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Aristoxenus set out to fashion a theory that would explain and support his position. He was a formidable scholar, author of some 453 published works, including biographies of Pythagoras and Archytas, and his Elements of Harmony
ristoxenus was the son of musician Spintharos in the city of Tarentum, home of Archytas a generation earlier. As a student of Aristotle, he learned the theories of Pythagoras, Archytas, and other Harmonists. However,
Aristoxenus became obsessed with the Pythagorean comma, the amount by which a series of 12 fifths was sharp of seven octaves. Aristoxenus subscribed to the notion that this series of fifths should be a cycle, without having to make the comma adjustment, and that this cycle should fit neatly into the octave. He proposed distributing the Pythagorean comma among the 12 fifths to make them fit. His proposed solution, however, put all the fifths out of tune, hardly acceptable to the singers of Greek choral music, the fifth being the most prominent of all musical intervals. The Harmonists were appalled, and as Harry Partch noted two millennia later, “the war was on.”
is probably the earliest extant treatise on Greek music theory.23 In laying the foundation for his theory of tempering the fifths, Aristoxenus claimed that the tones of the scale had a narrow range of acceptability that the ear would tolerate.
This, of course, is entirely subjective since the acuteness of the ear varies widely among musicians, and the ear might grow accustomed to anything. On the other hand, the Harmonists argued, a pure harmonic interval is either in tune or it isn’t.
A cycle of fifths that fit into an octave, however, required that these intervals be compromised, so Aristoxenus held firm to his conviction that the ear would accept such a compromise.
He gave new definitions for fourths and thirds and wholetones. He defined a wholetone as the interval between his tempered fourth and tempered fifth, or simply one-sixth of an octave. A semitone was a twelfth of an octave, and a fourth was a semitone and two wholetones. It almost seems that he was describing equal temperament, although he did not use any such term. Nevertheless, he abandoned harmonic ratios altogether, and described his scales in units that were 1/12 of a wholetone. Therefore his tetrachords were simply divided into 30 parts, and those parts subdivided into intervals. For example, two of his chromatic tetrachords were 4, 4, and 22 parts, or 6, 6, and 18 parts. He claimed that any interval smaller than a fourth was a dissonance, and therefore could be of any size. He allowed the wholetone to be divided “as melody admits of half-tones, thirds of tones and quartertones, while undeniably rejecting any interval less than these.”24 Thus he set a lower limit on the size of musically usable intervals, and a limit on how finely a wholetone could be divided.
Aristoxenus clearly understood the music of his time, and some of his tetrachords are fair approximations of those of Archytas. His scholarship was thorough. Still, his logic seems forced, somewhat circular, and his rationalizations are sometimes even contradictory. For example, Aristoxenus maintained that the ear, not numbers, should determine proper tuning. He faulted the Harmonists for relying on numbers rather than on the ear. On this point he clearly misinterpreted the history of harmonic tuning. The intervals of the Harmonists did not evolve
from mathematics. On the contrary, the intervals were recognized by ear, and the numerical relationships were discovered when theorists from Ling Lun to Archytas applied these intervals to pipes or strings. The ear clearly favors the simple
harmonic ratios (2/1, 3/2, etc.), the octave being the simplest and most obvious of these.
