You raise a valid concern. On the one hand, we often talk of periphrastic tenses (and other constructions); on the other, some insist that a tense should be confined to a single word. Others, again, hold that tense is a property of a sentence or clause, not of a word or phrase. Can this problem be solved at all?
The short answer is: there are different models; some models are incompatible with certain other models; and we are free to choose whichever model we prefer. The term periphrastic tense is useful in a model that allows for tenses that consist of more than one word, but not in a model that doesn't. The definition of "tense" is not an objective fact that exists independent of human analysis: it is ultimately a label of convenience created by the observer. Both kinds of models have merit.
Most language users happen to think of will do as the future tense. Some linguists use other models. There is no consensus, not even among linguists, about what constitutes a tense.
Even word boundaries are not objective facts
Perhaps the most fundamental issue you raise is that of word boundaries. What were once considered two separate words may fuse into a single, new word, as in cantare habeo => chanterai. At some point in its development, the status of this phrase-or-word must have been uncertain. This shows how relative the whole terminology is.
But in most cases, a reasonable case can be made for either one or the other, so that the fundamental issue temporarily recedes to the background; it should be noted, however, that what we consider a "word" is to some extent intrinsically subjective and a matter of convention. It is just a convenient demarcation. But let's move on.
Is tense determined by form or by function?
Let me illustrate the problem by means of Latin, where terminology has been fixed for a long time. Tense comes from Latin tempus, "time"; part of the oldest concept of tenses had to do with notions of time. However, there was never a one-to-one correspondence between tenses and temporal references. The pluperfect, for example, is normally used to refer to a time before a narrated time in the past, just as in English; and yet after postquam, "after", the perfect was used, not the pluperfect. Similarly, the imperfect and pluperfect could be used to refer to an hypothetical situation in the present, as in English if I was rich... (although subjunctives were far more common). And so on.
Si domi eram, pater me puniebat. = If at_home I_was, father me punished.
"if I were at home, father would punish me."
Postquam Galliam vidi, vici. = After Gaul I_saw, I_conquered_it.
"After I had seen Gaul, I conquered it."
And yet we still call the verbs in these examples imperfect and perfect, respectively, even though they do not have their usual temporal references. The reason we do this is that the form is named after its most common function, even though it can indeed have other functions. Latin and English do this and are by no means the only languages.
Do we then look only at the form of the verb, not at its function, when defining tenses in Latin? No. What we call the passive perfect is periphrastic/analytic/compound, just as in English:
Canis sum. = Dog I_am.
"I am a dog."
Visus sum. = Seen I_am
. "I am/was seen."
You could say this is not a special tense, but two words, one being a past particple, the other a present verb; and yet this is called the passive perfect. The reason is that it functions just as the perfect does—except that it is passive. Here function determines what we call it. This happens in English too when we say I will do it is in the future tense.
Humans like symmetrical systems
So then what constitutes a tense, if we can count neither on form, nor on function, at least not reliably so? The answer is probably symmetry. If there is a present active (video "I see"), a present passive (videor, "I am (being) seen"), and a perfect active (vidi "I saw"), we would like there to be a perfect passive. Because there was no such verbal form, a phrase was made to be equivalent, (visus sum "I was/am seen in the past"). We humans like our systems neat and symmetrical if possible:
Active Passive
Present video videor
Imperfect videbam videbar
Perfect vidi [visus sum]
Future videbo videbor
Now is this label "passive perfect" merely a convention? It may have been once, but, as people start believing in it, they start using it in ways that neatly fit the system, even if the meaning of visus sum was once somewhat different. It is in some ways a self-fulfilling prophecy. Whenever a sentence in the active perfect was passivated, instead of saying "oh, I can't do that", people started thinking, "this is the passive perfect; I will use it". The same applies to I will do it in English.
All three approaches have up-sides and down-sides
Is this a perfect system of terminology? No. There are serious disadvantages. But it has been in use for a long while, and most people think of "I will do it" as fitting within a neat system of past, present, and future, because that is the most convenient and obvious partition of our verb tenses, or so we feel.
Various branches of linguistics have proposed different systems and different terminologies in the past. This is a productive and beneficial approach. Some chose to focus on form and consider the English periphrastic future not a tense at all; they will only count affixes and endings as capable of forming tenses. This system certainly has merit.
Others have emphasised function; they have gone so far as to declare that, since many forms can be used for more than one function, as with si eram... / "if I was...", only foregoing form altogether leads to a consistent approach. Hence they treat tense as a property of a clause or sentence, not of a word or phrase. That way, only combined with a word like yesterday does was acquire a past tense; in if I was at work today, you wouldn't see me here, it is a present tense, because it refers to a situation in the present, be it an hypothetical one. This approach, too, has merit.
One could use several systems at once
As an alternative, we could invent new words for these two new approaches, such as *single-word tenses for the English simple present and simple past, and time-reference or temporality for the time-reference of a clause or sentence. Many different models are possible. Insisting on one model without considering the benefits of other models seems unwise. And saying "x is A" when you mean "I find the model in which x is called A most useful" is a simplification.
Suppletion as an illustration of a convenient choice
Some systems are uncontested, even though at some point in the past a fairly arbitrary choice must have been made.
