Tag: beginner piano

Leverage and Sound

Chopin, Etude in C# Minor from Opus 25:

Irving’s brother came today.  We wanted to get a rich cello-like / vocal-like tone out of the piano for the notes of the opening “baritone” melody for the left hand.  It is in single notes without accompaniment, so it is very exposed.  We need our entire sound/mechanical tool-kit to keep it resonant and sustained so there isn’t a moment’s break in the flow of the line.   Their softness shouldn’t belie their resonance.

Our first exploration was with leverage, the principle being that the greater the leverage you have over the production of each sound, the more that sound approaches the ideal piano-resonance.

The effectiveness of a lever is a function of how long the lever is and where you place the fulcrum on which to rest it*.  Leverage increases with the length of the lever and how remote the fulcrum is from the end of the lever that, from which in this case, the pianist initiates the motion of the lever.  If, for example, the lever is solely the length of a finger, and the third knuckle is where the fulcrum is, there is little mechanical advantage to depressing the key through the motion of that lever.  If the lever extends back into the wrist, and includes the finger, there is greater leverage on behalf of the movement of finger tip.  So the question is, how we can create the greatest leverage with the human body.

We ended up using a curious combination of several different levers, that ended up being connected one to the other.

The length of the arm, from shoulder to finger tips, while perhaps not the longest lever we can make of the body, is a conveniently long one that is still easily manipulated.

We started by his holding out both his forearms; straight out in front of him so that they parallel with each other and were horizontal to the ground.  We Left a comfortable distance between the two hands, about the same as the distance between the two shoulders.

We then had him move his arms up and down using just the shoulders as pivots.  At their highest points the arms were aiming well above the horizontal, at an angle of about forty five degrees.  At their lowest points the arms were just slightly below the horizontal.

Very soon, we changed it to an oscillating motion between the arms. One arm was at its lowest when the other was at its highest.  And they exchanged these positions.  We did this until he felt a sort of physical exhilaration from all that motion.

The next thing we did was to create a second, more imaginary, lever.  At the same time the arms were moving, we pretended there was the plank of a see-saw that connects the two hands (traversing the empty space between the hands), which, as a result of the arm motions, was itself going up and down as if two people were seated at each end of the see-saw.  The pivot of this imaginary see-saw was exactly half way between the hands, so that neither hand or arm had a mechanical advantage over the other – the advantages were equal.

I also had him imagine a secondary but similar see-saw between his two shoulders, as if an, albeit, small person was seated on each shoulder.  We continued exercising the combination of these levers until he felt a definite exhilaration from making these motions.

We then ‘elected’ his two index fingers as the sole ‘beneficiaries’ of all the motions he was making, so that the each index finger was backed up by the entire arm and contributing see-saws.

While continuing the oscillation of the arms he used alternating index fingers to play first the opening note of the second note.  The solo was no longer distributed solely to the left hand but alternately, from note to note, between one arm lever and the other.  If he played the first note with his left index finger, then he played second note with his right index finger.  Then back to the left index finger to sound the third note, the right again for the fourth note, and so on through the line.

During this procedure the fingers were to never loose their connection to the hand, and on to the wrist, the forearms, the elbows, all the way to the shoulders.

Sometimes the arms had to cross one another, but the more important thing was the swinging motion from one arm to the other regardless of which one was to the right or left of the other.

When he did this with physical abandon fervor, without thinking so much of the ‘proper’ or ‘usual’ way of pushing the notes down, the result, to our joint delight,  was an unusually rich sound, one that he was unaccustomed to getting on single notes.

Even when consecutive notes were ‘next door’ to each, only a half step or whole step away, we did not diminish the feeling of the widest possible see-saw between the arms.  In other words, while the objective distance between the consecutive notes might lessen, the subjective sense of how long that distance was always remained large.

The last step was to preserve the widest and most dynamic sense of an oscillating motion when going not just from one hand to the other, but from one finger of one hand to another finger of the same hand.

* The saying, concerning how levers work, as attributed to Archimedes, is: Give me the place to stand, and I shall move the earth.

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Microtonal Musings

Microtonal Musings:

1: “in tune” or “out of tune”:

The tonally trained ear expects to hear things in a certain way, and clings to that way in spite of gradually mounting evidence that what they are hearing is not tonal but microtonal.

I’ve demonstrated this with a sound experiment in which a major triad (such as C-E-G) is gradually transformed into a minor triad (C-Eb-G) followed by a reverse direction.  The root note and fifth remain constant; the third is gradually lowered over the course of a certain duration until it has fallen a half step, at which point the third starts rising in pitch, at the same rate as it the pitch was lowered, until it is back to its usual position as the third of a major triad.

