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Pythagorean Comma © Kevin Vincent Halpin This text is based on excerpts from Chapters 1 to 4 of Euphony. What is the Comma, and where does it come from? In the sixth century BC Pythagoras discovered that the harmonious relationships between sounds lie in the fact that they are based on ratios of simple whole numbers. The same single note from two identical instruments (unison, 1:1) is neither harmonious nor dissonant. But when two different notes are sounded together only certain notes achieve harmony. It is said that Pythagoras discovered this while walking past a blacksmith’s workshop. He noticed that the ringing of the different hammers on the anvils was producing simple harmonies, and the differences in sound were owing to the different sizes of the hammers. A hammer weighing half the weight of another rang at a frequency (or pitch) twice as high, which is an interval of an octave (ratio 2:1). A pair with weights in the ratio 3:2 sounded nice as well, which is an interval of a perfect fifth. Others say the differences in sound were because sheets of metal of different sizes were being hammered, and the hammers were the same. Others say the blacksmith story is a myth. Even if a myth the physics involved is true and such a discovery could have been made that way. Pythagoras then experimented. He observed that the same ratios exist in stringed instruments, pipes and bells. It is thus a principle which may be quantified and proven true. From this the physics was quantified: strings at the same tension but whose lengths were different in these simple whole numbered ratios, or strings of the same length but whose tension was different by the same ratios, produced the notes which are naturally related to each other harmonically. This is Pythagorean harmony, and the behaviour of a vibrating string will present these numbers phenomenally. A vibrating string is a ground note. In music theory today it would be called a tonic, and in harmonic analysis it would be called the fundamental (or prime). When a duplicate string is stretched tighter its tension is greater. It will then produce a higher frequency. When the duplicate string with that higher frequency is an octave above the first string it will vibrate in two equal parts (the ratio of the higher frequency to the lower is 2:1). A still point then appears on that higher tuned string—at the middle—where the vibration is displaced. Such a point of vibrational displacement is called a node. Tune another identical string even higher so that there are two still points and its pitch is then a perfect fifth above that octave (3:2 higher than the octave string); tuned at a quarter, there are three points of displacement and the sound is then a perfect fourth above the fifth (4:3 higher than the octave + fifth), making its tone a full two octaves above the ground note. It was discovered after Pythagoras’ lifetime that five vibrating parts is a major third again above the fundamental (5:4 higher again). kevinvincenthalpin.com 1 © Kevin Vincent Halpin Pythagoras was also a mathematician. Besides experimenting with strings, bells and pipes, he progressed the perfect fifth twelve times and identified all the unique perfect fifths. He also discovered that these notes would define a total interval of seven octaves, which is a compound perfect interval. Another discovery was that the notes identified by the continued progression are also the very notes in any 12- tone octave, though in a different sequential order. A third discovery was that in each arrangement, seven octaves or one, a comma manifested. The comma was a difference between what a sound should be by these whole-numbered ratios, and the sound actually produced when tuned by them. Mathematically, progressing 3:2 to the twelfth power (3:2 12 ) is 129.7463379 (1.5 x 1.5 x 1.5 and so on twelve times). But, the mathematical expression of an interval of seven octaves is 2 7, 128 exactly. This difference of 1.7463379 between 129.7463379 and 128 is the Pythagorean comma. The mathematics of the Pythagorean comma has a corresponding physical and aural expression. Tune all the Cs on a piano by 2 7 then play C and its octave notes kevinvincenthalpin.com 2 © Kevin Vincent Halpin over seven octaves. You arrive at a sound for the seventh octave of C (its name is C8 or B #7). Now start from C again by fifths, and tune them 3:2 higher than the other. The final note, the final fifth, must be the same note, C8 (enharmonically B#7), but now it sounds noticeably higher than C8 when tuned by 2 7. Two identical pianos, one tuned by fifths and another by octaves, would then not be in tune together. Commas would exist between the notes throughout both instruments. The comma is a serious problem because the differences in sound cause dissonance, an unpleasant-sounding clash between sounds which should be harmonious. Commas prohibited the development of tonal harmony for almost two thousand years after Pythagoras. The comma has similar manifestations across shorter intervals. When Pythagoras used his notes in a single scale they produced sounds which were not in tune the way they were expected to be, similarly. This is the ditonic comma, the same comma in seven octaves but within the interval of an octave. Name of comma Where it resides Mathematical Size as a ratio expression Pythagorean Between C1 and 3:2 12 :2 7 R = 3:2 12 ÷ 2 7 comma B#7 by octaves = 1.013643265 and C1 and B #7 by fifths ditonic comma Between any two 531441:524288 ( ≈ R = notes that should 74:73) 1.013643265, be the same ≈ 74:73 (1.013698630) In the second century AD Claudius Ptolemy (c.90–168) devised just intonation, a tuning system for the diatonic scale (Pythagoras’ scale, the doe–rei–me–fa–sol– la–tee scale). It too could not avoid manifesting the comma in any octave of any key, and the size of the comma was almost twice as great. It was almost half a semitone out. Not only was just intonation worse but now commas could be defined between the two systems. Not good when different instruments playing together could be tuned one by Pythagorean tuning and another by just intonation. Name of comma Where it resides Mathematical Size expression diesis Between any two 128:125 1.024 notes that should be the same syntonic comma Between a 5:4 81:80 1.0125 = 81 ÷ 80 = major third and the (81:64) ÷ (5:4) Pythagorean major third (81:64) schisma Between syntonic 32805:32768 1.00112915 comma and ditonic comma There are many other commas, some named after those in the ancient world who like Ptolemy tried to fix things, could not, and the remaining comma was named after them. There is for instance the comma of Didymus (or Didymic comma), which is found as a difference between A tuned by a 5:4 major third from F and the same A if tuned by a perfect fifth above D by 3:2. Today some people refer to any difference between any two notes which should be the same as a diesis . kevinvincenthalpin.com 3 © Kevin Vincent Halpin Additionally, a syntonic comma can be the difference between six tones and an octave, which should be the same but is separated by a comma. The difference between three major thirds at 5:4 and an octave (which is the same thing as six tones anyway) can be referred to as being either a syntonic comma or a ditonic comma. Since the Renaissance the problem of the comma has been addressed by temperament, methods of adjusting sounds to remove the unexpected differences. In the eighteenth century, ‘equal temperament’ was the method finally discovered to temper the comma effectively. Music written in the centuries since then relies on equal temperament to sound right, including jazz and rock music. But today there still remains a comma in any enharmonic note by equal temperament (for example C # can require a different fingering or breath than is required for D b even though they are enharmonically the same vibrating string or note from a wind instrument). Mathematically, 2 16 :3 10 = 1.10985715 and 9:8 = 1.125. The difference between the two ratios is only 0.01514285. But this is a noticeable comma aurally. Euphony tells the story of the comma and its impact on cultural and scientific history, outlines its solution, and surveys how nature is saturated with the principles underlying music. The book can be viewed or downloaded as a PDF from the link on the Comma page of kevinvincenthalpin.com. kevinvincenthalpin.com 4 .
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