Tag: concert 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.

1.1

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.

P.S.

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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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.

Intervals:

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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The Sad Decline of My Absolute Pitch

I have a love and hate relationship with my ear.

For a person with absolute pitch, it is often the case that each individual key (C Major, C Minor, C# Major, etc.) has its own special character and aesthetic, which strongly colors any music that I play or listen to.  What I don’t know is if others with perfect pitch experience the same thing, and perhaps more importantly, whether the character or aesthetic of a particular key matches those of mine.

I learned that I had absolute pitch in gradual stages.  When I was about three years old, I would listen to my older bother play pieces from “For Children” at his lessons.  I knew enough to know that there were qualities about each piece that didn’t vary from one time to another, and that these had to do more than with the particular notes and their sequence.

It was many years before I understood what absolute pitch was and that I possessed it.  Actually it was my friend Jeffrey Rothenberg who discovered it for me.  We were in Mme. F’s French class in our junior year at high school.  I remember two particular events in that class that year.  The first has nothing to do with absolute pitch but is just nice: in the middle of a class meeting, Jeffrey got up from his chair, said somewhat ecstatically “Spring is here, and the cherry trees are in blossom in the quadrangle”, at which point he drifted, almost floated, out of the classroom.

The other, was when my friend Jeffrey was trying to discover if he had perfect pitch.  He would lean over to Edward Goldstein on his right, sing a note into his ear, and ask him to sing it into my ear (I was to Edwards’s right) , and whisper into my ear: Jeffrey wants to know if you think this is an “A”.  The fact that I could do that somewhat surprised me.  I thought: so I guess I must have absolute pitch.

About one out of ten thousand people in the world have perfect pitch.  Most are not musicians and probably do not realize that they have perfect pitch.* I figure they just assume that everyone else in the world hears sound the way they do, and that includes a merger of the effect of the up and down-ness of pitch with the effect of a changing coloration to the sound.  Only if these people study music they will learn, perhaps to their surprise, that every time they hear a note, they are able to give it a name.

In  school I began a phase of showing off my absolute pitch.  I wasn’t good at sports, so this was my way of being “macho”.  For instance, I got a telephone call from my friend Linda who said.  She said: “Do you hear the piece I’m playing in the background, what is it,  I can’t identify it.”  I listened for a few moments.  I that point in my life I had never heard it before, but I knew it was by Bach, that it was a concerto, that there were two pianos playing,  and that it was in the key of C Minor.  So I said to Linda: Well I’ve never heard this before, but I would say it is the Bach Concerto for two pianos in C Minor, the first movement.  We hung up.  Ten minutes later, when they probably announced the piece over the radio, I get a call again from Linda.  She said, “show off!”.

One of my favorite spots during my High School years was the Brooklyn Botanic Gardens.  I probably spent more time there than in classes.  I even had one teacher who would ask one of the students: when you walk home would you look for Joe in the Botanic Gardens.  He is probably sitting by the stream.  If you find him, would you give him the homework assignment.

Yes, I was by the stream, bent over, listening intently to the gurgles of the water, and trying to figure out what the pitches were of this sound.  I never could get them right.  I would notate what was in effect a chord of many notes and would then try it out on the piano when I got home.  There was no similarity (even after allowing for the difference in the sound quality of a piano and a brook.   It wasn’t until another year or two that I learned what “white noise” was.   That the reason I could not notate the brook was because there were so many pitches, all at once, that there was no way for the ear to untangle them each from the other.  Additionally, at every moment the interval pitch make-up of the white noise would change slightly change, but in such tiny degrees that were measurable only in microtones.  Microtones are the unlimited number of pitches that exist, for instance, between a C and a C-Sharp – or a ‘distance’ called a half step or semi-tone.

My experiences at the brook awakened my interest in microtones and today I am using the computer to compose microtonal pieces.   I’ve even trained my ear to detect a difference of two hundredths of the distance between a C and C-Sharp.   But they had to be isolated tones and not in a mixture or hundreds or thousands of tones all closely ‘spaced’.

Which brings up the clarinet.   I had been playing the B-flat clarinet since the fourth grade.  The clarinet is a “transposing” instrument.  When it plays the note which the clarinetist identifies as  a C on the clarinet, it does not match the C on other instruments.   A C on the clarinet was a B-Flat on the piano.   Though I didn’t know it until I was a Junior or Senior in High School, I had developed unconsciously two separate but parallel senses of perfect pitch, one that names the notes as they were called on the piano, and one for the notes as they were called on the clarinet.