In this particular experiment the third is lowered (and later raised) at the rate of one hundredth of a semi-tone (a “cent”) every third of a second.

For many people, no change is noticed for a while.   Their ear continues to hear, or cling to hearing, a major triad – one albeit that is “out of tune” but still clearly intended to be a major triad.  The ear does not accept that it is perceiving a microtonal tonal triad that is neither major nor minor.   The microtonal change is considered an imperfection in the intonation.  There is no recognition of the triad as being of a new aesthetic species – neither major or minor.

Then a sudden switch occurs.   At a certain point in the migration downwards of the third of the triad, most interestingly a point that is closer to the eventual minor chord and further from the initial major chord, the sound, almost instantly, changes in the listener’s ear from being heard as an out of tune major triad to an out of tune minor triad.

This “inaccurate” minor triad persists until the third is close to its final value, at which point the sensation the minor chord at last is getting more and more “in tune”, until at the end it sounds very in tune.

The most interesting part of this sound experiment is that when the third starts traveling in the other direction, the location of the point where the ear ceases to hear the triad as an out of tune minor chord and flips over to hearing it as an out of tune major chord, does not occur at the same point as the similar position during the first phrase of the experiment.  This change in perception now occurs when the triad is closer to the final major chord and further from the minor triad.

2. A self-similar “fractal” chord:

I’ve made many experiments in discovering types of microtonal chords that have a distinct effect that is aesthetically interesting.

My aim was to create a microtonal analogy to a ‘self similar’ fractal design.   The results I got were extremely beautiful, and unlike in case number one, above, could not under any circumstances be heard as an ‘out of tune’ version of a more tonal chord.

I started with an arbitrary selection of a lowest and a highest pitch.  I then inserting a medium pitch that had the effect of dividing the overall range into two parts that bore a certain ratio (in my first experiment this ratio in pitch was 2 : 3).  I continued to divide up each of the smaller pitch intervals by the same ratio.  What started as just two pitches, became successively, hree pitches, five pitches, nine pitches, seventeen pitches, etc..*

*In computing the frequencies of the microtones I started with the unit of the “cent” (one hundredth of a half step) and then converted each cents value to a frequency.

3. Overtone series:

When an ear perceives a single tone or pitch from an orchestral instrument or the voice, an analysis of the sound vibration shows that there are actually a ‘chord’ of different pitches executing separate vibrations.  These additional tones are known as overtones.  If one could separate one overtone from the rest we would hear a sound at a different pitch from the one the ear first perceives.

One of the properties of the overtones is that they are the most spread out in pitch near the ‘fundamental frequency’ (the pitch that the ear perceives) and clump closer and closer together as they continue upwards in pitch.

There is a piece by Stockhausen called “Stimmung”* which has a group of singers each singling one of the upper overtones of a constant fundamental frequency.   A ‘range’ of overtones is chosen by the composer.  The fundamental is never sung, but a consecutive group of overtones is used.  A variable in this selection is what should the lowest pitched overtone in the overtone series that should be sounded and which is the highest pitched one, and how many overtones does that ‘interval’ contains.

In its application to microtonal music, such a group of overtones, can be used as a ‘scale’ of available pitches out of which the notes of the piece are formed.   However, one property of a scale is that it repeats over and over, usually at the octave.  We can make a series of overtones do this by taking just one octave of the overtone series and transposing its pitches up and down various numbers of octaves so as to form a continuous scale from bass to treble.

For most orchestral instruments the overtones are linear in frequency.  The first overtone is twice the fundamental frequency, the second overtone is three times the fundamental frequency, etc..   But the more three dimensional the instrument is the more it deviates from this simple linear pattern.  A bell, for instance, whose vibrating mechanism does not approximate a one dimensional line, has a different arrangement of overtones.

And if a four-dimensional creature were to suspend a four-dimensional bell from a string, and then set it into vibration, there would be an less linear overtone series.  It does not matter that we cannot construct such an instrument, for mathematics enables to predict what the overtones would be, and they can be reproduced exactly on an electronic synthesizer that is set up for microtones.   So we can form scales out of the overtone series for n-dimensional objects (where n goes beyond three).

Here is a list of sample possible constants for generating an ‘altered’ linear overtone series on a particular note:

In the following n is a whole number, and ff the fundamental frequency:

(pi) x (n) x (ff)

(e) x (n) x (ff)

Here are some other possibilities of generating a linear overtone series not based on multiplying the fundamental frequency by whole numbers.


