The Importance of What is Not Heard
Brahms: Intermezzo: Op 116 No. 4 in E Major
Often in a well constructed piece, the meaning of something lies in how it stands out in contrast, or in relief, to something else. Much of this has to do with memory, and what the listener may expect to hear at a certain time.
In the recapitulation of the Classical sonata movement, the second theme comes back in the in the tonic, not as we remember it, in the Exposition, in the dominant (or relative major). What happens at that moment is that an expectation is momentarily revived and enhanced by the composer but a new present reality is superimposed upon it. For a moment the two tenses interact*, but a moment or two later our ear has taken up its bearings in the new.
The ears of a sensitive listener will even prick up before the second theme, at the exact moment when the composer deviates from the harmonic path that led to the second theme in the exposition.
One of the things that makes late Brahms difficult to hear lucidly is that when something stands in relief with something else, we often haven’t had an opportunity to hear that something else earlier in the piece. So how does the pianist make a contrast with something that is not ever heard, but whose meaning lies entirely in its contrast to this unheard base or reference?
An example from the Brahms Intermezzo:
Consider the passage in measures 10 through 14. Contrapuntally, what is going on has less to do with the triplets in the right hand but in implied, but not literally heard, duplets, which are formed from the second and third triplet notes, if the first triplet note is put back onto the beat, omitting the first triplet note entirely, and playing the third triplet note as the second note of a duplet. If we do this, we suddenly hear a very conspicuous appoggiatura. In measure 12 for example the e5 is clearly heard as an appoggiatura to the d5.** As we shall see, this perception need not become vitiated by the delay of the restoration of the appoggiatura to its original position in the measure (one triplet eighth later than the sounding of the chord in which it functions as an appoggiatura).
The same relation of appoggiatura applies to the c5 to b4 and the a4 to g4. When performed successfully, this passage haunts the listener with the sustained feeling that something else is going on other than what is most obvious to the ear (delayed triplets). There lurks this implication of regularly arriving appoggiaturas on the beats. Similar appoggiaturas occur throughout the passage.
Brahms doesn’t stop there. Once he establishes to the ear that this comparison to the implied simplified counterpoint, he is able to take a further step to hide the actual appoggiaturas by attracting the ear, in measures 11, 12 and 13, to a descending scale in the top voice. But let’s pause for a second. Do we hear a scale? Almost. At least we get the feeling that there is a scale present. For here too, there is a layer of removal from what is heard to what one might call what is meant-to-be-heard. We hear a melody stopping and stopping in two note groups, which if there were no interruptions would be a coherent, fluid scale: b5 a5 g5 fs5 | e5 d5 c5 b4 a4 g4 | etc. The beauty of a melody arising from following this scale depends on the implication that our consciousness is able to pass lightly over the first the first of each group three triplet notes (a note that is merely part of an accompanying chord) so that the notes of the scale seem to flow connectedly one into the other.
I have my students leave out the first triplet note, and change the next two notes to regular eighth notes, putting the first of the eighth notes back onto the beat. The scale is now much clear to the pianist’s ears. Crucially, if that point, the student goes back to playing the written notes, the reference to the fluid duplet scale is not lost. It attempts to maintain itself in spite of the pauses. It haunts the image of the passage and changes a somewhat trivial passage in triplets to something more transcendent sounding.
Thus a passage can transcend itself. It becomes beautiful only in relief to something more basic, not literally heard, to which it yet can refer itself. Generally, in late Brahms, we often must try to make a passage sound like what it isn’t! (something clearer in harmony, clearer in rhythm, and clearer in voice leading and counterpoint).
* This momentary contrast, if it were prolonged would lead to a confusion in the sounds, like when a person accidentally takes a double exposure with a camera. If, however, the process could be frozen in time, and experienced just in space, we would have the equivalent of a biologist looking through a microscope that allows on eye to view one slide and the other eye view another slide, as for the purpose of noting what contrasts there are between them. A side by side comparison. In music it is more sublime. It is a a sound image from time past that melds with a sound image from time-present. The past isn’t gone it lives in memory, for many in the form of a sound-memory. The past sounds do not really sound in the glare of the light of present, but colors it. But a comparison is made.
