Artist Blog

Patricia Brennan: MAGIC SQUARE – Composition by numbers

I have always gravitated towards music by composers that mostly use unusual approaches for composition. Partly, because of my training as a percussionist, I was introduced to music that not only used rhythm as the core compositional element, but also I was introduced to music that was composed using unusual techniques. Some compositions even invited the performer to be a part of the compositional process by deciding on the specifics of the sonic landscape. 

Some of my earliest influences and inspirations during my percussion studies were composers such as John Cage, Steve Reich, Iannis Xenakis and David Lang. Each of them with a very distinct compositional voice. Also, individually using different approaches to composition through out their trajectory, mostly using unusual and unique techniques.

As a composer, my experiences as a percussionist always come through my music. Generally, rhythmic structures tend to be one of my main source of inspiration. Also, I like to explore different ways of creating rhythmic structures, either using numerical combinations, architectural patterns or in the case of the piece that this analysis is based on, mathematical formulas. 

This article focuses on the detailed analysis of my composition “Magic Square”. The origin seed for this composition is a mathematical formula by the same name as the composition.  The main characteristic of a Magic Square is that each layer adds up to the same number even though the numerical combinations are different. 

The layers of the Magic Square that I used as a base for this composition all add up to the number 15. This number became the base for the rhythmic structure of this composition. The time signature is 15/16, however there’s several rhythmic layers in use that build the rhythmic counterpoint of this composition. See image of the Magic Square used in this composition below: 

Image 1 – Magic Square where each layer adds up to 15.


The first layer that I wrote was the drumbeat. Generally, I like to write drumbeats for my compositions in order to support and enhance the rhythmic structure of a particular piece. In this case, the drumbeat became literally the heartbeat of the piece. 

For this rhythmic layer, I used the combination 9 + 5 + 1 which is the middle vertical layer read from top to bottom in the Magic Square. As the next step, I continued to break down the big numbers into smaller subdivisions. The number 9 became 3 + 6, the number 5 became 3 + 2, and the 1 stayed the same. These are all sixteenth note subdivisions. The breakdown for the number 9 (3 + 6), went through one more breakdown: 3 + 2 + 2 + 2. Essentially, each number from this layer was broken down to subdivisions of 2 or 3 with exception of the number 1.

Once I had the final breakdown of the drumbeat layer set, I proceeded to write a drumbeat that balanced the rhythmic consonances and dissonances. By consonance I refer to the “grounded” part of the beat and as the dissonance, the “unstable” part of the beat. In this case, the bass drum sits on the beginning of each of the bigger subdivisions: the beginning of the 9, the 5 and the 1. The snare enhances the circular feel of the micro subdivisions: the third partial of the ternary subdivisions. Also, it enhances the grounded binary feel by falling on the first partial of the binary subdivision. The hi-hat part works as a filler on the 8th and 9th partial of the number 9 (number from the original layer) and also on the 4th and 5th partial of the number 5 (number from the original layer). See image below: 

Image 2 – Drumbeat breakdown.


Before focusing on the pitch material, I started to write the rhythmic breakdown for the melody and the bass. I used different layers from the Magic Square for each melodic layer, and sometimes for each measure.

In section A, for the melody on measure one, I used the combination 2 + 7 + 6. Once again, I continued to break it down into smaller subdivisions: The number 2 stayed the same, the number 7 became 3 + 2 + 2, and the number 6 became 2 + 4. For the second measure, I used a different layer from the Magic Square: 8 + 3 + 4. The micro breakdown for this layer goes as follow: the number 8 became 4 + 4, the number 3 and number 4 stayed the same. 

As far as the bass line for measure one, I used the 9 + 5 + 1 breakdown, which is the same as the drumbeat. However, the micro subdivisions are different: the number 9 became 5 + 4 and the number 5 and 1 stayed the same. For the second measure on the bass line, the breakdown I used was 8 + 3 + 4, which is the same as the breakdown of the melody from the same measure. In this case the micro subdivision is also the same as the melody: the number 8 became 4 + 4, the and the number 3 and 4 stayed the same. The breakdown for the melody and bass line from measure one and two is the same breakdown as in measures five and six. See images below: 


Image 3 & 4 – Rhythmic breakdown of the melody on measures 1 & 2 from Section A.

As far as the breakdown for the melody from measure three, I used the same combination from measure one and five (2 + 7 + 6) but with slightly different micro subdivisions: the number 2 stayed the same, the number 7 became 3 + 2 + 2 and the number 6 became 3 + 3. For the melody on measure four, I used the same combination and micro subdivisions as measure two and six. 

As far as the bass line in measures three and four, I used the same combinations and micro subdivisions as in measures one and two, which is also the same for measures five and six. See images below: 

Image 5 & 6 – Rhythmic breakdown of the bass line on measures 1 & 2 from Section A.

Section B is based on a two measure phrase that develops harmonically as the section progresses. The rhythmic breakdown of this section is based on previously used combinations. For the melody, the combinations used go as follow: 2 + 7 + 6 for the first measure of the two measure phrase, and 8 + 3 + 4 for the second measure of the two measure phrase. 