2 to the 1/2 power

pi times e

sine of an angle

* There was work by Maurice Béjart’s modern ballet company which was set to the music of Stimmung which was sung on stage at Carnegie by the “Swingle Singers”.  Each singer intoned the pitch corresponding to one of the linear overtones of a single fundamental pitch.  So that the tones were not too widely separated pitch-wise, they used a part of the overtone series where there were approximately as many overtones within the scope of one octave as there are notes forming one octave of a more familiar scale.  For instance the following numbered overtones, in the fourth octave about the fundamental, span an octave and divide that octave into 8 parts: 7 8 9 10 11 12 13 14 15  

4. Graphing a mathematical function:

There is an aesthetic fallacy in trying to find a means of translation between something spatial (as a graph) and something temporal (as music).  However, if one is willing to experiment, one could try to derive the notes of microtonal chord from the y-values of some function f(x).  Each next note in the chord would be f(x) for each whole number value of x.  What would a parabola sound like?  A hyperbola?  A sine way?  We don’t know until we ‘hear it’.  We may stumble on a function whose sound as a chord is pleasing and unique aesthetically.

5. Expanding or compressing a tonal piece around a constant center of pitch:

This is more productive of interesting sounding tone groups.  Bach Chorales lend themselves nicely to this procedure.

Take each chord, translate it into cents, and then either increase or decrease each pitch in the chord relative to some stable frequency that is either be one of the pitches of the original chord or a pitch that is chosen randomly but which remains throughout the chorale as the center of expansion and/or contraction.   Or, another way would be to use the notes in one of the four voices as the “stable” pitch (even though it may change from beat to beat) and contract or expand the pitches of the other three voices relative to it.


For those of you who dabble in microtones would you let me know what methods you use or whether any of the methods described above have proven useful.  Thanks, Joe

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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.

An example:

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.


e5  d5



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.


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Playing With Authority, Intervals, and the Inner Heart of Music

Playing with Authority:

C.P told me at our last lesson: I am very soft spoken in my private life, and in my business life.  I am habitually quiet, but you have given me permission to speak out more, even though it is at the piano.  I can make more sound and command more attention.   Maybe it’s safer to do it on the piano first,  but nonetheless it an exciting change.

What I had been doing for the last few months with C.P. was to ask her to speak out her notes with more pride and more certainty.  She shouldn’t play it safe, be unassuming and be on guard for mistakes.  This was in her Bach prelude.  On the other hand, in her “Claire de Lune”, I said: here it less a matter of loudness or authority, and more about richness of tone, finding a deep and sensuous source for all your sounds; but that at heart it is the same thing as expressing yourself more fully.


Later in the lesson we were working on a new Bach Prelude (WTC I c minor). I pointed out to her the intervals that were formed between the two voices, particularly after the first sixteenth note of the measure and the first sixteenth note of beat three of the measure.  At first she asked a type a question that I had come to expect from her inquiring mind.  “What is the use of knowing intervals”?

First she gained facility in naming the intervals.  This led to her noticing how the sixth and the third (sometimes as tenths) were the most frequently used intervals between the hands.  I asked her if those two intervals had anything in common.  This led to the idea of inverting an interval and that thirds and sixths invert to each other.  This led to ask about seconds and sevenths, which meant we could discuss the role consonance and  dissonance in a tonal piece of music.

Perspectives leading to the inner heart of the music:

Then I put things in a broader perspective.  There are two ways of knowing something: from outside and from the inside.  From the inside is the goal. Often we cannot go directly into the inside of something unless we first take a series of perspectives on from the outside.  Intervals is one such perspective on the inner heart of music.  So are chords, rhythms, structural features, thematic development, listening awareness, and the list  proliferates.

I had a friend in High School, Stephen*, who sometimes took walks with me in Prospect Park in Brooklyn.  Once we were discussing the first of Emerson’s two essays on “Nature”, and how it is divided into sections, each on viewing nature from a series a different perspective.  He said this was like the bible story of Joshua.  Joshua’s goal was to get to the inside of Jericho.  So for seven days they walked around it getting, as it were, every possible perspective on it.  And on the last day the “walls came tumbling down”, or in other words, they now stood on on the inside of the city, just as the musician’s goal is to live in the inner heart of the music.

