Category: Music Theory

Chopin Etudes Advice

“Chopin Etude in C Major, Op 10, No 1.


The way up:

Use the left hand as a ‘helping hand’ for the right hand, by placing
it under the right elbow. As you get to the pinkie note in the right
hand on the way up (measure 1, 3, etc.), have the left hand raise the
elbow of the right hand in an arc-like motion through space, and then
drop it back down on the keyboard, in such a way that the right hand
thumb makes contact with its next note. This action should be
repeated each time the pinkie is followed by the thumb; every time the
thumb begins another terrace of upwardly directed pitches.

The way down:

As you get to the thumb note on the way down (measure 2, 4, etc.),
have the helping hand lift the right elbow as before, up and over to
place the right pinkie on its next note. This action should be
repeated each time the thumb is followed by the inky; every time the
pinkie begins another terrace of downward directed pitches.


Op 10 / 1 continued: General: Three things:


There is an exaggerated action of the thumb and second finger of the
right hand causing them to move apart from each other – further apart
than they can normally diverge without additional help from the rest
of the hand, such as placing the third finger adjacent to teh second


In conjunction with number one, I allow each finger in the right hand
to make contact with its intended key at the most comfortable location
along the longitudinal axis of that piano key. For instance, given the
current physiognomy of my right hand, I placed the tip of my right
second finger near the lip of the white keys to compensate for a
strange twist that has become more exaggerated as I get older.


On the way ‘up’, I make a movement to the right, after I play the
pinkie, a movement that draws along with it the right thumb,
depositing it near to where the thumb needs to be next. In analogy to
this, on the way ‘down’ (usually in the next measure), the thumb moves
off to the left after it plays its note, which gently brings the
pinkie along with it to the left, to where the pinkie needs to be in
order to begin the next terrace of four descending pitches.

“Chopin Etude in A Minor, Op 10, No 2



My right hand feels awkward in the beginning of measure one (when it
plays the triad c4-e4-a4), especially if I am in any way already
conscious of having to move the right hand upwards from that triad to
to proceed to bf4. I did not notice this discomfort prior to today
when I practiced, but it was clear that it has always been there. I
found an interesting way to get rid of this discomfort. I dissect the
triad into two parts. At first I play just the upper two notes (e4
and a4(. Then almost immediately I tuck in the c4 with the thumb.
This makes it easier for me to transition from the triad to the ascending
notes bf4, b4, c4..

In contrast to this strategy, when the notes start back down, as in
measure 3, instead of first playing the middle and top notes of the
triads, I play the bottom and middle note and then almost immediately
tuck in the top note followed by the descending sixteenths.

“Chopin Etude in C Sharp Minor, Op 10, No 4


In case if I forget to pay attention, try to ensure that the palp of
the finger is the part of the finger that makes contact with each and
every sixteenth note regardless of the finger being used.

The thumb is not aligned with the other fingers in a way that makes it
is easy to have its palp come into contact with the key. I use some
‘stand ins’ for the regular palp. For instance, I can use the side of
the thumb adjacent to the side of the nail (on the right side of the
right thumb and on the left side of the left thumb). This is sort of
OK, but feels less palp-like than if I act as if I can twist the thumb
radially. and make contact with the key with the section of the side
of thumb that runs from the side of the nail to about half way to the
first knuckle.

Apart from to this attention to the palp of the fingers, I find it
useful, for each finger vibrate back and forth along the longitude of
the key, but just with an ambit of a half an inch or so, alternating
being nearer then farther from the fall-board, until after a fraction
of a section the finger settles down on the spot with which depress
the key lever.

`Op 10 / 4: Measure 1, et. al.

When I use a rotational action of the forearm, I tend to classify it
under a very different category of motion. For lack of a better term
I call the category “blob and deflect”. The blob parts means that the
palm of the hand sinks into the white notes of the keyboard further
from the fall board than the lips of the black keys. Once I’ve
established the presence of my entire palm on a continuous, unbroken
segment of the keyboard, I act as if I’m going to rotate the forearm,
but actually do something else. Let’s say it is the right arm and the
rotation is clockwise. I try to make a forearm rotation, but no
matter how strongly I try to make the rotation, all five finger tips
remain fixed in their place of contact with each key. The torque of
the forearm changes from one finger to the next but enough pressure
downwards remains that no finger tip looses its firm contact with its

Blog and deflect is a very powerful and balanced, or stabilized,
motion when we want to change the point of contact of the hand on the
keyboard from the thumb to the pinkie, for the purpose of playing a
very fast sequence series of notes that belong to a portion of scale
or an arpeggio.

