API: make 'freq' mean the same thing for PeriodIndex as DTI/TDI

[jbrockmendel]

ATM PeriodIndex.freq tells us about the index's dtype, while DatetimeIndex.freq tells us about the difference between any two consecutive elements.

Changing PeriodIndex.freq would let us share more code.

update If we do this, we should also deprecate PeriodIndex.is_full, which seems to be a kludge for checking if the index is freq-like in the DTI sense

[simonjayhawkins]

p = period_range("2022", periods=9, freq="M")
print(p.freq)
print(p[::3].freq)

<MonthEnd>
<MonthEnd>

p[::3].freq would be <3 * MonthEnds>?

we would also not raise in infer_freq and return the frequency of the Index, xref #6771

[jbrockmendel]

That's what i have in mind, yes.

[simonjayhawkins]

So would also add the freq to the PeriodIndex repr?

PeriodIndex(['2022-01', '2022-02', '2022-03', '2022-04', '2022-05', '2022-06',
             '2022-07', '2022-08', '2022-09'],
            dtype='period[M]', freq='M')

and for p[::3]

PeriodIndex(['2022-01', '2022-04', '2022-07'], dtype='period[M]', freq='3M')

[jbrockmendel]

Looking at what this might take, I think for consistency if we deprecate PeriodIndex.freq we'd probably need to do the same for Period.freq, which would in turn likely require deprecating freq in the Period constructor. That's pretty invasive and I don't have any good ideas for what to call it instead. "date_offset" is accurate but not that informative.

Another option would be to deprecate 'freq' on DTI/TDI and rename it to something else, though again i don't have a good idea of what that something else would be. "interval" came to mind but that would be confusing with pd.Interval.