I go.
I went.
Do these two forms belong to the same verb? Yes, you will, say, because that is what you were taught, and because they "feel" like the same verb, just with odd forms. But, in the past, there were two verbs, both meaning something like going (although there were no doubt some differences between them). At some point the present form of a verb resembling go was taken, its past forms discarded (or not, if such never existed), and the past form of a verb resembling went.
We could say, "there are two defective verbs in modern English, one lacking a past form, the other a present form"; but we choose not to do so. That is to some degree arbitrary, but in this case it is just very convenient. If certain linguists would prefer to treat them as two different verbs, then let them do so, if this is somehow more convenient in a certain linguistic analysis. Or they could just say "this verb consists of two different roots", as they no doubt do.
Of course they're possible.
As to whether you'd want to use them, that's another question entirely. Each new auxiliary verb you use further narrows the temporal interpretation of the verb. After a certain point, it just doesn't matter 99% of the time. In other cases, it just sounds weird.
For the perfect continuous, that's likely because of the double be (has been being) that, like had had in pluperfect, even though it may be perfectly justified, often sounds better simplifying it (has been). Depending on the verb you need, you can partially avoid this slight cacophony by using a different auxiliar for the passive, like get, although that doesn't solve the convulatedness.
Continuous with modals will and would work exactly the same:
| It will have been getting written for hours |
| pres. bare inf. past.part. pres.part. past.part. |
| mod.fut. aux.perf. aux.cont. aux.pass. main |
Here I have a sentence, with the morphological form written on the second line, and the purpose of the verb (to generate the future, auxiliary to the passive, etc)
Modal will (or alternatively would) accepts any non-defective verb. Have fits the bill, and merely requires a past participle. That's been, which is used to form the continuous and simply requires a present participle. That's getting, which is one of the auxiliaries that can form the passive (you could also use being here). It needs the past participle of transitive verb, and written is just that.
Of course, if you use another structure other than modal will for the future, like going to it's even crazier:
| It is going to have been getting written for hours |
| pres. pres.part. bare inf. past.part. pres.part. past.part. |
| aux.cont. aux.fut. aux.perf. aux.cont. aux.pass. main |
I suppose if it's really important to emphasize that the fact that it is currently preparing to be in the process of being marked upon at some point in time prior to some time posterior to now, that works great, but really, it's overkill :-)
Best Answer
What is a tense?
In linguistic terminology, "tense" is a part of verbal paradigm that refers specifically to the time of an utterance. It is impossible for any language to have more than three tenses in this sense, since any action is either past, present, or future.
In English, we do the basic tenses this way:
But what is that with the word will there in the future tense example? It turns out that while English can refer to present and past time using inflections on the verb itself, the future tense always requires another word. Furthermore, there are multiple ways of doing this:
So while English has plenty of ways to refer to future actions, in terms of base verbal morphology there are only two tenses in English: present and past.
So what about perfect, progressive, and the rest of that stuff?
Linguists refer to these as aspect. A verb's aspect refers to its duration, frequency, or completeness. English has three core aspects:
Plus, we can combine progressive and perfect together as follows:
Unfortunately, the way that these forms interact with meaning is very complex. In particular, we often use the simple present ("I walk to the store") to refer to habitual actions, and the simple progressive ("I am walking to the store") to refer to currently ongoing actions.
Now you've made me upset
That's because of mood, the other major component of the English verbal complex. Mood refers to the speaker's attitude towards the action, whether the speaker thinks the action is necessary, obligatory, inevitable, hypothetical, etc. We have a lot of moods in English, indicated by our modal verbs:
Here, again, the form interacts with the meaning in a complicated way. The modal verbs will and shall tend to indicate future time more than anything really "moody", and there are constraints on which moods can be used in which tenses. Just to keep you on your toes.
Really we have 4 modal verbs which occur in present/past tense pairs: will/would, shall/should, can/could, may/might, and then must which can only be present-tense.
And don't forget about voice
Because we also have active voice and passive voice in English, which refer to the subject and the object are assigned to the verb.
These have nothing to do with tense, but they are still part of the verbal paradigm.
Putting it all together
If you multiply all of those together you get eighty-eight possible combinations.
Don't try to memorize them all. Just try to remember the way the pieces interact, and you should be able to construct and interpret any combination that you come across. And remember that many verbs, like the past debitive perfect passive about to appear in this sentence, should rarely have ever been used by anyone.
But it's not that simple
It never is. The preceding elements are the core verbal paradigm, but there are a lot of other things that English does with its verbs to indicate elements of mood, aspect, or tense. Just to name two, we have:
There are lots of other combinations of helping verbs, adverbs, and prepositions which are sometimes used to express tense-like or aspect-like things in English. Merely knowing how the core verbal paradigm fits together doesn't necessarily help you interpret these kinds of utterances. Rather, these idiomatic verbal constructions have to be learned one at a time.
Also, please do read the comments on this answer, as the commenters have brought up numerous other subtleties and distinctions which I didn't get into the main post. The final takeaway of all this discussion is that English verbs are complex and you probably can't count how many forms they have.
Have fun!