I use just such a powerful deflection, anchored by the elbow*, for the
four notoes starting with bs4 (the third note of the measure), the four
notes starting on cs4 (the third sixteenth note of beat 2), et. al. It
is as if a mouse trap has suddenly sprang shut rightwards, the purpose
of the motion being to compress the pinkie against its note without
loosing contact with the notes the other fingers are playing en route
to the pinkie. The rotational motion is counterbalanced by a force
directed downwards.


Op 10 / 4 : measure 1, et. al.

Small rotations can always be treated as if parts of a larger, more
comprehensive rotation, which if allowed to continue would complete a
360 degree motion.

In the previous example the right hand deflected to the right to get,
for instance, from bs4 to e5. But one can widen that motion, whose
center is the second and third fingers adhering to cs5 and ds5, by
having the thumb start the upwards motion from an a4, and have the
pinkie end the rotation not on an e5 but perhaps an f5 or g5, or even
an a5. Rip the hand from left to right to play the series of ascending notes that now cover as much as an octave.

It is useful to demonstrate to the hand what a 360 degree rotational
motion is like. Start with the right palm resting on the keyboard but
facing upwards and the right elbow extending out to the right of the
torso, Then gradually rotate the arm until you execute as close to a
360 degree rotation of the hand as possible so that the palm is again
facing upwards. To do this requires great mobility on the part of the
elbow and the shoulder. The point of this exaggerated motion is that
any smaller rotation executed during the piece breathes the air of
this enlarged motion, making the smaller motion feel less constrained
in its potential of degree of rotation.


Op 10 / 4: measure 3 et. al.

Sometimes changing the order of the notes simplifies the execution of
a group of notes. We can then carry over that feeling of comparative ease into the original order of the notes.

In this measure we have four note groups, starting with this one:
cs5 gs5 e6 a5

Going up to the e6 (with the pinkie) from gs5 (with the second finger)
and then coming back down from e6 to a5 (with the third finger) can be
tricky. This awkwardness disappears if the re-order the notes as
follows: cs5 gs5 a5 e6. One can simplify the physical action even
further by completing the pitch shape, by adding notes after the e6 so
that there is a symmetric descent of pitches. This is done by
playing: cs5 gs5 a5 e6 a5 gs5 cs5. Sort of completing a circle.


Op 10 / 4 : measure 4 et. al.

Broken octaves in the right hand.

A sharp, slight, tremorous, jerking motion of the tip of the elbow
along an arc in space that brings the elbow inwards towards the
abdomen. Though this leftwards motion by the elbow is in the opposite
direction of the intended motion of the pinkie which is rightwards to
the higher note in the broken octave, one action coordinates with and
supports the other. Imagine that there is a pivot point located
midway down the forearm and underneath it, that translates any
leftwards motion by the elbow, and with it the upper half of the
forearm, into rightwards motion on the part of the lower part of the
forearm near the wrist.

This same motion can be used when it is desirable to move the right
pinkie rightwards away from the other fingers including the fourth


Op 10 / 4 : measure 5

Play through the gs2 and as2 but then simply hold onto the as2. Play
the next four notes very fast (fx2 gs2 as2 b2 with fingers 4 3 2 1).
Bring in the right hand chord, printed on beat two, but not where it
is supposed to be (which would be in conjunction with the as2) but
delayed by a sixteenth so that it comes out together with the b2. Do
a similar “mis-coordination” between the hands with four note groups
starting gs2 (the third sixteenth of beat 2), the as2 (the third
sixteenth of beat 3), etc.,

Then run the measure as a whole, always withholding the right hand
chord so that it coincides with the highest pitched note of the four
ascending sixteenths. Then, as an afterthought, restore the right
hand chords to their original position, on the quarter note beats.

Fingers 4, 3, and 2 launch themselves towards the goal of the first
finger, as if the first three notes are trivial grace notes to the
thumb note.