*An interesting thing about Stephen.  He was born with only short stubs in the places where the fingers emerge from the hand.  When you are a teenager everything seems possible.  So one day Steven asked if I could teach him to play piano.  Without hesitation I said yes.  We chose the first prelude from the first book of the WTC.  By the rotation of his forearm, and thinking of his hand as a wheel, and thinking of the stubs of the fingers as teeth of a gear wheel, we found a way not only to make sounds on the pianos with the his virtual fingers, but gradually gained a sophistication in the control of the rotation, together with the possibility that at any moment the arm could lift the wheel of the hand off the piano so that when the wheel came back down on the piano the virtual finger ajacent to the one that just sounded a note, could land on any key regardless of its distance on the keyboard from the previous note.  Steve went on to Cornell, and I wish somehow I could be in contact with him again.

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Further Italian Concerto Progress!

A.B.  was here for his lesson yesterday.  We were working on the third movement of the Bach Italian Concerto.  We brought to the next level his ability of bringing things under the control of the ears.

I was  reminded of medieval philosophers when they talk about god’s  abilities: that god merely needs to think something and it becomes actual in the real world.  So in piano performance the true controller over how a passage sounds is not based on intentional or controlled physical motions, but simply the ‘ear of god’ (actually the ear of the pianist) noticing how things are sounding – which, miraculously, transforms what is heard from potential to actual.

The more I was able to get A.B. to focus on his ear, the more contented he was to practice just a small chunk of the music and not, as is his wont, to continue on and on regardless of what happens in the passage.  We should first ‘frame’ the chunk of the music being undertaken.  That you will  find that the smaller the chunk size, plus, the slower the tempo, the more the ear naturally takes over for the body.

Some other things that I said during the lesson to keep A.B. focused on what he heard rather than what he felt:

1) the notes never escape the reach of your ear.

2) wherever your hand goes, the ear follows.

3) the physical action of making a note often occludes the ear’s ability to hear the same note.  This is an important reason why is it not such an easy matter to “just listen”.

Some of our work had to do with specific spots in specific measures:

In measure 2: the last two quarter notes plus the first notes of the next measure (in the left hand).

The principle here is, in order to get clear and crisp parallel sixths, don’t be content thinking of the three written sixths as being the “complete story”.  I extended the passage by having him play a scale an entire ascending octave of parallel sixths using the notes of the F Major scale.  “This is the ‘larger’, the more complete ‘whole, of which we have but a limited section being quoted.  Once you conceive the part as representing the whole, then no matter how few sixths you play they will come alive.  The listener will have a sense of where the sixths came from before the first one to be played (c3-a3) and where they are going to go if allowed to continue beyond the a2-f3.  It is the “gestalt”, this organized whole, one that is greater than its parts, that should be the object of our perception, and be that which our hand wants to “embrace” when playing.


In measure  3:  a2-f3 then f2, in the left hand.

Even though the thumb releases the f3 before the f2 is played, let the thumb nonetheless act to balance the pinkie.


Also in measure 3: the fifth eighth note in the left hand – bf2.  No matter how he tried physical to control and balance the sound of the bf2 from the surrounding notes of the F major scale, he could never get it to sound how he wanted … until, that is, he recognized that the b-flat, though far  removed from the right hand, functioned as the 7th of a third inversion C dominant-7 chord (bf2–e5-g5-c6).  This allowed the bf2 to find its destiny as enabling a brief assertion of a dominant chord, in an unstable  inversion , in the midst of an ascending F major scale.

Relating this to today’s major theme, if not by engaging with the ear, no matter how you to try to play something, it will always sound wrong.  Which leaves the pianist to try one after another physical  experimentation, all the time completely missing the sound-reason for the note.


In measure 5: the notes on beat one and the following eighth note.  A.B. was having difficulty separating the two voices in the right hand.  I made a suggestion that, agreeably, seemed to have nothing to do with the issue at hand.  Listen, I said, to the f4 in the left hand and hear it meld into the f5 an octave higher (in the right hand’s lower voice). Sometimes we have to think ‘across the grain’ and find the solution to something in a different geometrical dimension than the one in which we first located the issue that required our attention.


Measures 30 and 31: the left hand

“Throw” the left thumb rightwards as if it would separate itself from the rest of the hand.  Do this with more energy and momentum than would seem to be warranted by the physical distance the thumb has to travel away from the other fingers of the hand.

The principle here, is analogous in a way to the “gestalt” thing we mentioned concerning measure 3, when we spoke of completing the implied whole, not being content with only the notes that literally sound or are literally there.  In these measures the distance the thumb has to travel is expresses a larger distance (subjectively) than the pitches of the notes seem to indicate (objectively along the keyboard).  We sometimes have to ‘overreach’ in order to ‘reach’.


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