The Fusion of the Hands

A.B. playing Albeniz: Orientale

#1

As a general principle the left hand should always be playing with and encouraging the right hand.  When nothing is written for the left hand in a particular measure, then, for practicing purposes, the left hand can either provide notes that support the right hand harmonically, or make gestures as if playing these notes but without sounding them – as long the physical effort involved is tantamount to or greater than the effort that would be made to sound the notes.

In the section where A3 is held and the remaining fingers play a series of parallel triads in inversion, AB’s right hand feels insecure; he says that it doesn’t feel balanced; the fingers feel awkward trying to play the exact notes of the triad. I asked him to play the octave a2-a3 in the left hand, and to re-play with each triad in the right hand.  “Miraculously”, his right hand no longer felt out of balance.  The reason that it is best when both hands are lending mutual support to each other is because we are bilaterally symmetric creatures – our arms and legs are mirror images of each other.

If we interlace the fingers of our two hands and then move our hands conjointly around in space (up and down, sideways, it doesn’t matter), we are no longer automatically conscious of what one hand is doing versus what the other hand is doing. They have lost their individual identities once fused together in a larger, single, natural entity. Starting with this larger unit, we can then farm out assignments to each hand.  There is a ‘pulse’ generated by the center of the body that travels like an electric current down both arms in concert.  This pulse can also cross from arm to arm in analogy to how the optic nerves crisscross on the way from the eyes to the brain.  We should assume, in both cases, that each gains support from the other.

The hands form a unity such that each hand suffers when that unity is broken.

#2

A chord is the same regardless which hand plays it:

In the same section of the piece, where a sequence of parallel triads occur over a held a3, A.B. says that if he uses his right hand to play all three notes of each triad, his ear is more able to be aware of the chord that is formed by the three notes.  I said that ideally, we want to reach a point where what we hear is not dependent in any way on which hand is playing which notes of the chord.  The chord exists as a single sound unit regardless of which notes in the chord are played by the right hand and which by the left hand – it’s always the same chord with the same sound. Physical differences are secondary.

The Principle of Nested Parentheses

In simplifying algebraic expressions, one starts with the innermost
pair(s) of parentheses, simplifies and reduces it, then works
gradually outwards to the outermost parentheses, i.e. the full
expression.

Today, I want to advocate the opposite process for piano
practicing, one that begins, figuratively speaking, with the outermost
parentheses and works inwards until all the details are presented.

A.B. is playing a piece called “Orientale” by Albeniz. We set up our
first, our outermost parentheses, to surround 5, 6 and 7 of the piece.   We left the parentheses empty except for the first chord of measure five
and the first chord of measure seven. Everything in between was omitted. We tried to effect a connection between just these two chord/islands in time, a connection that was crafted to make those two chords in sequence sound musically self sufficient and meaningful. Bear in mind that, as with any good parenthetical statement, the words (notes) inside the curved brackets are of less importance than what lies outside the brackets.

We next subdivided the outermost parenthesis into two two nesting
parentheses. The first nesting parenthesis goes from the beginning of
the fifth measure to the beginning of the sixth measure, the second
from the latter to the beginning of the seventh measure.

Each of the new, nested parentheses is of less importance than the original, surrounding parentheses. Thus the chord at the beginning of measure six is of less importance than either the chord at the beginning of measure five or the chord at the beginning of measure seven. The presence of the sound of the chord at the beginning of measure six should in no way interfere with the way that the chord in measure five connects with the chord in measure seven. This inner chord is not quite trivial, but it is at a different order of magnitude than the other two. This difference in magnitude should be noticeable both in terms of the amount of physical action and exertion used to sound it and in terms of its musical importance.

We continued the process by further dividing each sub-parentheses into
more numerous shorter parentheses. This process continued until, at
the last stage, every note in the original passage is present and
sounding. Gradually all the chords and melody notes appear.* At each
stage the full, or final, picture becomes more and more fleshed
out. The new material added by way of detail is, as in the stage prior
to it, stepped down in terms of the magnitude of physical action and
exertion made to execute it.

