CLN: small clean-up PeriodIndex (easy parts of #23416)
xref #23416
Hello @jbrockmendel! Thanks for submitting the PR.

 - There are no PEP8 issues in the file [`pandas/core/arrays/datetimelike.py`](https://github.com/jbrockmendel/pandas/blob/1573d21f6bc43e671ec792627d34042821171e40/pandas/core/arrays/datetimelike.py) !

 - There are no PEP8 issues in the file [`pandas/core/indexes/period.py`](https://github.com/jbrockmendel/pandas/blob/1573d21f6bc43e671ec792627d34042821171e40/pandas/core/indexes/period.py) !

 # [Codecov](https://codecov.io/gh/pandas-dev/pandas/pull/23423?src=pr&el=h1) Report
> Merging [#23423](https://codecov.io/gh/pandas-dev/pandas/pull/23423?src=pr&el=desc) into [master](https://codecov.io/gh/pandas-dev/pandas/commit/7191af9b47f6f57991abdb20625d9563877370a2?src=pr&el=desc) will **increase** coverage by `0.02%`.
> The diff coverage is `100%`.

[![Impacted file tree graph](https://codecov.io/gh/pandas-dev/pandas/pull/23423/graphs/tree.svg?width=650&token=eZ4WkYLtcO&height=150&src=pr)](https://codecov.io/gh/pandas-dev/pandas/pull/23423?src=pr&el=tree)

```diff
@@            Coverage Diff             @@
##           master   #23423      +/-   ##
==========================================
+ Coverage   92.19%   92.21%   +0.02%     
==========================================
  Files         161      161              
  Lines       51192    51173      -19     
==========================================
- Hits        47197    47190       -7     
+ Misses       3995     3983      -12
```

| Flag | Coverage Δ | |
|---|---|---|
| #multiple | `90.65% <100%> (+0.02%)` | :arrow_up: |
| #single | `42.23% <100%> (-0.01%)` | :arrow_down: |

| [Impacted Files](https://codecov.io/gh/pandas-dev/pandas/pull/23423?src=pr&el=tree) | Coverage Δ | |
|---|---|---|
| [pandas/core/arrays/datetimelike.py](https://codecov.io/gh/pandas-dev/pandas/pull/23423/diff?src=pr&el=tree#diff-cGFuZGFzL2NvcmUvYXJyYXlzL2RhdGV0aW1lbGlrZS5weQ==) | `96.11% <100%> (+2.1%)` | :arrow_up: |
| [pandas/core/indexes/period.py](https://codecov.io/gh/pandas-dev/pandas/pull/23423/diff?src=pr&el=tree#diff-cGFuZGFzL2NvcmUvaW5kZXhlcy9wZXJpb2QucHk=) | `92.15% <100%> (+1.11%)` | :arrow_up: |
| [pandas/io/feather\_format.py](https://codecov.io/gh/pandas-dev/pandas/pull/23423/diff?src=pr&el=tree#diff-cGFuZGFzL2lvL2ZlYXRoZXJfZm9ybWF0LnB5) | `77.14% <0%> (-8.58%)` | :arrow_down: |

------

[Continue to review full report at Codecov](https://codecov.io/gh/pandas-dev/pandas/pull/23423?src=pr&el=continue).
> **Legend** - [Click here to learn more](https://docs.codecov.io/docs/codecov-delta)
> `Δ = absolute <relative> (impact)`, `ø = not affected`, `? = missing data`
> Powered by [Codecov](https://codecov.io/gh/pandas-dev/pandas/pull/23423?src=pr&el=footer). Last update [7191af9...1573d21](https://codecov.io/gh/pandas-dev/pandas/pull/23423?src=pr&el=lastupdated). Read the [comment docs](https://docs.codecov.io/docs/pull-request-comments).
 Thanks!