Pythagoras had added an eighth string to the kithara to assist in modulation and changing mode. In 457 B.C. Phrynis won the music award at the Greek Panathenaic competition with a kithara to which he had added a ninth string to further facilitate changing mode. However, when Timotheus of Sparta added three more strings to make twelve, he evidently stepped beyond the bounds of good taste, for he was viciously mocked by the poet Pherecrates for introducing “weird music,” and unceremoniously booted out of Sparta.19 From this little incident we
B
lyra, and also by the aulos, an oboe with a double reed and up to 15 holes. The aulos often had two pipes, one being a drone (the fundamental 1/1) and the second playing the melodies. Thus we see that harmony was indeed a part of the early Greek music that we think of as “monophonic.” It is true that the primary structure of this music was melody, but harmony was understood, sung, and played.
oth Greek music and music theory flourished in the third and fourth century B.C. At the theater, the chorus was accompanied by the stringed kithara and
see that the cultural inertia resisting innovation in musical instruments is nothing new. The “traditionalists” always forget that their tradition was once a novelty.
A Greek kithara. Pythagoras added an eighth string to accommodate various scales and modes, Phrynis added a ninth and won the Panathenaic music award, but when Timotheus added three more strings to make twelve he was
kicked out of Sparta for making “weird music.”
At this time a group of Pythagoreans began to call themselves the
“Harmonists.” In Greek mythology Harmonia was the daughter of Venus and Mars. Cupid was her brother. She had a son and four daughters with her husband Cadmus, but her grandmother, Juno, mother of Mars, despised Venus and took it out on poor Harmonia by persecuting her children. Harmonia and Cadmus were so Timotheus later settled in Athens where his innovations were tolerated, and where he received encouragement from another innovator, the great dramatist Euripides.
distraught by this that they asked to be sent away to the Elysian fields, resting place of the virtuous. Harmonia personified order, perfection, and peacefulness.
The Harmonists took her name to symbolize those qualities in the pure harmonies of music.
The greatest scholar among the Harmonists was Archytas from the Greek colony in Tarentum, Italy. He was the first in history to notice that the interval of the third that singers naturally sang was not the sharp Pythagorean / Ling Lun third derived by the series of fifths (81/64) but was rather the pure harmonic tone represented by the relationship 5/4. Archytas was not aware of the harmonic series, but his pure third was the fifth harmonic in that series.20 The singers instinctively sang this harmonic tone unless forced sharp by an instrument tuned to the Pythagorean third. Archytas noticed this, calculated the relative string lengths on a monochord, and recorded for history the pure just intonation third.
Archytas did not stop there with his observations and experiments, and later added a tone based on the seventh harmonic, an alternative wholetone about a quarter of a semitone sharp of the Pythagorean 9/8, a tone represented by the ratio 8/7. Thus, he is credited as the first in history to recognize and document the natural tones based on the fifth and seventh harmonics.
Greek musical scale theory at this time was based on the concept of the tetrachord, the interval of the fourth, and the division of this interval into three subintervals by the placement of two other tones between the tonic and the fourth.
These four tones were called the hypate, parhypate, lichanos, and the mese. The first tone was always the fundamental 1/1, and the mese was always the pure harmonic fourth 4/3. The four tones created three intervals, and these were always represented as ratios of whole numbers since the Greek Harmonists understood this to be the true expression of a harmonic interval. The two internal tones gave the tetrachord its particular quality. When the three intervals were two wholetones and a semitone the tetrachord was called diatonic. When the three intervals were a minor third and two semitones the tetrachord was chromatic. A major third and two quartertones created an enharmonic tetrachord. There were many different
sizes for a “wholetone,” for a “semitone,” or “quartertone,” and also for either a
“major third” or a “minor third.” These terms were general, not specific. For example, a wholetone could be 9/8, the sharper 8/7, or the flatter 10/9. A typical diatonic tetrachord might be composed of the three intervals:
16/15 10/9 9/8
These are a semitone, a small wholetone, and a Pythagorean wholetone.
