Tag: music theory

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.


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.


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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The Importance of What is Not Heard

Brahms: Intermezzo: Op 116 No. 4 in E Major

Often in a well constructed piece, the meaning of something lies in how it stands out in contrast, or in relief, to something else.  Much of this has to do with memory, and what the listener may expect to hear at a certain time.

An example:

In the recapitulation of the Classical sonata movement, the second theme comes back in the in the tonic, not as we remember it, in the Exposition, in the dominant (or relative major).  What happens at that moment is that an expectation is momentarily revived and  enhanced by the composer but a new present reality is superimposed upon it. For a moment the two tenses interact*, but a moment or two later our ear has taken up its bearings in the new.

The ears of a sensitive listener will even prick up before the second theme, at the exact moment when the composer deviates from the harmonic path that led to the second theme in the exposition.

One of the things that makes late Brahms difficult to hear lucidly is that when something stands in relief with something else, we often haven’t had an opportunity to hear that something else earlier in the piece.  So how does the pianist make a contrast with something that is not ever heard, but whose meaning lies entirely in its contrast to this unheard base or reference?

An example from the Brahms Intermezzo:

Consider the passage in measures 10 through 14.  Contrapuntally, what is going on has less to do with the triplets in the right hand but in implied, but not literally heard, duplets, which are formed from the second and third triplet notes, if the first triplet note is put back onto the beat, omitting the first triplet note entirely, and playing the third triplet note as the second note of a duplet.  If we do this, we suddenly hear a very conspicuous appoggiatura.  In measure 12 for example the e5 is clearly heard as an appoggiatura to the d5.**   As we shall see, this perception need not become vitiated by the delay of the restoration of the appoggiatura to its original position in the measure (one triplet eighth later than the sounding of the chord in which it functions as an appoggiatura).

The same relation of appoggiatura applies to the c5 to b4 and the a4 to g4.   When performed successfully, this passage haunts the listener with the sustained feeling that something else is going on other than what is most obvious to the ear (delayed triplets).  There lurks this implication of regularly arriving appoggiaturas on the beats.  Similar appoggiaturas occur throughout the passage.

Brahms doesn’t stop there.  Once he establishes to the ear that this comparison to the implied simplified counterpoint,  he is able to take a further step to hide the actual appoggiaturas by attracting the ear, in measures 11, 12 and 13, to a descending scale in the top voice.  But let’s pause for a second.  Do we hear a scale?  Almost.  At least we get the feeling that there is a scale present.  For here too, there is a layer of removal from what is heard to what one might call what is meant-to-be-heard.  We hear a melody stopping and stopping in two note groups, which if there were no interruptions would be a coherent, fluid scale: b5 a5 g5 fs5 | e5 d5 c5 b4 a4 g4 | etc.  The beauty of a melody arising from following this scale depends on the implication that our consciousness is able to pass lightly over the first the first of each group three triplet notes (a note that is merely part of an  accompanying chord) so that the notes of the scale seem to flow connectedly one into the other.

I have my students leave out the first triplet note, and change the next two notes to regular eighth notes, putting the first of the eighth notes back onto the beat.  The scale is now much clear to the pianist’s ears.  Crucially, if that point, the student goes back to playing the written notes, the reference to the fluid duplet scale is not lost.  It attempts to maintain itself in spite of the pauses.  It haunts the image of the passage and changes a somewhat trivial passage in triplets to something more transcendent sounding.

Thus a passage can transcend itself.  It becomes beautiful only in relief to something more basic, not literally heard, to which it yet can refer itself.   Generally, in late Brahms, we often must try to make a passage sound like what it isn’t! (something clearer in harmony, clearer in rhythm, and clearer in voice leading and counterpoint).

* This momentary contrast, if it were prolonged would lead to a confusion in the sounds, like when a person accidentally takes a double exposure with a camera.  If, however, the process could be frozen in time, and experienced just in space, we would have the equivalent of a biologist looking through a microscope that allows on eye to view one slide and the other eye view another slide, as for the purpose of noting what contrasts there are between them.  A side by side comparison.  In music it is more sublime.  It is a a sound image from time past that melds with a sound image from time-present.  The past isn’t gone it lives in memory, for many in the form of a sound-memory.  The past sounds do not really sound in the glare of the light of present, but colors it.  But a comparison is made.


e5  d5



Clearly there is a D Mjor chord trying to fully form and as an e5 yields to the partially formed chord and resolves to the chord note d5.


