infer_freq() doesn't recognize monthly output if the time dimension is the middle of each month

What happened?

CESM used to write the time dimension of its output files at the end of the averaging period, so for monthly output the following would hold:

• January averages would have a time dimension of midnight on February 1
• February averages would have a time dimension of midnight on March 1
• etc

The version currently being developed uses the middle of the averaging period, so

• January averages now have a time dimension of noon on January 16 (15.5 days into a 31 day month)
• February averages now have a time dimension of midnight on February 15 (14 days into a 28 day month)
• etc

Some of our diagnostic packages (https://geocat-comp.readthedocs.io/en/latest/user_api/generated/geocat.comp.climatologies.climatology_average.html) require uniformly spaced data and rely on xr.infer_freq() to enforce that. infer_freq() recognizes Feb 1, March 1, April 1, ... as monthly but does not do the same for January 16 (12:00), Feb 15, March 16 (12:00), April 16, ...

What did you expect to happen?

It would be great if infer_freq() could recognize a time dimension of monthly mid-points as having a monthly frequency

Minimal Complete Verifiable Example

import numpy as np
import xarray as xr

month_bounds = np.array([0., 31., 59., 90., 120., 151., 181., 212., 243., 273., 304., 334., 365.])
mid_month = xr.decode_cf(xr.DataArray(0.5*(month_bounds[:-1] + month_bounds[1:]), attrs={'units': 'days since 0001-01-01 00:00:00', 'calendar': 'noleap'}).to_dataset(name='time'))['time']
end_month = xr.decode_cf(xr.DataArray(month_bounds[1:], attrs={'units': 'days since 0001-01-01 00:00:00', 'calendar': 'noleap'}).to_dataset(name='time'))['time']

print(f'infer_freq(mid_month) = {xr.infer_freq(mid_month)}') # None
print(f'infer_freq(end_month) = {xr.infer_freq(end_month)}') # 'MS'

MVCE confirmation

• Minimal example — the example is as focused as reasonably possible to demonstrate the underlying issue in xarray.
• Complete example — the example is self-contained, including all data and the text of any traceback.
• Verifiable example — the example copy & pastes into an IPython prompt or Binder notebook, returning the result.
• New issue — a search of GitHub Issues suggests this is not a duplicate.
• Recent environment — the issue occurs with the latest version of xarray and its dependencies.

Relevant log output

>>> print(f'infer_freq(mid_month) = {xr.infer_freq(mid_month)}') # None
infer_freq(mid_month) = None
>>> print(f'infer_freq(end_month) = {xr.infer_freq(end_month)}') # 'MS'
infer_freq(end_month) = MS

Anything else we need to know?

I'm not familiar enough with xarray to be able to offer up a solution, but I figured logging the issue was a good first step. Sorry I can't do more!

Environment

Start at xarray's infer_freq() entry point and reproduce the provided MVCE with the mid_month and end_month arrays. Trace why the monthly mid-point sequence returns None, then add regression coverage showing that it is recognized while the existing end-of-month behavior remains unchanged.