Clearly there is a D Mjor chord trying to fully form and as an e5 yields to the partially formed chord and resolves to the chord note d5.
The Connection and Disconnection of Notes
I’ve had an idea lately that it would be nice to do a lesson and then post a blog post about it right afterwards. I think this will gain in spontaneity and insight, despite what it might loose from lack of editing and proofreading.
A.B. Was playing WTC I C f (which is my short hand for Well Tempered Klavier, Book One, C Major (C is uppercase), the fugue and not the prelude (f).
This is a new piece. The first thing he said, was how hard it was to read a fugue. It poked at his sore spots as a reader and a player. I said, forget all of that. Play the chord on the first beat of this measure, and ask yourself what it is the most natural and comfortable of playing it. By starting there, it is as if you were starting the piece, just from a different measure than measure one. So your hand had no allegiance to what it may have done a moment earlier if it had played the last part of the previous measure.
Now, before you go on any further, DON’T TRY connect the present arrangement of the fingers in the hand with the next one. Playing Bach clearly is not a matter of figuring out a fingering, or getting used to making certain connections in the hands and fingers. One never goes from “here” to “there”. All there is are “here-s”. Each one is undiscovered until right now. It is always as if you are playing the piece for the first time.
With each new note, or if not that frequently then at least with every new eighth note’s worth of the piece, pause and ask yourself “what is the most natural and comfortable of playing these notes”, especially if the hand need no longer “remember” where it was a moment earlier.
Playing through the piece is discovering, as if for the first time, a new position for every moment’s new notes. In doing it this way you enter into the joy and spontaneity of the fugue; the experience is wonderful, and in no way a chore.
A.B.: So what do I do the next time I play this spot, wouldn’t it help if I gradually got to know, through repetition, where my hands go next? Me (waxing poetic and philosophical): No, the only thing you have to remember is to forget. A.B.: But doesn’t that sometimes mean I get further and further into trouble with my fingers and dig myself into a hole from which I cannot get out. J.B.: There is a simple solution to this. When you are least sure where to go next with your hands and fingers, when you feel you’ve gotten stuck in the mud and don’t where to go, that is the time to take your hands off the keyboard. Remove the hands from the piano, even if briefly. Start with a new slate, for by removing you hands from the keys, you have let go of the immediate past, you can discover, as if for the first time, the most natural and comfortable position for the two hands together on the next notes. So, if you never know where you are going to get into fingering problems, remove your hands from the piano.
He tried it. It was a fine sounding connection. He said: but if I remove my hands from the piano there will be a break in the sound continuity, things will not sound connected. I said: then how comes what you just did, which involved letting go of the keyboard and removing the hands from the keys, ended up sounding more flowing and more connected than I have heard it before? By removing the hand, you have no choice but to find a new position, a new and most comfortable position, for the next notes.
Be careful, I said, of sixteenth notes (or eighths) in one voice that are moving in steps. That can lead you down a perilous path. You will stop looking for a new hand position for each sixteenth, until the fingers get caught in the keyboard and get bogged down because you have “run out of fingers”. No, you never run out of fingers, there are always five new fingers in the hand for each new note.
When I say “find the most comfortable position” I mean one in which no finger ‘remembers’ where it was a moment earlier. Nothing about its position in the hand is biased or coerced.
To save time in writing, I am using the convention of having:
|: Ernie is the name of my cat 😐
To mean that I have gotten trapped in an endless loop and am saying the same thing over and over. And in such a way as if I never had said it before, but rather someone recorded me, quickly hit stop, rewind, and play. The idea is that it becomes a spoken “mantra” whose efficacy is in its being repeated, until the mind’s state becomes transcendental, a state in which one does not connect things physically SO THAT they can connect sound-wise (sorry I’m sounding a little to “new age” for a cynical Jew from Brooklyn).
