Leverage and Sound
Chopin, Etude in C# Minor from Opus 25:
Irving’s brother came today. We wanted to get a rich cello-like / vocal-like tone out of the piano for the notes of the opening “baritone” melody for the left hand. It is in single notes without accompaniment, so it is very exposed. We need our entire sound/mechanical tool-kit to keep it resonant and sustained so there isn’t a moment’s break in the flow of the line. Their softness shouldn’t belie their resonance.
Our first exploration was with leverage, the principle being that the greater the leverage you have over the production of each sound, the more that sound approaches the ideal piano-resonance.
The effectiveness of a lever is a function of how long the lever is and where you place the fulcrum on which to rest it*. Leverage increases with the length of the lever and how remote the fulcrum is from the end of the lever that, from which in this case, the pianist initiates the motion of the lever. If, for example, the lever is solely the length of a finger, and the third knuckle is where the fulcrum is, there is little mechanical advantage to depressing the key through the motion of that lever. If the lever extends back into the wrist, and includes the finger, there is greater leverage on behalf of the movement of finger tip. So the question is, how we can create the greatest leverage with the human body.
We ended up using a curious combination of several different levers, that ended up being connected one to the other.
The length of the arm, from shoulder to finger tips, while perhaps not the longest lever we can make of the body, is a conveniently long one that is still easily manipulated.
We started by his holding out both his forearms; straight out in front of him so that they parallel with each other and were horizontal to the ground. We Left a comfortable distance between the two hands, about the same as the distance between the two shoulders.
We then had him move his arms up and down using just the shoulders as pivots. At their highest points the arms were aiming well above the horizontal, at an angle of about forty five degrees. At their lowest points the arms were just slightly below the horizontal.
Very soon, we changed it to an oscillating motion between the arms. One arm was at its lowest when the other was at its highest. And they exchanged these positions. We did this until he felt a sort of physical exhilaration from all that motion.
The next thing we did was to create a second, more imaginary, lever. At the same time the arms were moving, we pretended there was the plank of a see-saw that connects the two hands (traversing the empty space between the hands), which, as a result of the arm motions, was itself going up and down as if two people were seated at each end of the see-saw. The pivot of this imaginary see-saw was exactly half way between the hands, so that neither hand or arm had a mechanical advantage over the other – the advantages were equal.
I also had him imagine a secondary but similar see-saw between his two shoulders, as if an, albeit, small person was seated on each shoulder. We continued exercising the combination of these levers until he felt a definite exhilaration from making these motions.
We then ‘elected’ his two index fingers as the sole ‘beneficiaries’ of all the motions he was making, so that the each index finger was backed up by the entire arm and contributing see-saws.
While continuing the oscillation of the arms he used alternating index fingers to play first the opening note of the second note. The solo was no longer distributed solely to the left hand but alternately, from note to note, between one arm lever and the other. If he played the first note with his left index finger, then he played second note with his right index finger. Then back to the left index finger to sound the third note, the right again for the fourth note, and so on through the line.
During this procedure the fingers were to never loose their connection to the hand, and on to the wrist, the forearms, the elbows, all the way to the shoulders.
Sometimes the arms had to cross one another, but the more important thing was the swinging motion from one arm to the other regardless of which one was to the right or left of the other.
When he did this with physical abandon fervor, without thinking so much of the ‘proper’ or ‘usual’ way of pushing the notes down, the result, to our joint delight, was an unusually rich sound, one that he was unaccustomed to getting on single notes.
Even when consecutive notes were ‘next door’ to each, only a half step or whole step away, we did not diminish the feeling of the widest possible see-saw between the arms. In other words, while the objective distance between the consecutive notes might lessen, the subjective sense of how long that distance was always remained large.
The last step was to preserve the widest and most dynamic sense of an oscillating motion when going not just from one hand to the other, but from one finger of one hand to another finger of the same hand.
* The saying, concerning how levers work, as attributed to Archimedes, is: Give me the place to stand, and I shall move the earth.
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
Playing With Authority, Intervals, and the Inner Heart of Music
Playing with Authority:
C.P told me at our last lesson: I am very soft spoken in my private life, and in my business life. I am habitually quiet, but you have given me permission to speak out more, even though it is at the piano. I can make more sound and command more attention. Maybe it’s safer to do it on the piano first, but nonetheless it an exciting change.