In my twenties and thirties, if I was scheduled to teach a lesson, and I felt like I was coming down with a cold, I would protect the student by sitting on the other end of the room from the student.  It somewhat freaked out the student when they noticed no difference in my interaction with them, as when I would say something like: “Irving” you just played an F natural instead of an F sharp” (yes Irving existed even back then).

So, everything was going along swell between me and my absolute pitch, until the  invasion of original instruments.   The difference is: why listen to a clarinet play, in tune, the solo in the slow of movement of Beethoven’s 3rd Symphony, when we could hear it played out of tune on an instrument created during the early 1800s.    I grant that this is just a humorous way to describe the early music movement, but there was something more sinister for people with perfect pitch.   The orchestra tunes the “A above middle C”.  That ‘A’ would vary in pitch through the centuries.  In Bach’s time, the A was almost a half step below normal today’s concert pitch.  Thus began a process that was sully my pristine world of pitch.

At the beginning, when I heard a performance on original instruments, I would say “this is a piece in B Major”.  The piece had all the aesthetic qualities of that were characteristic of the key of B Major.  At the end of the performance I was of confused to hear that it was a piece in “C” Major.  Sometimes it was even a piece I knew but which I suddenly could not identify because it was in a different key.  However the worst thing was that after decades of original instrument performances, my “B” started sounding like a C.   And I was too old apparently to develop a second sense of perfect pitch to go along the first.   Talk about being confused.  I could not really tell any more if the piece I was hearing was in C major, tuned down, or C as I grew up with it.

And so performances on original instruments spread like a virus over my entire nexus of absolute pitch.  This was the beginning of the sad decline of my perfect pitch.

But the next step in this sad story totally befuddled me.  I was in San Francisco giving a lesson over the phone to a student in Oregon.  I did a lot of long distance phone lessons in those days – now I use skype.  She was playing the C-sharp minor fugue from Book One of the Well Tempered Klavier.  I got tired of holding the phone to the same ear (my right ear), so I switched quickly to my left ear.  And lo,  the pitch of the piece dropped by about an eighth tone (25 cents).  At first I thought I was imagining the difference, but wasn’t, on further experimentation the difference persisted.

I wondered whether the ears, like the eyes, consist of a dominant one and non-dominant one. I knew that with my eyes, if I closed one eye and then the other, an object in the near ground or mid ground, would change its alignment with the objects in the far ground. When I used both eyes, what I saw was what I had seen through my dominant eye.   I splendid musician I know, Wendy Loder, has confirmed having the same experience, with an even larger pitch difference than I experience.

Now I was faced with something similar with my ears.  Two pitches, one in each ear, but the higher of those two pitches was the one I head when I was hearing with both ears.  In my case the pitch that I heard through my right ear alone was the same as the pitch I heard with both ears.  That was freaky because I wondered where did the other pitch go.  It must still be in my brain somewhere.

I was offered this explanations.  The cochlea, in the inner ear, shrinks as one ages.  The cochlea in both ears might be aging at different rates.  Analogous single nerve endings in the two cochlea, that had always responded to a middle C still, in a sense did so, but now responded to pitches near middle C, but not exactly at the same.

As I write this, I am seventy-one years of age.  My original perfect pitch has survived through the years in only one case: notes coming from the piano.  Only occasionally for the other instruments of the orchestra.  But at least I’m never off my more than a semitone.

So, things couldn’t get too much worse – right?

Recently, the next nail in the coffin of my absolute pitch occurred in the form of how I was hearing octaves.  I used to object to the “stretching” of octaves that many tuners did when tuning the higher range of the piano.  I used to hate tuners who would tune the high octaves sharper than the mid range octaves.  Suddenly, though, I was now experiencing a distortion in the pitch of the high notes of the piano that made me wish I could stretch the octaves.  If I played a lower C, in the octave of middle C or an octave lower, together with one of the highest C-s on the keyboard, the higher C sounds a half step lower than the lower C.  It was like hearing a C and a B.  To be honest, this phenomenon had been creeping up on my over the years.  At first it was a curiosity.  Now it was intolerable.  The string for the higher C would have to be stretched tighter, almost up to a C-sharp, for it to sound like the same note as the lower C.  Now i know why some tuners stretched octaves.**

To be honest, I would have much rather had my absolute pitch go away entirely rather than in agonizing stages.  But there was always enough left of the absolute to know that something was amiss in my perception.  It was a more benign form of when a patient is consciously able to trace the course of her illness.  Now I am starting crave the bliss of ignorance of not having absolute pitch at all.  I can sense that my ability at relative pitch is asserting itself in situations where absolute pitch made relative pitch unnecessary.

