Fingering Explained Part 1 (What All Pianists should Know)

[1piano4joe]

31. That's write THIRTY-ONE. Add these numbers 5+10+10+5+1. I figured this out for myself. I haven't seen it any where else. It's original and I am the author. Although I'm sure I just reinvented the wheel again. Why this isn't in every method book ever written I have no idea.

5c1=5

5c2=10

5c3=10

5c4=5

5c5=1

Yes, those are mathematical combinations. The First number which is five in every case is the number of fingers on your hand. The Second number is how many of them you are going to use.

Detailed Analysis:

5c1=5 means you can finger any  single key 5 different ways. Your thumb(1),Your index finger(2),Your middle finger(3),Your index finger(4) or your pinkie(5).

5c2=10 means you can play any interval (think 2 note chord ) as many as 10 different ways. Some will be obviously physically impossible such as an octave with fingers 4 & 5.

These 10 are:

1,2
1,3
1,4
1,5
2,3
2,4
2,5
3,4
3,5
4,5

5c3 Three note chords have up to 10 fingerings and they are:

1,2,3
1,2,4
1,2,5
1,3,4
1,3,5
1,4,5
2,3,4
2,3,5
2,4,5
3,4,5

5c4 Four note chords can only be fingered five ways (not including the thumb playing two keys) and they are:

1,2,3,4 Just don't use your pinkie

1,2,3,5 Your not using you ring finger

1,2,4,5 No middle finger here

1,3,4,5 No index finger

2,3,4,5 No thumb

5c5 only 1 way

1,2,3,4,5

These are all used extensively all the time. These are your "palette of colors" from which you choose. Knowing which to choose and why will be addressed in Part 2.

In bowling the number is 1024

10c1
10c2
10c3

etc. My yet unpublished book titled "spares" will have these listed and how to convert them.
See if you can verify these numbers for yourself with your scientific calculator.

I just wanted to give a little back to all the fine people on here.

[zolaxi]

Sorry, but you lost me after "31."

[birba]

You can say that again.  Either he's completely off his rockers or I'm as dense as the london fog.

[nystul]

Hmmm.

2^5 = 32 (the 32nd combo being the one where 0 fingers are used). 5 unique fingers with 2 states each.

2^10 = 1024.  10 unique fingers or bowling pins with 2 states each.

The person who does math in binary can count from 0 to 1023 (1111111111b) on his fingers.  But knowing the number of combinations isn't particularly important for determining the right one to use.

[nyiregyhazi]

31. That's write THIRTY-ONE. Add these numbers 5+10+10+5+1. I figured this out for myself. I haven't seen it any where else. It's original and I am the author. Although I'm sure I just reinvented the wheel again. Why this isn't in every method book ever written I have no idea.

Because it has zero relevance? What is specifically aided by knowing these things? A chord is fingered by considering what is comfortable based on objective consideration of how the hand physically fits the intervals, coupled with consideration of context among adjacent notes. Are students supposed to start with a checklist to slowly narrow down to something for every chord- rather than look at blatant context, for an immediate fingering? Do you have obsessive compulsive disorder (that's a sincere question- not an intended insult)? Also, there are situations where the order of the fingers might be reversed. Sometimes the thumb will come in the middle of a chord, say, with the second finger over the top. So 31 isn't strictly accurate as the number of possibilities, meaningless as it is anyway.  And permutations cannot be defined according to cold mathematics, as there are physical constraints that will render many permutations impossible.

[perfect_pitch]

I'm just going to ignore this thread... there's something bizarrely stupid about it.

I figured this out for myself. I haven't seen it any where else. It's original and I am the author. Although I'm sure I just reinvented the wheel again. Why this isn't in every method book ever written I have no idea.

Probably because it has no real relevance to piano playing.

[ted]

While this idea admittedly has little relevance to fingering prepared pieces I have used it for years to form exercises which use the whole range of finger combinations and sequences. I have also found it most fruitful in cultivating textural variety in my improvisation.

The same combinatorial approach in the field of harmony occurred to me back in the seventies. I wrote an article about it but music publications didn't want it and it ended up in the New Zealand Mathematical Magazine.

https://www.box.com/shared/xau7zbj2ss
https://www.box.com/shared/vhsc8a176c
https://www.box.com/shared/a6b9hqrris

As conventional musical theory has never interested or concerned me, I made my own based on combinatorial notions and it has proved very fertile for the last forty years, in getting harmonic vocabulary into my brain so it can emerge spontaneously later on in improvisation.

Combinatorics of one sort or another are deeply embedded into many aspects of playing piano music. Therefore, while the intended use of the original poster's thoughts seem peculiar, the underlying idea, it seems to me is of the utmost interest.