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Technical Challenges in Moszkoski’s Etude in F and Beethoven’s Sonata in E Major, Op 14.

A.J’s lesson today.  Two works he is preparing for a competition.

-From: 15 Etudes de Virtuosité, Op.72 No. 6 (by Moritz Moszkowski), “Presto” in F major/.

#1. The “ear” as the abstract creator of the figurative shapes of  sounds:

#2. The balance of sound between the two hands.

#3. A ladder falls apart if there are no rungs connecting the sides.

#4. The undulating patterns of three-note groups in the right hand:

-Beethoven: Sonata in  E Major: Op 14 / 1 :  I : The left hand sixteenths in the development section.

#5. “Additive” Clusters as a unit of pulsation through the passage  with                    sixteenth notes.

#6. The desired effect in sound does not always follow upon a logical  or                   teleologically designed set of causes.

-From: 15 Etudes de Virtuosité, Op.72 No. 6 (by Moritz Moszkowski), “Presto” in F major/.

#1. The ear as the abstract creator of the figurative shapes of  sounds:

The ear, as the observer of the flow of sound content through time, may seem at first to be but a passive instrument.  It listens, it notices, and only with a slight lag as the sound has already been physically produced.

If not distracted by the physical actions we make to start the sounds, there is an exact in knowing in the accuracy of the notes.  The awareness of sounds in the ear and the more it is divorced from any muscle movements that physically give rise to the sounds–the more accurately and subtly it  judges the sound characteristics of the music being played.

Through a miraculous confusion of tenses, the ear as a passive listener, after the fact that the sound has begun, yet can be the most effective force in controlling our sounds.  In this regard it is far more efficacious than consciously controlling and gauging the quality of quantity of our muscle movements.  This present tense in consciousness is not a mathematical instant, a point of zero duration.  It contains, according to the French philosopher Henri Bergson, the spilling over of the past into the present and the impetus of the present to be on the verge of becoming the future.  A duration, though recognized as the present, has yet the efficacy of having what has just happened to have an undefinable but definite effect on what is just about to happen.

#2. The balance of sound between the two hands.

A.J. is having difficulty coordinating the two hands in the Moszkowski Etude in F.  His left hand seems to be pursuing its own course–not blending with the right hand, but merely showing up at the same time as the right hand – at the beginning of every third note in the right hand.

I suggested that as he played one of the left hand chords, hold it for a few moments along with the first of the three triplet notes in the right hand.  He should see if his ear could spread its attention over the sounds from the right hand as well.  And then determine quickly whether together, the sounds of both hands formed a cohesive whole.

With just a little attentiveness, just a split second after the notes start sounding, one will notice whether the sounds from both hands seem to reach out towards each other eclipsing the distinction between them and creating a larger sound-whole than either hand’s sounds alone.  And this fusion takes place as he listens to the sounds.  It takes but an instant for this synthesis to occur.  At the first instant there are disjointed sounds from two sources, but a mere instant later these sounds have instinctively reached out towards each other to form a synthesis in consciousness.  Or, to put it another way, it takes just a bare moment for the ear to note and to form a larger whole out of the sounds of both hands.

To his surprise, A. had no difficulty in causing these sounds to fuse together though physically they were made by separate physical acts pertaining to a coordination of the individual sides of his body.  He was surprised since in as much as he wasn’t aware that had done nothing physical to effect this balance, but merely remained attentive to the sounds for more than a split second.  This synthesis had nothing to do with any physical effort to make the sounds be simultaneous.  Nor was there any specific mental ‘effort’ involved.

The combined power of his ear and his brain focusing on the notes, brought the sound together. His previous preparation and physical muscle memory came through in a moment where his head may have easily gotten in the way–thus is the power of the ear.

#3. A ladder falls apart if there are no rungs connecting the sides.

As an analogy for what had just happened, I suggested that the left and right hands were like the two vertical sides of a ladder.  They can remain upright only if there are rungs crisscrossing between them.  These sides had nothing to keep them together without the ear forming the rungs.  Without the attentiveness of the ear those side pieces would fall apart.