If you are a draughtsman, and you use the same writing implement over and over again during an hour of work, do you always consciously try to pick it up in the same way as before. Do you have to think of its position before taking it into your hand. No, it only becomes natural to do if you allow the body to learn unconsciously, so that the 100th time you take the pencil in your hand, it is consciously just as unplanned and spontaneous as the first time. From the conscious point of view (and not for the unconscious, which is busy learning and practicing) you are always finding something ‘new’ (not ‘old’) and finding it for the ‘first time ever’.
All of this started falling into place when I physically caused A.B. to remove his hands from the piano after each current sound. When he resumed, the next sound and all the newness of freshness of the morn: is it ever really the same sun that rises the next day (Thoreau says something about this in “Walden”. Every hand position is “discovered” spontaneously. So I sometimes started using the repeating mantra |: every position is new and discovered spontaneously :|. or just, |: find a new position 😐
Don’t be afraid to let go, for that is the only true way for the body to find what’s next. It is the opposite strategy that common sense tells us to follow. Consciously you forget it even happened before. You think you can’t do this “A”, but you can. You just need to keep an experimental mind, and prove it to yourself over and over with the freshness of every new sound.
And, by the way, when you find the ‘new position’ it always for both hands together, never for just one hand or the other. Let the body, let the ear, always synthesize together every new sound in the piece. And the listener has no desire to complicate the wholeness of the musical experience by knowing which part of what they hear came from your left hand or your right hand.
After a while, all I was saying to him was “let go” … “let go” … “find the new position”, “find a new position”. There was one moment I could tell that he was trying to figure out the best fingering for a series of consecutive notes. I said: that was not a new position, it was a ‘trying to get there from the old position to the new’. There is never a ‘there’ to which to get, everything is a ‘here’.
A.B. said, how can I have a totally new position in my hands when I am required to hold over one of the notes (holding down a note in one voice while the notes in the other voices change). I said: I agree that you have some issues with what I might call, by analogy, if it were spatial more than of time, “negative space” (E.G. is it two profiles or is it a vase). A held note is not due to a finger that holds tightly to its position on the keyboard. It is do to a new position that that finger assumes every time another voice voices to a different note. The fact that the finger remains on the same key is secondary and incidental from a physical point of view. There is no difference between writing a half note, in a score, and writing the same note as four eighth notes, each tied to the next.
Negative space also involves things like, the action of when to release a note in a voice after the finger playing has gotten inured to holding it down when it has been held for a while. Another example are rests, in general in a particular voice, which must be incorporated into the “sound” continuity of the piece.
So, abandon any noble effort by the left or right hand to connect the notes in the fingers. Don’t do that! Let it go.
I would love feedback regarding the usefulness of this type of blog entry. It probably suffered from repetitiveness but it did not loose my original excitement about discovering these things, and in keeping pace while writing how things evolved through time during the hour of the lesson. Thanks for reading.
Technical Challenges in Moszkoski’s Etude in F and Beethoven’s Sonata in E Major, Op 14.
A.J’s lesson today. Two works he is preparing for a competition.
-From: 15 Etudes de Virtuosité, Op.72 No. 6 (by Moritz Moszkowski), “Presto” in F major/.
#1. The “ear” as the abstract creator of the figurative shapes of sounds:
#2. The balance of sound between the two hands.
#3. A ladder falls apart if there are no rungs connecting the sides.
#4. The undulating patterns of three-note groups in the right hand:
-Beethoven: Sonata in E Major: Op 14 / 1 : I : The left hand sixteenths in the development section.
#5. “Additive” Clusters as a unit of pulsation through the passage with sixteenth notes.
#6. The desired effect in sound does not always follow upon a logical or teleologically designed set of causes.
-From: 15 Etudes de Virtuosité, Op.72 No. 6 (by Moritz Moszkowski), “Presto” in F major/.
#1. The ear as the abstract creator of the figurative shapes of sounds:
The ear, as the observer of the flow of sound content through time, may seem at first to be but a passive instrument. It listens, it notices, and only with a slight lag as the sound has already been physically produced.