I can now sit and contemplate what might be the next stage in the sad decline of my absolute pitch.

* Research at the University of California in San Diego found that while many may be born with it, discovering the gift is likely more the result of nurture than nature.Sep 18, 2012 (from a Google search)

** About ten years earlier I was offered another more ‘scientific’ and objective reason for stretching octaves.  In physics the string is often considered as a one dimensional object.  This allows the math to be simpler.  But a string is three dimensional.  It has length, width in a horizontal plane, and width in a vertical plane.   There is a “nodal” point at the half way point along the string   which as result divides the string into two parts, each part sounding an octave above the string at full length.   A nodal point is a place along the string where, under certain circumstances no vibration takes place.   But if the nodal point is three dimensional, rather than a nodal ‘point’ we have a nodal ‘sphere’.   This causes each of the remaining, vibrating halves of the string to be slightly less than half the length of the full string, and thereby have a pitch that is slightly higher than one octave above the pitch of the string vibrating as a whole.

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Harmony in Late Brahms

Harmonic wonders in Brahms’ Intermezzo Op 117 No 3 in C# Minor

#1: The key

What is so compelling about C Sharp Minor?  Perhaps it is due to the effect of certain pieces that were written in that key: the opening fugue of Beethoven’s late quartet, op 131; the 5th prelude and fugue from the first book of the Well Tempered; and, Brahms’ Intermezzo Op 117 No 3.

The spell of this key is both obvious and subtle.  Tragic but not overly so.  Deeply reflective of the human condition, but without overstatement. or wallowing.   Notes that enter this solar system of retain reveal the opposite effects of stoicism and great sympathy.

#2. The theme.

As in many of Brahms’ his late pieces, the motivic material out of which the piece is woven are themselves terse and simple.  Nothing startling in itself.  In Op 117 No. 2, just two notes, descending in a step, suffice to create the entire varied panoply of music effects heard throughout the piece.  In the case of Op 117 No. 3 it is woven out motives of three notes, rising in pitch by the steps of a scale.

Sometimes the distance between the second and third note is enlarged to three half steps.  Sometimes the middle note stands as a passing tone between the other two notes which are chord tones.  Sometimes the middle note stands as a lower neighbor note connecting two identical chord tones.  Out of these motives a long theme emerges which takes most of the entire first line of the score.

Every time the theme returns it does so in one way – invariable: exactly the same sequence of pitches (C# D# E …).  There is a growing sense of ineluctability about it, an effect that is progressively offset however by changes to the chords that set the notes of the theme.  No matter how these chords lead us away from the tonic, C# minor, all eventually leads us back to that tonic.

At the beginning the theme appears without simultaneously sounding chords.  When it appears again there are implications of triads (chords having just root, third and fifth).  At the next statement, the chords embracing the theme have become 7-chords.  And even later they have become 9-chords.  This is done is such a way that, a particular note of the theme, let us say the third note, E natural, sounds first as the third of a chord, then as the seventh of a chord, and ultimately as the 9th of the chord.  This means that the chords, rather than being built on the same root note, are built on changing root notes: a more revelatory way of enlarging upon the chords, so as to be always expanding the harmony.

We go on to trace in somewhat more detail these changes of harmony, changes that are always put held in check by the constancy of the theme notes:

– At the beginning we hear the theme as unisons, amplified by sounding in three octaves once at once, an effect made starker by the absence of vertical chords.  The harmony is there. but remains adumbrated by just the melody notes (which are sensed as chord tones and which as tones of embellishment.

– When the theme appears again at the end of measure 5, it is almost as if Brahms wants to keep the harmonic implications as Spartan as possible.  There is a hint of the tonic chord (C# minor) and a dominant chord (G# major).  The third is missing in the dominant chord, though, so we hear it as major only through the implication of the melody notes.   The effect here is one that when I played the piece today I described to myself as tragic inevitability tempered with patience and nobility.

