Applying what I learned from playing the third and fourth octaves on saxophone, my causal experience found the instructions given for whistles mostly fantasy, hallucinations, and myths. Much of what is said may work for "true believers" but I found it much simpler to use something different.

The pitch that comes out of the whistle is strongly affected by the "pitch" of the air volume inside your mouth. If you can "tune your head" to match the note for the fingering you use, the note played will generally be produced easily.

If you set the pennywhistle (or whatever) aside and just "pucker up and blow," even if you're not an accomplished "mouth whistler" and only get a sort of "raspy noise" you should be able to detect a pitch. Raising the middle of your tongue slightly inside your mouth will reduce the air volume inside your mouth and should raise the pitch of the wheeze/whistle. If you can "match the tuning" of your head to the note you finger on the p'whistle, the note you want should be played most easily.

The first two octaves should be fairly easy. There is a slight increase in pressure required for the second octave, but very little change in the the volume of air needed.

Practice by playing (G whistle fingerings?) a series of D d D d D d notes without moving anything but the inside of your mouth. Then go on to E e E e E e, F# f# F# f# etc., up the first octave.

Getting the top d' (the +1 of the "2 octave +1" range) may be aided by "leaking" or even opening the topmost hole (with the other 5 closed) but individual whistles vary in whether this is needed. Especially for higher notes, a "sharp attack" will start the notes more reliably than "wheezing into them," although with practice one should be able to "soften" the attack to avoid making everything staccato.

Once you are able to "blow the note" that you finger, on some whistles you may find that "fine tuning in your head" can pull mistuned notes to proper pitch, and/or "bend" notes without changing the fingering - but this takes at least a half-hour of practice once the method is understood.

(The myth that playing excessively high notes requires squeezing down your head volume until your brains squirt out has never been found to be accurate. There is also no - verified - truth to the claim that this is only true because p'whistlers only have "half a brain" to begin with.)

There are numerous other "instructions" that claim to work, but this is the simplest, and most accurate, method I found.

The matching of mouth (and for larger instruments chest) volume to the pitch you want is universal for many wind instruments, although for more sophisticated instruments it's more critical to getting "tonal quality" than just pitch.

John