The Granularity of Palindromes
Introduction
Palindromes are a fascinating tool for composers, writers, and linguists alike. I find myself utilizing them in many of my own works as a composer, in my notated music. In electronic music, I often use a reverse sampling of sounds within my compositions. Using both of these compositional tools side-by-side an idea came to be: What would it take for me to compose a reverse function using acoustic instruments? Is such a thing possible? This led to me diving deep into palindromes utilizing my tools as a composer, theorist, and technologist to discover the finer points of palindromes in composition which led to the conclusion that palindromes are far more complex than we realize, and we are just scratching the surface of possibilities. From the jovial palindromes of Haydn, to the reversed waveforms of electroacoustic music, the breadth of possibilities is immense as we ponder the finer points within the Granularity of Palindromes.
What is a Palindrome?
A Palindrome is defined as a word, phrase, or sequence that reads the same backward as forward. Examples of this might be the word racecar, phrases such as nurses run or - my personal favorite – taco cat. In music, it is a similar concept except we often say it is a series of notes that sound the same played forward or backward. Consider Example 1 as an example of a melody that has yet to become a palindrome:
Example 1
To expand this excerpt into a traditional palindrome, we simply notate the original materials from measures 1 and 2 and write it backwards immediately after the original musical excerpt – last note first, first note last, and so on until the entire melody is written backwards note-by-note. This can be thought of as a retrograde for those familiar with pitch-class theory. See how this occurs in Example 2 below:
Example 2 | A Simple Palindrome
You’ll notice that each note is reflected perfectly after the halfway point, articulations included. To finish the effect, we would need to re-incorporate dynamics and even reverse our crescendo into a decrescendo. This has been done in Example 3.
Example 3 | Dynamics Integrated
The excerpt is now a fully realized musical palindrome with notes reflecting onto themselves after the halfway point including all the original articulations and dynamics.
Historical Context
It may be a surprise to find J.S. Bach in a paper about palindromes, given their nature. While he never penned an official palindrome, he did compose what was called a crab canon or retrograde canon. This canon was written as part of his “Musical Offering” for Frederick the Great of Prussia. It is the composer’s intent that the performer play the notes – as written – forwards and backwards simultaneously. It is not a palindrome by definition, but the canon does take into considerations many elements of what a good palindrome must including, among other items, how the musical line sounds performed backwards as well as forwards. Though not specifically mentioned, the performer(s) could in theory perform their part forward (or backward in case of the second part) and then in reverse which creates a sort of mobius strip[1] effect of a never-ending piece that continuously loops onto itself. This, of course, leads to tabletop duets composed by many composers including Bach and even Mozart with his Der Spiegel which, while not palindromes, share many similar traits with both featuring a reversal, or retrograde, of the original material although the tabletop duets also feature an inversion of the original material.
The composer Franz Joseph Haydn, known widely as a jokester in the classical canon, famously wrote many jokes into his many symphonies and string quartets over the course of his career. Among these musical jokes is his Symphony no. 47 in G Major which, as its’ third movement, includes a minuet and trio. However, with a 1-reprise minuet, and 1-reprise trio – this third movement appears a bit lacking in length until we observe the text “Menuet al Roverso” for the minuet, and “Trio al Roverso” for the trio. Haydn intends for the performers to – after each repeat, perform the reprise backwards as the second reprise for both the minuet and trio. This is a great early example of a musical palindrome from the common practice period.
Composers such as Haydn, Schubert, Schoenberg, and even contemporary composers such as Hans Zimmer employ them to some degree or another. In serialism, we often utilize the retrograde which is a practical use of palindromes in composition if immediately preceded by the prime form of said retrograde row. In electronic music – particularly fixed media – we often employ a method of reversing a sound in an attack-decay pattern. So, what is it that defines the scope of a palindrome? Is it enough to simply play a melody backward? How do we treat ornamentations? Another option is to, as in Zachary Daniels’ A Minimal Journey[2], write an entire piece as one long palindrome with a turning point midway through. When doing so do we do a note-by-note reversal? By phrase? By smaller rhythmic groupings? What about articulations and dynamics? Do we reverse their order of appearance? Do we completely invert our attack-decay pattern and dynamic contour? This is where our acousmatic terminology might help us. In electronic music we often use a type of manipulation called granular synthesis[3] which, by definition, allows us to select how granular we want our synthesis to be. By the millisecond? 17 milliseconds? Just 5? A lengthy 100 milliseconds? It is here that we establish our most important tool when talking about palindromes in music: The Granularity of Palindromes.