In this system the final details, including all the individual notes
in the score which we insert at that last stage are, strangely
enough, the least important. At each stage we discover that we can
make a convincing musical phrase out of just the material constituting
that stage.* Though eventually there will be more notes present, the
notes that are there in each each level sound entire and musical, as
if nothing is being left out – no note or chord missing, each should
note in no way depends for its musical character on any implied notes
we will hear in the future.**

At the final stage, when all the notes are sounding, all the other steps which we have previously enunciated are still “there” in some sense, enriching the overall texture of the passage.

* In one possible stage, we discover, for instance, that playing just the first two of each group of right hand triplet notes, creates its own independent melody without requiring the third note.

** Generally speaking, it is too easy to make connections between two
things that come one right after the other in time. It is their very
proximity that calls our attention to the relation between them. But
who is to say that the current music note in time should not form a
relation with a note that occurs two or more notes later, or later
still. And if there are such medium and long distance relationships,
they are the building blocks of nascently growing organizational
units of the piece until the whole piece is interconnected. As these
units grow longer in time the beginning of the unit is only partially
retained in memory, first as an ‘after image’, and then deeper and
deeper in memory, until they back to mind if they are in some way
reiterated or altered.

Tidbits From Recent Lessons: Shostakovich, Chopin, Mozart, Bach

R.M: Shostakovich: Prelude # 10

Its syntax is filled with sonic miscues, altered expectations.  Like a peptide chains that have been snipped apart into separate amino acids in order to form unexpectedly new peptide chains.

Each time something unexpected happens can you find your new harmonic footing before the minimum possible number of notes has passed.

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A.J. Chopin Waltz in C# Minor

Joe: Sometimes you are not sounding all the written notes.  How do you  know if every note sounds, for instance, in an interval or chord?

Ideally your ear knows or can quickly take stock of every note.  Otherwise you can try this: Play the chord and release all but one note.  Is that note sounding?  Is it sounding in a way that you think will balance well with the rest of the sounds in the chord.

Repeat process for each other note in the same chord.

More mechanical based approaches:

-tap each note separately once or twice before sounding the chord.

-have the illusion that you are not playing the chord notes simultaneously but that you are articulating them one at a time.

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J.M. Mozart: C Minor Fantasie

#1

No matter what you do in the opening two measures, when you get to the B-flats at the beginning of measure 3, forget any connection with the immediate past, the only note that it should connect from is the C-naturals at the beginning of measure 1.  Similarly with the next forthcoming groups of measures until you reach A-flat.

#2

When the right hand settles down into repeating ds4-fs4 as sixteenths, don’t let any of those thirds escape your attention regardless of what the left hand is doing or is in the midst of doing.

#3

The ending of one phrase and beginning the next.  How you start the  next phrase, musically and physically, can be strongly  influenced and controlled by the way you release the last note in the first phrase.  How you end something is a big detriment of how you begin what’s next.

#4

The B-flat major section.

How to create a coherent and flowing melodic line in spite of the variations in the rhythm.

Before playing the melody as a dotted eighth followed by two sixteenths and a quarter note, play those four notes as a triplet followed by a quarter note.  In that form, the descending steps of the B-flat major scale (d5 c5 bf4 a4), assert their simple melodic flow and harmonic coherence.  Then, right away, “capture” what you just heard – but add in the extra parameter of the rhythm.  If done with a calm mind, the melodic flow of the triplets will not be lost in the written rhythm.  It happens ‘automagically’.

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A.B. First prelude from book One of the Well Tempered.

Liberating the expressivity in the bundled chords.

#1

Choose one note from the measure you are about to play.  Sing and hold that note from the beginning of the measure to the end of the measure while playing at the keyboard the measure as written.  In the next, and next…, measures do the same, either 1) choosing as the note to hold the note that is in a similar place in the measure as the one you held in the previous  measure, or 2) purposefully switching at random to some other note in the next measure (singing and holding that note from the beginning to the end of the measure).

#2

For evenness.

