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The Birth of By Stuart Isacoff At left, from Gregor Reisch’s Arithmetica (1503), Boethius and Pythagoras in an imaginary competition using Arabic numerals and an abacus. Above left, Albrecht Altdorfer’s Birth of the Virgin (1525) and above, Masaccio’s fresco The Trinity (c.1428). Opposite page, Piero della Francesca’s The Flagellation (c.1460). 24 Winter 2005 Early Music America Temperament and Perspective In the cradle of the Renaissance, new ideas about proportion affected the development of painting and music A often reflect dramatic shifts in a society’s take on reality. Indeed, when aesthetic issues generate high emotions, it’s a good bet that ideological or religious sensibilities are also at play. These become clear only by examining a multitude of cultural forces – going beyond narrow musical, literary, or painterly issues to develop a broader context. When I set out to learn about historical tunings, it was easy enough to locate material about the mathematics of temperaments but more difficult to discover why people were at each other’s throats over the matter. Why, for example, did Giovanni Battista Doni attempt to discredit Girolamo Frescobaldi in 1640 with charges of drunkenness, stupidity, and worse when they disagreed about the tuning of an organ in Rome? And what of Gioseffo Zarlino’s assertion, a generation earlier, that Vincenzo Galilei’s rejection of sanctioned tuning theory was the equivalent of committing an immoral act? These stories carry the weight of philosophical and societal currents beyond matters of mere personal taste or artistic inclination. The particular bone of contention in these cases, of course, was just how far musicians could deviate from music’s pure number harmonic ratios. The parameters had been established long before. Pythagoras, in the sixth century B.C.E., was said to have discovered in a blacksmith shop, as the smithies hurled their hammers against anvils, that a pure octave is created by RTISTIC DEVELOPMENTS two tones, the higher of which is vibrating twice as fast as the lower, that a pure fifth is produced by two tones vibrating in the ratio 3 to 2, and that a pure fourth by two tones vibrating in the ratio 4 to 3. These musical concordances sounded so right, so harmonious, that they were believed in Ancient Greece to be nothing less than signposts for the eternal order of the universe. By the second century, Clement of Alexandria would gather music’s proportions into the Christian fold by proclaiming that Christ himself had decreed them. Yet another “divine” ratio gained prominence in the 13th century. This was the pure third, beloved particularly by the Renaissance-era English, and constructed, in its major form, by two tones vibrating in the proportion 5 to 4. The simple-ratio musical concords are beautiful – one might say magical. And yet they embody a distressing paradox. Tune a fixed-pitch instrument so that from any note you can find a perfect octave, and you will be unable to do the same for perfect fifths. Octaves are based on multiples of 2 (formed by the proportion 2 to 1) while fifths are based on multiples of 3 (formed by the proportion 3 to 2). Both 2 and 3 are prime numbers; and powers of prime numbers can never be equal. Thus their progeny are incommensurate. Indeed, pure octaves, fifths, and thirds, when multiplied across a keyboard, all produce differing versions of the scale tones. Therefore, except in extremely limited circumstances, the harmonic building blocks they form are not very practicable. Cultural influences In the 15th century, grappling with such intricate problems involving proportion spurred techniques that would radically alter the course of all Western arts. In painting, the outcome was artistic perspective. In music, it was temperament. Though it has been little noticed, the ideas that gave birth to each emerged from the same cultural wellspring. A number of factors made the time ripe for these developments. Michael Baxandall points out in his book Painting and Experience in Fifteenth Century Italy (Oxford University Press, second edition, 1988) that one of the most important skills taught in Florence schools was that of gauging. “It is an important fact of art history that commodities have come regularly in standard-sized containers only since the nineteenth century,” he writes; “previously a container – the barrel, sack or bale – was unique, and calculating its volume quickly and accurately was a condition of business.” Indeed, in this respect painters and merchants had much in common; to the commercial man, “a pile of grain [could be] reduced to a cone, the barrel to a cylinder.” Piero della Francesca, a highly gifted perspectivist painter of the era, actually wrote a mathematical handbook for Early Music America Winter 2005 25 merchants entitled De abaco. So the conscious exploration of proportion was ingrained in everyday life to an extent we can little appreciate today. The tool used most often in commerce was called the Rule of Three (also known as the Golden Rule and the Merchant’s Key). Piero della Francesca described it in this way: “The Rule of Three says that one has to multiply the thing one wants to know about by the thing that is dissimilar to it, and one divides the product by the remaining thing.” To demonstrate, he applied it to the following problem: if seven bracci of cloth are worth nine lire, how much will five bracci be worth? Renaissance merchants used this rule in dealing with questions of pasturage, discounts, the adulteration of commodities, and currency exchange; even with it, things could get horribly complex, because each city had not only its own currency but also its own weights and measures. In any case, proportion was in the air. As Baxandall states, “Piero della Francesca had the same equipment for a barter deal as for the subtle play of intervals in his pictures.” Indeed, Leonardo da Vinci used the Rule of Three for a problem about weight in a balance and came up with the terms 6, 8, 9, 12 – the Pythagorean harmonic scale as discussed in 15thcentury musical and architectural theory. (The numbers 6 and 12 formed an octave, the interval between 6 and 9 and between 8 and 12 generated a fifth, and so on.) Pietro Cannuzio’s Rules of Music’s Flowers placed precisely this mathematical notation at the top of its title page: an invitation, says Baxandall, “to the mercantile eye.” Other aspects of this culture also influenced the direction art would now take. The Aristotelian vision of a universe immutable in form, with all things striving to fulfill their singular purpose in a great chain of being, began to crumble. Renaissance minds shifted away from an abstract sense of formal order and toward the lessons of concrete experience. Thus Filippo Brunelleschi’s radical experiment in perspective art – through which he created the illusion of three-dimensional space on a flat surface – arose because 26 Winter 2005 Early Music America How orthagonals appear to converge at a point in the distance, from Leon Battista Alberti’s De pictura (1435). the personally “subjective” had gained philosophical legitimacy. Leon Battista Alberti, who canonized the technique in the first published method utilizing the “vanishing point” – that imaginary spot in a painting where receding parallel lines beginning at the front (called orthogonals) appear to converge in the “distance” – explained that the effect depended entirely on the idea of seeing things through a “window,” that is, from a particular (not universal) point of view. An event that also helped push things in this direction was the arrival in Italy, shortly after 1400, of copies of Ptolemy’s long-forgotten Geographia, which had been written around 150 C.E. Brought from Constantinople in the form of Greek manuscripts, it provided a lesson in how to depict a It was easy enough to locate material about the mathematics of temperaments but more difficult to discover why people were at each other’s throats over the matter. curved map on a flat surface. An important printing of the work, with additional maps, was issued in Florence in 1482. Ptolemy’s maps helped spur a cartography craze that would continue through the 16th century. Altering proportions There were moral and religious implications to it all. Plato had condemned the beginnings of perspectival art in his own day for distorting the true proportions of things, replacing reality and nomos (law) with subjective appearance and a resulting arbitrariness. (The proportions of music’s simple concordances were, of course, similarly regarded as eternal law; St. Augustine even declared that they should be used as a template in the construction of churches.) Now, however, Renaissance artist Lorenzo Ghiberti would include in his Commentaries translations of Alhazen, the Islamic natural philosopher whose optics sought a balance between mathematical certainty and direct experience. Perspective was being transformed into an emblem of spiritual truth. Peter of Limoges’s Treatise on the Moral Eye, which appeared in an Italian translation in 1496, declared, “a consideration of the eye and of such things as appertain to it is a very useful means of knowing more fully about divine wisdom.” The most significant and useful aspect of what Brunelleschi and Alberti accomplished was that they described the subjective world according to a rational and repeatable procedure. Erwin Panofsky, in his Perspective as Symbolic Form (Zone Books, 1997), calls it the “objectification of the subjective,” or the “carrying over of artistic objectivity into the domain of the phenomenal.” Quoting from Pomponius Guaricus’s De sculptura, Panofsky writes that there was now a mathematically justifiable rule to determine “how far two things ought to stand from another, or how closely they ought to cohere, in order that the intelligibility of the subject matter is neither confused by crowding nor impaired by sparseness.” Of course, as Panofsky also reminds us, “it is essential to ask of artistic periods and regions not only whether they have perspective, but also which perspective they have.” Once the genie was out of the bottle, it took on a life of its own. Visual artists like Leonardo and Piero, not content to depict the world “realistically” (in