Permutations and Embellishments
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Glossary/Tags: #embellishment # permutation # practice #goal tone
This exercise is intended to serve a number of the most important needs of a musician
Increasing the depth, breadth and duration of musical attention
Becoming aware of our mannerisms and exploring alternate material
Developing the ability to ‘distribute’ the attention
Integrating our different sensitivities and intelligences: thought, feeling and movement
Clarifying and working practically with the distinction between structure and ornament
Analysis of musical continuity
Reading music at sight as well as hearing internally
Enhancing musical memory
A suggested practice regimen is included
Working with permutations is a very old practice. It was recommended to enhance creative thought, mostly perhaps, for becoming aware of a larger field of options in creative work. Especially in improvisational work, decisions are made so quickly that there is often no time for choosing from among several options and consequently much improvisational behaviour tends to follow habituated musical decisions. Listeners may not realize why the music has no ‘spark’ or ‘life’ and the musician also senses that the music is ‘old’, traveling well-worn patterns. This website offers many possibilities for arousing new thinking and new hearing and this exercise of permutations is a prime example of a way to generate awareness of variations. Permutations are everywhere.
In the popular puzzle Rubik's cube invented in 1974 by Ernő Rubik, a permutation of the surface colors is created by each turn of of the puzzle faces. It is properly called a permutation because the colour of each surface remains the same; there are no substitutions, no new ones are added and none are taken away. Therefore it is only about changes in the ordering or sequencing of the colours, and so finding new configurations is a matter of permutation.
As a youngster, I worked with putting these 15 tiles in an intentional sequence: 1 - 15 or 15 - 1 or some other order. There is no number 16, because a space is needed to facilitate the movement of the other tiles.
Each horizontal row below shows the same three colours in a different order. There are only six permutations of three items. Working with only three positions will make a practical starting point for this work with permutations and embellishments.
It is useful to try this yourself. Without looking at the image below, try to visualize six different orders for the three colours. Or, without writing them down, imagine seating three guests at dinner in all the different configurations (let’s say: Stacy, Lakshmi, Jing). You can try this with three pieces of paper or three different coins. But perhaps the most interesting is to try singing the six permutations of a triad’s three tones (e.g., do-mi-sol, do-sol-mi, etc.).
The tendency to return to past solutions, past actions, is something we all do. It requires less expenditure. The permutation exercises may give certain people the means to begin finding previously unsuspected solutions. begin in a very easy manner and gradually introduce more nuanced and demanding material. And then you can see for yourself how to extend it even further.
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This exercise addresses both musicianship and improvisation.
Each four-note permutation will occupy four beats.
Each beat begins with the “goal” tone and is followed by a melodic gesture which carries the melody to the next goal tone. If you are able, you can certainly practice and perform in this continuous manner: each permutation with its “traveling” pitches takes only four beats. However, this leaves no time to prepare for the upcoming pitch and the ornaments you will use.
So it is strongly suggested that, at least at the beginning, you take 2 or 3 beats after each full permutation to catch your breath and prepare for the next phrase. As you work more intensively with this exercise, the following advice may become relevant:
To lighten the load of finding the four pitches you intent to play or sing there must be a clear line of thought.
1 First you isolate the lowest note and assign it the role of “prefix,” so you will find all the permutations that begin with that lowest note.
Permutations
One of the habits we each have as musicians (and pretty much everything else we do) is to rely on a few favourite turns of phrase. The expression “turns of phrase” is not every clearly defined but here is a simple example.
I would sometimes ask students how many different ways does a particular composer order the notes in a four-pitch grouping? Well composers tend to be very conscious of alternative choices but they are also interested in maintaining an economy of means, so that every composition doesn’t sound like an inventory of their stock and, more importantly, so that limitations in the use of variant elements contributes to a focused character for the piece.
With four pitches (or four elements) there are 24 possible ways to order the notes, and that results in 24 different melodic contours. But improvisers tend to stay with only a few favourite shapes. There’s nothing wrong with that but then often a player feels stylistically stuck and wonders how to enlarge their creative scope. So simply practicing the 24 shapes is one very simple exercise, and the increased familiarity gradually creeps into one’s playing even without trying.
It is cumbersome to use words to help others with their musical practice, so be patient. These ideas are transferable to different situations so you can try the out elsewhere.
Before you begin to sing, visualize the shape of the four tones. In the same way that we dial a friend’s number by automatically hitting the right numbers (when we could not actually say the numbers from memory), we can know the shape of the four tones. In a separate place I’ll show the two main ways that people tend to visualize a shape of four items but I’ll summarize it here.
Here are some good suggestions for practicing permutations.
Having a preferred value—a default value—to use as a reference.
As a preparation for the work with permutations, we begin with the idea that when working with two pitches, one will be higher or lower than the other. To make my own work simpler, I have a default—a kind of preference—where the lower one will usually be the starting point. So with the two elements—pitches, in this instance—my personal inclination is that I will first take them first as ascending; then as descending. It’s arbitrary but it’s a definite preference; it means that I don’t have to think about that again.
