Tag: music theory

More Beautiful Sounding Octaves: for the Medium-Size Hand

When I play octaves, there is a tendency, at least in my-sized hand, to have the pinkie and the thumb move towards each other when contact the keys.  But it is worth sometimes practicing in way so that the tip of the pinkie as well as that of the thumb should move in a line along the longitude of their key.  This requires my attention, because the hand is already spread for the octave, and the first and fifth fingers moving slightly towards each other happens naturally.  Especially for the thumb it is a more natural movement.  So, just once in a while, practice octaves so that those fingers move in a plane so that they go directly and horizontally towards the body in an extension of the longitude of their keys.

The muscles needed to move the thumb and pinkie in this direction move in these constrained directions require first, in the case of the right pinkie, an extreme flexion of the third knuckle, down and aimed to the right as it moves in the direction of the body, aided also somewhat by a flexion in the right side of the wrist.  In the case of the right thumb it should practice its motion by slowly tracing  over an imaginary straight line extending beyond the lip the key aimed towards the body.  The third knuckle, where it attaches to the wrist, is prominent in keeping the thumb congruent with this line.  As the motion is made the thumb is always compensating for the desire to move outwards and away from the second finger.

 

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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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Deciding What is Controllable. Also: Transforming the Polyphony of a Gugue

Well Tempered Klavier, Book I, C Major, Prelude

#1

After finishing the piece I simply asked A.B. what he liked and disliked about his rendition.  He mentioned several negative things and then struck on the one thing that I had primarily noticed: that he stopped the harmonic flow of the piece every time he went from the end of one measure to the beginning of the next.   I missed sensing that inexorable connection that energetically pushed me from the chord in one measure to the chord in the next measure.

He said that he had previously tried condensing the piece into a chorale of whole-note chords (each chord took the place of one measure).  However, he had trouble because he couldn’t do it with any speed.   I said that the speed was of the essence of the procedure.  Ideally, each eighth note’s worth of the piece, in terms of its duration, became the duration of one of the chords of the chorale.

Since it was difficult to shift chords that quickly, I recommended to him that he play just the chord a single measure followed by the chord of the next measure, and then stop.   I asked him to play the first chord as if it were a grace note going to the second chord – the latter being held longer.

This he could do.  We repeated the process for each measure going into its next measure.

Now that we had merged two amino acids into a somewhat longer chain of molecules, I asked him to play as written, while I, in the higher treble, waited until he was near the end , but not at the very end, of one measure only then played the chord of that measure as a grace note followed the next measure’s chord.  Only I would get to the second chord before he finished playing the current measure.  That anticipation of my chord connection gave him the necessary push and energy to keep the piece’s harmonic flow going across the bar line.  Then I would remain silent at the second piano until he reached nearly to the end of the new measure at which point I would break in with the chord of the new ‘current’ measure played as a grace note to the chord that governed the measure that was about to start.  And so on.

#2

A.B. remained worried in particular that the pinkie sixteenth note in the right hand at the end of each measure does not connect smoothly to the next sixteenth note in the left hand at the beginning of the next measure.

His default solution was to figure out  exactly how he wants his pinkie to play that note.  I solved his issue by stepping entirely around his approach.  As he played the piece I sang “la la la…”, but starting with 6th 7th and 8th sixteenth notes of one measure and ended with the fifth sixteenth note of the next measure. on notes to the first note.  I then waited too the 6th note of the new measure and began the process of again singing along for 3 + 5 or 8 notes.

In the form I was singing it, with the way I was grouping the notes that I sung versus those I did not sing, I purposely glossing over the connection between the two notes on either side of the bar lines..  It happens automatically.  By spreading a solder, or flux over the end of one measure and the beginning of the next, I effectively made less important the connection of the bar line.

I noticed in my singing that I helped things along by making misplaced crescendo starting on my first note and ending towards the eighth note.  This helped smooth over A.B.’s faulty connection over the bar line.

At this point we moved on to the fugue: transforming the polyphony of a

fugue in C Major, Book One, Well Tempered:

A.B.: why do I find it so difficult to not hold a voice note longer than it is supposed to last, when the note is already meant to continue sounding through a certain number of the next notes in some of the other voices.  For example if the target voice is a quarter note, or longer, and the other voices are enunciating sixteenth notes.

I gave a brief answer: remembering when to lift a sustained note in one voice is the requires the opposite of everything you do right when knowing when to start a note.  It’s the “dark side” of piano technique: it requires doing everything the wrong way; or is it now the right way?

A.B.: why did you do that?  Why was it working?

Joe: I think it is important to have a distinct pre-vision, pre-conception, of what the beginning of the next measure is going to sound like before you get to it.  It is a strange balance of knowing what’s coming and still being surprised by it.

#2

Can we transform the sound the sound of the fugue in the student’s ear?

We experimented using two pianos with re-registering one of the voice of the fugue.  He would choose one voice to play, and transpose either an octave higher or lower than  it was written, while I played the remaining three voices (without the fourth) at the other piano.

Results:  A.B. said:

My voice sounded different than before.  I head it saying and meaning other other things than I had before, but then realizing that it was the same voice with the same names to its notes, just transposed, and that there was at the same time an abiding identity between both versions of the voice, an identity which was preserved, was eternal and fixed  and was impervious to change of octave.   The new stuff that suddenly I heard, in how that voice combined with the other voices, must have been there latent to the note’s names themselves alone, or to say it in another way were just as present as aesthetic and sonic effects when I played that voice in its original octave.

In the future we will have A.B. play just one voice, but in the octave higher than written and the octave lower than written without playing it in the octave is written.  Later again we can transpose one voice by more than one octave.  If it is the soprano voice we can have it sound in the tenor voice’s range or even below the bass voice.  In the case of the bass voice, we can have it sound in the alto voice’s range or so that it is the highest sounding of all the voices.  At any time I can choose to play all four voices and not just three, leaving one voice to him.  Or, he can choose at random to play just the voices, while I play the other two.  Or, three voices.

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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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Microtonal Musings

Microtonal Musings:

1: “in tune” or “out of tune”:

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

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

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

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

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

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

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

2. A self-similar “fractal” chord:

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

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

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

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

3. Overtone series:

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

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

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

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

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

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

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

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

(pi) x (n) x (ff)

(e) x (n) x (ff)

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

1.1

2 to the 1/2 power

pi times e

sine of an angle

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

4. Graphing a mathematical function:

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

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

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

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

P.S.

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

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