For the bass line in this section, the combinations used go as follow: 9 + 5 + 1 for the first measure of the two measure phrase, and 8 + 3 + 4 for the second measure of the two measure phrase. Another aspect about Section B is that it’s rhythmically influenced by the structure of a piano montuno. See images below: 


Image 7 & 8 – Rhythmic breakdown of the melody line on measures 1 & 2 from Section B.

Lastly, in section C, both the melody and the bass line are based on the layer 8 + 3 + 4. However, I used the retrograde of this layer for one of the measures from the two measure phrase. For example, the first measure of the melody in section C starts with the retrograde combination 4 + 3 + 8. And then, on the second measure goes back to 8 + 3 + 4. As far as the bass line, I used the opposite process: on the first measure of Section C, the bass line starts with 8 + 3 + 4 and then, on the second measure of the phrase, it flips to 4 + 3 + 8. See image below: 


Image 9 & 10 – Rhythmic breakdown of the bass line on measures 1 & 2 from Section C.

Once the rhythmic structure was set, I continued to write the melodic and harmonic structure of the piece. Some measures explore different micro moments of tension and release, and other larger phrases in the piece are based on a particular tonal center or scale. For example, the first and second measure from Section A, go through the following harmonic changes: Csus2, AbMaj7 #11, Asus7, Bbsus7, Absus7/Gb, Absus7 and Dm9. And on the other hand, the first two measures of section B are both based on the F Spanish Phrygian scale. See images below: 

Image 11 & 12 – Score excerpt: measures 1 & 2 from Section A and measures 1 & 2 from Section B.

Another example is the contrast between the third and fourth measure from Section A and the first and second measures from section C. The third and fourth measure from Section A focus on the micro tension and release concept and are based on the following harmonic structure: BbMaj9 (F/BbMaj7), C+, F, D7 #9 (F/D7), G, Absus2, Asus2, Bbsus2, BMaj7 and Bm7. On the other hand, the last four measures from Section C are based on the F Spanish Phrygian scale. See images below: 

Image 13 & 14 – Score excerpt: measures 3 & 4 from Section A and last four measures from Section C.

As far as the interpretation of this piece, it varies depending on the context. The first recording of this piece was a solo version. This version starts with an improvisation that hints at elements from the composition. Once the composition starts, each two measure phrase becomes a playground for improvisation while at the same time keeping elements from the rhythmic, harmonic and melodic fabrics. The second recorded version of this piece, called “Square Bimagic”, features a quartet. In this version, the composition is played a bit more straightforward. The A section functions as opening material, the B section becomes a solo form and features a bass solo. The first part of section C functions as transitional material and the second part features a drum and percussion solo. This version also made use of dynamics in order to shape the energy buildup within each section. 

Generally, the idea with this piece, as far as interpretation, is that of flexibility. Not only flexibility within the approach but also instrumentation. Even though the structure of the piece is based on very specific rhythmic phrases and harmonic changes, the goal is openness. 

This composition can be heard on my solo record “Maquishti”, released under the label Valley of Search (2021).

And, the quartet version of this composition can be heard on my record “More Touch” under the name “Square Bimagic”, released under Pyroclastic Records (2022). 


About the Author:

Vibraphonist, marimbist, improviser and composer Patricia Brennan “has been widely feted as one of the instrument’s newer leaders” observed The New York City Jazz Record. 

Growing up in Veracruz, Mexico, Patricia’s childhood immersed her in the rich musical culture that developed her artistic voice. Patricia’s search for freedom in her musical expression led her to extending the range and sonic possibilities through the vibraphone and mallet percussion in improvisational music and composition. 

Patricia’s first album, the solo album “Maquishti”, was #4 in the New York Time’s top 10 jazz albums of 2021. Patricia received the rising star vibraphonist award on Downbeat’s 2022 Critics Poll and was listed #4 vibraphonist of the year on the 2023 Critics Poll. Her newest album, “More Touch,” released in November 2022, is already receiving accolades from critics. “More Touch” was included in several Best Jazz Albums of 2022 lists such as NPR Best Music of 2022, Bandcamp and PopMatters. The opener track “Unquiet Respect” was listed under NPR’s 100 Best Songs of 2022, a list featuring music from all genres. 

Patricia works both as a leader and side person for renowned artists such as Grammy winning Arturo O’Farrill’s Afro Latin Jazz Big Band, the Grammy nominated John Hollenbeck Large Ensemble and other bands including the Michael Formanek Ensemble Kolossus, Matt Mitchell’s Phalax Ambassadors, Tomas Fujiwara’s 7 Poets, Mary Halvorson’s Amaryllis amongst other groups and collectives. She has performed in venues such as Newport Jazz Festival, SF JAZZ, and Carnegie Hall, as well as several international venues including venues in Germany, Finland, Netherlands, Mexico, and Argentina. 

Patricia is a Valley of Search artist, Pyroclastic Records artist, Audeze artist and a BlueHaus Mallets artist. She is currently on the faculty at the New School for Jazz and Contemporary Music, the Jazz Studies program at NYU Steinhardt and has given lectures at several colleges including Manhattan School of Music and UCLA.