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When I was taking lessons as a kid I know my scales, arpeggios and chords were never as good as they should have been, so now as a teacher I've been harping on my students to get the correct fingerings on these according to their scale books. Well, I finally had some free time and decided to practice these myself and discovered the published fingerings really are awkward in some cases, in particular when it comes to using the fourth finger on four-note chords/arpeggios and diminished seventh chords.

For example, LH E major chord is fingered 5421. In solid form this feels fine, but in broken form and the arpeggio the 3rd finger is much easier because it reduces the stretch from the 5th finger.
Diminished 7th chords are also all fingered 5421, but with certain black-white key combinations I find it difficult to get the fourth finger in, and often prefer to play the 4th finger instead of the 5th finger if the bottom two notes are black keys.

What do you recommend for fingering? If I do end up changing the fingering I'll have the awkward task of saying, "sorry, I goofed. You're right." to my students after having argued with them about it in their lessons. (A side comment - I never expected students to challenge me on fingering, even in pieces they started learning with me)

I'm not a teacher, but I'm curious about which scales & arpeggios book you're using.

I was brought up on ABRSM, where scales & arpeggios are part of the exam syllabus, and I've always used 3 on black keys in D, E, A, B major arpeggios.

I love it when students challenge me...I am not perfect, and being humble and teachable actually increases respect.

Fingering does have some flexibility... to a certain extent depending upon the person's hand size, technique ability, and what exactly is the fingering under scrutiny...what precedes it, and what comes after it, and what is the tempo, to name a few things.

I would just say to the student the following, which is true:

That sometimes there are several ways to finger things, and good teachers teach a basic "standard" way to beginners, who, upon gaining some knowledge and ability, can adjust it, as long as the "adjustments" are not due to lack of technique, or some willy-nilly ad-hoc changes on the fly, but rather are sensible changes that fit the piece and the player who gains an advantage because of the changes.

And that the student does not deviate from the changes once they are determined to be helpful.

Suggestion:

Finger it this way: 5 ? 2 1, then let the student choose the missing finger.

I believe most students will choose 3. Either 3 or 4 will work.

Tell your students there are two solutions, 3 or 4, and that there is simply not one right answer.

Then remember that eventually we have to deal with the dim7 chord with the octave, and then we use every finger:

B D F Ab B = 5 4 3 2 1 in the LH, reversed in the RH, so obviously reversed in RH.

I'd be interested in what other players use as default. I usually use 3, but always on 4 with second inversions with the octave (LH E G C E = 5 4 2 1)

rocket88 has exactly the right principles here.

I've seen those fingerings in Czerny lots of times. 5421 for the LH in root position and 1245 for the RH in 2nd inversion. I prefer to teach 5421/1245 only for 1st inversion in both hands (the 2-4 gap corresponding to the gap of a 4th in the middle of the chord). So I routinely cross out 4s and write 3s in Czerny exercises.
1245 might make sense for diminished 7ths because the notes are so equidistant, but I prefer to teach 1235, probably for the same reasons as you.

I say you're the teacher and you can set their fingerings for exercises, while acknowledging that this isn't the only way someone might finger it. (In repertoire the students should have more input as long as their fingering choices aren't foolish and can be played consistently.)

But you should also know the curriculum you're assigning so you don't get into awkward positions.

In root position, I've always used 5-4-2-1 for C major and similar in LH. And of course, in first inversion. (Similarly, in C major 2nd inversion RH, I use 1-2-4-5).

I only use 5-3-2-1 if the third is major and on a black key.

I read somewhere that a major reason why some pianists have weak 4th fingers is because it's rarely used in arpeggios, when they're taught to always use 3 if possible.....

If using 3 causes a player to avoid 4, that's not good. But using 4 as a default for any "white 3rd" could be terrible too.

Consider (LH)

G B D G, G Bb D G, alternating. If you are locked into a theoretical fingering you will use 3 on B and Bb, but changing back and forth, either in arpeggios or block chords, you may want to alternate between 3 on B and 4 on Bb.

There are other factors. Some teaching try to blanket all future problems by covering all skills in advance. Then when something comes up that is not covered by those rules and exercises, something else needs to be "invented".

My position is that fingering is an art, so you develop a sense of what to use through playing a huge amount of music, then discovering how to finger the really hard things.

I've just tried your alternating major/minor arpeggios, and have no problem playing the whole lot 5-4-2-1, even at high speed . (In fact, I find it slightly easier than alternating 3 and 4).

The middle finger gets used a lot in chords, trills and other ornaments. Not so the fourth finger - very few pianists would choose to trill using it, if 2-3 or 1-3 is a possibility. To me, it makes sense to deploy 4 where the shape of the hand fits, as in the C major chord and arpeggio in root position (I assume you also use 5-3-2-1 for the chord as well as arpeggio) - presumably that's why Czerny uses that fingering, as does the ABRSM in its scales & arpeggios books.

After all, when one starts playing lots of diminished (and major 7th) arpeggios/chords, not using 4 is no longer an option......

I too have no problem playing the whole lot 5421 at speed. But I would use alternating 4 and 3 myself. It just feels better to me. And that's the point, really. I don't avoid using the 4th finger (I need all the fingers I have!) and I don't consider my 4th to be weak, but I grew up playing all (root position) arpeggios with 3. The exam system here (AMEB) suggested fingering but didn't impose it, and there were no penalties for not adhering to the suggestions. My teacher (bless him) was always one for not fixing anything that wasn't broken, so as long as the results were musical and I was moving freely and comfortably, he let me go, using his (considerable) experience and judgement.
Later on I used 4 in arpeggios far more often, with no problems in adapting. Now I flip between the two, depending on my mood.
This.

I have never had a student criticized for using a fingering that is a bit different than what may be normally called for.

I'm very picky about first inversions of minor and major triads in the RH (1 2 5) and second inversions in the LH (5 2 1) because those are basic hand movements. I give a simple rule for them. Use 3 when the middle finger is very close to the middle. Use 2 when the note in the middle is closer to the thumb, 3 when closer to the pinky (F Ab Db is 1 2 5 in the RH). But I don't see a clear advantage one way or the other for four notes when thumb and pinky are always used with 2, and notes in the "inside" of 2 and 5 may be doing anything.

Here is an example of a pattern that would literally force 3 and 4 to be used, LH:

C D E G C, A G E C, not the best example in the world, but far more unusual things happen.

Sorry, bennevis, I missed your question the first time around. I checked both "The Brown Scale Book" and the Technical Requirements books published for the RCM exams.