Dec 18, 2020
Music and mathematics share more in common than is typically realized of them. Many of these connections were made by Pythagoras, a Greek philosopher, who was frequently alluded to in many images about or including to Music in the Renaissance period in many famous works such as Raphael’s The Ecstasy of Saint Cecilia and The School of Athens and Pinturicchio’s Musica and more.
Music and mathematics share more in common than is typically realized of them. Music theory and harmony revolves around the mathematical relationships of pitch, sounds and intervals. Many of these connections were made by Pythagoras, a Greek philosopher, who was frequently alluded to in many images about or including to Music in the Renaissance period. I will be discussing Raphael’s The Ecstasy of Saint Cecilia and The School of Athens and Pinturicchio’s Musica as well as the studiolo of Federico da Montefeltro and Pietro Cannuzio’s Regule florum musices, while highlighting mathematical aspects and symbols to illustrate this visual connection was not just an homage to Music’s history, rather it was a symbol of the knowledge needed to understand music – knowledge that perhaps was more ubiquitous than is normally understood. Commissioned by a patrician of Bologna known for her piety, Elena Duglioli dall’Olio, Raphael’s The Ecstasy of Saint Cecilia (Fig. 1) depicts the moment of Saint Cecilia’s ecstasy from the songs of the angels above her, while holding an organetto, a handheld organ, surrounded by broken instruments as well as St. Paul, St. Joan the Evangelist, St. Augustine and St Mary Magdalene.
While it may seem bewildering for the patron saint of music to carelessly handle a hand organ, Saint Cecilia, who was well-known and recognized in Bologna, is often associated with deep spiritual change which can be seen in Raphael’s painting.  The painting depicts St. Cecilia in a state of ecstasy from the choir of angels, thereby elevating her soul and mind from the earthly delights and into the realm of heaven. The choir of singing angels represent musica angelica, celestial music, which is not only praise for God but also the expression of happiness of redeemed souls and it was thought that Musica angelica could be accessible on earth to the people in a state of ecstasy which can been seen in Saint Cecilia’s expression. After attaining spiritual elevation and understanding, St. Cecilia lowers the organetto which has helped her achieve a higher mind but is now no longer necessary. Similarly, the scattered in disorder and party damaged instruments at her feet, represent secular music. St. Cecilia, standing over these instruments, while being lifted into ecstasy by the singing angels, becomes the personification for religious music’s domination over secular music. Raphael painted The Ecstasy of Saint Cecilia in accordance with the wishes of the patron, representing several philosophic and religious ideas. However, he was also able to incorporate popular ideas of the time, allowing the painting to speak about music in more ways than one. While St. Cecilia’s ecstasy speaks to the triumph of religious music and the superiority of vocal over instrumental music, the composition of the painting speaks to the Pythagorean belief that ratios and small integer numbers are the structure of the universe. Raphael painted six angels which was considered primus numerus perfectus, the first perfect number, and was privileged by God who created the world on the sixth day. In addition, the angels are placed in a group of three and then joined by a fourth angel who sings from a separate sheet of music, but still holds the same book from which the other three angels sing. This group of six angels is then completed by a duo of angels on the right. The sequence of numbers 2,3, and 4, and 3,4 and 6 are common within the Pythagorean theory, when applied to music, give perfect music intervals. For example, 2 and 4 are in the ratios of 1:2 which is the most perfect music consonance, the octave. When divided by any of the other numbers, the result are also perfect consonances, the second, fourth and fifth.  Raphael also proved his knowledge of the Pythagorean theory of proportion and mathematic principal of harmony in his School of Athens (Fig. 2) where the same musical motif and progression is shown in a diagrammatic design of a lyre on his tablet with the numbers: VI, VIII, IX, XII (Fig. 2a). The connections between the numbers expressed the Pythagorean Musical Scale and various intervals of an octave. 