Op 10 / 4 number 4 : m8, et. al. (right hand)

This is a place where I use a procedure which I call “extreme” speed.
It involves playing faster than the fastest I seem to be able to play.
It bypasses, literally hops over, the barrier I encounter when I
gradually speed a passage up, little by little, or by bigger chunks,
until I’m playing as fast as I can. It is at that point that I need
to say to myself: “now play it much faster!” The previous limit I
approached like a curve in math that works its ways towards a vertical
line but never gets there. I seem to get nearer and nearer to the
speed that I have set as a goal, goal but the increments by which I
get nearer to it shrink in size, so I never get to it.

The disadvantage of getting closer and closer method is that the same
muscles are always being called into action, they are just being asked
to flex more rapidly. What if these muscles cannot simply flex faster
and faster? What if muscles that one would never have thought of
using need to come into play? How do I discover what they are? How
do I learn to activate them? We can answer these questions simply by
attaining our limit and then determining by sheer dint of will, to a
good deal faster. I know of no other way to discover this new
arrangement among the muscles unless they are already functioning.

I gave a long distance lesson to a student at the Oberlin Music
Conservatory. It was Op 10 / 8 in F Major. She had to play it in
twenty minutes at her piano class, but she could not get it up to the
suggested quarter note equals 178. I asked her if she had a metronome
with her. She said yes. I said: set it to 220! She said, Joe, if I
can’t play it at 176 how am I supposed to be able to play it at 220?
I said: humor me; just try it. One page into the etude she stopped
and said: Oh, I see what you mean. I said: it is actually easier to
play it at that speed than at the slower 176, because you have
spontaneously engaged, without planning it ahead of time, different
muscles…in fact you had no choice but to do that.

In a few days I will publish the second part of my advice for solving
technical difficulties inn the Chopin Etudes. Please be patient.

* Which may as a result make a spasmodic motion in the general
direction of the torso.

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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.


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.


J.M. Mozart: C Minor Fantasie


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.


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.


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.


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’.


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

Liberating the expressivity in the bundled chords.


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).


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.


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


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).


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.


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How to Tackle Difficult Pieces, Practiced Simply

A.B.’s lesson on 4/3/19 on the first prelude from Book One of the Well Tempered Klavier

Balancing memory with freshness:

Be surprised and delighted with each new chord (which is to say each new measure).  This is to balance out the impregnation of the piece by memory, from having heard and/or played the piece many times.   Instead create a “beginner’s mind” for whom the new chord is fresh, unexpected, and bathed in morning light.  You just don’t know what’s coming.  Memory doesn’t go away but a proportional balance is attained between memory and the unforeseeableness of the future.

The persistence of a single chord through an entire measure:

In this piece it helps that you were formerly an organist, for as long as you hold the keys down on the organ manual the sounds continue unabated, persistently, and without the piano’s ‘decay’.  Hear in your “inner” ear of imagination the five different notes of each measure as a simultaneous ensemble, which continues unbated as a totality from the beginning of the measure to the end of the measure.

A.B. is not satisfied with his control over the evenness of the sounds in a measure:

Take a single measure out of the flow of the piece.  Reiterate the first note of the measure over and over until it “sounds like you want”.  Do this without thinking of the other notes and whether they will match the first note in sonically – in other words this is not yet about evenness between notes).  Then switch to the second note.  Play it ever and over, until, as before, it sounds how you want.  Repeat this procedure for each further note in the measure.  When you play the measure as written you will notice in retrospect that all the notes were even, although you were in no way trying to match them, but instead having each note have its ‘ideal’ sound.  A musician with a  good ear will always be able to tell when a sound has reached a certain ideal perfection, but not through analysis, through an intuitive sense of the sound.

For evenness when one note, occurring between two other notes, is not balanced sound-wise with the others:

In the measure that begins : f2 f3 a3 c4 e4, the c4 was not balanced with the a3 and e4.  I suggested that he hold down the a3 and e4, and while they are being held, repeat c4 over and over.

Another path to evenness: the written notes are part of a larger whole:

In measure one, for example, turn the measure’s notes into a rapid arpeggio that starts, with the highest pitch, e5, descends through the notes of the chord until reaching the bottom note (c4) and without pause re-ascends to the top note.  This creates a more cohesive and integrated motion in your hand.  Once you have this gestalt, you can remain silent during the first part of this arpeggio and start playing in the middle of it, at the note that is supposed sound first in the measure.  Eventually there is no need to pause or mark time for the first half of the arpeggio, it can occur in the inner feelings of the body in just a split second.

