Our harmonic and melodic system is based on half steps and other systems are based on quarter tones. When we listen to the quarter step system, we are uncomfortable because of this unusual pitch relationship. Our instruments are not capable of playing quarter tones. So why am I posting information on how to perform the imposable? I think you might find the exercise interesting, or even mind altering.
Our brass instruments are based on an overtone system which overlaps each fingering series, i.e. open fingering followed by 2, 1, 12, 23, 13, and 123. If you think of each series overlapping the next, you will realize that as we ascend the scales, this overlapping will create additional false fingerings. False fingerings are different fingerings for the same note. In the lower range, fewer notes have alternate fingering but as we ascend to the higher notes, more and more fingerings become available. Some of these alternate fingerings may be new to you and whether they have any use or value will be something for you to determine. The first page of material includes all of the notes and their alternate fingerings that I have used. The second page was written to utilize these fingerings in a series of exercises. Not every pitch will be exactly one quarter steps for each instrument is made differently just as every player will perform them differently.
As you play through the exercises, try to simulate a true quarter step difference. You will be tempted the first few times to play in half step intervals which will force your ear to incorrectly perform the exercises. With practice, you will eventually be able to relax your ear enough to get with the system.
If nothing more is gained than added knowledge of alternate fingerings, you will have not wasted your time.
If you would like to listen to true quarter tone scales, I encourage you to listen to the Don Ellis Orchestra which was equipped with true quarter tone trumpets. These instruments were designed to play true quarter steps and the unusual sound can only be described as bizarre. Each was designed with an additional fourth valve which lowered the horn one quarter step. How the trumpet players were able to adjust their ears to this system is a wonder.