(Figure 2a) These musical aspects and theory of mathematical proportions were no stranger to Raphael due to his education as artist and his involvement with the court. In the atmosphere of the papal court, it was not difficult to acquire and practice music and theory as it was considered the duty of the elite to learn and know many skills, outlined in Baldassare Castiglione’s “Book of the Courtier”, one of which was music. Music, according to Castiglione, had “…such varied melodies, that the listener’s spirits are moved and inflamed, and thus entranced seem to be lifted up to heaven…because he softens and penetrates our souls by placid means and full of plaintive sweetness, gently stirring them to sweet emotion.”  Music was not just a skill to be learned just merely entertain. It was a tool to calm the soul and lift the mind closer to heaven. Baldassare Castiglione also credited Pythagoras to finding the key or “rapture” to “drive from [the] minds those painful thoughts and grievous troubles with which our life is filled.” Music was also depicted in secular contexts as it was one of the Seven Liberal Arts. In the Studiolo of Federico da Montefeltro’s ducal palace in Gubbio (Fig. 3), many objects illustrated, as part of the tromp l’oeil effects of the shelves, peaked out from the half-open latticed wooden doors (Fig 4), that highlighted the many interests of Federico da Montefeltro, including music.
Both string and percussion instruments were featured on the shelves, in sections 1,4,5 and 13, which include a horn, harp, tambourine, drum, lutes, flutes, fiddle and an organ. All these various instruments are attributes to the personification of Music (Fig. 5), believed to have been made by Joos van Ghent.
Music is depicted as a woman, seated on a throne with a portative organ, similar to St. Cecilia, and holding a book, while a man kneels before her. Music, similar to the other personifications of the Liberal Arts, was hung above the marquetry of the room. The placement of Music in the studiolo as well as the other Virtues and Liberal Arts was meant to urge Federico to a contemplative life, enabling him to take cognizance of the eternal affairs of God. The studiolo was a room used for the pursuit of knowledge and the decoration of it meant to inspire those inside – elevation thoughts studies to a higher truth. These instruments of music are often found close in proximity to objects that could represent mathematics as well as the sciences. For example, a horn is found on the same shelf as a book and scroll and a cittern, a stringed instrument, is on the same shelf as a compass which is common tool in mathematics and other liberal arts. These instruments of various studies are speak to the interests of not only Federico da Montefeltro but also of any well-respected man of the court. Pinturicchio created the first complete cycle of the liberal arts in 1493 for Pope Alexander VI in the Vatican palace. Each liberal art in the fresco cycle was personified as a female who was centered in the composition, shown holding or using objects related to the art form depicted, similar to Joos van Ghent’s Musica. Because the personification figures of the liberal arts are surrounded by up to 10 figures, Pinturicchio transforms the figures themselves from the personification of academic studies, into altarpieces themselves. The composition of Pinturicchio’s Musica (Fig. 6) contains four putti, Musica herself, and seven other figures. Together, the putti and other figures symbolize three separate ensembles with Musica as the solo performer.
(Figure 6) The trio on the left, the first ensemble, represent secular entertainment. At the far left, a figure holds a stringed instrument identified as a vihuela. Original from Spain, the vihuela was integral to secular entertainment in the Borgia court of Alexander VI. Behind the vihuela player is a singer, noted by his open mouth and gaze towards the harpist suggesting he is being accompanied by both instrumentalists. The harp of the third figure in the trio has twenty-six countable strings which is consistent to surviving instruments of the time. In addition, the landscape to the left of Musica, contain two large medieval buildings which symbolize man-made structures, complimenting the man-made instruments featured in the trio, emphasizing the secular and man-made representation of this section of the fresco. The group of four, the second ensemble, on the right represent sacred music. Two men read from an oblong octavo book and although it is not clear what is written in the book, both men have their mouths open, suggesting they are likely both singing. It was believed the vocal music was of a higher standard than music made by instruments, because the voice was a gift from the heavens. A singer reads from an oblong octavo sheet of music which was used for a sacred setting, accompanied by the two men, also singing. The landscape behind this group show large rock formations, greenery, and trees – all things created by God. The third ensemble are the putti. The two putti playing