Yet another path to evenness:

When a baton twirler causes the baton to make a circle, it is the result of a sequence of different motions all blended together in a one overall fluid motion.  I’m ignorant of the breakdown of those motions, but you can still imagine, yourself as twirling a baton, one cycle every half measure (as the note pattern repeats).

I would sing a sustained line for A.B.:

Sometimes I would sing a sustained melody, one note per measure, starting at the beginning of each measure, made up of the top note of each measure.  Maybe I thought of doing this because I Gounod’s Ave Maria flitted through my mind.  That Gounod may have felt that the Bach begged for a continuous line (adumbrated by Bach made tangible  by Gounod).  The effect that my singing had unconsciously on A.B. was each note of the measure was instinctively made to balance, or fuse sonically, with the sustained note I was singing.

How to bring out the dramatological curve of a piece, even though it was originally played on an instrument of a constant degree of loudness: 

There are not many overtly dramatic moments in the piece that stand out from the monotonous (sic) patterns that repeat every half measure.

And even if we become aware at a certain time of these moments, they will afterwards fade into the background due to the abrasion or erosion of constant playing of the piece.  So  make the most of these moments.

Here is one example.  Chords outlining diminished chords, for instance, happen only a few times in the piece, but each time it does, try to react to the sound of the chord as being jarring, intense, dissonant.  This effect can be gained even without making any change in the loudness of those measures versus the surrounding measures. One can intimate a dramatic curve merely with intent and adumbration in the flow of the notes.

One of my other students, while playing through the Adagio from  Beethoven’s Op 13, came across of a few measures of diminished chords in the passage leading back to the second A section of its ABA form.  She said “diminished chords are ugly”. I said: that’s great, can you make them sound as ugly as possible!

Another example. When an interval of a minor second in the left hand, treat it as an astonishing, unexpected dissonance.

One more example, this time a longer passage:

In the second half of the page there is a long dominant pedal point in the left hand playing g2 (lowest line of bass clef).  As he went from one measure to the next I repeated: “long … long endeavor … never stops … we’re not ‘there’ yet”.

Matching two sounds that are separated in time:

When you play the first half of a measure and get to the highest note, consciously hold its sound in your ear’s memory, so that when you play the same note in the second half of the measure you can match it with the first.

Sometimes a “group” of notes is just one note:

In the last few measures of the prelude, I find that it is not useful to think of groups of four notes, or even two notes, the measures are too ambiguous compared to what has preceded it throughout the piece. My way around this is to play these last measures in “groups of ONE” note.  To promote this I say out loud as i am playing: “One”, “one”, “one” …. “.  Every note bears little allegiance to every other note except when though of in retrospect.

Remember that your pinkie is part of your hand, not a separate appendage:

Often your pinkie seems to be out in right field, detached from the rest of your hand as if it were a separate appendage.  Hold the pinkie in the unity of your whole hand.

Isolating Variables: the sequence of fingers as against the sequence of pitches:

This is in line with what we just said about the pinkie being “held” in the hand.  In measure three A.B. is using fingers 1, 3 then 5 to play g4 d5 and f5.

I asked him to cover the notes g4-a4-b4-c5-d5 with the five fingers of his right hand.  Play it as a cluster and hold it.  And while holding all five notes try to lift the thumb and replay the G, then again while still holding all the notes, raise the third finger and replay the d5, and similarly with the pinkie for f5.  Just focus on an awareness of the identity of which finger you are playing, as if to say “these are the fingers I’m going to use: 1 3 and 5”.  Then use the same fingers but for the written notes (g4 d5 f5).  You hopefully will feel an interesting transference of the awareness of which fingers to use, now mapped onto a different set of fingers.

Isolating Variables: The sensation of evenness as against any physical actions taken to instill evenness, especially when there is a new set of notes:

There is an ’emotional’, a generalized physical sense in the body as a whole, of ‘balance’ among the notes of the keyboard that are played together and in close succession.  As with any feeling, this emotional state can be reproduced at will under different circumstances.  Rather than the details of how to play the next measure evenly, try to reproduce the experience of having this feeling.

