I’ve touched on this concept before, but today we’re going to have a look at Boethian Proportioning in some detail. What we need to remember is: once Boethius wrote his treatises about Philosophy, Mathematics, and Music, he became the default source for such information for about 1000 years. Boethius didn’t follow Vitruvius’ theory of Proportion. Instead, Boethius based his Mathematical theory on the much older Pythagorean Greek modes. Granted, Vitruvius also worked on Greek Mathematical Theory, but his was based on the later model of the perfect natural human form as derived by Polycleitus in the 3rd Cen BCE.
Being as the culture he lived in more accepted the Ancient Pagan Greek Philosophers and Mathematicians to the more “modern” Pagan Romans, and that Boethius himself was a Christian, his writings on Mathematics, Philosophy, and Music replaced Vitruvius’ proportional modes. The nearly contemporaneous writings of St Augustine of Hippo that argued against the perfection of Vitruvius’ standard (6:10:16, which simplifies to 3:5:8) with a more theologically supportable 6:12 (the number of days of Creation, and the number of Disciples, being twice the number of days of Creation).
Boethius became the mathematical theorist that everyone learned of in European (Christian) Aristocratic Education and later in the University System in the West. It wasn’t until Vitruvius’ “Ten Books” was rediscovered and retranslated in 1414 that the return to the Roman Golden Age began. That’d be that era we refer to as the Renaissance, though the real swing didn’t get going until the next infusion of Classical Thought in 1453… but that’s a wholly different interest of mine.
By the Fifteenth Century, the traditions of the trades had been so calcified in Boethian theory that it took another couple hundred years for Vitruvian Theory to bleed into common usage. Even then, it was modified by the innovation of the decimal point… But that’s much higher order stuff than we need to look at here.
Boethius’ Musical Theory was outlined in his “Fundamentals of Music”, published sometime around 520 CE, by pulling together the essential strands of musical theory as described by Pythagoras, Plato, and Aristotle. Book I of his five-book tract became the primary textbook for the study of music theory (Fundamentals of Music/De Institutione Musica). The series of proportions that Boethius described for music are the same as he described for Architecture and Design, as well as the movement of the heavens and order of the Natural World (Of Mathematics/De Arithmetica, The Consolation of Philosophy/De Consolatione Philisophiae). And we can see through the extant examples of Western Design in furniture and architecture from the Middle Ages, it is the Boethian modes that were maintained in the artisan trades. Don’t mistake illiteracy for ignorance. These artisan tradesmen might have been “uneducated” but they were hardly ignorant.
So what are these modes? Well, we have the Unison, the Double, Sequitertian, Sesquialter, the Triple, Quadruple, and the Sescuioctave. Or more commonly, there is a Note (1:1), an Octave (2:1), Perfect Fourth (3:4), Perfect Fifth (2:3), Octave plus Fifth (3:1), Double Octave (4:1), and a Tone (9:8). The musical Third (5:4) and the Major Sixth (5:3) weren’t given a Latin name by Boethius. Of all these “Notes,” almost everything in Medieval European design is covered with just four of these intervals. 1:1, 2:1, 4:3, and 3:2. Seem familiar? It’s similar to that “four chord” (I-V-vi-IV) pop song from the previous blog post. Each interval can also be easily counted on one hand, simplicity matters.
Going back to the Greek Era and maintained by Boethius, this series of intervals were typically expressed visually as such:
Does we need to understand the Mathematical Theory and Philosophy behind these intervals to design and make furniture? Not at all. All that is required is to know that these are the “pleasing” proportions. This is how out artisan antecedents laid out their designs, and the “why” behind it was often unknown and irrespective of their trade. Most simply: they did things this way because the Master Craftsman who taught them, told them to do it this way and they in turn taught their apprentices the same way.