My best advice is, given your propensity for on the spot evaluation and analysis of what you just heard yourself play a moment ago, don’t react to anything;  don’t think, don’t be upset, with anything that has happened, just notice it in passing.  When you do analyze it provokes an attempt on your part to physically alter what you will try to do to sound the next note.  You quickly trap yourself into an endless series of corrections, in anticipation of what may go wrong with each next note, because it went wrong with the current note.  The result is that no note is played in a fresh and unencumbered way.

Stay in the present.  If you don’t, one of the things that will worry you is how you will be able to sustain any evenness you have already achieved for so many more measures to come.

The piece plays itself – without much help from you.

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A.B. First fugue from book one of the Well Tempered

#1

There are some crazy sections in this fugue, harmonically.  Let things wax expressive when Bach has demanded this by the way out notes and modulations he has written.  If it helps, think that Bach and not you is demanding this heightened expressivity.  It’s his fault (sic).

#2

You say that when you listen to a recording of the fugue things often go by too fast for your ear to pick out each and every theme entrance regardless of in what voice or voices it occurs.  Especially in the stretto sections.

I suggested this procedure:

Listen to your favorite recording.  Mark in the score the first four notes (only)* of each and every theme entrance.  Play along with the recording but only at the moments in the score that you marked; just four notes.  For the rest of time just listen to the sound of the music flow by.

 

* Playing more than four notes can lead to technical difficulties if the tempo of the recording is faster than you are playing the work.  It will also confuse things in the strettos.

 

The Connection and Disconnection of Notes

I’ve had an idea lately that it would be nice to do a lesson and then post a blog post about it right afterwards. I think this will gain in spontaneity and insight, despite what it might loose from lack of editing and proofreading.

A.B. Was playing WTC I C f (which is my short hand for Well Tempered  Klavier, Book One, C Major (C is uppercase), the fugue and not the prelude (f).

This is a new piece.  The first thing he said, was how hard it was to read a fugue.  It poked at his sore spots as a reader and a player.  I said, forget all of that.  Play the chord on the first beat of this measure, and ask yourself what it is the most natural and comfortable of playing it.  By starting there, it is as if you were starting the piece, just from a different measure than measure one.  So your hand had no allegiance to what it may have done a moment earlier if it had played the last part of the previous measure.

Now, before you go on any further, DON’T TRY connect the present arrangement of the fingers in the hand with the next one.  Playing Bach clearly is not a matter of figuring out a fingering, or getting used to making certain connections in the hands and fingers.  One never goes from “here” to “there”.  All there is are “here-s”.  Each one is undiscovered until right now.  It is always as if you are playing the piece for the first time.

With each new note, or if not that frequently then at least with every new eighth note’s worth of the piece, pause and ask yourself “what is the most natural and comfortable of playing these notes”, especially if the hand need no longer “remember” where it was a moment earlier.

Playing through the piece is discovering, as if for the first time, a new position for every moment’s new notes.  In doing it this way you enter into the joy and spontaneity of the fugue; the experience is wonderful, and in no way a chore.

A.B.: So what do I do the next time I play this spot, wouldn’t it help if I gradually got to know, through repetition, where my hands go next?  Me (waxing poetic and philosophical): No, the only thing you have to remember is to forget.  A.B.: But doesn’t that sometimes mean I get further and further into trouble with my fingers and dig myself into a hole from which I cannot get out.  J.B.: There is a simple solution to this.  When you are least sure where to go next with your hands and fingers, when you feel you’ve gotten stuck in the mud and don’t where to go, that is the time to take your hands off the keyboard.  Remove the hands from the piano, even if briefly.  Start with a new slate, for by removing you hands from the keys, you have let go of the immediate past, you can discover, as if for the first time, the most natural and comfortable position for the two hands together on the next notes.  So, if you never know where you are going to get into fingering problems, remove your hands from the piano.

He tried it.  It was a fine sounding connection.  He said:  but if I remove my hands from the piano there will be a break in the sound continuity, things will not sound connected.  I said: then how comes what you just did, which involved letting go of the keyboard and removing the hands from the keys, ended up sounding more flowing and more connected than I have heard it before?  By removing the hand, you have no choice but to find a new position, a new and most comfortable position, for the next notes.