[nyiregyhazi]

While this idea admittedly has little relevance to fingering prepared pieces I have used it for years to form exercises which use the whole range of finger combinations and sequences. I have also found it most fruitful in cultivating textural variety in my improvisation.

The same combinatorial approach in the field of harmony occurred to me back in the seventies. I wrote an article about it but music publications didn't want it and it ended up in the New Zealand Mathematical Magazine.

https://www.box.com/shared/xau7zbj2ss
https://www.box.com/shared/vhsc8a176c
https://www.box.com/shared/a6b9hqrris

As conventional musical theory has never interested or concerned me, I made my own based on combinatorial notions and it has proved very fertile for the last forty years, in getting harmonic vocabulary into my brain so it can emerge spontaneously later on in improvisation.

Combinatorics of one sort or another are deeply embedded into many aspects of playing piano music. Therefore, while the intended use of the original poster's thoughts seem peculiar, the underlying idea, it seems to me is of the utmost interest.

In the context of composition, I'd be far more inclined to agree. It's the idea that original post is supposed to mean anything of use to a performer (regarding fingering) that I'm bemused by. Arguably it might be a little synthetic to start from something numerical- but a good composer can see where interest arises and capitalise. It makes sense that something creative could come of what starts as a mathematical process. Once a pool of possibilities has been created, the process of drawing from it would be entirely human. However, I cannot see ANY obvious respect in which the original post could provide something of creative or practical use to a performer. It's just maths for the sake of maths. Having a generally inquistive mind, I did find it mildly interesting to think about the possible combinations. But I have no idea what I could possibly use that knowledge in aid of. For it to reveal anything about musical texture, we'd have to include both hands. And even if we take 1024 possibilities, it still says far more about something physical than about anything musical. We can at least square that figure, merely to bring in issues of how different textures are created by different spacings and intervals (which is still the tip of the iceberg-considering that it refers to one lone chord).

[ted]

In the context of composition, I'd be far more inclined to agree. It's the idea that original post is supposedto mean anything to a performer about fingering that I'm bemused by. Arguably it might be a little synthetic to start from something numerical- but the composer can see where interest arises. It makes sense that something creative could come of what starts as a mathematical process. Once a pool of possibilities has been created, the process of drawing from it would be entirely human. However, I cannot see ANY obvious respect in which the original post could provide something of creative or practical use to a performer. It's just maths.

Yes, I agree exactly, "pool of possibilities" is a very good description. The idea that pretty patterns directly translated into music will produce pretty sounds is silly to me too. Mind you, it was anything but silly to an awful lot of famous composers in the twentieth century. I don't understand why though. Heuristic algorithmic composition, on the other hand, is interesting, and I have had a go at that, but I'm getting too old to spend time on it when I can sit at the instrument and immediately record hundreds of ideas which transport me in a direct, human sense.

Probably these things did not occur to the original poster and I am reading more into it than could possibly have been intended.

[cjp_piano]

While this idea admittedly has little relevance to fingering prepared pieces I have used it for years to form exercises which use the whole range of finger combinations and sequences. I have also found it most fruitful in cultivating textural variety in my improvisation.

The same combinatorial approach in the field of harmony occurred to me back in the seventies. I wrote an article about it but music publications didn't want it and it ended up in the New Zealand Mathematical Magazine.

https://www.box.com/shared/xau7zbj2ss
https://www.box.com/shared/vhsc8a176c
https://www.box.com/shared/a6b9hqrris

As conventional musical theory has never interested or concerned me, I made my own based on combinatorial notions and it has proved very fertile for the last forty years, in getting harmonic vocabulary into my brain so it can emerge spontaneously later on in improvisation.

Combinatorics of one sort or another are deeply embedded into many aspects of playing piano music. Therefore, while the intended use of the original poster's thoughts seem peculiar, the underlying idea, it seems to me is of the utmost interest.

Your analysis is similar to set theory, have you studied this? I remember it from grad school. We analyzed some atonal music this way.

[1piano4joe]

Pointless, irrelevant, useless and stupid my posts have been called.

 I am a tutor of organic chemistry, calculus, physics and biology. I went to medical school. I scored on the physics section of the MCAT (Medical College Admissions Test) in the top 3% of the country. This includes students considered the best and the brightest with their 4.0's from Harvard, Yale and MIT.

I organize and process information differently. I have over 10 years of formal college education costing over $200,000 in tuition and have attended around a dozen or so universities. This included many lab hours involving the "scientific method" where you state a hypothesis. etc. So yeah, my posts will be intelligent and unusual to say the least.

One poster said, "Either he's completely off his rockers or I'm as dense as the London fog."

Maybe I should stop posting or you should stop reading them as Scientific American isn't for everybody.

O.K. I feel a little better now.

Oh, I forgot. I also play the piano.

Most method books teach "Correct Fingering".