This distinction applies to many situations in playing.

For instance: there is the sensation we get of playing an ascending set of pitches.  This feeling can be conjured up even if we are playing a descending set of pitches.  Sometimes doing this is very useful in a Bach fugue to help homogenize two different voices, so that what a second voice is doing does not sound too dissimilar from what a first voice is doing.

Or, a sense of enlarging and getting louder can overlay a series of notes that are getting softer.

Or, a sense of wide space between the fingers in the hand can overlay a passage that involves a series of notes only one half step apart from each other.

Or, the sense of energy that we get from one very dynamic piece or passage from such a piece, and overlaying that feeling of energy onto all passages, slow or fast, loud or soft.

Making a clear connection between two non-adjacent fingers:

There is a measure in the first part where the pianist plays this sequence of notes: b3 c4 e4 g4 c5 … .

Notice that I tapped your fourth finger when you went from your third finger on g4 to the fifth finger on c5, It was meant to show the hand the focus of the ‘connection’ between the fingers playing g4 and c5, more at being located at the connection between the 3rd and 5th fingers.

At another point in the lesson I slid a pencil between his second and fifth finger.  The pencil passed over those two fingers but passed underneath the fingers in between them.  This helped him sense that those two fingers don’t act separately, but more at being the two ends of the plank of a see-saw, and thus the result of one single action.

More about see-saws:

Regardless of what two fingers play one after the other, and regardless of the distance between the notes they play, always an imaginary see-saw plank between the current note’s finger and the next note’s finger.  Add to this image an almost felt, pivot point, midway between the two fingers.  Now pretend you are a very strong person who  can make the two ends of the plank move reciprocally move up and down just by leaning first on one side and then the other side of where the pivot.

Once you are on the second note resulting from the first see-saw, move the see-saw’s location so that it connects this second note with the note that follows it.

To develop the sense of this see-saw, and the ability to relocate it quickly, it may help (using measure one as an example) to do this exercise:

Go back and forth between c4 and e4 (something which I notate as |: c4 e4 :|.  Once that see saw is functioning organically do the same for |: e4 g4 :|, and so on.

Addendum to the previous section:

It is your tendency, when you encounter a problem in a measure, to  just play ahead for quite a long time, and then tend to the problem later.  It is good to balance that tendency out with the ability to not move ahead, maybe only as far as the end of the current measure, and then focus in on tiny details.  Focusing entails a greater degree of  awareness of what is  happening physical and sound-wise, plus  reiterating that tiny detail until it sounds how you want it to sound.

Don’t rob the last note of each measure of its full duration:

A.B. usually tries to rush into the new hand position at the beginning of the next measure.  He feels that he may not have enough time to do it in, and compensates by holding the last note of the current measure a little shorter than the other notes of the measure.  I said “it is always good to try to hold longer whatever note sounds just before a leap, a skip, or a change of hand position.  One can deal with this near the end of the note by continuing to hold it when your hand tells you it is time to let go of it.  There is another way that is just as effective, that is more at being located time-wise at the beginning of the note rather than near the end.  Start the note with the “intention” of holding it longer.

We reached the goal of evenness:

Joe: in general today we have accomplished one of your goals: the sound is now even throughout.  During the attempt to make each note sound clear and close to its ideal sound, you were finding it easier to do this when playing all the notes a little louder than usual.  Often two variables get tied together, “entangled” as it were.  On the hand playing more evenly, on the other playing more loudly.  The latter helps achieve the former, only at some point, you want to separate the  former from depending on the latter.  Once you have effected this separation, the evenness and clear-speaking-ness of each sound, no longer depends on loudness and can occur at any dynamic you choose.

General comment #1:

 Notice that while you tend to try to solve things with specific actions of specific fingers, I almost never suggest a solution that involves the fingers, but relies instead on a more integrated motion of all the parts of the arm from shoulders to hands.

General comment #2:

I think you are evolving from one species of musician into another species: from an organist to a pianist.

