Coming and Going: A Summary of my Research on Medieval Music

Overall, my research ended up taking about 100 hours to complete, including the initial reading and writing the paper. The paper, itself, was much longer than I imagined it would be, at a length of about 28 pages. I ended up only analyzing three of the four pieces that I previously listed. I cut the motet Tuba sacre / In arboris / Virgo sum by Philippe de Vitry in order to avoid redundancy in my analysis. In the end, I believe my paper was an appropriate length to culminate and predict based on the information I learned. I have really enjoyed completing this research project this summer. It has been eye-opening to look into the connections between music and mathematics in this historical sense. I am looking forward to hopefully completing future music research. Below is a summary of my paper, “Coming and Going: Plotting Time, Space, and Direction in Medieval Music.”


Composers throughout the Ars nova were able to demonstrate the pervasiveness of numbers and mathematics through the utilization of isorhythm and retrograde motion. The Ars nova (“new art”) was a period in 14th century France in which many changes to society, particularly to the arts and music, occurred. One overall change came in the form of increased secularity, resulting in dissatisfaction with the church. In music, the main changes of the time came with the addition of imperfect (duple) mensuration and the use of red coloration of notes to distinguish between the perfect and imperfect divisions.

Isorhythm was a process in which the tenor (lowest) voice of a 3 or 4-part motet contained repeating patterns of rhythm and pitches that worked together to create a structured and mathematical basis for the piece. The repetitive pattern of rhythm is called the talea while that of pitches is called the color. While not new to the Ars nova, this process was expanded and improved upon during this time. Another process, retrograde motion, through which a musical phrase or structure is repeated in reverse, was also used frequently by composers. Numbers could be displayed in a representative manner through the use of Medieval numerical symbolism. See examples of this here:

Although debated by musicologists, the composer and poet Philippe de Vitry is credited with writing the treatise Ars nova, from which the time period gets its name. Vitry wrote many works, including multiple motets for the allegorical poem the Roman de Fauvel, which provided a satirical account of political events at the time. One of these isorhythmic motets was titled Garrit gallus / In nova fert / Neuma and discussed the events related to King Philip IV’s counselor, Enguerran de Marigny. The Latin text contains animal allegory, specifically proportioned line numbers, and both Biblical and mythological allusions. The structure of the piece as a whole (seen below where the blue bars represent the taleae and the green bars represent the colors) is further mathematically accentuated through palindromes (retrograde motion) in the talea and phrase structures.

garrit gallus talea and color

Guillaume de Machaut was another noteworthy composer and poet prevalent during the later part of the short time period. His isorhythmic motet Quant en moy / Amour et biaute / Amara valde depicts both a spiritual and a lover’s journey. References to both secular love and Biblical texts give the piece what musicologist Anne Walters Robertson calls a “polyphony of meaning.” As with Garrit gallus, the talea and phrase structures contain palindromes or repeating patterns. Machaut utilizes “diminution” to proportionally shorten the talea and color of the second section of the piece, thus further showing the prevalence of math in the structure (as seen below where the blue bars represent the taleae and the green bars represent the colors).

quant en moy talea and color

Machaut’s rondeau Ma fin est mon commencement, while a different form than the motets, continually used numbers and specific structural components throughout. The piece demonstrates retrograde motion in the text’s structure and meaning, the rondeau’s structure, and the music. The top two voices alternate and sing the retrograde of each other in the second section of the piece while the tenor sings the retrograde of itself in the second section. See a visual and audio representation of this here: The large amount of retrograde motion in the piece suggests, as the text translates, “my end is my beginning.” Going further, this mathematical-like structure can represent such cycles as the recursive journey of Christ, the lover’s journey, and the hero’s journey.

Overall, all three pieces demonstrate the extensiveness of numbers and mathematics during the Ars nova. The pieces span the time period in order to make the most effective argument given the length of this research project. Further research could be done on more pieces, composers, and forms of pieces during the Ars nova in order to determine the broader impact of the techniques of isorhythm and retrograde motion on music.


  1. cdjones03 says:

    Hi there!

    Your project was super fascinating to read through, and you definitely taught me a lot more about the detailed structure of medieval music. I’ve been able to study music history in school before, even last semester, but never before had I heard about the talea, color, isorhythms, or use of retrograde motion in medieval music specifically. Listening through the Machaut rondeau, it was really eye-opening to notice how the voices specifically interact and exchange, and how some verses are just reverses of the previous one. I’ve studied retrograde motion with serialism/12-tone row before, but it’s cool to see the procedure has existed for so long.

    Considering how the medieval era saw the rise of metrical organization of music, this definitely makes sense why music would begin to be organized in this style of repetition and numerology. I’m not sure if any texts have survived or even exist for this, but it would be interesting to read exactly what was going through these composers’ minds when they represented these certain religious ideas in their music. Although their music was primarily secular, it really illustrates how impossible it was during the medieval era to truly be free of the influence of Christianity and the church. Thanks for the new knowledge!

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