The Granularity of Palindromes
For the sake of notated music not always being measured in milliseconds, a new definition is needed to define the granularity of music. We often speak of rhythmic grids when teaching and performing music, which is based on dividing a bar or beat into partitioned units of music. Applying this to palindromes seems to be a natural extension of this technique used to teach rhythm and analyze our music. This technique requires a term in order to be more easily addressed, to which I have begun to use the term notational granularity. Notational Granularity is the note-length by which a musical idea is divided into for transformational or analytical purposes, using common and uncommon note divisions, regular/irregular divisions of the measure, or even the length of a musical phrase in its’ most extreme use cases.
To demonstrate this concept, we will again utilize our first example, but in order to simplify the excerpts I will focus only on the melody for the next two musical examples. Additionally, I will repeat the measures for clarity of analysis and discussion. We will write a palindrome with the proposed notational granularity of an eighth-note beat unit.
Example 4 | Notational Granularity of an Eighth-Note
It’s different, but still highly similar what we would expect to a certain degree. The sixteenth-note groups stick out, however, due to their fitting within our grain-size of an eighth-note rhythmic unit and therefore retaining their localized contour of descending step-wise motion. Now, what if we were to alter the granularity and think of the quarter-note rhythmic value as being my new grain length?
Example 5 | Notational Granularity of a Quarter Note
Here, the note groupings seem all too familiar, with the ascending eight-note pairs on the off-beat now becoming the beat and the eighth plus two sixteenths figures once on the beat are now on the off-beat.
By tweaking our rhythmic grid and selecting a smaller or greater value for our notational grain length, we are left with a series of palindromes that each have their own “feel.” This application of differently-sized rhythmic grids aids us in breaking down the individual musical moments and preserving some or none of the originally composed intervallic relationships. This could also be used as a device by which we can compose, develop, or analyze musical themes.
Palindromic Melodies
A Palindromic Melody is a melody that itself is a palindrome. Such melodies might be standalone, act in juxtaposition to a simple bass-line, or act as a palindrome with static harmonies underneath. These melodies are often constructed with either the original melody or the combined palindrome in mind. Sometimes these are constructed with articulations, dynamics, and other considerations in mind that may or may not carry over into the palindrome of the original melodic content. An example of the aforementioned can be found here:
Example 6 | Palindromic Melody
Of course, this palindrome can be wildly different feeling if we add things such as articulations, slurs, and other musical elements in addition to the expected pitch and rhythmic values. Take the following example, where there appears to be a shifting rhythmic feel in the melodic content due to the shortened staccato in addition to the dynamics and accented notes:
Example 7 | Heavily Articulated Palindromic Melody
Rhythmic Considerations
When writing palindromes, a composer must take great care to write a good set of pitches that sounds just as engaging backwards as it does when heard in the originally intended direction. Often times, though, composers give little care to how it is that we differentiate the ending of a palindrome from the beginning. An easy method of doing something to make for a more interesting palindrome might be to cut a long note off early such as the following proposed palindrome:
Example 8 Hindemith-Like Violin Excerpt from Minimal Journey
On Rests
Silence is just as much a part of the music as the notes themselves. It’s important then, to observe them and think about how you wish to treat them when looking into palindromes in your own music. Looking at one of our first examples, the phrase “Nurses Run.” If we were to do a literal, character-by-character palindrome of this phrase, and account for the space between the words, we would get the phrase “nur sesruN” and yet most of us, when asked for the palindrome of “Nurses Run” will repeat “Nurses Run.” This emphasizes the importance of accounting for rests, note lengths, and even cut-offs or individual attack-decay patterns in music when writing palindrome music.
Will you use them as they are written? Or perhaps you will add rests where you previously found they were missed.
Palindromic Length and Forms
If we were to tweak the notational grain length to be even larger, we could even argue that many musical forms are already in a vaguely palindrome format, albeit in a very simplified form. In ternary form, there is an A section, a B section, and finally another A section. This, if the A section is exactly as it sounds before can be thought of as a long, clunky palindrome. A rondo might also be considered palindromic in nature if we consider the typical Rondo Format, ABACA, and ensure that a second half of the Rondo follows that is the reverse formal design we end up with ABACA ACABA. This would create a slightly more complex, but still clunky palindrome due to the granularity being too wide. A True palindrome form would be one that has a central point after which the musical content repeats itself in a mirrored fashion. The fulcrum is the point at which the original material meets the retrograde, or the center-point of the palindrome.