What I had been doing for the last few months with C.P. was to ask her to speak out her notes with more pride and more certainty. She shouldn’t play it safe, be unassuming and be on guard for mistakes. This was in her Bach prelude. On the other hand, in her “Claire de Lune”, I said: here it less a matter of loudness or authority, and more about richness of tone, finding a deep and sensuous source for all your sounds; but that at heart it is the same thing as expressing yourself more fully.
Later in the lesson we were working on a new Bach Prelude (WTC I c minor). I pointed out to her the intervals that were formed between the two voices, particularly after the first sixteenth note of the measure and the first sixteenth note of beat three of the measure. At first she asked a type a question that I had come to expect from her inquiring mind. “What is the use of knowing intervals”?
First she gained facility in naming the intervals. This led to her noticing how the sixth and the third (sometimes as tenths) were the most frequently used intervals between the hands. I asked her if those two intervals had anything in common. This led to the idea of inverting an interval and that thirds and sixths invert to each other. This led to ask about seconds and sevenths, which meant we could discuss the role consonance and dissonance in a tonal piece of music.
Perspectives leading to the inner heart of the music:
Then I put things in a broader perspective. There are two ways of knowing something: from outside and from the inside. From the inside is the goal. Often we cannot go directly into the inside of something unless we first take a series of perspectives on from the outside. Intervals is one such perspective on the inner heart of music. So are chords, rhythms, structural features, thematic development, listening awareness, and the list proliferates.
I had a friend in High School, Stephen*, who sometimes took walks with me in Prospect Park in Brooklyn. Once we were discussing the first of Emerson’s two essays on “Nature”, and how it is divided into sections, each on viewing nature from a series a different perspective. He said this was like the bible story of Joshua. Joshua’s goal was to get to the inside of Jericho. So for seven days they walked around it getting, as it were, every possible perspective on it. And on the last day the “walls came tumbling down”, or in other words, they now stood on on the inside of the city, just as the musician’s goal is to live in the inner heart of the music.
*An interesting thing about Stephen. He was born with only short stubs in the places where the fingers emerge from the hand. When you are a teenager everything seems possible. So one day Steven asked if I could teach him to play piano. Without hesitation I said yes. We chose the first prelude from the first book of the WTC. By the rotation of his forearm, and thinking of his hand as a wheel, and thinking of the stubs of the fingers as teeth of a gear wheel, we found a way not only to make sounds on the pianos with the his virtual fingers, but gradually gained a sophistication in the control of the rotation, together with the possibility that at any moment the arm could lift the wheel of the hand off the piano so that when the wheel came back down on the piano the virtual finger ajacent to the one that just sounded a note, could land on any key regardless of its distance on the keyboard from the previous note. Steve went on to Cornell, and I wish somehow I could be in contact with him again.
I’m back! Revving up your engine. Change of register within a theme. State “A” and state “B”
I haven’t blogged for a while. It’s been a rough month health wise and mood-wise. But here I am again. I’ve nothing too organized to write about today, so please indulge me is I am desultory in this blog entry.
#1 Revving up your engine for a moment before playing a difficult passage.
When confronted with a rapid passage that that moves in a series of notes of equal duration, let us say eighth notes in the right hand, covering several bars of 4/4 time, it is useful to rev up your engine (like a race car driver awaiting the flag to drop to start the race) and then overflow those four notes as the race begins and you cruise through the passage. This ‘revving’ up can consist of playing the first four notes over and over again in a loop, until the thrust of your “jet engine” has increased to the amount when you can then let off the breaks and sail down the runway.
#2 Change of register within a theme statement
When a melody is transferred by the composer from one octave range to another, it is important that the pianist “carries” the sound of the note from one octave to the other. Sorry to mix metaphors, but the listener has to be “led by the hand” from one range to the other, so that the new destination note sounds as alike as possible to the starting note, but for the accident of pitch range. Usually changing the octave of a note causes a major change in the quality of a note. But in this case we want to stress the sameness of the note despite its appearance in different ranges on the piano.
We want the listener to feel that it is the same sound that has taken off one outfit and put on a second, while still being able to recognize the person wearing the clothing.
#3 State “A” / State “B”
Solving technical hurdles, simplifying a passage, If you are not already familiar with the idea of state A and B, see:
do a ‘search’ on the front page for “practice technique”
With my students, I often use the terms “state A” and “state B” when referring, in the first case, to some altered way of playing or approaching a difficult passage that sheds new insight on its meaning or which unlocks the technical difficulties involved in the passage. State B, which follows upon state A, is playing the passage again, but this time as written in the score (in its performance form). The idea is that the insight gained in state A carries over into state B.
