ/ respectively. However, these vowels have a very close relationship in the language. For example, the following words have the /eɪ/ vowel:

grate ful

ful sane

inflame

The following words, on the other hand, even though they have the same root, usually have the vowel /æ/:

grat itude

itude in san ity

ity inflammatory

This change from long to short vowels usually happens when there are extra syllables added to the base or root of the word. The more syllables there are in a word the more short vowels and the fewer long vowels we are likely to see. We can think of this as a kind of phonological rhythmic clipping. It helps to reduce the amount of time needed to say that syllable in order to accommodate the other unstressed syllables. The following long and short vowels have this same relationship:

eɪ / æ

i: / e

aɪ / ɪ

əʊ / ɒ

aʊ / ʌ

These relationships can be seen in the following pairs of words respectively:

chaste / chas tity

/ tity pen al / pen itentiary

al / itentiary wise / wis dom

/ dom joke / joc ular

/ ular south / southern

The Original Poster's Question

The /aɪ/ diphthong that we find in words such as Michael, Christ and wise has a phonological relationship with the short vowel /ɪ/ in English. Because English is stress timed, it has systematic methods for reducing vowel lengths in order to accommodate unstressed syllables without radically disrupting the natural rhythm of the language. When we add syllables to base words containing the PRICE vowel, /aɪ/, it is likely to change to a KIT vowel /ɪ/. This is what we see in the Original Poster's examples:

'maɪkl /'mɪklməs

'kraɪst / 'krɪsməs

'waɪz / 'wɪzəd

'saɪkl / baɪsɪkl

Transcription note

I have used a standard British English transcription, as used by John Wells in Longman Pronunciation Dictionary