If not distracted by the physical actions we make to start the sounds, there is an exact in knowing in the accuracy of the notes. The awareness of sounds in the ear and the more it is divorced from any muscle movements that physically give rise to the sounds–the more accurately and subtly it judges the sound characteristics of the music being played.
Through a miraculous confusion of tenses, the ear as a passive listener, after the fact that the sound has begun, yet can be the most effective force in controlling our sounds. In this regard it is far more efficacious than consciously controlling and gauging the quality of quantity of our muscle movements. This present tense in consciousness is not a mathematical instant, a point of zero duration. It contains, according to the French philosopher Henri Bergson, the spilling over of the past into the present and the impetus of the present to be on the verge of becoming the future. A duration, though recognized as the present, has yet the efficacy of having what has just happened to have an undefinable but definite effect on what is just about to happen.
#2. The balance of sound between the two hands.
A.J. is having difficulty coordinating the two hands in the Moszkowski Etude in F. His left hand seems to be pursuing its own course–not blending with the right hand, but merely showing up at the same time as the right hand – at the beginning of every third note in the right hand.
I suggested that as he played one of the left hand chords, hold it for a few moments along with the first of the three triplet notes in the right hand. He should see if his ear could spread its attention over the sounds from the right hand as well. And then determine quickly whether together, the sounds of both hands formed a cohesive whole.
With just a little attentiveness, just a split second after the notes start sounding, one will notice whether the sounds from both hands seem to reach out towards each other eclipsing the distinction between them and creating a larger sound-whole than either hand’s sounds alone. And this fusion takes place as he listens to the sounds. It takes but an instant for this synthesis to occur. At the first instant there are disjointed sounds from two sources, but a mere instant later these sounds have instinctively reached out towards each other to form a synthesis in consciousness. Or, to put it another way, it takes just a bare moment for the ear to note and to form a larger whole out of the sounds of both hands.
To his surprise, A. had no difficulty in causing these sounds to fuse together though physically they were made by separate physical acts pertaining to a coordination of the individual sides of his body. He was surprised since in as much as he wasn’t aware that had done nothing physical to effect this balance, but merely remained attentive to the sounds for more than a split second. This synthesis had nothing to do with any physical effort to make the sounds be simultaneous. Nor was there any specific mental ‘effort’ involved.
The combined power of his ear and his brain focusing on the notes, brought the sound together. His previous preparation and physical muscle memory came through in a moment where his head may have easily gotten in the way–thus is the power of the ear.
#3. A ladder falls apart if there are no rungs connecting the sides.
As an analogy for what had just happened, I suggested that the left and right hands were like the two vertical sides of a ladder. They can remain upright only if there are rungs crisscrossing between them. These sides had nothing to keep them together without the ear forming the rungs. Without the attentiveness of the ear those side pieces would fall apart.
Once such a sound-synthesis has been effected at any point in the piece, the possibility then exists to ‘mold’ and shape the forms of these connections. It now became possible to mold how one of these composite sounds morphed into the next one. Though intangible in nature, the pianist now has a focal point to help steer all the course of all the individual notes of the composition through the medium of time.
Though to the body, the sounds originating from the right and left hands seemed to exist spatially apart and separate, from the point of view of the attentive ear they were (already) fused together. It is more the ear, something intangible, than the body, something tangible, that ’causes’, these sounds to meld and form a resonant four-note chord. We need only seek whether they do.
It is only after the fact of their fusion into a single sound that we can, for analytic purposes, speak of these fused sounds as having two spatially distinct origins.
#4. The undulating patterns of three-note groups in the right hand:
Next we turned A’s attention to just the right hand’s stream of notes, a rapid stream of triplets. I suggested that each and every group of three such notes comes to life in a molten state, which the pianist can then form into a well-rounded shape. Despite their melodic and harmonic differences, all such three-note groups should cast into the same shape. This creates a form ‘texture’ that holds the entire piece together.