The important question is what to do after doing state A followed by state B. Many students will do state A, realize the benefit of doing it as they then play state B, but if they play the passage a second or third time, simply in its state B form, the benefits from having done state A gradually wear off and the passage begins to resume the state it had been in originally. When the student has completed the cycle state A – state B, she should resist the temptation to try the passage again in state B, almost as if to test whether the benefits previously gained are still showing in state B, or perhaps to try to improve the passage even more. Unfortunately the benefit from state A though it normally carries over automatically into the first iteration of state B, by simply following state A closely in time, becomes lost and diluted if you simply replay the passage in state B, over and over, without going back in between to state A again.
Always go back to state A, before doing another try at state B, for state A stands to state B as a going back to the well, the fountain, the source of the inspiration and insight that enlightens the passage.
Thus concludes a series of scattered thoughts. Let me know what you think or have questions about in the comments, and also tell me if you would like me to write about something specific next time. Some health stuff has burdened me, so the posts might be a little scattered. But stay tuned, I’m here.
Ear Training, an Introduction: the What, the Why, the How
#1 Why do ear training?
Ask a student or performer if they listen as they play, and the answer which they give, without much pause to think, is generally “yes”. Yet the ability to hear clearly while playing, and to understand what one is hearing, is the principal things that sets a good player apart from others. The good player does not only have a good technique, but they have as strong an ability to listen completely and objectively to the sounds they are making. In the hands of a master, technical matters are brought under the control of the ear.
It is a surprise to most musicians if you tell them that they are not really listening attentively when they play. That too much of what they consider listening is actually physical sensations generated in the muscles causing notes to sound. At the moment of an attack of a new note, there is often more tactile and kinaesthetic feeling going on than listening.
When the physical action stops but the note continues to sound, it is easier to focus purely on the sound. Ideally there is a way to how to isolate sound from any muscular feelings or other sensations than that of hearing.
There is a way for the ear alone, whether that of the pianist or a listener, to learn to identify and distinguish among the many relations into which notes can combine.
Each such relationship produces for us a quality, and it this quality that forms the basis of ear training.
Being a good musician means having a mastery over the medium in which music exists, I.E. sound. When possessed of such mastery, one can mold the medium of sound to one’s will.
No prior experience is needed to begin to develop the sensitivity of the “ear”.
#2 Sound is a quality.
The experience of sound is a quality and not a measurable quantity.
How notes combine into a single conscious experience is not a dividable into half steps or ticks of a clock. Being a quality, there is no way of describing the quality of sound using words. We must experience it. If we try to ‘describe’ it to someone else, it is useless unless the other person has also experienced it directly.
The sounds we hear may result from combining notes in some measurable way, but we do not “hear” these measurements. The quality of a chord, for example, is like a perfume. It impresses us directly and unmistakably. We do not need anything extra, such as the chemical makeup of the perfume, to fill ourselves with its aroma.
As I walk I may identity a certain scent in the air as that of a “rose”. But unless one has already experienced this aroma and then also learned to associate it with the same word that I use, it is of no use to say the word “rose” to another and expect that they will know what scent we are talking about.
Associating an aroma with a word does not alter the aroma in any way. We can study and examine the rose, but all we gain is knowledge (facts, quantitative measurements, etc.). But all the while the fragrance persists calmly in our consciousness apart from anything visual, descriptive or analytical.
It is easy to stray from just the quality. We are apt to substitute for it a symbol in the form of a name or an image.
A ‘rose expert’ can tell us while blindfolded what the name is of the specific type of rose they are smelling. And though an ‘ear training expert’ would be able to give separate names to different patterns of sounds, it is more important that we simply have the ability to recognize when sound qualities are the same, or just similar, or vary more considerably. Thus, while we could say: ah, that’s a perfect fifth sounding, or that’s a major chord in the first inversion, or those melody notes all belong to this or that scale, the important thing is that when you hear a perfect fifth and then a perfect forth, you can “smell” the difference. If we never heard of a ‘half step’ we would still be aware of the difference in quality.
