Pythagoras c. 570 – 495 BC
Pythagoras was an Ionian Greek philosopher and founder of the religious movement called Pythagoreanism. Most of the information about Pythagoras was written down centuries after he lived, thus very little reliable information is known about him. He was born on the island of Samos, and may have travelled widely in his youth, visiting Egypt and other places seeking knowledge. Around 530 BC, he moved to Croton, a Greek colony in southern Italy, and there set up a religious sect. His followers pursued the religious rites and practices developed by Pythagoras, and studied his philosophical theories. The society took an active role in the politics of Croton, but this eventually led to their downfall. The Pythagorean meeting-places were burned, and Pythagoras was forced to flee the city. He is said to have ended his days in Metapontum.
To Pythagoras, music was a way of physically experiencing mathematics (ratios).
It was one of the four arts in the Quadrivium.
The quadrivium comprised the four subjects, or arts, taught in medieval universities after the trivium. The word is Latin, meaning “the four ways” or “the four roads”. Together, the trivium and the quadrivium comprised the seven liberal arts. The quadrivium consisted of arithmetic, geometry, music, and astronomy. These followed the preparatory work of the trivium made up of grammar, logic (or dialectic, as it was called at the times), and rhetoric. In turn, the quadrivium was considered preparatory work for the study of philosophy and theology.
According to legend, the way Pythagoras discovered that musical notes could be translated into mathematical equations was when one day he passed blacksmiths at work, and thought that the sounds emanating from their anvils being hit were beautiful and harmonious and decided that whatever scientific law caused this to happen must be mathematical and could be applied to music. He went to the blacksmiths to learn how this had happened by looking at their tools, he discovered that the note the hammers made when they hit the anvil, is directly proportional to their weight. LINK
Pythagoras and his followers developed ratios to represent intervals. The general principle is: if the frequency of a tone is X, the frequency of its octave is 2X; the frequency of the fifth (above the octave) is 3/2X, etc. The frequency of vibration of a string is inversely proportional to its vibrating length. Frequency ratio:
2:1 octave (twice the vibration rate; 1/2 the vibrating length
3:2 fifth
4:3 fourth
5:4 major third
etc.
By using the ratio 3:2, the pure (beatless) fifth, Pythagoras enabled construction of the diatonic scale through successive fifths when reduced to a single octave. Chromatic pitches may be obtained simply by extending the succession of fifths. At this point, a rather significant problem arises. Because of the Pythagorean scale derivation utilizing the pure fifth, enharmonic pitches are not equivalent! In other words, F# does not sound the same as Gb, for example. As we will discover under equal temperament, our present-day tuning system utilizes a fifth which is slightly out of tune … Pythagorean tuning becomes less palatable as music becomes more chromatic. LINK
 
- Play the pure fifth, from C to G. Compare to equal temperament tuning, which is more narrow by two cents. The equal temperament fifth is phasing slightly.
- Other fifths are pure as well, except for the wolf. Play the wolf, from G# to Eb. Pretty nasty, eh..
The subject of music within the quadrivium was originally the classical subject of harmonics, in particular the study of the proportions between the music intervals created by the division of a monochord. A relationship to music as actually practised was not part of this study, but the framework of classical harmonics would substantially influence the content and structure of music theory as practised both in European and Islamic cultures. LINK
The quadrivium may be considered as the study of number and its relationship to physical space or time: arithmetic was pure number, geometry was number in space, music number in time, and astronomy number in space and time. Morris Kline classifies the four elements of the quadrivium as pure (arithmetic), stationary (geometry), moving (astronomy) and applied (music) number.
Pythagoreans elaborated on a theory of numbers, the exact meaning of which is still debated among scholars. Another belief attributed to Pythagoras was that of the “harmony of the spheres“. Thus the planets and stars moved according to mathematical equations, which corresponded to musical notes and thus produced a symphony.
