The Origins of Consonance

Understanding how consonance was defined is key to understanding the choices medieval composers made when they began to sound multiple pitches simultaneously.
At its most basic level, consonance is defined as a combination of tones that satisfies the ear. It was further understood that consonant intervals (the name for the distance between two pitches) were stable and pleasant-sounding. Other intervals were thought not to be and were known as dissonances. For medieval composers who wanted to create pieces in which more than one note sounded simultaneously, consonances were necessary for all points of departure and closure in a polyphonic composition, that is a composition with a multiple equally important musical lines.
The concept of consonance derives from Greek notions, particularly ideas advanced in Plato's Timaeus and the teachings of Pythagoras, the Greek philospoher of the 6th century B.C. As none of Pythagoras' writings survive, there is debate as to how Pythagoras decided what is a concord and what is a discord. Likely the empirical observations narrated below were allied with Pythagoras' idea that Natural (counting) numbers were the basis of much of humanity.
For Pythagoras, the music of the spheres, that music responsible for all processes of the earth, and to which Plato called "one visible living being, containing within itself all living being of the same natural order" was understood to be in the same ratios as those of the anvils described below. While the story speaks of anvils, similar proportions could be found in vibrating strings (as in a violin) or vibrating columns of air (as in a clarinet). The following account of Pythagoras's discovery comes from the writings of Boethius, a fifth-century Roman statesman, philosopher, and mathematician who was primarily responsible for transmitting the ancient Greek view of music.

An excellent reference on Pythagoras can be found here.

Pythagoras and Proportion

"It was then principally for the reasons set forth in the previous section that Pythagoras abandoned the judgement of the ears and transferred attention to measuring scales, having no faith in the human ear, which can suffer change in part through its own nature and in part through external accidents. It can also vary with age. He had no confidence either in musical instruments from which are often produced great variation and instability. If, for example, you wish to consider strings, more humid air will weaken the vibrations while dry air strengthens them. The large size of the string will produce a tone of lower pitch, while a thinner string will produce a tone of higher pitch. Or in some way, the original state of uniformity may change. Since the same situation prevailed with all other instruments, Pythagoras thought all these unworthy of consideration and had little faith in them. So for a long time he sought assiduously for other means by which judgements concerning consonance could be firmly established. In the meantime, while he was passing a smith's shop, by the pleasure of the gods, he heard the hammers when struck produce in some way out of the diverse sounds a musical harmony. Astonished at this, which had long been a subject of inquiry to him, he went into the shop and after long consideration decided that the diversity of sounds was due to the force of the blows. In order that he might solve this problem decisively he ordered the men to exchange hammers. But it was found that the properties of the sounds did not depend on the strength of the men, but the same properties were found to exist with the interchanged hammers. When he had observed this he examined the weight of the hammers. Of five hammers, two were found with weights in a ratio of 2 to 1 and these produced sounds an octave apart . He found that the one which was double the weight of the other had a weight four-thirds that of another and produced a sound higher by a fourth. One hammer, which had a weight three halves that of another, produced the consonance a fifth above.... Even before Pythagoras the musical consonance of octave, fourth and fifth were recognised, but Pythagoras was the first to find by the way just described the proportions associated with these musical harmonies. In order to make clearer what has just been said, let us, for example, assume that the four hammers (the fifth being disregarded) have weights represented by the numbers 12, 9, 8, 6, respectively. Then the hammers with weights 12 and 6 were found to be an octave apart. The hammers with weights 12 and 9 (ratio 4 to 3) are a forth apart, and the same is true of the hammers with weights 8 and 6 respectively. The hammers with wights 9 and 6, respectively are a fifth apart. (1)

"On his return home from the smith's shop Pythagoras attempted in various ways to find out whether the whole theory of consonant sounds resides in these proportions. He now turned to strings attaching equal weights to them, and judged their consonances by ear. On the other hand he also varied the lengths of reeds by doubling and halving them and by choosing other proportions, and thus by differing observations developed a complete faith in his results.... Led on by these earlier results he examined the length and thickness of strings. And thus he invented the monochord, concerning which we shall have something to say later. The monochord acqired this designation [Latin: regula] not merely because of the wooden scale by which we measure the dimensions of strings and the corresponding sounds, but because any particular investigation of this kind made with a monochord [regula] is so firmly established that no investigation can any longer be misled by doubtful evidence."

1) In reality the hammer size has nothing to do with the pitch of a struck anvil. The anvil size is responsible.

Boethius' account has been quoted from

Pythagoras as depicted by Boethius

All text © Todd Tarantino 2002-2012.
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