A root position C major triad is "correctly fingered" by the right hand thusly

C E G
1 3 5

Is is not? Are we all not taught that?

The 1st inversion is "correctly fingered"

E G C
1 2 5
 
Apparently, my piano teacher was a retard. Right?

Scales in octaves are played with fingers 1 and 5 taking the black keys with 1 and 4.

Is this not so?

Scales in thirds can use 1,3 then 2,4 then 3,5 etc.

Do they not?

Lastly, consider in Chopin's prelude Opus 28 No. 20 in C minor the 3rd measure, beat 2, the five note chord of the right hand

E  G  Bb  C  E
1  2  3    4  5

I will "stick to my guns" and say this is the one and only way to finger this chord.

You disagree. I implore (beg you) to post your "bizarre fingering" for this chord.


As to set theory, one should be able to see all  31 fingerings as a subset of this one chord.

E  G  (An interval of a minor 3rd)
1  2


E  Bb (An interval of a diminished 5th)
1  3


E  C (An interval of a minor 6th)
1  4

E  E (An octave)
1  5


I will not post all 35 subsets with explanations.

But I will add some more useless, pointless, nonsense here but first

WHY ARE YOU STILL READING THIS?

The way I see it is like this.

Any one note can be played by any of five fingers.

A two note melodic interval or chord has at most 10 fingerings. Some are physically impossible while others are uncomfortable to say the least. My favorite poster like many others on this web site is a fellow by the name of Bernard. So like he so often says, "Try It".

A three note chord (or the 6 derived melodic arpeggios) has 10.

I will explain my term "6 derived melodic arpeggios".

1 3 5  

1 5 3

3 5 1

3 1 5

5 1 3  The left hand beginning of Mozart Sonata in C

5 3 1

I am tired now and this post is already long enough.

Besides this should be enough "insignificant nonsense" for posters to ridicule and poke fun at for now.

But as Arnold once said, "I'll be back".

[nyiregyhazi]

Most method books teach "Correct Fingering".

A root position C major triad is "correctly fingered" by the right hand thusly

C E G
1 3 5


Is is not? Are we all not taught that?

The 1st inversion is "correctly fingered"

E G C
1 2 5

Yes- because it has applicable practical use. Are you going to clarify some form of practical use for your calculations? What is it, exactly? What am I supposed to do with them? A genuinely interesting insight into fingering is if we analyse WHY the fingerings are done the way you say above. If you look at the positions, you'll see that the four fingers functions as a unit, whereas the wider physical reach is covered between 1 and 2- which are the easiest fingers to open up. The same principle explains why we again have 135 for 2nd inversion. Although the fingers are the same as the root position, the hand is opened up in the same way as 1st inversion. There we have a PRACTICAL analysis- that helps a student understand the purpose of the fingering, where it comes from and why it the most physically natural solution. Not just a series of disconnected observations from which nothing can be drawn about how to make fingerings. If you're interested in analysis- why don't you add some context and reason that would make it apply to something?

A two note melodic interval or chord has at most 10 fingerings.

No, more than that. There may sometimes be good reason to reverse the order of the two fingers, when using thumb and the 2nd or 3rd- perhaps even the 4th of 5th. I wouldn't wish to rule it out.

If you're interested in both physics and piano playing, you might find my blog interesting. I've been using basic mechanics to make sense of how it can be a key can be moved with mechanical efficiency and how impact can objectively be prevented etc. But the point is that it all relates to something I can actually use to improve upon the ease of playing. It's not analysis for the sake of analysis.

[cmg]

While this idea admittedly has little relevance to fingering prepared pieces I have used it for years to form exercises which use the whole range of finger combinations and sequences. I have also found it most fruitful in cultivating textural variety in my improvisation.

The same combinatorial approach in the field of harmony occurred to me back in the seventies. I wrote an article about it but music publications didn't want it and it ended up in the New Zealand Mathematical Magazine.

https://www.box.com/shared/xau7zbj2ss
https://www.box.com/shared/vhsc8a176c
https://www.box.com/shared/a6b9hqrris

As conventional musical theory has never interested or concerned me, I made my own based on combinatorial notions and it has proved very fertile for the last forty years, in getting harmonic vocabulary into my brain so it can emerge spontaneously later on in improvisation.

Combinatorics of one sort or another are deeply embedded into many aspects of playing piano music. Therefore, while the intended use of the original poster's thoughts seem peculiar, the underlying idea, it seems to me is of the utmost interest.

I think all the ideas presented here are incredibly intriguing but they speak to my total inability to think like a mathematician.  And I curse every lousy math teacher in my years of education!  Like Ted, I improvise but struggle mightily to wade through a logical, harmonic discourse in my own head.  It's exasperating and causes me to lose wonderful musical ideas to the ether.  They just evaporate leaving little trace of harmonic/theoretical meaning I can grasp.  I just don't speak that language and that you people can impresses me mightily.  Any suggestions for remedial work on my own to get harmony/theory into my dense brain would be appreciated.

