Fast Chromatic Scales
The standard fingering for the chromatic scale, starting from C, is 1313123131345 for one ascending octave of the RH (the top is fingered for a return). This fingering is difficult to play fast because it is composed of the shortest possible parallel sets and therefore contains a large number (5) of parallel sets. Its main advantage is its simplicity which makes it applicable to practically any chromatic sequence, starting from any note. One variation of this is 1212123121234, which enables a little more legato.
In attempts to speed up the chromatic scale, several sequences using longer parallel sets have been devised; all of the "accepted" sequences avoid the use of the thumb on a black key. The most commonly used is, starting from E, 123123412312 (Hauer, Czerny, Hanon). One complication with this fingering is that the starting sequence should be changed depending on the starting key in order to maximize velocity. Also, the RH and LH are different; this sequence uses 4 parallel sets. You can shrink it to 3 parallel sets by playing, starting at C, 123412312345. With good TO technique, this scale might be playable, but even with TO, we rarely use a 51 or 15 transition, which is difficult. Clearly, the restriction of avoiding the thumb on a black key limits the choice of fingering.
If we allow one thumb on a black key, a good scale is, starting from C, 123412341234, with the thumb on G# for both hands and 3 identical parallel sets - the simplest possible configuration. I call this the "4-finger chromatic scale". The advantage is simplicity; you use the same fingering no matter where you start, and the fingering is the same for both hands. With good TO technique, this scale can be useful; you only need to pay special attention to the 14 or 41 where 1 is on G#. With the liberation of the thumb by using TO, perhaps it is time we used a chromatic scale with the thumb on a black key and enable playing incredibly fast chromatic scales with ease.
In summary, although most exercises are not helpful, exercising scales, arpeggios and the 4-finger chromatic scale have a special place in piano technique acquisition. Because you can use them to learn so many fundamental technical skills, they must be part of a pianist’s daily practice program.