Tag: Bach

Even More Thoughts on How to Play a Bach fugue

A.B.’s playing of the first fugue of book one of the Well Tempered has improved by leaps and bounds.  Due to the high quality of his mind he can contemplate and at the same time be in wonderment at the amazing things, small and large, going on in the piece.

Here is what arose on Thursday, May 16, at his latest lesson.

#1.

When he is physically tense, the first place it shows up is in the form of movements with his lips and mouth.  He usually makes one such motion per note .

Last week we worked on doing away with these mouth motions.  Sometimes such motions help generate pulse and flow but just as often they force the piece to come out uttered in little tiny pieces.  A phrase cannot flow through time if it is comes to a stop and then resumes with each new note.  Frequent mouth motions can cause unintentional separations between one note and the next.  A note should be like each new bead on a necklace.  Without gravity and the string holding the beads together the necklace looses its shape and meaning.

He was able to control this for a measure or so before the mouth motions obstinately crept back in.

We worked out a compromise.  If he is to make a separate mouth motion for every note, let that motion be that of the expelling of puffs of air.  Later on the air can be let out more continuously.  The continuing flow of air is the physical equivalent of the flow of sound in a phrase – just ask any singer.  The piano, and many other instruments, model their flow and expressivity on the human voice.

#2

Joe: If you think of the physical actions you make while playing, now that they are not the cause of the sounds.  Nor are you yourself the cause of the sounds.  Sounds just “pass you by”, flowing by your consciousness.

#3

The general question arose of how do we stay on course if we make a mistake and deviate from the printed score.  We have to find a way of getting back on track as rapidly as possible – hopefully the the next note.  An important component of the alacrity with which you get back on track lies in the answer to the question: how do you react, both morally (I’ve made a mistake and a mistake is bad thing) as well as emotionally (what does it to our self confidence , our self worth).   Any negative reaction of either type makes it more difficult to find your way back onto the tracks, and makes it harder, in space-wise in terms of finding where we are in the score, and time-wise, to resume the correct flow.

Here is another way of stating the problem of getting back on the tracks.  How quickly can we begin at any random point in the piece (whether at the beginning of a measure or even at an arbitrary point within a measure) and resume the ease and flow that we  have at the place if we started the piece from the beginning.

It is good to lard the piece with a plethora of random spots from all of which you want to learn to be able to start up the piece, and ideally take no time to get on board the moving train and flow ahead with the correct notes and rhythms.

Just like coming in at the middle of a conversation and quickly figure out what is being talked about, every note in a piece is (or can be) the beginning of that piece.  B.A. summarized how hard this was for him to do: sometimes when I start from a random point in a piece it doesn’t even sound like it is from the same piece.  And, where did these notes come from and where are they going … how quickly can one become aware of the answers to these.   The answer to the last part: as instantaneously as possible.  This reminds me of the famous Gauguin painting “D’où Venons Nous / Que Sommes Nous / Où Allons Nous” (Where are we coming from, what are we, where are we going?”.   To put it in another way: how very important it is to spot the common character and individuality of a piece even just within a single minute detail from that piece.

When you practice starting the piece from a random point, just play until you are back on track, don’t keep playing to the end.   You want to leave practice time for starting from other points in the piece.

#4

Fingers getting tangled:

There are times when the hands draw very near to each other, touching, overlapping, interfering with each other.   In particular the thumbs (and even the second fingers) will cross over each other and afterwards uncross.  This situation must be coordinated down to tenths of a second.  It is a “pas de deux” between two fingers / hands, wherein the bodies of the ‘dancers’ need to fuse as much as possible into one entity that is constantly changing shape as a whole.  Every motion on the part of one dancer must be fused with a simultaneous motion from the other dancer.  It is as if there is a common consciousness among the two.

#5

The general question arose as to where does one phrase ends in a Bach fugue and another phrase begins.

This can become marvelously complicated because, in a fugue, two or more voices may be in the process of sounding the main theme, yet, at a given moment one of these voices may be at the beginning of the architectural shape of the theme through time, while another somewhere in the middle of the architecture of the theme, and a third voice might be in the midst of concluding the end of its theme statement.

How does the pianist simultaneously  make one voice sound like it is ending while another is beginning when the two voices are clearly both stating the same theme.  B.A. had a nice way of putting this: how does a voice say that it’s ending.

Put in terms of the chords that underlie the passing notes in the voice melodies – frequently the shift from one such note governing chord in the harmony of the fugue to the next such chord, does not occur simultaneous in all the voices.   One voice may enters the domain of the next chord before the others.  They are harbingers of the next chord; pathfinders.  Another voice may arrive into the new chord not until the other voices have clearly established the chord.

#6

Situation: one finger is holding an extended note while other fingers in the same hand are enunciating a series of changing notes.   This requires that the finger holding the note be very flexible and can change its overall stance in response to changes in what the other fingers are doing.  The key to clear articulation often lies less in the equalization of the fingers playing the changing notes, and more in the ability of the finger holding its note to suddenly change it’s alignment with the keyboard, and its stance relative to the other fingers of the hand while, at no instant, losing its the overall equilibrium.