The most recalcitrant triplets, the ones that would most resist such shaping, occur when the right hand is playing a chromatic scale. No group of notes yields up so little harmonic value to a repetitive pulsation, The chromatic scale is most innately without a shape. If started on a C Natural, and if accompanied in the other hand by a C Major chord, the scale tries to break down into uneven units of, first four note (C C# D D#), then three notes (E F F#), then five notes (G G# A A# B). This is too much of a strain on the scale which therefore yields up little by way of harmonic implication. It is the changing size of the harmonically influenced note groups that render the scale inchoate rather than redolent with harmony. In this etude, the smithy of the mind resists this falling apart of the chromatic scale and obstinately takes every three note group, regardless of its harmonic implication, and shapes those notes into a three note melody without reference to harmony. If I had to express this using a spatial analogy, each three notes would be, in its unformed state a straight line, which the agency of the ear then coerces into the shape of a letter ‘U’.
Beethoven: Sonata in E Major, Op 14 / 1 : I : the development section:
#5. “Additive” Clusters as a unit of pulsation through a lengthy passage with sixteenth notes:
The chord with which the left hand commences, c3-e3-a3-c4, is a first inversion A Minor chord. For many hands this is an uncomfortable arrangement that promotes flitting moments of tension. There are two ways out of this dilemma. One is for the hand to change its overall shape as each finger takes its turn enunciating its note, removing if necessary the other fingers from the previous notes they played. The other way does away with all the physical difficulties by having the ear take on a constructive role, building up, one note at a time, the eventual cumulative sound of the 4-note chord (c3-e3-a3-c4). When doing this, each single note, in its turn, prepares the eventual and cumulative sound of the four notes occurring at the same time. It is only through the first iteration of this four-note sequence of tones that the full chord does not sound until the fourth note. But after that, and with the pedal down, the simultaneous sound of the four notes is continuous. The evenness in the balance of this four-note is not the result of mechanical manipulation but the result of the expectation of an ear focused on the simultaneous sound of the four notes.
#6, The desired effect in sound does not always follow upon a logical or teleologically designed set of causes:
A.J. didn’t see how such a passive, ear-based, technique could possibly effect the evenness and balance of the four sixteenth notes. This prompted the following conclusion from him, his most significant realization of the lesson though at the same time not a logical one.
“Mechanically what I did makes sense as a way of achieving the sound effect that I want in sound. And yet … the result is the sound which I desire. This apparent disconnect between cause and effect is a normal sign of a sudden breakthrough technically. The physical means of doing something, when considered in and of itself, may or may not seem to be capable of logically producing the sound effect that the ear is after.
Yet that effect is what is achieved. So it makes no sense. It takes bravery to abandon physical/logical sense of consistency between cause and effect and be accepting of what in “Big History” is called an “emergent form”; or a form that is not contained in the some of its parts.
You find that the way to the newly emerging form is not foreseen in its physical and mechanical causes. The means happens to produce the ends, but cannot predict the effect.
Where Does Sound Come From?
Stranded on a dessert island.
Imagine a person born blind, living alone, on a proverbial deserted island, out of touch with society, surviving through what she can grasp with her hands. Sight has never had an influence on her notion of reality.
From a hidden vantage point we notice that a bird is singing near where she is standing. We assume she hears it; but cannot see it. For her there is only a certain quality of sound, to which only we can give the name bird. For her, it never gets beyond being just a sound, although she can distinguish one sound from another on the basis of its quality.
Thus she is someone who 1) has never seen, 2) never seen a bird, and 3) wouldn’t be able to conceive that there is something called a bird. There are simply no past connections between the sense of sound and the sense of sight. There is nothing linking the sound of the bird with the sight of a bird.
If the question “why” arises in her mind, probably in the form of “why this sound and not another sound”, the question can only be posed by her within the domain of time and not space: “why do I hear that sound now and not at another time.”
At this moment, a miracle occurs.
Our subject can now see. One of the first things that happens is that she sees a bird, although it is not at that moment singing. Thus at this point there is no reason for her to form any sort of link between the sound of the bird and the image of the bird in front of her.