So the lowest unused pitch will always be used first. This helps no matter what method I use. (In other circumstances this idea of default preference might translate as darker to lighter, softer to louder, more bland to most spicy, etc. It can make it easier to recall things. I know that I put the larger bill denominations toward the inside of my wallet. Things like that.
By the way, making use of ordered elements also works when they are not quite in order. If
the two items are NOT in the preferred order, then that fact can itself make it easier to remember. If you try to remember this sequence [1 - g, 5 - z, 8 - f, 3, etc.] the randomness is makes it difficult for some. But an item displaced from a known or preferred order can be easy to recall: [1, A, 2, B, 3, C, 4, W, 5, …]. We tend to notice change: the last iteration of the fugue theme was two octaves higher than expected; the last ii-V-I turnaround in the trumpet solo used a substitute dominant: ii-bii-I. Items that confirm the pattern are easier to recall and items that deviate can also become easy to recall.
no matter the size of the intervals, you can visualize them as 1+3 or 2+2.
First the body-intelligence forms a clear image of the three notes that you want to ornament. Each successive three-note group is only a permutation—a reordering of those same three notes. They are used as the basis for the improvisational work to come. I showed you that it was possible to play those six shapes without the participation of thought—without relying on thought at all. You should practice this by occupying your thinking mind with something: counting by 2s or 3s or naming the days of the week or reciting the alphabet while you play the six shapes ... anything so that you see clearly that your sense of movement can, unassisted by thinking, know these six shapes: low-middle-high, low-high-middle, etc.
Remember that in this exercise you are not embellishing notes but intervals, that is, the travel from one note to the next. This “next” tone is often called an arrival tone or a goal tone and, without knowing that tone, you will not know the interval. Without knowing the interval you cannot make an expressive shape in your travel from one to the other.
So the first step is about body intelligence. We all have a high degree of this and use it to ride a bike, straighten when we trip, walk from one building to another while talking to a friend or listening to music, making coffee, remembering where we last used the cell phone, …
Next, we have to know if we are about to play (or sing) an interval of a third or a second: the big one or the little one. Making that kind of distinction is a kind of low-level action of thought (though the body can also do this) and the mind must also be aware if the interval is ascending or descending. So there is a small element or involvement of the moving intelligence (to make the six successive shapes) and an element of mental comparison to actually identify the interval.
Lastly we need to remember (or, later, to decide on the spot) which embellishing gesture we want to use. This is a matter of expression, and matters of expressiveness are essentially best guided by the intelligence of feeling. Each embellishing figure will begin to carry for you a kind of emotional tag of some kind. That is, it will have a palpable meaning for you. I don’t mean an meaning that can be verbalized but a feeling-meaning.
So, as with many of the exercises I give, they demand a fairly complete involvement of attention, of different kinds of attention. And this work then begins to feel like a collaboration of different intelligences and sensitivities. Remember that when you sing, you name only the three main tones (1, 2, 3 or do, re, mi or you can name the notes) and you do not name the embellishing tones in between. That is done so as not to confuse the main tones (the tones of arrival) with the tones of expression. If you do conflate the two levels, the result will usually be some kind of aimless melodic meandering (“noodling”) or, at best, melodies that don’t seem to go anywhere, that never reach any points of conclusion, that are not able to create artistic tensions and then release them. So have a good practice with all this.
This should be accompanied by some notated examples of three simple ornaments with intervals of a second (steps) and intervals of a third (two steps).
Permutations of Three Stepwise Tones: (to be realized with voice and also on piano)
Filename: 2a. Permutation & Ornamentation of 3 tones. rtf
The downward stems show the simple three-note permutations of G-A-B (do-re-mi). The upward stems show three simple embellishing figures for the intervals of seconds and of thirds. While they are written out in rather ‘square’ rhythms, these are among the most ubiquitous gestures in composed and improvised music everywhere.
This exercise has almost nothing to do with “memory” but is concerned with the ability to stay located in a continuous process—in this case, as with most of the exercises—a multi-tiered process which demands several different kinds of attention. So besides continuity of attention it also focuses on breadth and depth of musical attention.
The text-based notation below shows the same three figures for seconds and for thirds are are shown in standard notation above. The capitalized solfège syllables indicate the ‘goal’ tones. Those are tone of “arrival” and they are always on the beat. The syllables are to be spoken when sung. The syllables written in lower case are embellishments are to be sung melismatically, that is, by extending the vowel from the previous goal tone syllable. It is easy to get confused what is a main (goal) tone and what is ornamental (your improvisations, eventually), so the system of naming is important.
The examples below show only one of the 6 shapes you’ll be practicing: Do-re-mi for the first three examples and then Do-mi-re for the next three.