Ear Training is most successful when you work with qualities; when you use your innate, rapid, intuitive faculty of directly perceiving even the most subtle differences in quality between one combination of notes and another. At first maybe we may only notice the most obvious differences, as between a chord and a melody. In the world of odor, this would be like only being able to tell difference in quality between the smell of a lilac and that of a rose. Later though we will be able to notice the difference in quality between various types of chords (various types of roses), and still later the subtlest differences between chords that arise from the intervals between the notes in the chord, their inversion, the number of notes they contain. Our ability to distinguish between similar qualities in sound gets finer and finer.
If we hear a fast melody, we can tell from its overall quality through time just how many notes were in it (without counting as we hear them).
Eventually we become like the rose expert and can detect slight variations in quality between two roses on neighboring bushes. We will be able to tell the difference between two chords that have the same root note, same ‘quality’ (major, minor…), and the same number of notes, whose only difference lies in the arrangement on the staves of where the root notes are, the thirds, and the fifths. We will be able to single just one note from the chord with our ears and say whether it is a root note, third, or fifth.
We can be just as expert with intervals, melodies, and any other abstract relation between pitches (what I call “Sonic Geometry”). We just want to avoid the temptation of applying some sort of musical ‘ruler’ to the sounds, by which we can measure the distance between two notes by a sense of their distance on a staff, along a piano keyboard, or along a violin string.
#3. Resolving ‘complex’ ear training abilities into an amalgam of simpler abilities.
In looking for a starting point for ear training, we might be tempted to start with something like : what is this chord that I just heard? However, this is already a fairly complex ability. It entails separate skills: is it a chord I’m hearing; how many notes are sounding; is it in root position or in an inversion; if I wanted to can I single note with my ear each individual note; can I tell what the intervals are between these notes; which of these notes are root notes, thirds, fifths, etc..
To come to realize that the original question involves an amalgam of simpler abilities, we can learn to ‘refract’ through a ‘musical prism’ the original ability to see if it resolves into simpler component. Nor should we be surprised if these simpler abilities, in turn, if each is put through another prism, do not resolve into even simpler abilities. All ear training questions ultimately boil down to: 1) which is higher in pitch of two notes? 2) which is longer in duration of two notes? 3) how many sounds just sounded? Then we can work our way backwards to our original question: what chord did I just hear?
#4. Ear training is fun to do when there are two people together.
If you do not have access to a computer program*, or to a class being offered locally, Ear Training can be easily practiced with the help of just one friend and a room with two pianos (one will do also but it is a bit more cumbersome logistically).
The two people go back and forth presenting “questions” or answering questions. The questions are always some combination of sounds. The answer is either given in words or by reproducing the sounds on the other instrument.
Some examples on the simpler side:
Play two notes in a row: ask which one was either higher in pitch or longer in duration. You can do something similar with three or more notes in a row (which was highest in pitch; which was longest in duration).
Play two or more notes at the same time: ask how many notes were sounding. Play two or more notes one after the row: ask how many notes sounded.
Play a series of notes, one at a time, from an agreed upon range. Have your partner try to match each one. This range can after a while be expanded when agreed upon. Later, let it be two simultaneous notes from an agreed upon range.
Agree that all the notes will be, for example, C-naturals, then play C-s in different ranges of the piano and have your partner match it in the correct octave.
Some examples of something with moderate difficulty:
Play examples of intervals (harmonic or melodic) but limited to only two possible answers (major third / minor third; perfect fourth / perfect fifth …). Your partner provides the name of each that you play. Later, there can be three possible correct answers (and eventually more).
Did the two chords just heard contain the same notes, or was one or more different (one being much harder than several).
The same principle of starting with two correct choices, then adding a third, fourth, etc.. can be applied to most ear training situations: distinguishing among types or aspects of chords, three-chord harmonic progressions, types of rhythms, etc.).
As things advance, and the recipes become very gradually more complex.
Here are some examples of things of harder difficulty:
Which steps of a common scale did you just hear and in what order? Or match the same notes (given the first note).
Was the chord in root position, first inversion or second inversion …
Listen to two chords: by how many half steps (and whether up or down) did the root notes move.
How many of the notes in one chord were also in the next chord.
Here is the scale of a particular key (play it one octave up and then down). Then play a series of chords. Ask on which scale step each is built. Complicating factors can be whether the chord is in root position or inversion; whether non-diatonic chords are allowed; whether altered steps of the scale can be used for root notes.**
* You are welcome to request a copy of the “Joe Bloom Ear Training Program” which runs on PC-s but unfortunately not on Macs.
** For a more complete list of ear training activities, just send me a request.