Aristoxenus also says that “the subject of our study is the question, in melody of every kind, what are the natural laws according to which the voice in ascending or descending places the intervals.”25 Left to itself, the voice places the intervals at the pure harmonics since this is the easiest and most natural. We are again reminded by Aristoxenus’ comment that Greek music was primarily monophonic, with choirs perhaps doubling at the octave. However, we would be mistaken to assume, therefore, that the Greek musicians and theorists did not understand harmony as two tones being sounded simultaneously, for surely they did. That is what the monochord of Pythagoras and Archytas was all about. The earliest notation is melodic, so we can only speculate what singers may have achieved harmonically. It is hard to imagine that with all of the choral singing going on, no one ever sang a fifth or third harmony part, or droned a tonic while the melody was sung, if not by design, then at least by experiment or even by accident. It is equally hard to imagine with all those strings being added to the kithara that no one ever plucked more than one at the same time or sang and played different parts. Almost certainly they did, and in any case, they knew what a simultaneous fifth or fourth or third sounded like. We tend to think of the
evolution of knowledge, in this case musical knowledge, as being linear - monophonic, homophonic, polyphonic, harmonic - but in fact the evolution of knowledge is never quite so neat. Clearly, early Greek music was monophonic in execution, perhaps with the octave doubling (which is itself a harmony, albeit the simplest possible harmony), but it is equally clear that these early Greek musicians fully understood more complex simultaneous harmony, learned the musical intervals from such harmony on their monochords, and built their melodic scales from these natural harmonic principles.26
Finally, in some cases, Aristoxenus is just simply wrong. He claimed that the Harmonists “fabricated rational principles, asserting that height and depth of pitch consist in certain numerical ratios and relative rates of vibration - a theory utterly extraneous to the subject and quite at variance with the phenomena.”27 Here, Aristoxenus goes too far. He attempts to throw out the discoveries of Ling Lun, Pythagoras, Archytas, and others. His attempt to temper the fifths to fit a cycle of twelve into an octave is not in itself wrong; it is a subjective approach to music, and has some practical value, but to base this approach on the assertion that the harmonic ratios are “extraneous” and “at variance with the phenomena” of acoustics is wrong. All tempered music is a sacrifice of harmony for a gain in simplicity. Furthermore, to base everything on the octave, 2/1, as Aristoxenus does, and then reject all other harmonic ratios is entirely inconsistent.
Meanwhile in third century China, musical theorists were struggling with these same issues. Rather than temper the fifths, King Fang (“King” is a name, not a title) extended Ling Lun’s series of 3/2 fifths looking for the point at which it would match an octave (an impossibility we know will never happen). He calculated the lengths for a series of 60 lü, bamboo pipes like those of Ling Lun.
He observed that the fifty-fourth lü was only a tiny bit sharp of a higher octave of the fundamental. The comma between the two was infinitesimal, about 36- thousandths of a semitone. Western music theorists did not discover this until Nicolas Mercator proposed a similar system in the 17th century, and later the 53- notes-per-octave scale was used by some composers as a good approximation of just intonation.
In China, however, the notion of tempering the fifths, although considered, was not practiced. The practice may have even been forbidden by some Chinese rulers who considered such a compromise to be risking the wrath of the gods.
There was a belief that altering the harmonies violated natural principles and would lead to social decay. There are stories of emperors traveling the country, requiring local musicians to play for them, to ascertain the stability of the regions.
In fairness to Aristoxenus, and in the spirit of co-existing musical systems, his proposed tempering of the fifths was an inevitable theory as later systems of temperament attest. Aristoxenus erred in discounting the pure harmonic ratios as extraneous. Had he been more precise in his evaluation perhaps he would have viewed the tempering for what it was, a practical compromise for the benefit of fixed-tone instruments.
It was during this same time, in Alexandria, that another musical innovation was made that would influence music even until our time. Ktesibios of Alexandria designed and built the first keyboard instrument, a pipe organ called a hydraulos.
The wind for the pipes was supplied by hydraulic pressure from water enclosed in a container that was fixed with a hand pump.
Today, music can be said to be divided into two separate streams. On the one hand there is keyboard and fret based music which has traditionally required temperament, most notably equal temperament. On the other hand, there still exists today pure harmonic music as performed by a cappella choirs, barbershop
quartets, certain chamber groups, string and brass ensembles, and as written by many modern composers. These two streams of music are isolated from each other because they use different fundamental tuning systems. The third century B.C. can be seen as the point at which the streams began to diverge. The theories of
Aristoxenus, and the invention of the keyboard set music on two separate courses, In Greece, the tempering theories of Aristoxenus met an immediate challenge by Euclid (c. 300 B.C.), the founder of western geometry. Euclid exposed Aristoxenus’ most obvious errors, and asserted correctly that the
harmonic ratios were natural, not theoretical. He demonstrated that six wholetones (six 9/8s) were sharp of an octave (Theory 9 in his Section of the Canon), and therefore showed that describing a wholetone as “one-sixth of an octave” was arbitrary. Euclid showed how to divide an interval geometrically, and
demonstrated that the Aristoxenean ideal of dividing a wholetone into “halves,”
“thirds,” and “quartertones,” was impossible, since 9/8 cannot be divided into aurally equal parts by ratios of rational numbers. Indeed the war was on.
and although many musicians since have temporarily succeeded in merging the two streams, they remain asunder. The 21st century may see these streams merge once and for all.