– The next statement of the theme occurs after a Spartan interlude.  This interlude begins with an inexorable march notes of equal value (eighth notes) to which, at the end, sixteenths are added so that there is a sense of reprise of the rhythm of the main theme (which uses the rhythmic germ of sixteenth, sixteenth, eighth.   The measures of the interlude repeats, but with the magical addition of an extra voice appearing in the left hand which creates a rhythmic counterpoint to the steady eighths, but whose beauty is largely the result that these attempts at rhythmic variation are still imprisoned by the constant eighth notes.

As the theme occurs throughout the piece, it does so unaltered in terms of pitches (a series that always begins with the notes C# D# E.  However the chord that is woven around theme has expanded into a 7-chord, an F# Minor-7 chord.

The piece is in three broad parts, the first of which is brought to a close with a statement of the theme that retreats into its initial harmonic simplicity: there is a tonic chord, there is a dominant chord, but but the two are blurred together by the retention in the dominant chord of the C# from the tonic chord, an effect added to by the absence of a third in the dominant chord.  We are being reminded, though subtly, that we are in C# Minor, so that we more fully appreciate the modulation to A major in the second part.

The second section of the piece a contrapuntal and harmonic miracle brought down to earth from the celestial harmony of the spheres.  I want to hold off describing what Brahms does there until we have followed the remaining statements of the main theme which occur in part three.

When the theme next appears in its entirety is at the beginning of the third section.  The 7-chord has been expanded by a D#-9 chord (the ninth being a minor ninth above the root note).  It is as if Lear asked the of his three daughters: “what {harmony} can you say to draw a third more opulent than your sisters? Speak.”*.  Unlike Cordelia’s answer, which is most understated and matter of fact, Brahms’ answer is a 9-chord compared to the previous “sisters'” ‘5’-chords and 7-chords .

In the coda, at the penultimate entrance of the theme, the melody is accompanied by a chord whose root note is now A# which, with the other chord tones in the melody, forms an A#-half-diminished 7 chord.

At the very last, we hear the theme one more time and, as if surveying the field of a great harmonic battle which has taken place over just a single day, and which now looks deserted and barren: the original theme returning one last time accompanied by just a tonic and dominant chord, bringing the piece to rest on the C# minor chord, which, which for the first time in the piece is heard alone, complete in time and unencumbered.’

#3. The middle section of the piece

The middle of the piece deserves special analysis.  Let us start by trying to “uncover” its ‘main theme’, or at least that, which by default acts in the place of a principal voice line.  In itself it is not the most melodious of note sequences,  it is devoid of any rhythmic personality, and moves seemingly randomly from one note to another – sometimes in skips, sometimes in steps, and sometimes in capricious jumps: seeming somehow in between insipid and random.  However, this seemingly undernourished melody is is able to usher in, with each new note, a new “chapter”, a new harmonic vista, allowing us to see further and then further to the harmonic horizon, as if from higher and higher vantage points.

In its most stripped down form, which we never get to hear literally, the theme consists of the succession of the notes E F E B G# A C.  Upon this Brahms performs a series of transformations and eventually metamorphoses.

The first transformation seemingly makes things worse rather than better:  a random yanking around of the melody notes from one octave range to another.  In its base form, all the notes would sound in the octave of middle (we shall call this octave number “4”).  But this remnant of pitch stability is dislodged so that the first E sounds just in octave 4; the F simultaneously in octaves 4 and 5; the following E just in octave 5.  This is followed by B in octaves 5 and 6 simultaneously, G# in one octave only – octave 5, and the last two notes, A and C, each heard in two octaves (octaves 4 and 5).  What began as uninspired is now wonky** as well.  It is like being on a roller coaster and leaving out the parts that connect the low points and the high points.

To this Brahms adds a single bass tone, A, modestly appearing at the beginning of each measure of 2/4.  It is a first attempt at establishing a tonal center for all the meandering of the theme.

But the stability is further broken by shifting each melody note (which lasts a quarter note, or four sixteenths), to the “left” so that each note comes in a sixteenth too early, just before each beat.