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The “Spiral” of Fifths – An Alternative to the Circle of Fifths

The “Spiral” of Fifths – a “self similar fractal”:

If you start a new major scale on the fifth step of any current major scale, the new scale will have one more sharp than the previous scale.  Sometimes this new sharp is in the form of a flat from the previous scale that has been “sharped” into a natural in the new scale.*  Or a sharp from the previous scale that has been ‘sharped’ into a double sharp.  The new sharp always appears om the seventh step of the new scale.

If you start a new major scale on the fourth step of a current major scale, the new scale will have on more flat than the previous scale. Sometimes this new flat is in the form of a natural in the previous  scale which was been “flatted” into a flat in the new scale.  Or a flat from the previous scale has been ‘flatted’ into a double flat.

These two processes (adding one sharp at a time or one flat at a time) can go on forever.  If one keeps adding a sharp, the new scales will begin to contain double sharps, then triple sharps, etc..  If one keeps adding a flat, the new scales begin to have double flats, then triplet flats, etc..

The correct drawing of these two interlocked, and inverse procedures is not a circle of fifths (or fourths) but a spiral of fifths.  Just as interestingly, there is no beginning or ending to this spiral.  There is always another layer of the spiral outside the last layer currently displayed in any representation of this spiral, and so on indefinitely.  Similarly, there is always another layer of the spiral compressed inside the smallest layer of the spiral currently displayed, and so on.

Most of us learned about key signatures using the diagram of the circle of fifths.   Yet, a lot is left unanswered by this image.  It does not explain, for instance why, when you got about half way through the circle sharps somewhat arbitrarily turn into flats.   Why just then and not sooner or later in career of the circle?

Just as a C# means something different than a Db, and just as Cx means something different than D, theoretically there is nothing to prevent us from having a C-triplet-sharp which is different than a D# or Eb.  We may never see a triple sharp throughout our playing lives but it exists as sure as ultraviolet and infrared extend the boundaries of the visible portion of the electromagnetic.**

In the spiral of fifths, if we are located at C Major, then the next seven positions (or “o’clocks”) on the spiral will bear the addition of one more sharp, until at 8 o’clock double sharps start to appear in the scale.  And seven hours later than that “triple sharps” will start to appear, and so on every seven hours.

Similarly with flat.  If we go in the other direction from C Major, the next  seven keys bear the addition of one more flat, until at “4 o’clock’ double  flats begin to appear, and so on every seven o’clocks further inwards around the spiral.

Advantages to the spiral shaped diagram:

We no longer have to treat as arbitrary the point when we switch from flats to sharps, or vice versa.  We can also see the dynamic relationship of sharps and flats to each other, not as having two separate, essential identities, but both as being an expression of the same, single reciprocal principle of flatting and sharping.

If you draw a straight line from any key in the spiral inwards towards the center of the spiral, all the keys that lie along that line are all enharmonic equivalents of each other, moreover the letter of the alphabet in the name of the tonic progresses one letter at a time through the musical alphabet.

The letter “C” is always special:

C major has the distinction of being the only key consisting of only naturals and no sharps.   C sharp major has the distinction of consisting of only  sharps.   C double sharp major has the distinction of consisting of only double sharps.   C-triple sharp major consists only of triple sharps.   Etc..

Going in the other direction: C flat major has the distinction of being the only key consisting of only flats.   C double flat major has the distinction of being the only key consisting of only double flats.  Etc..

* For instance, the key of F major stands to the key of C major as having “one more flat” than the later, the note B-Natural in the C Major Scale being flatted into a B-Flat in the F Major Scale.  Going in the other direction, the “new” sharp in the C Major scale is the B-Natural on the seventh step, which had been, in the F Major scale, a B-Flat.

** A curious thing happens when we have ‘sharped’ C-natural twelve times, so that we now have  C-dodectuple-sharp.  We find that C is its own dodectuple sharp one perfect octave higher.  In the same way we can ‘flat’ C-natural twelve times, and find that C is the dodectuple-flat of itself.   And then the process could continue until we find ourselves beyond the range of frequencies covered by the piano keyboard, going past vintuple sharps or flats (or would it be vigintuple), then centuple, and so on indefinitely.  For there is no lowest or highest frequency that a sound can have in theory.  A frequency of a billion vibrations per second is just as possible theoretically as a frequency of one billionth of one vibration per second.   Whether they are audible is another question.  And at some point when we get down to the size of molecules of air, perhaps there is no higher frequencies physically possible (got to think that one through).

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