So for intervals of a second there are these three ornaments:
1. Anticipation
Do - - - Re - - - Mi - - - - - - - these goal tones become
Do - re - Re - mi - Mi - - - - - - - this is what you sing
Do - o - Re - e - Mi - - - - - - - this is how you name them
2. “Going away” (from the goal tone)
Do - - - Re - - - Mi - - - - - - - these goal tones become
Do - ti do Re - do re Mi - - - - - - - this is what you sing
Do - o o Re - e e Mi - - - - - - - this is how you name them
3. “Going past” (the goal tone)
Do - - - Re - - - Mi - - - - - - - these goal tones become
Do - re mi Re - mi fa Mi - - - - - - - this is what you sing
Do - o o Re - e e Mi - - - - - - - this is how you name them
For intervals of a third there are these three ornaments:
1. Passing Tone
Do - - - Mi - - - Re - - - - - - - these goal tones become
Do - re - Mi - - - Re - - - - - - - this is what you sing
Do - o - Mi - - - Re - - - - - - - this is how you name them
2. “Going away” (from the goal tone)
Do - - - Mi - - - Re - - - - - - - these goal tones become
Do ti do re Mi - - - Re - - - - - - - this is what you sing
Do o o o Mi - - - Re - - - - - - - this is how you name them
3. “Going past” (the goal tone)
Do - - - Mi - - - Re - - - - - - - these goal tones become
Do re mi fa Mi - - - Re - - - - - - - this is what you sing
Do o o o Mi - - - Re - - - - - - - this is how you name them
Remember that you must work with all six permutations, not just the two shown above.
Also remember that, for “going away from” and “going past” the goal tone, you must choose the direction of the ornament according to whether your next goal tone is higher or lower than where you are at the moment. If the next goal tone is higher, than “going past” will require you to go higher and beyond; if the goal tone is lower, than “going past” will require going lower and beyond.
You must be able to link each of the embellishments for seconds with any of those for the thirds. Therefore, there are nine possible combinations. Trying to memorize these would be useless; it is only a matter of understanding, imaging, and running a seamless process while realizing the pattern with your voice or your instrument.
So there are the six embellishment gestures;
A. Seconds: anticipation
B. Seconds: going away (from goal tone)
C. Seconds: going past (the goal tone)
a. Thirds: passing tone
b. Thirds: going away (from goal tone)
c. Thirds: going past (the goal tone)
and hence there are nine possible combinations of these six gestures:
Aa, Ab, Ac, Ba, Bb, Bc, Ca, Cb, Cc
You will have to sing and play any of all 9 combinations above.
(It is important to acquire real facility with this material because we will soon add a fourth tone, allowing us to work with 24 permutations of a tetrachord. I already posted material using four tones. When we finally work with four stepwise tones (called a tetrachord) it will give the required level of work for this topic.
The tempo of your practice is really important. The work with permutations continues as you work with four and then five tones.
Click on the following page to see the next three pages.
Once you’re familiar with the six permutations of three tones, you can easily add a fourth tone by taking it as a ‘prefix.’ That’s written on the third staff of each system below. But it’s just one way. Here are two other methods to discover twenty-four possible shapes with four tones.
There are a number of methods to find all 24 melodic shapes (contours) of four tones. The point is not to find the easiest or your favourite, but to work with different methods to keep your attention focused. I call the first one the 2+2 method because it divides the four tones into two diads. The third staff shows the 1+3 method, which uses a ‘prefix’ each one of which is followed by the si permutations of the remaining three tones. The middle staff shows a method that was played on a Nonesuch Explorer Series LP, which highlighted the music of ancient Greece.
The 24 permutations are notated below: one permutation in each measure. They are written using d-e-f-g—a minor tetrachord, but the embellishing figures remain the same. The upper staff of each system uses the following embellishments: going ‘past’ the goal tone on seconds; going away from the goal tone on the thirds; passing tones filling in the fourths. The lower staff of each system uses the embellishments: going ‘away’ on seconds, going past on thirds, passing tones on the fourths.
For something considerably more challenging try working with the following page which shows 120 permutations of five notes:
G, Bb, C, D, plus the Bb an octave higher. It is very interesting to develop a grammatical (or syntactical) sense of order to help yourself keep track of where you are. Note that both the lower Bb and the C are always followed by a note a P5th higher. And while each tone is an eighth-note in duration, the F and G which follow the low Bb and C are only a 16th-note. That give a syncopation to the whole measure and as the permutations unfold, that syncopation moves around the three quarter-notes. And this means that those two rhythms (8th + 16th) are also subject to the permutative process. (Take a look at the study page which follows. If you can follow the logic, it will be very helpful.) Difficult exercise, but quite interesting! And it will never grow old.
This final page (below) is a ‘study’ page to help you work with the 120 permutations. Notice that the C and low Bb are given a different note-head to remind you that they each comprise two notes. Also notice that there are 20 measures, each of which will itself have 6 permutations of the final three units. So 20 main permutations times six internal permutations equals 120 total permutations. I have rewritten the rhythms in 6/4 to make the reading a bit easier for some players. Good luck!
