Tag: piano technique
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).
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
The Balance Between Hands
B.A.’s Lesson on 3/21/19
His piece: Mozart: Adagio In B Minor:
Sound and time:
Though you are playing the piece, there is no physical intent on the body’s part at any time. The piece just flows through time as if carried along by the inner pressure and necessity of time itself. No note that sound wants to ever stop sounding!* This is true of short and very short notes as well as long notes. Every note wants its day basking in the sunshine of listener awareness.
Balance of sound between the hands:
A.B. is concerned that his left hand isn’t dexterous (sic) enough to balance with what the right hand is doing. The only solution that he could think of was that he should practice the left hand alone until it is the way he wants it to be. But I felt that there is no way of knowing what the left hand should sound like until it is heard together with the right hand. The sounds of one hand color the contemporary sounds in the other hand. There is no way of observing how the left hand will sound in ensemble with the other hand, when it sounds alone.
The balance of sounds between the hands has its mechanical side. Imagine a point in space midway between the hands and on the keyboard. For the hands to sound balanced, everything having to do with one side of the body needs to be balanced with everything having to do with the other side of the body. The imaginary point midway is the balance point to regulate the two sides. Or you can think of it as the imaginary center of gravity of the two hands. Sometimes it helps to imagine that it is the point at the center of gravity, and not the separate actions of the hands, that is going up and down to produce the sounds, and when you do this the sounds will occur absolutely simultaneously and in balance. All this hands, without, or because of avoiding trying to do anything special to regulate one hand or the other.
Balance of sound within a single hand:
A.B. had to play an Alberti-like bass where the following notes are repeated in the left hand |: d3-fs3 a3 :|. I said you will never know how to balance the a3 with the other two notes until you have already heard the a3 sounding with the other two notes – before you first play the a3. This is “gestalt-ing” the chord (in this case d3-fs3-a3 or even a grander D major chord spread over many octaves). Though time fragments it, the whole is nonetheless always there; both in your hand and in your ear.
Control of the fingers comes from further up the arm (who controls whom):
There was one place where B.A, said, no matter how he tried, he couldn’t balance a certain two notes. They were a third apart, and were played together in the left hand. My solution was eclipse what the individual fingers were trying by putting the hand into a loose ball or fist. With the fingers thus neutralized in the presence of the entire hand, flex and un-flex the fingers, all ten at once. Now, at this point, without any other preparation or intent, play the third that is troubling you.
If the piano mechanism has a center in the torso and then has interconnected parts leading away from that center to a periphery at the fingertips, then the controller of each segment of that mechanism is the next segment closer to the center and further from the fingers. When things are not coming out how you want, seek further up the arm (forearm, then elbow, then upper arm, then shoulder…).
Fusing the arms together – putting them into another plane of action:
To demonstrate to him that control of one part of the mechanism often lies in another location, and in particular how this principle applied to the behavior and activity of the hands and fingers, I had him fold his arms in front of his chest (right hand to the left and left hand to the right). With the arms thus fused, and lying along a horizontal plane, take particular notice of the two elbows. Gently and weightlessly transport the elbows to the keyboard, with the help of the leaning over the piano. Now start moving the fused mass of the arms in a way that causes the elbows to push down random clusters if sounds on the piano. Then, without further thought, without planning anything that your fingers are going to do, play the current passage in the piece. The difference was striking. The piece moved in a stately and even flow, which manifested the very flow of time itself. Every note was subsumed in this inexorably moving flow that brought along with it every note – every note in its right place.
Fusing the arms together – so the hands act as one:
Another means to the same end, that of making the sounds cohere within the flow of time, is to have two hands move absolutely together as if fused, even if there is a separation in space between them. Have them play random notes that imitate the feel of the rhythmic coordination of the passage. “But what about rests in one hand”, he asked. There is no reason to stop the motion of the hands, though at one moment or another, one hand, though moving, does not produce a sound.
Where did your pinkie go?:
Sometimes your right pinkie, gets detached (figuratively speaking) from the rest of the hand and this causes it to play a note without good control over how it sounds. Try placing your pinkie silently on the note it is to play. Now see if, by using mostly the muscles in the pinkie, you can get your entire hand, and even your entire arm, to move around in space. This will help reestablish an equilibrium between the pinkie and the rest of the hand. And the entire hand will control how the pinkie makes it sounds.
The persistence of a chord:
Sometimes a chord (or even just a single note of a chord), that sounds at the beginning of a measure, wants to persist through the entire measure as if that measure was nothing more than a comment upon the existence and persistence of that chord.
* Unamuno, the Spanish writer and philosopher, in his book “The Tragic Sense of Life” refers to a passage in Spinoza in which the latter says something to this effect: every being, in that it is a being, strives to persist in its own being. And that this is the essence of that being (to persist as such through time).