Every aspect of playing that is important in the lower and middle register is also important above the staff. ie tone, phrasing, smooth slurs, clean tonguing. We learn these by playing music like the Concone studies. Well if you want those same skills in the upper register then play music in the upper register. A month of playing those same Concone studies an octave up will do more for your playing than a year of arpeggios, scales, pushups or anything else. Yes it is HARD to do. It is NOT fun and since you will not perfect it in a day or two then you don't get that " feeling good about yourself as you hit your first squeeky high r."
The arpeggio approach builds muscle but not control, not tone, no tonguing skills, it stiffens and causes a loss of flexibility . But it does make you "FEEL good" after all you can see that you moved a note.
The arpeggio / scale exercises are all gross muscle building. These will help you to pound out a high note at the end of a chart but they will not help you to play musically up there. They were designed to take you to a certain point and no further. Even the names imply their goals Stevens costello triple high c.. double high c in 10 minutes.. double high c in 27 weeks. They shoot for the student to be able to ----------------------- hit ------------------ a certain note.
My goal is for you to play it.
Let's look at how most people develop their range. From low c to g on the staff they played scales, etudes, SONGS, ect. From high c to super c they played arpeggios holding the top note. So the lower register was developed by making music and it IS musical. While the upper register was 'developed' by making noise and it is NOT musical. I hope that my point is clear. Arpeggios, scales and slurs are only for power. To make music you must play music.
Let me expand on playing musically in the
upper register. I've known players who developed range by arpeggios only.
It works to a point. They worked on it as weight training. (Every other
day. The problem was that they increased the stiffness in their lips to
a point where they lost flexibility. This approach also easily leads to
requiring an embouchure shift. Think about it as you play your 1 octave
arpeggio or even a scale the starting note gets higher and higher. You take a breath and play the next series. You take another breath .... There is an almost overwhelming desire to make subtle changes on each breath. Here is a test start on high c and play an arpeggio up then play it down to low c. Was it slow to respond or of a poor tone quality? Then you are playing with an embouchure shift. There are some ways to avoid this.
Always set your chops for a g on top of
the staff. It is only an octave to high g and only an octave and a half
to low c. When you do practice arpeggios or scales up for range always
play them back
down to low c or below on the same breath. This will help you to learn to play all registers with one embouchure. The reason to play simple songs one or two octaves up is to learn to play musically. Even if it is Mary had a little lamb there IS phrasing. Take a group of melodies that you already have ie Concone (legato or the lyrical studies), Hering 32 etudes, old H.S or Jr. High solos, allregion music, Beethoven transcriptions. Play each exercise 2 times the first time as written and the second time 8va. I used all of these plus The Lazarus Method for Clarinet. Hey clarinet parts move around and if you practice this stuff no lead book will ever throw you.
So you say that you have tried all of this
yet still the notes don't come out. Then you are not letting the sound
out. There are several ways that this can happen. Too much lip compression
will roll the lips in so far that the air can NOT come out. Too much pressure
on your top lip can pin it and again hold in the sound. Finally Too much
lip curl will prevent the notes from coming out. The difference in
embouchure set between low c and second line g is almost none. After all it is only a fifth. So why do people make faces and strain when going from high c to high g. It too is only a fifth. Very little difference in embouchure setup just more airspeed.