Once such a sound-synthesis has been effected at any point in the piece, the possibility then exists to ‘mold’ and shape the forms of these connections.  It now became possible to mold how one of these composite sounds morphed into the next one.  Though intangible in nature, the pianist now has a focal point to help steer all the course of all the individual notes of the composition through the medium of time.

Though to the body, the sounds originating from the right and left hands seemed to exist spatially apart and separate, from the point of view of the attentive ear they were (already) fused together.  It is more the ear, something intangible, than the body, something tangible, that ’causes’, these sounds to meld and form a resonant four-note chord.  We need only seek whether they do.

It is only after the fact of their fusion into a single sound that we can, for analytic purposes, speak of these fused sounds as having two spatially distinct origins.

#4. The undulating patterns of three-note groups in the right hand:

Next we turned A’s attention to just the right hand’s stream of notes, a rapid stream of triplets.  I suggested that each and every group of three such notes comes to life in a molten state, which the pianist can then form into a well-rounded shape.  Despite their melodic and harmonic differences, all such three-note groups should cast into the same shape.  This creates a form ‘texture’ that holds the entire piece together.

The most recalcitrant triplets, the ones that would most resist such shaping, occur when the right hand is playing a chromatic scale.  No group of notes yields up so little harmonic value to a repetitive pulsation, The chromatic scale is most innately without a shape.  If started on a C Natural, and if accompanied in the other hand by a C Major chord, the scale tries to break down into uneven units of, first four note (C C# D D#), then three notes (E F F#), then five notes (G G# A A# B).  This is too much of a strain on the scale which therefore yields up little by way of harmonic implication.  It is the changing size of the harmonically influenced note groups that render the scale inchoate rather than redolent with harmony.  In this etude, the smithy of the mind resists this falling apart of the chromatic scale and obstinately takes every three note group, regardless of its harmonic implication, and shapes those notes into a three note melody without reference to harmony.  If I had to express this using a spatial analogy, each three notes would be, in its unformed state a straight line, which the agency of the ear then coerces into the shape of a letter ‘U’.

Beethoven: Sonata in E Major, Op 14 / 1 :  I : the development section:

#5. “Additive” Clusters as a unit of pulsation through a lengthy passage with sixteenth notes:

The chord with which the left hand commences, c3-e3-a3-c4, is a first inversion A Minor chord.  For many hands this is an uncomfortable arrangement that promotes flitting moments of tension.  There are two ways out of this dilemma.  One is for the hand to change its overall shape as each finger takes its turn enunciating its note, removing if necessary the other fingers from the previous notes they played.   The other way does away with all the physical difficulties by having the ear take on a constructive role, building up, one note at a time, the eventual cumulative sound of the 4-note chord (c3-e3-a3-c4).  When doing this, each single note, in its turn, prepares the eventual and cumulative sound of the four notes occurring at the same time.  It is only through the first iteration of this four-note sequence of tones that the full chord does not sound until the fourth note.  But after that, and with the pedal down, the simultaneous sound of the four notes is continuous.  The evenness in the balance of this four-note is not the result of mechanical manipulation but the result of the expectation of an ear focused on the simultaneous sound of the four notes.

#6, The desired effect in sound does not always follow upon a logical or teleologically designed set of causes:

A.J. didn’t see how such a passive, ear-based, technique could possibly effect the evenness and balance of the four sixteenth notes.  This prompted the following conclusion from him, his most significant realization of the lesson though at the same time not a logical one.

“Mechanically what I did makes sense as a way of achieving the sound effect that I want in sound.  And yet … the result is the sound which I desire.  This apparent disconnect between cause and effect is a normal sign of a sudden breakthrough technically.  The physical means of doing something, when considered in and of itself, may or may not seem to be capable of logically producing the sound effect that the ear is after.

Yet that effect is what is achieved.  So it makes no sense.  It takes bravery to abandon physical/logical sense of consistency between cause and effect and be accepting of what in “Big History” is called an “emergent form”; or a form that is not contained in the some of its parts.

You find that the way to the newly emerging form is not foreseen in its physical and mechanical causes.  The means happens to produce the ends, but cannot predict the effect.


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