These three intervals would create a tetrachord scale with the following four tones:
1/1 16/15 6/5 4/3
These are the tonic, a semitone, a minor third, and the fourth. Musicians unfamiliar with seeing intervals and musical tones represented this way may find the fractions (or ratios) confusing, but it is worth attempting to understand these harmonic expressions, because they are the very foundation of the musical arts. In a more familiar language, the four note scale above, in the key of C is:
C C# Eb F
The intervals are then clearly recognized as a semitone and two tones. In our modern equal temperament all the semitones and tones are the same size, but this is not the case in just intonation, or pure harmonic music, and was not the case with the early Greek Harmonists. The tetrachord above (with various sizes of tones and semitones) is found in Babylonian, Greek, Balinese, East Indian, Arabic, and African music. For musicians interested in deepening their understanding of this fundamental scale theory, or expanding their use of scale options, see Divisions of the Tetrachord by John Chalmers.21
The Harmonist Archytas was the first to define tetrachords in all three classical genera, the diatonic, chromatic, and enharmonic. Octave scales were
constructed by sticking two tetrachords together with a disjunctive tone between them. For example, if we took the tetrachord above, added it to itself with a wholetone in the middle, we would have:
C C# Eb F - G Ab Bb C
This is the Greek Dorian (Ecclesiastical Phrygian), and also the East Indian Hanumat Todi, and the Arabic Ishartum. In modern, western style these tones would all be equal tempered tones and semitones, but in the other traditions the precise harmonic intervals can vary. In any case, this is the fundamental form of scale construction upon which our entire western musical tradition was founded.
In terms of just intonation harmonic intervals, the above scale could look like this:
1/1 16/15 6/5 4/3 3/2 8/5 7/4 2/1
The Greek modes were created by shifting the tonic within a given scale. In the Greek Dorian mode in C above, if we shift the tonic to the Ab, we have the Greek Lydian, our modern diatonic major scale in Ab.
Archytas was also the first to describe the difference between the arithmetic and harmonic division, or “mean,” in the construction of scales. The arithmetic mean was simply the equal division of a string as performed by Pythagoras. This division is spatially equal but not aurally equal. By dividing the wholetone harmonically, Archytas introduced the semitone 16/15 which has stood the test of time as a popular just intonation semitone of choice.22 This harmonic semitone is about a quarter of a semitone sharper than the Pythagorean version derived from a series of fifths.
Archytas and the Harmonists can be credited with liberating harmonic music from the Pythagorean series of fifths. They recognized that the semitone, the thirds, and the sixths were harmonic tones in their own right. By instinctively hearing and recognizing the harmonies of the higher harmonics, they set pure
harmonic music on its way. Archytas himself must have had an extraordinary ear.
He literally picked these pure harmonic tones out of the air without any tradition or aid to guide him, and some of his enharmonic tetrachords suggest a keen ear that was able to discriminate among a variety of tiny intervals.
Meanwhile, in the far east, the harmonic tradition of Ling Lun had evolved along lines very similar to the developments in Greece. Chinese musicians had, like Archytas, recognized the tones based on the fifth and seventh harmonics.
During the fourth century B.C. a bronze kin (small koto), called the “scholar’s lute,” was tuned as follows:
1/1 8/7 6/5 5/4 4/3 3/2 5/3 2/1
In our nomenclature, in the key of C, this scale would be:
C D Eb E F G A C
The harmonically interesting feature here is that the intervals all appear in Greek music at this time, although the scale style is entirely distinct from anything one would see in Greek music. This shows that the harmonic fundamentals are universal, but that the musical application is cultural. The distinguishing feature in the scholar’s lute scale is that the tetrachord, the interval of the fourth (4/3), is divided into four intervals rather than the Greek three. This allows for both the
Composer Harry Partch has commented that “in a healthy culture differing musical philosophies would be coexistent, not mutually exclusive,” and certainly the era of the Greek Harmonists witnessed this sort of musical openness and curiosity. Both Plato and Aristotle extrapolated harmonic theory and attempted to apply the sense of order and perfection to human morality and governance.
Perhaps they carried the harmonic concepts too far, but in any case we can see how powerful was the influence of early music theory on mathematics, philosophy, and political theory.
major and minor third in the same scale. The wholetone (8/7) is supported by the seventh harmonic of the fundamental, and is slightly sharp of the wholetone of Ling Lun and Pythagoras (9/8) which is based on the third harmonic, the pure fifth (3/2).
These harmonic discoveries were made almost simultaneously in both China and Greece, and it is safe to say they were made independently. Thus we see that the ears of the musicians in both the east and west were in complete agreement as to the precise harmonic tones that were considered pleasing. Without knowledge of the harmonic series or other physical qualities of sound that would be discovered later, these musicians independently found by ear the just intonation ratios of pure harmony.
However, the practical matter of building fixed-tone instruments that allowed free modulation was about to raise its head, and everything would be thrown into question. The Pythagorean notion that a cycle of fifths should fit into an octave would not go away.