"​My experience with worn cutlass bearings is that when they wear, the prop shaft creates a pulsing vibration that varies with speed, not a high pitch whirring sound. Do you get the same sound when running the prop in reverse? Maybe a line fouled on the prop shaft? Do you get any sound while running under sail with the transmission in neautral and engine off (letting the prop freewheel)?"
Paul, I will try running the prop in reverse and see, we just had the bottom cleaned so I do not think a line is fouled, but will check. We will also allow the prop to freewheel and check that out as well, thanks for suggestions!

"​ My theory is the shaft is hitting its resonant frequency setting up a vibration that can be heard but not felt and a slight change in RPM makes the sound disappear. "
Gary this is what I think is occuring but the Admiral's hearing is better than mine and so is her intuition so we wanted to make sure. I appreciate your comments, thank you!

I believe I have found the answer to our "Singing Propeller" . After several searches on Google I found several comments from other sites. Here are some of the comments:

I think I can help you relax on this one. You have what is called in the trade a "whistling propellor". We have the same problem, appearing in the very same rev band you have encountered. What happens is that a harmonic resonation is set up by the propellor presenting itself as a high pitched whine as you experience. The good news is that it apart from the irritating noise it is not harmful in any way, other than minutely diminished efficiency. The cure requires a lift ( I would do it next time you lift the boat for the winter rather than lift just for this) It's not a DIY job - you need a prop repairer - what they do is attach a small edge to one of the blades of the prop to alter its harmonic characteristics. Sometimes requires dropping back in/trial/adjustment. 
The marine engineer I had look at it told me to use the boat normally for the season and have it looked at next time she was out. This was a second opinion by a professional as I wasn't really convinced by the first one!
Read more at
 
Is it "prop singing" - ie a rather misnamed screeching noise that occurs at certain narrow range of prop rpm (caused by too sharp trailing edges on the prop blades) when the prop is clean, but not if there is a bit of weed / polybag on it? Our new boat had it and it sounded god awful until the builder took a grinder to the prop, now its super quiet and smooth. Noises generated by the prop or shaft often sound a lot worse inside the boat than outside (because the sound propagates through the steel) - is this the case for you?
 
 
Some propellers in service produce a highpitched noise, often referred to as singing. This sound typically is a clear harmonic tone much like that of a ringing wine glass. (Other sounds, which have been attributed to singing, have been characterized as “grunting” or “sawing”. These sounds are most likely different in nature than the ringing tone considered here.) More of an annoyance than anything harmful, the causes of singing are not completely understood. This report will try to introduce what – in our opinion – is the likely cause of singing, as well as to offer some guidance regarding ways to eliminate singing from a propeller.
Mitigation of singing These relationships also tell us that singing is a function of propeller diameter and rpm, boat speed, and trailing-edge size (thickness) and “roundness”. We cannot do much about diameter, rpm or speed, but we can modify the edge geometry. This has been the strategy for all efforts to eliminate singing. Most propeller professionals (and others) are familiar with the “anti-singing edge” – a chamfering of the trailing edge, typically on the suction side. The intent of this shape is to avoid the creation of curving flow eddies by cleanly separating the flow off of the blade. The following graphic illustrates the desired geometry of an anti-singing edge, where points of flow separation are spaced both in thickness and in flow-stream position  http://hydrocompinc.com/wp-content/uploads/documents/HC138-SingingPropellers.pdf
 

 I cleaned the prop and applied prop paint to the shaft, strut and prop when we last had the boat out. We had our bottom cleaned by a diver recently and the singing started the first time we took the boat out for a sail after the hull was cleaned. I believe that when the diver was cleaning the prop the paint my have been scrapped off, enough to cause the singing.