#7

Sometimes a student is confused when the main theme starts on a different note compared to the opening of the fugue.   If the change of starting note represents a change of harmonic region, then it makes makes sense to the modern player.   However, it is harder situation to make sense out of when when the theme entrance is still in the original harmonic region.  Thus a theme entrance, instead of starting on the original series of notes at the beginning (C D E F …) begins instead with D E F G, or E F G A, etc..  That instead of representing a modulation, it represents the desire of the theme to enter on a different note of the C Major scale but cling obstinately to the same scale.  Some of us may think of this as a hark back to the Catholic Church modes of the middle ages, in which case  D E F G is simply the beginning of the “Dorian” mode, E F G A the beginning of the “Phrygian: mode, etc..  But it is not always clear that this was how Bach may have been thinking.  Perhaps the underlying constant is the C Major scale (or tonic of another harmonic region of the fugue) and how it stubbornly controls things even a theme entrance tries to start on a different note of the scale than the tonic.

#8

A part of fugue technique is to instantaneously move one finger left or right, from one note to another, regardless of how far apart those notes are on the keyboard.  This is not something mastered by gradually practicing such a motion faster and faster.  It is more the absolute determination ahead of time to be on the second note zero seconds (zero fractions of a second) after the first note ends.  In other words: for the finger to find itself already on the new note, without any travel time in between.   This is quite possible.  The body is capable of doing this if one insists this be the case, a determination that starts before one starts moving the finger at all.  Such instantaneous change of by just one fingers promotes a greater clarity and crispness in the consecutive notes of a voice.  The goal is that no connection of one note to the next be any more sluggish than any other.

This itself is a component of the general ability of the entire hand snap from one hand position to another position.   Sometimes fingering alone will not provide a sense of connection (even if allows for singer substitutions).  It may require an action like the triggering a mouse trap: with little or no preparation, no anticipation, and seemingly no time at all taken to make the change in position.

To achieve such alacrity in changing the shape of the hand it is necessary for the arms as well as the hands to be weightless, and the muscles in the hand being ‘at attention’  but when the moment comes for the change in the shape of the hand, offers no resistance to the onset of that motion.  It as if the muscle is passive and is being moved from an external source of power.  Even the forces that initially raise the arms to the keyboard can be felt in the body as if the arm was being moved not by its own muscles, but a force external to the entire body.  This feeling can be induced by imagining the arm belongs to a puppet, and an unseen puppeteer moves the arms upwards by pulling on the strings that connect the puppeteer the puppet’s arms.

At a lesson the teacher can literally provide this external force.   For instance supporting the student’s hands so they will feel to the student as if they are floating on the keyboard rather than pressing down on the keys.  Additionally, should their be any pressure downwards (other than to activate a key) it is more easily detected by the student if they are pushing down on another person than an inanimate object like the keyboard.

#9

We noted a connection between the technique of finger substitution  on a held note (in anticipation of using a more convenient finger on the next note) and the technique exercise found, as in “Hanon”, of using the fingering 4 3 2 1 (in the right hand) to repeat the same note four times in a row, and then to do the same on other notes, throughout the exercise. Though the overt purpose of this exercise is to learn fast repetition of the same note (on the assumption that changing from one finger to the next is faster than using the same finger over and over again) it also prejudices the hand for doing a quick substitution of one finger for another on one note without re-sounding the note.

#10

A.B. brilliantly put many of the above points into a common perspective by saying: it is all about who is doing what to whom and when.

We concluded the same lesson by working briefly on the companion prelude in C Major from Book One.

#1

Part of A.B.’s quest has been to play the notes in the prelude as evenly as possible.  So much of this depends the balance between the notes of the common chord that is outlined by the succession of notes in each measure.

To make these chord more obvious to the ear let the player while playing,  “densify” each chord.  For instance, if there is an opening between the written notes for an additional note of the chord, add that note to the chord and play all the notes that now belong to the chord all at once as a vertical sonority.  For instance in measure 2, there is room for an f4 between the d4 and a4, so that we create a five-note chord: c d f a d.  Or taking it a step forward we can also add a c5 between the a4 and the d4, forming a six-note chord.  The chord, has been a D Minor-7 chord the entire time, but the additional chord tones just make the chord stand out more clearly to the ear.  This can be done, at one time or another, for every chord in the Prelude when Bach’s written notes allow for such additions.

Note that the additional notes mentioned so far all lie in the range defined by the lowest note of a measure and the highest note of the same measure.  An equally valid technique, and one more vivid to the ear, would be to add additional notes belonging to the same chord that are lower than the printed lowest note and the same for the highest note written in the measure.

This way you can generate chord of 8 or more notes, and, if you add the use of the pedal. chords of any number of notes (culling notes from the bass range of the keyboard and the high treble).  If you play such a chord then play the chord made up out of just the written in the measure, you will gain a sudden sense of how the written chord is a just a part of the larger chord.  And whatever the sound and mood characteristics of the larger chord, they are transferred into the more compact form of the chord without any loss resonance and character.