The pythagorean scale c : d : e : f : g : a : b : c is given by 1/1 : 9/8 : 5/4 : 4/3 : 3/2 : 5/3 : 15/8 : 2/1. The most pure interval aside from the octave is hence the pythagorean fifth, given by a frequency ratio of 3/2. A comparison of progressions of pure intervals reveals that it is (mathematically) impossible to arrive at exactly the same point: A circle of twelve pythagorean fifths is about 74/73 sharp of seven pythagorean octaves, the difference is known as the pythagorean comma. Likewise four pythagorean fifths are 81/80 sharp of a pythagorean third two octaves up, known as the syntonic comma
The classical music traditions of Oriental and Western music are all based on the same musical theories of scale building credited to the ancient Greek Pythagoras (570 -495 BC). As was the case in other artistic and scientific fields, Arabs translated and developed Greek texts and works of music and mastered the musical theory of the Greeks (i.e. Systema ametabolon, enharmonium, chromatikon, diatonon, quadrivium). Such inter-influences can often be traced in language; for example, the word Shî’ir (poetry in Arabic) bears much similarity to its equivalents in other Semitic languages (such as Shûr in Aramaic and Shîr in Hebrew), and Shîro in Babylonian. Over the centuries the three traditions followed a separate path of development, each of which is now recognized as a form of high art, but each with a distinct musical ‘dialect’. Arabic classical music went through an important period of early development during the 9th through the 12th centuries when the Arabs ruled large parts of the Middle East, North Africa and southern Europe. Arabic scholars made significant contributions in studying and interpreting the works of the ancient Greeks; the Arabic system of modes known as maqamat came out of these early studies. In Arabic maqamat, the octave is divided into 24 equally spaced quarter-tones. Classical Arabic composers show skill in the development of these quarter-tones not through harmony or polyphony as in the West, but through melody. To Western ears trained in 12 tone equal temperament, these quarter-tones can sound odd at first and are sometimes referred to as micro-tones. Western classical music, by the time of J.S. Bach, had developed into a system of tuning known as equal temperament, where the musical octave is divided into 12 equally spaced half-tones. These tones are easily visible on any piano or fretted guitar today. Equal temperament enables Western composers to create works using complex harmonies and polyphony. However it eliminates quarter tones, as they do not work well in harmony / chords. Before J.S. Bach pianos did have quarter tones – image. The high point in the development of the Turkish classical style is during the Ottoman Empire period from the 15th through the 20th centuries. Turkish makams closely reflect Pythagorean thinking in the use of proportional tuning. The eighth-tone is equal to 1 Pythagorean Comma (approximately 23 cents), which plays a crucial role in micro-tonal pitch development within any mode.
The earliest surviving fragments of the writings on music by the fourth century BCE Greek peripatetic philosopher and writer on harmonic theory, music and rhythm, Aristoxenus of Tarentum, are papyri found at Oxyrhynchus.
“Perhaps the most amazing papyrus fragment is a large excerpt from Aristoxenus’ Rhythmica, a part of which was first published in 1898 as fragment 9 of the Oxyrhynchus Papyri. In 1968 it was revealed that fragment 2687 of the Oxyrhynchus Papyri completed columns 2-4 by supplying fourteen or fifteen lines at the bottom; this same fragment added substantially to columns 1 and 5. Nearly one hundred lines of the text have now been uncovered in papyrus dating from the third century C.E. But this is not all. Fragments 667 and 3706 of the Oxyrhynchus Papyri preserve in characteristic Aristoxenian language an analysis of conjunct and disjunct scales and of genera. These fragments, too, date from the second or third centuries C.E. and may very well contain parts of the sections of Aristoxenus’ Harmonica missing in the manuscript tradition” (Mathiesen, “Hermes or Clio? The transmission of Ancient Greek Music Theory”, Palisca, Baker, Hanning [eds.] Musical Humanism and its Legacy. Essays in Honor of Claude Palisca [1992] 5-6).
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