Can the modal auxiliary verbs be considered to be in the present tense (form, not meaning)?

Question

According to Practical English Usage by Michael Swan, modal auxiliary 
verbs do not normally have past tenses:

The modal auxiliary verbs are will, would, shall, should, can,
  could, ought, may, might, and must. Their grammar is different from
  that of other verbs: for example, they have no infinitives,
  participles or past tenses [...] Modal verbs do not have infinitives or
  participles (to may, maying, mayed do not exist), and they do not
  normally have past forms.

So can each modal auxiliary verb be considered to be in the present tense 
(in form, not meaning)? Since the modal auxiliary verbs are finite and they
do not have past tense forms (according to Michael Swan's Practical English
Usage), each must be in a present tense form.

Or are will, can, may and shall the present tenses (in form, not 
meaning) and would, could, might and should the past tenses (in
form, not meaning)?

What about the modal auxiliary must?

Allen S. · Answer

Modals have neither a past tense nor a present participle.  Let's take the example of "can", in the sentence, "I can play the piano."  The past tense of can doesn't exist;  there is no such word as "can-ed".  To express the intention of "I 'can-ed' play the piano", you must paraphrase:  "I have been able to play the piano."  Similar workarounds are required for "may" ("I was allowed to play the piano.")  The lack of a present participle for modals poses the same problem, and requires the same tactic of paraphrase: "Can-ing play the piano makes me very happy" becomes "Being able to play the piano makes me very happy."

"Must" has no past tense. "Must-ed you have missed your son's ball game?" is expressed by, "Did you have to miss your son's ball game?".

There was a time when the Germanic origins of American English indeed regarded "could", "might", "should", and "would" as past tense forms of modals, but no longer (except for some uses of "could" as the past tense of "can").  SEE, http://www.ling.upenn.edu/~beatrice/syntax-textbook/box-modals.html

Can the modal auxiliary verbs be considered to be in the present tense (form, not meaning)?

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