the pipes turn inward to Musica, identified by ‘MVSICA’ inscription, who is seated while holding a three-stringed lira da braccio, but their gaze is towards the ground, suggesting that while they are accompanying her, they hold high regard for her. The accompaniment of these putti is more symbolic than realistic as the lira da braccio, the instrument Musica holds, was a solo instrument. Two other putti hold a draped cloth behind Musica, simultaneously uniting and transcending the man-made world on the left and the realm of nature on the right. There is one figure that stands apart from the rest. In the front of the composition, to the right, a man holds two hammers, appearing in the midst of striking the floor, symbolizing the story by Nicomachus of Geras in his Manual of Harmonics which tells of Pythagoras passing a blacksmith’s shop only to hear hammers of different weights ringing with dissonant intervals which led him to discover the system of music ratios. This story is also told by Boethius in his text that was the core music textbook for the next millennium, De institutione muisica. Musical intervals, as we have seen, have been related by the Pythagorean School to arithmetical ratio of not only intervals within music but length a string and pitch. Pythagoras stands as a symbol for the mathematical aspect of music.  Music, in addition to being a tool of elevation of the soul and widening of the mind, is also in many ways, synonymous to harmony and by extension, mathematics. Although the examples so far have come from artists whose work was for the nobility of the court or of a similar standing, those working in the commercial industry, had the means of understanding the visual cues described previously. The knowledge necessary was accessible to more than the aristocracy because of the problems of proportions and ratios those in the commercial industry had to solve on a regular basis by using a universal arithmetical tool called “The Rule of Three” also known as “The Merchant’s Key” or “Golden Rule”. The Rule Three stemmed as far back as the thirteenth century with Leonardo Fibonacci, another mathematician, and it was how the people of the Renaissance dealt with problems of proportions and ratios. While it may seem that the only people who would care about proportion problems would be artists, the problems that dealt with the comparison of numbers greatly exist outside the realm of sculptors and paintings such as the brokerage, discounts, tare allowance and currency exchange. They all deal with proportions in turn, they are dealt with the Rule of Three. The Rule of Three also intersected in many ways with Pythagorean harmonic scale – the basis of western harmony. Pietro Cannuzio’s Regule florum musices (Fig 7) was able to illustrate the Rule of Three as it relates to western harmony in the harmonic scale at the top of his title page.
All of this is to say, Quattrocento people became adept at mathematics and complex numerical solving, through daily practice by reducing diverse information to geometric proportion so much so that the painter’s study of proportions of human anatomy paled in comparison with the mathematics merchants were used to. While the details of harmonic series of intervals in these works were used by musicians and sometimes painters and sculpture, everything was but unfamiliar to those who worked as bankers and tradesman due to their skills by commercial education. That is to say, the knowledge needed to understand the meaning behind the scales were perhaps more ubiquitous than is normally understood for their knowledge and use of mathematics was a large part of their formal intellectual equipment.  Music does not belong to any one place and the knowledge to understand it does not belong to any one type of person. It is tempting to think, given the examples by Raphael’s paintings, Baldassare Castiglione’s studiolo and Pinturicchio’s fresco, music is only for the court and highly educated individuals. However, while the writings of Baldassare Castiglione in his “Book of the Courtier” suggest that knowledge of music must be grasped to be a good courtier, the knowledge he describes is not all that different than the knowledge a tradesman or banker has gained over the many years of daily practice. Due to their expertise in conceptualizing, visualizing and manipulated ratios and forms, Quattrocento citizens were more sensitive to images the symbols that carried the marks of a similar process they have been conditioned to see. Perhaps most important aspect of mathematics and music is the continuity – many intellectuals, artists and commercial tradesman alike could argue that mathematics and music are two sides of the same coin. The frequent visual connections to mathematics in depictions of music was a nod to possibility that truly anyone could grasp the true meaning and affect of music on the mind and soul. Music, because of the fundamental mathematics behind it, exists everywhere and it gives equal opportunity for enlightenment and elevation – all you must do is look and learn.