their subjective fashion), experimented with proportion and placement to create a myriad of illusionistic effects. The results could be awe-inspiring: Masaccio’s fresco The Trinity conveyed depth so realistically that viewers believed they were viewing real holes in the image. The analogy to temperament is striking. Here, too, the inviolable universal proportions of music – what earlier musicians might have labeled “reality” and “law” – were altered through a rational, repeatable procedure, governing how far pitches ought to stand from one another or how closely “they ought to cohere” – for purely subjective reasons. The first text that mentioned the process was by Leonardo’s friend, Franchinus Gaffurius. In his Practica musicae (completed in 1483), he noted that organists were slightly diminishing their fifths. The reason is clear: if an instrument is tuned in the normal Pythagorean practice, stacking pure fifths to arrive at the pitches to be used, the resulting thirds will be unpleasantly wider than pure. By adjusting the fifths – diminishing each by one fourth of the amount by which those thirds would be too wide – those narrower stacked fifths will now yield a third that is pure. Gaffurius, though a strong advocate of Pythagorean tuning, was nonetheless reporting on the emergence of early meantone temperament: the musical equivalent of perspective. By tampering with proportion, temperament (among its meanings are “to tamper with,” “to mix ingredients proportionately,” and “to regulate”) creates an aural “window” through which musical relationships can be shaped. In many temperaments (and this is notable to a high degree in the later non-equal circulating kind), melodies and harmonies can, through the formulaic alteration of pitch placement, project shades and contours to the ear corresponding to the way foreshortening and other techniques project a very particular visual “angle” to the eye. (For a fascinating account of Bach’s Brandenburg Concertos No. 2 and No. 5 performed in Werckmeister III, a classic non-equal circulating temperament, listen to the CD Early on the Pitch label, produced by the American Festival of Microtonal Music under the direction of Johnny Reinhard (available at www.afmm.org.) Deciding the temperament that is most appropriate depends, of course, on many factors, including the musical material at hand and the medium to be used (in the case of harpsichords, for example, this involves such issues as string material and scaling – the length of each string in relation to its pitch). Certainly no single approach fits all cases, either in painting or in music. For instance, Piero della Francesca’s The Flagellation (c.1460) uses conflicting sets of proportions: the floor tile pattern in the area that features Jesus is based on incommensurable ratios, while the floor pattern beyond is based on a simple arithmetical division. Albrecht Altdorfer, in the Munich Birth of the Virgin (1525), writes Panofsky, created an “absolute oblique space” – that is, a space in which there are no discernable orthogonals or frontals. This corresponds in time roughly to Adrian Willaert’s Quid non ebrietas of 1519, which Edward Lowinsky demonstrated was a theoretical argument in favor of equal temperament (“Adrian Willaert’s Chromatic ‘Duo’ Re-Examined,” Tijdschrift voor Muziekwetenschap 18). And equal temperament, in which the octave is divided into 12 equal parts, contains, like Piero’s floor under Jesus, irrational proportions, and like Altdorfer’s Birth of the Virgin, is often described as presenting an oblique or non-directional frame. There are advantages to oblique space and to equal-tempered instruments, however (provided the instruments can accommodate the roughness or “grit” inherent in this tuning – and not all can). As Jean-Philippe Rameau suggested in arguing for its adoption, equal temperament achieves its incredible utility by allowing the imagination of the listener to create the proper musical hierarchy intended by the composer. Rather than being, as it is usually described, a tuning of all “grays” – that is, of no real perspective – equal temperament actually provides a ground for simultaneous multiple perspectives. The listener can discern from the musical context what role a particular pitch is meant to play, and he or she can do this no matter what the key or musical texture. Thus, as music, like all art, continued to stretch free of its old moorings, this tuning – though admittedly hard to achieve and vilified by many – provided an unobtrusive canvas on which composers were free to play. From the Theorica Musicae (1492) of Franchinus Gaffurius: Pythagoras discovering the harmonic ratios in music. Stuart Isacoff is the author of Temperament: How Music Became a Battleground for the Great Minds of Western Civilization (Vintage, 2003) and editor of Piano Today magazine. In Practica musicae, Gaffurius noted that organists were slightly diminishing their fifths. Though a strong advocate of Pythagorean tuning, he was nonetheless reporting on the emergence of early meantone temperament: the musical equivalent of perspective. , Early Music America Winter 2005 27