Tube Filament layout in AA5

I think they can whistle and howl for the same reason.

Cliff, You have it all wrong.
Radios hum because they don't know the words to the songs.  

Serious note:
Typically the filaments on the higher voltage tubes will burn out first also. The 35W4/35Z5/45Z5, 50C5/50L6/25L6/50EH5's.
 Be sure the center tap on the rectifier tubes also is intact. If not it causes a current imbalance in the rectifier fil. ckt, and can cause problems. Replace burned out pilot lights immediately also. They are typically tied to the rectifier tubes fil.; replace with the SAME TYPE BULB also. The old current balance thing. A bad pilot lamp can drastically shorten the life of the rectifier tube filament.

One of the problems that confronts most radio designers, is HUM.
Since hum has to be minimized in tube radios, the order of filaments in a string plays an important factor. 
Filtering of course takes place in the power supply.

The 117 volt hot lead starts at pin 4 and out at pin 3,
 repeats in series for the next 3 tubes.
First is the rectifier 35W4, then in order by the following tubes 50C5 (A-F OUTPUT) 12BA6 (I-F), 12BE6 (CONV), then reverses the filament string hot side to pin 4 then out at pin 3 of 12AV6 (DET 1ST A-F) to common negative, the cathode pin 2 of this last tube is also tied to common negative of supply.
The reason is the detector is the most sensitive to hum. It is the first stage of audio amplification, then any of that amplified hum is past on to the final audio tube which amplifies even more. So the cathode and filament of the detector have the common tie point to reduce any hum. However there will always be some neglible hum, but not annoying. Especially as the volume is increased it will override that hum.