The Need for New Classifications and Terminology
By taking a closer look at palindromes of composers over the years, and into today – and introducing the idea of granularity to our palindromes, we are given a wide spectrum of possibilities regarding palindromes in music. We have note-by-note retrogrades at one end of the spectrum, and the fully reversed waveforms of acousmatic music at the other, as well as everything else in-between. When discussing these two extremes, and much of what lies in the middle, we are left with the need for new terminology, notational and auditory palindromes.
A notational palindrome is a palindrome that focuses on what is notated on the page. One example of this would be the piece by Dr. Zachary Daniels, A Minimal Journey. An older example, from the classical cannon, would be Schubert’s [Piece Name]. In the twentieth and twenty-first centuries, composers such as Kyle Gullings and Phillip Glass utilize notational palindromes in works such as Dirge for the New Sunrise and Music in the Shape of a Square respectively.
An auditory palindrome is a palindrome which focuses on the end result heard by the listener, instead of what is written on the page. On the furthest end of this proposed spectrum, the resulting soundwave itself would end up reversed. This can be done in most Digital Audio Workstations with relative ease for the purposes of fixed media and other types of acousmatic music. For traditionally notated music, this often requires we alter the given notated music of a palindrome in order to focus on the sounds an audience (hopefully, ideally) hears.
Transforming Notational Palindromes to Auditory Palindromes
Of course an important part of music is the dynamic, articulation, and phrasing of each given note or musical idea. If we look at measures 35 and 36 of the first violin part in A Minimal Journey, and observe the palindrome that happens after our axis of reflection at the 36/37 bar-line, we see the following notational palindrome:
Example 10 Minimal Journey mm. 35-38 1st violin
Most people would agree that this functions as a palindrome. If we wanted to cross over from a notational to an auditory palindrome however, we must take into consideration the articulations given, primarily the staccato. What if we notated the staccato as a sixteenth note in the palindrome to represent the space between the F# and A, and C# and E?
Example 11 Minimal Journey mm. 35-38 1st Violin Version B
This is an example of transforming a notational palindrome into an auditory palindrome.
Dynamics in Palindromes
Dynamics are a problem when talking about auditory vs notational palindromes. Take the example below, where a passage of music is piano before a subito forte in measure 4. If we treat it as a purely notational palindrome, we are left with a passage that starts forte, and transitions to piano in a relatively subito manner.
Example 12
If, however, we consider the auditory results of the palindrome, we might place a subito piano immediately before the forte so that after the fulcrum the passage would be piano as intended. This results in the following example:
Example 13
While neither interpretation is correct, or incorrect, each listener/performer might have their own preferences as to which sounds better or feels more “accurate.”
In Conclusion
While palindromes are by no means a new concept, they are garnering further interest from a variety of performers, composers, and academics alike. This is possibly because we enjoy using clever music-making tools, or even because we are looking for new methods of taking a compositional concept or idea further than before. The proposed terminology and tools are my take on what to do with palindromes and their various idiosyncrasies. Applying techniques of granular synthesis from the acousmatic music realm to that of notated music has many advantages when discussing palindromes as it allows us to differentiate between that which is written, a notational palindrome, from that which we hear, an auditory palindrome. We do this with concepts such as notational granularity, and being mindful of dynamic markings in our music.
Whether we are writing palindromes as composers, or analyzing them, proper terminology is important to establish a baseline and the terminology proposed in this paper attempts to establish a set of terms for us to use when talking about The Granularity of Palindromes.
What’s Next?
To further this study, I’m going to write palindromes that intentionally explore the extreme ends of the spectrum of auditory to notational palindromes. I believe this will help me further understand these unique compositional tools that so many of us are fascinated with. My plan is to further develop the terminology/given definitions within this paper and refine the process by which we talk about palindromes. Ultimately, I hope to understand more of my own fascination with them.
References
Appendices
Figure 7.1 | Bach, A Musical Offering BMV 1047
Figure 7.2 | Haydn, Symphony no. 47 in G Minor
[1] The Möbius strip is a geometric surface with one side and one boundary. First described in 1858, it is formed by giving a half-twist to a rectangular strip and joining the ends. Its properties have found applications in mathematics, engineering, physics, and art.
[2] A Minimal Journey, Zachary C. Daniels | Appendix A | https://zachdaniels.com/music/minimaljourney
[3] A form of synthesis based on a process called granulation. Granulation involves breaking down an audio sample into tiny snippets of audio called “grains.” These snippets are typically between 1 and 100 milliseconds long.