[1piano4joe]

It should be apparent from the original post that those are 31 right hand fingerings as the notes are strictly monotonically increasing. If you reverse the order (read them backwards) then you end up with 31 left hand fingerings. The resulting list of left hand fingerings could be considered yet additional right hand fingerings where "crossing" is permitted.

Mathematically

Permutations consider the order to be important.

Combinations do not consider the order to be important.

I chose to not further complicate the issue with permutations.

Thank you nyiregyhazi for pointing out the distinction between the two and the resulting additional fingerings produced when considering permutations. There will always be more permutations than combinations. And for pointing out this as a "pool of possibilities" as that was all it was intended to be.

BTW I do have OCD. I have to learn all the scales. I have to learn all the pieces in the book even the ones I don't like. I eat my way through the menu at the Chinese Restaurant since my OCD demands this. It is quite annoying and tiring at times but sometimes has its advantages.

Many thanks to ted for pointing out how useful a tool this knowledge can be for forming exercises and improvisation.

Finally, not everything has an immediate practical purpose but I have learned to cultivate an interest in things as an end in themselves.

[nyiregyhazi]

BTW I do have OCD. I have to learn all the scales. I have to learn all the pieces in the book even the ones I don't like. I eat my way through the menu at the Chinese Restaurant since my OCD demands this. It is quite annoying and tiring at times but sometimes has its advantages.

  That makes sense. Quite honestly, I think I have a mild version of it myself. I've been very specific about categorising scales and arpeggios based on fingering patterns and have a number of exceedingly systematic time-saving approaches to analysis.

https://pianoscience.blogspot.com/2011/10/scale-fingering-made-easy.html

I still find it baffling that scales are typically entirely learned by reading and rote- without such significant trends being used as a basis for understanding.

Why not find a way to focus it into something directly productive? The thing about fingering is that it's chosen based on physical and musical context. Having a checklist of all options contributes nothing- as you still have to come back to that same physical and musical context, to have any means of deciding which one to use. Running through combinations for a process of trial and elimination is far slower than contextual logic- which is ultimately what fingering must be determined by. Why not look for something else to analyse that will lead to something you can use in playing?

[ted]

Your analysis is similar to set theory, have you studied this? I remember it from grad school. We analyzed some atonal music this way.

Not specifically, I just realised that a chord type is characterised by a partition of twelve and thought that might be a good way to systematically go through all types, getting their sounds into the unconscious to use later on in improvisation.

I think all the ideas presented here are incredibly intriguing but they speak to my total inability to think like a mathematician.  And I curse every lousy math teacher in my years of education!  Like Ted, I improvise but struggle mightily to wade through a logical, harmonic discourse in my own head.  It's exasperating and causes me to lose wonderful musical ideas to the ether.  They just evaporate leaving little trace of harmonic/theoretical meaning I can grasp.  I just don't speak that language and that you people can impresses me mightily.  Any suggestions for remedial work on my own to get harmony/theory into my dense brain would be appreciated.

I don't actually use all that during improvisation in the way a jazz pianist uses strings of complicated chords. I just used it as a learning tool to get a large vocabulary of sounds. I am worse than you as I completely lack understanding of conventional harmony. I have never felt any inclination to follow one particular chord with another, cannot understand why things should go ABAC in lumps of twos and threes and so on. Structure and rules have no relevance at all to the sort of music which moves me the most deeply. I tried, of course, as a kid, but could never see the "why" of it, why people bothered with it. At the intellectual level I could understand the "how to" aspect, even put it into practice, but I desire more than making a Meccano model in sound according to an instruction booklet. More fundamentally than that, I don't really like most classical or jazz any more, so there seems even less point in trying to produce it using rules. Probably end up like Louis Wain with his cat pictures I suppose.

So unfortunately I have given the opposite impression to that I intended and I assure you the whole thing is as much of a mystery to me as it is to you.

[werq34ac]

I'd like to further complicate the issue with the possibility of playing a single note with more than one finger, which does have practical purposes like getting more sound out of a single note or allowing the hand to move into a better position while holding the note. According to this, there are not 5 ways to play a single note, but 31 ways to play a single note. And it get more complicated with 2 notes does it not? We're ignoring physical possibility since you yourself said that it's obvious that some configurations are impossible. Though I will say that 12345 on a single note is not quite impossible.

Also one can play more than one note with one finger. For instance, the thumb often plays two adjacent notes at once on some large chords.

And the correct fingerings part is total bogus. 135 may be most comfortable on it's own, but in a larger context (which a major triad usually has in a piece) your fingering can change based on it's practicability. I won't expand on this because it's "what all pianists should know."

Correct me if I'm wrong?