Pythagoras was the first documented scholar to establish the notion of musical ratios, believing that music could be represented as pure mathematical ratios, which he believed were the ratios of the cosmos.
The Renaissance recycled his ideas, where architects, artists, and craftsmen used these musical ratios in the built environment because they believed that, as twentieth century architecture historian Rudolf Wittkower maintains, “as man is the image of God and the proportions of his body are produced by divine will, so the proportions in architecture have to embrace and express the cosmic order, citing ideas of Pythagoras and Plato's mathematical cosmos.”
Renaissance architects and artists like Francesco Giorgio, Leon Battista Alberti, Serlio, Palladio, and later Blondel, are among those whom scholars have researched to use such proportions in their work and writings.
Proportions in Music
Baroque musical theorist Jean-Phillippe Rameau asserts that the reason for the pleasant sonic tonalities in diatonic scales is because their intervals are easily proportioned when extracted from a length of string. As documented in his Treatise on Harmony, a length of string that produces a certain pitch could be easily divided up to create different intervals of said pitch.
Nicomachus of Gerasa (c. 100 A.D.), the neo- Pythagorean, tells tales of Pythagoras pondering these questions about pleasant intervals, with respect to the harmony of the spheres and planetary bodies. Nicomachus says that Pythagoras once passed a crew of blacksmiths at work, and found that as their hammers struck the anvils, common intervals – the octave, fifth, fourth, and major second – were sonically produced.
Pythagoras thought the hammers’ different weights caused the interval soundings. His investigations showed that the weights of the hammers were respectively 6, 8, 9, and 12 pounds - when the 12 pound hammer struck with the 6 pound hammer, a musical octave sounded (6:12 = 1:2); when the 12 pound hammer struck with the 9 pound hammer, a perfect fourth sounded (9:12 = 3:4).
Ever since then, Pythagoreans have assumed that these principles can be applied to the music of the planetary spheres, with their relative weights and sizes producing different interval soundings as the planets orbit and rotate.
To further embellish on these ideas, let us assume that we have a piece of string held at two points that produces that pitch C. If the length is divided into two equal parts, the pitch of each side of the string also produces a C pitch, except an octave higher. If one were to divide the string into three equal parts and pluck the string at the point where the resultant length is 2/3 times the original length. The pitch produced would be the 5th note in the scale (Dominant), in this case, the G note.
If the string is divided into four equal parts, and the string is plucked at the point where the resultant length is 3/4 times the original length, the pitch produced would be the 4th note in the scale (Subdominant), the F note.
If the string were divided further, the 6th (Submediant – ratio 3/5), 3rd (Mediant – ratio 4/5), 2nd (Supertonic – ratio 8/9) and 7th (Leading note – ratio 15/16) degrees would be produced.
This Week’s Challenge: Translating Your Favorite Tunes into Geometric Shapes
When the degrees of the scale are rendered in ratio form, one can start to associate shapes that have similar proportional qualities. The tonic degree, ratio 1:1, could be represented as a square, and the Dominant degree, ratio 2:3, could be represented as a rectangle with the same width: length ratio.
These ratios are not limited to the notes of the diatonic scales; they could be applied to the whole chromatic scale. Please refer to the image attached to this article to show the relationship between the Tonic note (shown as C in the image) and its proportional relationship to the rest of the chromatic scale.
After you have acquainted yourself with the geometric representation of each tone within the chromatic scale, pick a musical piece that you like. Your actual choice is irrelevant, but just make sure that it has enough of a harmonic melody line and chord progression to sustain your involvement in this exercise – and also make sure that you can get your hands on the complete score.
Now, it's time to translate the entire musical score into a series of shapes; translate each musical segment / melody on its own, and figure out a way to line them up so they can still be read as a score. You may want to overlap the instruments on top of each other while giving each a color of their own to differentiate them, or diagram them all in a subsequent manner underneath each other (much like how a traditional score works).
The former allows you to see the visual harmony between the shapes, while the latter still retains a rhythmic quality to the flow. You may run into a few problems with trying to diagram chords – which my suggestion would be to think three dimensionally.
When you find yourself more comfortable expressing your musical self through shapes and mathematical proportions, try to read and create music that way. Dabble in composing via mathematical fractions or via a composition of shapes and patterns. Ask yourself what this type of composition adds to your musical lexicon that differs from traditional norms, and how you may utilize that to your advantage.
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References
Hersey, G. L., Architecture and Geometry in the Age of the Baroque, (Chicago: University of Chicago Press, 2000)
Rameau, Jean Philippe, Treatise on Harmony, circa 1722, translation by Philip Gossett , (Dover Publications, 1971)
Wittkower, Rudolf, Architecture Principles in the Age of Humanism, 4th edition (W.W. North & Company, New York, 1971)
Mahmoud Riad - Mahmoud Riad is an Architect / Musician who graduated from the School of Architecture, Planning, and Preservation at the University of ...
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