Be careful, I said, of sixteenth notes (or eighths) in one voice that are moving in steps.  That can lead you down a perilous path.  You will stop looking for a new hand position for each sixteenth, until the fingers get caught in the keyboard and get bogged down because you have “run out of fingers”.  No, you never run out of fingers, there are always five new fingers in the hand for each new note.

When I say “find the most comfortable position” I mean one in which no finger ‘remembers’ where it was a moment earlier.  Nothing about its position in the hand is biased or coerced.

To save time in writing, I am using the convention of having:

|: Ernie is the name of my cat 😐

To mean that I have gotten trapped in an endless loop and am saying the same thing over and over.  And in such a way as if I never had said it before, but rather someone recorded me, quickly hit stop, rewind, and play.  The idea is that it becomes a spoken “mantra” whose efficacy is in its being repeated, until the mind’s state becomes transcendental, a state in which one does not connect things physically SO THAT they can connect sound-wise (sorry I’m sounding a little to “new age” for a cynical Jew from Brooklyn).

If you are a draughtsman, and you use the same writing implement over and over again during an hour of work, do you always consciously try to pick it up in the same way as before.  Do you have to think of its position before taking it into your hand.  No, it only becomes natural to do if you allow the body to learn unconsciously, so that the 100th time you take the pencil in your hand, it is consciously just as unplanned and spontaneous as the first time.  From the conscious point of view (and not for the unconscious, which is busy learning and practicing) you are always finding something ‘new’ (not ‘old’) and finding it for the ‘first time ever’.

All of this started falling into place when I physically caused A.B. to remove his hands from the piano after each current sound.  When he resumed, the next sound and all the newness of freshness of the morn: is it ever really the same sun that rises the next day (Thoreau says something about this in “Walden”.   Every hand position is “discovered” spontaneously.  So I sometimes started using the repeating mantra |: every position is new and discovered spontaneously :|.  or just, |: find a new position 😐

Don’t be afraid to let go, for that is the only true way for the body to find what’s next.  It is the opposite strategy that common sense tells us to follow.  Consciously you forget it even happened before.  You think you can’t do this “A”,  but you can.  You just need to keep an experimental mind, and prove it to yourself over and over with the freshness of every new sound.

And, by the way, when you find the ‘new position’ it always for both hands together, never for just one hand or the other.  Let the body, let the ear, always synthesize together every new sound in the piece.  And the listener has no desire to complicate the wholeness of the musical experience by knowing which part of what they hear came from your left hand or your right hand.

After a while, all I was saying to him was “let go” … “let go” … “find the new position”, “find a new position”.  There was one moment I could tell that he was trying to figure out the best fingering for a series of consecutive notes.  I said: that was not a new position, it was a ‘trying to get there from the old position to the new’.  There is never a ‘there’ to which to get, everything is a ‘here’.

A.B. said, how can I have a totally new position in my hands when I am required to hold over one of the notes (holding down a note in one voice while the notes in the other voices change).   I said: I agree that you have some issues with what I might call, by analogy, if it were spatial more than of time, “negative space” (E.G. is it two profiles or is it a vase).  A held note is not due to a finger that holds tightly to its position on the keyboard.  It is do to a new position that that finger assumes every time another voice voices to a different note.  The fact that the finger remains on the same key is secondary and incidental from a physical point of view.  There is no difference between writing a half note, in a score, and writing the same note as four eighth notes, each tied to the next.

Negative space also involves things like, the action of when to release a note in a voice after the finger playing has gotten inured to holding it down when it has been held for a while.  Another example are rests, in general in a particular voice, which must be incorporated into the “sound” continuity of the piece.

So, abandon any noble effort by the left or right hand to connect the notes in the fingers.  Don’t do that!  Let it go.

I would love feedback regarding the usefulness of this type of blog entry.  It probably suffered from repetitiveness but it did not loose my original excitement about discovering these things, and in keeping pace while writing how things evolved through time during the hour of the lesson.  Thanks for reading.

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.