“The proportions of the voices are harmonies for the ears; those of measurements are harmonies for the eyes.”

1. Mathematics of Music in Renaissance Architecture “The proportions of the voices are harmonies for the ears; those of measurements are harmonies for the eyes.”

2. Amphion Apmhion Building the Walls of Thebes with his Music. Abraham Jansz. van Diepenbeeck (1596-1675), illustrator; Cornelis Bloemaert (1603-ca. 1684), engraver.

3. “If two strings are made to vibrate under the same conditions, one being half the length of the other, the pitch of the shorter string will be one octave (diapson) above that of the larger one. If the length of the strings are in the relation of two to three, the difference in pitch will be a fifth (diapente), and if they are in the relation of three to four, the difference in pitch will be a fourth (diatessaron).” Rudolph Wittkower, Architectural Principles in the Age of Humanism. Pythagorean Musical Ratios

5. 1 2 3 4 9 8 27 Plato’s Lamda:“Plato, through Timaeus, says the creator made the world soul out of ingredients, and formed it into a long strip. The strip was then marked into intervals.”

6. 1 2 3 4 9 8 27 Monad Point First Even & Odd Number Line Squares Plane Cubes Solid

7. Arithmetic Mean: the sum of n integers, divided by n ex. (1 + 2) / 2 = 3/2 • Harmonic Mean: the reciprocal of the arithmetic mean of n digits’ reciprocals ex. For digits 2 & 1: ([1/2]+[1/1])/2 = ¾  4/3 Filling in the Music

8. Filippo Brunelleschi (1377-1446)

9. Cathedral of Florence

10. Architectural elements • Homogeny of wall, space, light, articulation • All attributed to certain ratios • Set of geometric proportions based on squares • Proporzioni musicali – 6 : 4 : 3 : 2 Bruneschelli’s Dome

11. Dufay’s Motet

12. Four sections of motet have proporzioni musicali Inclusion of tenors is directly related to the thickness of architecture Utilization of the number 7 “… both dome and motet reduce gothic dimensions to classical symmetry and proportion, both extricate themselves from the harmonic incommensurability of older systems and emerge self-contained, self-sustaining forms” Dufay’s Motet

13. Leon Battista Alberti (1404-1472)

14. Santa Maria Novella - Florence

15. Dimensions are an octave “Rhythmic, geometric unity” Façade as a perfect square Santa Maria Novella

16. TempioMalatestiano - Florence

17. Composed of whole number multiples • 50, 40, 36, 30, and 18 Roman feet • Evident in composition of outside of the Tempio TempioMalatestiano

18. “ You see whence the measures and proportions of the pilasters arise, what you would change brings discord into all that music” - Alberti

19. “The very same numbers that cause sounds to have that concinnitas, pleasing to the ears, can also fill the eyes and mind with wondrous delight. From musicians, therefore, who have already examined such numbers thoroughly, or from those objects in which Nature has displayed some evident and noble quality, the whole method of outlining is derived.” - Alberti

20. Compare and contrast Brunelleschi and Alberti’s architectural works with a few bullet points in your journals Using Geometer’s Sketchpad, create your own façade mimicking that of the Santa Maria Novella Homework!