Some scientists now enter the scene.
They introduce themselves, and present her with a series of pictures. Included is a picture of the same species of bird that she has been hearing. She is asked to choose from among the pictures the one she thinks would be most closely associated with the sound she already knows.
This request perplexes her. She cannot even understand the general form of the question. At this stage of the story, sight is still new to her. She knows of no reason why a sight and a sound should be related to each other, even that they could be related to each other. While the sound, for the scientists is the “sound of a bird“, she has no need to make, or even conceive, such a statement.
She has no grounds for choosing one picture as against another. This makes it arbitrary which picture she chooses. If she is “artistic” by nature, perhaps she may form an aesthetic comparison: which sight feels like it goes with this sound.
Her judgment in this matter cannot yet be based on cause and effect. And even if she has a notion of cause and effect from her previous experiences in which there was no sight, sound as far as she can tell, needs no cause.
“Excuse me”, she asks, “are you saying that a sound requires a sight to cause it? That among all the random lines and shapes I see, which seem aimlessly distributed in space, there are certain lines and shapes that for a reason I cannot conceive ‘belong’ to each other, stand out from the other lines and shapes because of a mysterious relationship, which in turn you call the cause of the sound I have been hearing – not just now, but whenever I hear it.
When you speak of this mysterious connection between just certain lines and shapes, you use the strange word ‘object’, as if by saying that word it should be obvious to me why just those lines and shapes clump together with each other. And then, now that I supposedly believe in something called ‘the object’ whatever an object is, it is also the cause for the sound – something that never seemed necessary to me for the sound to occur. Why should there be such a complicated and seemingly arbitrary way of connecting things in my mind, based on an invisible (at least to me) concept called ‘object’, without which, you say, I would not hear my sounds. Furthermore that I have to choose among several of these objects, and pronounce the words ‘this object is the cause of what I hear?’ That sounds like an enchanter’s spell. My universe was full and complete without sound requiring a cause. Being sighted is sure a complex thing.”
At last she picks one of the pictures. “If I pick this picture today can I pick another picture tomorrow to be the cause of this particular sound in my head? I ask you this because for now, none of the pictures that you show me bear any inner resemblance to the sound that I know.” The psychologists say: “No, you must believe that a sound arises in your consciousness because of a certain event happening in space, which something has to do with a particular object that you see, and always that object and not another.”
She comes to her “senses”.
She is left alone for a few days to ponder this perplexing situation, a situation that until now, without sight, had no reason or necessity to exist.
During one of these days she just happens to hear the sound of the bird at the same time that she is looking at a bird. This may have happened on the preceding days, but this time she notices that the beak of the bird moves in tandem, in time, with the occurrence of the sound. She knows this much more because of time rather than space. The togetherness of the sight and the sound is based on a common moment in time.
This forms the basis of a series of ongoing experiences by which the sound of the bird is gradually linked in her mind to the image of the bird.
As with the pictures of shown by the psychologists to the girl, sights that are visible to a growing, young child at only at certain times, during for example a concert, are only gradually coordinated by that child with something seen in space in the concert hall. It turns out that the people who are holding musical instruments in their hands seem to make motions that are most consistently synchronous in time with the changes of the qualities of the sounds.
Here’s the first important point. Once such an association is made by the child, he or she forgets that there was a time when no such association had been made.
The second point is: was either the woman on the island, or the person in the concert hall, missing anything crucial when they was unable to relate the object ‘bird’ or the object ‘violin’ with a certain specific sound quality of sound? I say no. Nothing essential to our understanding and appreciation of sound is added to by the tacking onto the sound a relationship with sight. And in the concert hall, it adds nothing important to essential qualities of the music as sound alone.*
For those of us who do not need such distractions as sight offers, and can remain glued to the sounds of the piece, we enter an ideal realm of pure relationships between pure sounds, closed off from everything else, and not lacking a thing.