 Lucia Marchi, “For Whom the Fire Burns: Medieval Images of Saint Cecilia and Music,” Receracre, Fondazione Italiana per la Musica Antica (FIMA), Vol. 27, No. ½, 2005, p. 1.
Tomas Connolly, Mourning into Joy: Music, Raphael, and Saint Cecilia, London, Yale University, 1994, p. 2
 Stainislaw Mossakowski, “Raphael’s ‘St. Cecilia’. An Iconographical Study.” Zeitshrift fur Kunstgeschichte,Deutsher Kunstverlag GmbH Muchen Berlin, Vol. 31, 1968, pp 4,6
 Marchi, Lucia. “For Whom the Fire Burns: Medieval Images of Saint Cecilia and Music.” Receracre, Fondazione Italiana per la Musica Antica (FIMA), Vol. 27, No. ½, 2005, p. 7.
 Stainislaw Mossakowski, “Raphael’s ‘St. Cecilia’. An Iconographical Study.” Zeitshrift fur Kunstgeschichte,Deutsher Kunstverlag GmbH Muchen Berlin, Vol. 31, 1968, pp 7,11-12. 2
 Baldesar Castiglione, The Book of the Courtier: Book One. New York, Charles Scribner’s Sons, 1901, p.50
 Baldesar, Castiglione. The Book of the Courtier:Book Two. New York, Charles Scribner’s Sons, 1901, p. 90
 Stuart J. Robb, “Pintoricchio’s ‘Musica’ of the Sala delle Arti Liberali, palazzi Pontifici, Vatican.” Early Music, Oxford University Press Vol. 42 No. 2, 2014, p. 167.
 Marcin Fabianski, “Federigo da Montefeltro’s ‘Studiolo’ in Gubbio Reconsidered. Its Decoration and Its Iconographic Program: An Interpretation.” Artibus et Historiae. Vol. 11, No. 21, 1990, pp. 200-201, 208.
 Cecil H. Clough, “Art as a Power in the Decoration of the Study of an Italian Renaissance Prince: The Case of Federico Da Montefletro.” Artibus et Historiae, Vol. 16, No. 31, 1995, p. 29
 Stuart J. Robb, “Pintoricchio’s ‘Musica’ of the Sala delle Arti Liberali, palazzi Pontifici, Vatican.” Early Music, Oxford University Press Vol. 42 No. 2, 2014, p. 168 -
 Baxandall, Michael. Painting and Experience in Fifteenth-Century Italy. New York, Oxford University Press, 1988, pp. 95-96, 99-101
Baldesar, Castiglione. The Book of the Courtier. New York, Charles Scribner’s Sons, 1901.
Baxandall, Michael. Painting and Experience in Fifteenth-Century Italy. New York, Oxford University Press, 1988.
Clough, Cecil H. “Art as a Power in the Decoration of the Study of an Italian Renaissance Prince: The Case of Federico Da Montefletro.” Artibus et Historiae, Vol. 16, No. 31, 1995, pp. 19-50,
Connolly, Thomas. Mourning into Joy: Music, Raphael, and Saint Cecilia. London, Yale University, 1994.
Fabianski, Marcin. “Federigo da Montefeltro’s ‘Studiolo’ in Gubbio Reconsidered. Its Decoration and Its Iconographic Program: An Interpretation.” Artibus et Historiae. Vol. 11, No. 21, 1990, pp. 199-214.
Marchi, Lucia. “For Whom the Fire Burns: Medieval Images of Saint Cecilia and Music.” Receracre, Fondazione Italiana per la Musica Antica (FIMA), Vol. 27, No. ½, 2005, pp 5-22.
Mossakowski, Stainislaw. “Raphael’s ‘St. Cecilia’. An Iconographical Study.” Zeitshrift fur Kunstgeschichte,Deutsher Kunstverlag GmbH Muchen Berlin, Vol. 31, 1968, pp 1-26.
Robb, Stuart J. “Pintoricchio’s ‘Musica’ of the Sala delle Arti Liberali, palazzi Pontifici, Vatican.” Early Music, Oxford University Press Vol. 42 No. 2, 2014, pp. 167-174.