So far in this analysis we see how octaves have been changed, almost capriciously, melody notes brought in ahead of time, one modest note in the bass to remind the listener of where the first beats of the measures actuakkt are.  Not a good state in which to leave things.  There is however one more step to the transformation,  Each melody note is accompanied by an voluptuous figure of four sixteenths that swoop down and then soar back up.  Somehow this makes everything else make sense to the ear.  If we examine these four note groups, by making chords out of their notes, we get this succession of triads: e-cs-e, e-b-d, e-cs-e, e-d-f, fs-d-fs, forming just by themselves, as it were, a single “thick” melody.

There are still other startling details before this section finishes***.  But it is the last one that is the most stunning.  Something that the ear believes and disbelieves at the same time.   It occurs after the second double bar, where the key signature changes back to 4 sharps.  In other words a point when we would expect either a return to the original theme, or first, some transitional passage linking the the end of the second part, the one we have just been discussing, to the beginning of the third part.  What happens instead a polyphony worthy of the forty-voice Renaissance motet “Spem in alium” by Thomas Tallis.  Little, nascent, voices, appear and then disappear into the overall harmonic firmament, each one living just for three notes, each modeled exactly on the first three notes of first statement of the theme at the beginning of the piece.   Moreover each ephemeral voice makes its appearance in such a way as to partially obscure (or perhaps “occlude” is a better word) the end of the previous one.****  This process continues in a cascading fashion working its way through an elaborate dominant-like (G#) harmony that barely succeeds in stabilizing the whole affair.   I say barely because it is intentionally destabilized  by coercing a cadence to end it based on a B Minor-7 chord.   And then, to completely throw the listener off, a similar cascade begins, seemingly without reason or preparation, tracing over the first cascade but with each note two half steps above the similarly placed note in first cascade.   The first was built around the dominant of the original tonic key of the piece.  That seems to make sense if we looking for a transition back to the original C# minor key.  The second is just ‘quasi-dominant’ in nature, but its root note is A#!   Now this unexpected A# turns out to be the dominant of D#, which is the dominant of G# which is the dominant of the original C#.  Thus it only becomes clear why this shift of two half steps had occurred between cascades when the original theme comes suddenly comes back in its full form (starting with the usual notes: cs4 ds4 e4) but this time surprisingly surrounded by a D# Major (9) chord.   This chord then transforms itself until, in a fairly short amount of time, we are clearly back in the original C# minor.  Thus the previous six measures have prepared two separate but parallel things: the longer range goal of an eventual return to the tonic key of C# Minor, and a shorter term goal of preparing the D# chordal sonority that will underlie the return to the original theme.

#4. The end of the piece

In the last two measures of the piece we have a stable, lasting C# Minor chord.  So, in the end, all is drawn back into this tonic key, which may have been at times out of mind, but which never lost its grip on the piece.  At the end there is only the solace that no matter hard we try to get away from fate, we never free our self completely from its somber embrace.  As is the case with Brahms – perhaps the greatest worker of harmony – for all the restless harmonic movement towards or away from the tonic, no note, no chord, no modulation is ever away from the tightest control of the original key.  There is never a chord, be it ever so remote from the tonic, that is not perfectly clear to the listener as to its relationship with the tonic, in spite of as many as several key regions that we would have to travel through to get to it from the tonal center of the piece.

These are just some of the harmonic and thematic wonders of Brahms’ Op. 117 No 3.

* King Lear, Act I, Scene I

** Searching google produced this definition for wonky:

– (of a thing) unsteady; shaky…                – synonyms: wobbly, unstable…                                        – not functioning correctly; faulty.

***

In measure 4 and 5 of this middle section there are two flowing voices, one in eighths, and one in sixteenths, that chime with each other in the presence of a D# followed by an F#, and a B followed by an A.   This all occurs as the ‘main’ melody settles down to rest on a long C.

In the fourth and fifth bar of the section that follows the next double bar, the simplest kind of canon is utilized, but with ranks closed, the imitative voice starting but one sixteenth after the imitated voice, but also with the imitated voice sounding a sixth lower than the imitative voice.  A simple idea structurally but one which has the effect of creating near chaos with where the principal beats are supposed lie.  The ear wants to be thrown a lifeline, and Brahms does so, but with the least clarification that will still shed some light on the situation.

****

The first such contracted theme motive is on the notes bs4 cs4 ds5, snd when the ds5 occurs, it is hidden in the middle of a triad, whose bottom note is the beginning of the next, furtive, momentarily flickering motive entrance (whose notes area4 bf bs4).

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