* It is for some but not all of the concert goers, that visual impressions can serve as a distraction or refuge from just having to listen to sound from one moment to the next. For them there are the distractions of the appearance of the concert hall. For them, too, there is the all important information in the program notes, which they are relived to believe captures something essential that they miss in the progression of the sounds alone. But thanks to the program notes, they are able to go up to someone at intermission and say, wisely: “wasn’t it wonderful how the composer used the brass section in the second movement of the symphony to create a delicate halo of sound around the rest of the orchestra!”.
Simplifying A Difficult Passage
A simple example of the procedure.
Irving is a late beginner. He is playing just the right hand of one of the easier Bach pieces. He thinks it will be too hard to put the hands together. I suggest that the next time he plays just the right hand, he lay his left hand down on the keyboard hand and let it rest there passively. He says: I don’t see any advantage in doing this, it certainly is not going to make playing the two hands together any easier. But he tries it. He is immediately struck by the fact that the right hand seems harder to play when the left hand is simply present on the keyboard. He says the right hand feels different. I agree: what one hand does is influenced by the other hand. Still, he said, this is easier than playing both hands together.
What we had created is an in between point between playing just with one hand and playing with both hands. Instead of one larger ‘step’, going directly from playing with one hand to playing with both hands, we created two smaller steps: 1) right hand alone 2) right hand with left hand running interference, 3) both hands playing the written notes. Instead of going directly from step one to step three, all that’s left for Irving is to go from step two to step three.
However, what if going from step two to step three turns out to be too big a jump? We simply divide that jump into two smaller parts. Between step two and step three we insert this: the left hand, though not yet playing its part, now is moved around while the right hand plays. Not moving any place particular but just so he is conscious of motion occurring in the left hand while trying to concentrate of the right hand.
We were following in the foot steps of one of the great thinkers in the Western tradition. The philosopher René Descartes (1596-1650), in his “Discourse on Method” suggested that one follow a four-step plan for seeking truth in the sciences. The second of these was, in his words “to divide up each of the difficulties which I examined into as many parts as possible, and as seemed requisite in order that it might be resolved in the best manner possible.”
We interpolate new steps as needed. As soon as the transition from one step to the next is not easily accomplished, we interpolate another step. The beauty of this procedure is that one can interpolate as many steps as one wants between the starting state and the final state we are aiming for. The more intermediate steps there are, the smaller becomes the change in difficulty from one to the next. In its ideal form, when using this process, the pianist will not be aware of any increase in difficulty when going from one step to the next, and the final state, the one desired all along, will, when attained, seem no harder than the first step.
A careful analysis:
The trick is how to define the first state. It must bear a direct relation to the last state. In Irving’s case it was easily found: “I’ll start with one hand at a time before trying both hands together”. Other technical and musical difficulties require a more penetrating analysis. Often the starting state is found at the end of a reverse process that starts with the final state, and gradually simplifies it, step by step, each time by removing what is most difficult to execute from what is left, and/or what is least essential musically, until a simplest state is left or revealed, one that is simplest to play yet still bears a resemblance to the final state.* During this process, each stage, while simpler than the last, should still contain the essence of the previous stage.
We operate like a grammarian who diagrams a sentence in a manner that reveals its more essential and less essential parts and clauses. Or like a chemist who by analysis reduces a complex compound into smaller and smaller molecules until at last the atomic structure of the compound is revealed. Or like a philosopher who seeks into an issue to find the more basic principle on which it rests. In my case, it is a procedure I use especially when teaching ear training: taking an ear training ability that is presented as being simple and whole, but then refracting it through a prism and showing that there are other, simpler component abilities that underlying the target ability. And even some of the simpler abilities turn out to be, themselves, complex in relation to even simpler abilities. Start with the simplest abilities and build back up.
* By removing notes without compromising the general meaning of the passage. By removing melodically unessential notes from a melody leaving only the salient notes. By simplifying a complex rhythm into simpler rhythms (simpler in nature and simpler in execution). By simplifying a complex harmonic progression to a simpler one that is easier for the ear to track, and easier for the fingers to play.