#2

In terms of this grouping every note of the measure into the unified sound of a single chord (versus hearing just separate notes), it is the pinkie note in the right hand that is “furthest” from the left hand note that is the first note of the measure.  And not so much in space as measured on the  along the keyboard but in time that has passed since the first note.   For some this creates a feeling of the pinkie being a dangling participle after the previous four notes .  The feeling can occur even more so when the pinkie plays the last note of the measure prior to the unseating of the current chord and succession by the next chord.   Some pianists have a tendency to have their pinkie ‘separate’ from the rest of the hand when an articulating a note that is beyond a certain distance from the thumb, with the result is that there is less rather than more control of how the pinkie notes fits together with the notes the other fingers are playing.  There is sometimes a poker “tell” observable by the teacher when the student is singling out the pinkie and feeling like it is not part of the hand.  It is if the pianist raises the pinkie higher off the keyboard than the other fingers before playing its note – an attempt on the student’s part to gain better physical control over the pinkie but usually with the result that the pinkie sounds disconnected from the other fingers.

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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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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.

 

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Complex counterpoint in Bach

Some of the most complex wonders of Bach’s counterpoint, such as the F minor Sinfonia or the first movement of the B minor flute sonata, result from a procedure that is structurally, surprisingly simple.

There are three voices in the Sinfonia and three in the flute sonata (two in the keyboard and one in the flute).   At the same time, there are several distinct motives present, any one of which can appear in one voice or another at any time.  There are often three different motives appearing simultaneously in the three voices.

This suggests a creative procedure similar to making a mosaic.  In a mosaic, there may be only several differently shaped pieces, or tesserae.  From these few pieces, the entire structure of the work is created.

Though there is great variety in the sound of the piece, the parts making it up at any one moment are just one of the severally shaped pieces of the mosaic.

The counterpoint is most effective when these basic pieces are as different from each other as is possible.  The F Minor Sinfonia is the perfect example of this.  There are just three motives:  one is a series of slow, chromatically descending quarter notes; the second is a series of three eighth notes, a movement up of a third and then down of a second; the last is faster, moving motive in sixteenths and thirty seconds.  Bach creates an entire universe, aesthetically full and complete, with these three motives that each take up abode in one voice and then another.  What genius!

While artists may follow the procedure of “separating” the voices in their playing, sometimes the greatest separation effect comes from following a single motive as it migrates from voice to another.  I first discovered this principle in the five voice fugue in C-sharp minor in Book One, especially after the eighth notes begin.

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Bach’s counterpoint – who’s on “top”?

The amazing thing about Bach’s counterpoint is that the musical meaning of a piece does not change if you transpose one or more voices to different octaves or simply rearrange the order of the voices from highest to lowest.  The voice that is originally on top does not have to be on top for the counterpoint to have the same effect.   It is as if each voice in a Bach fugue is transparent to all the others.  No voice, regardless of its pitch range, is opaque in that it blocks or occludes any other voice.   The voices shine through each other.  Figuratively speaking, every voice is on top.

Being the top voice loses its automatic prominence to the ear.  All of them, in effect, are on top, or none of them are.*  You can think of it either way.

It is a mistake to concentrate our attention when playing on the top voice, or to whichever voice currently “has the theme.”

Of almost no other composer is this true.  In a Beethoven sonata, for instance, only rarely can you move the right hand notes into the bass and the left hand notes into the treble and have a result that ‘works’ as well musically, that aesthetically resembles the un-switched version.   In Beethoven, depending on which octave a note sounds in, it creates an impression of sound that is different than the same note sounding in a different octave.  Each octave range has its own sound-personality.

One of the ways of determining the unique properties of the musical space of a great composer is to experiment subjecting the piece to certain specific types of distortions and then noting the results sound-wise.  One such attempt at distortion is the rearrangement of the voices described above.  Here are some others:

– how much can you change the tempo of the piece without distorting the meaning of the piece or its character?

– how much can you change the rhythm of the melody without changing the essence of a melody?

– how much of the essence of the music is changed by changing the instrument or instruments playing it?

– to what extent will the integrity of the piece be ruined by starting to switch around parts (what comes first, what comes second, etc.)?

The answers to these questions will be obvious to the ear, and will vary from one composer to another.  They reveal to us what some of the basic, “geometric” properties of that composer’s “musical space” are.

These experiments are similar to those in the mathematical subject of topology, where a common question is to ask how much one can distort a shape and still have it retain certain basic properties.

*This is why Bach’s fugues for organ work as fugues even though it is often the case that each voice sounds in several octaves at once because of linking an eight foot stop with a four foot stop and/or a sixteen foot stop.  The same applies to Bach on a harpsichord when ‘couplers’ are used to cause a note to sound in more than one octave at once though we are only pushing down one key.

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