Demonstrating the Pythagorean Intervals

os sa yeyeark:l charges RACTEY is se diols d py NETT: WOUZEL, Uy rans op LIE of a deep-level mathematictCapea nced world is soothed to serenity by means exoerie io students as zation of reality. But it is highly abstract. and can easily appear dwelling on avoid ors instruct many Hence mere numerological mumbo-jumbo. ihis aspect of early Greek philosophy. preierming to stress the more Demonstrating the Pythagorean Intervals A] and credible mundane Pythagorean theorem. This seems a pay, for at least two reasons. grace about Firs. the rotion of a musical/mathematical universe has a charm and aspect general ihis Second, tion. it which can be quite attractive to the imagina EVE BROWNING COLE tical demon - of early Greek thought, in which the force and potential of mathema University of Minnesota, Duluth One ui the discoveries of the early Pythagorean schooi which seems to have proven tremendously exciting to its initiates concerned the relationship between musical harmonic intervals and the first four natural numbers. The basic harmonic intervals can be understood in terms cf ratios, and the numbers involved in those ratios display interesting features. The octave, achieved by stopping a string at the half. way point along its vibrating length, thus consists of the ratio 2:1; che fifth, 3:2; the fourth, 4:3. These numbers added (1 + 2 + 3 + 4) equal 10. The number I0, figured in the pyramid-of-dots shape bv which it would have been represented. therefore itself contains (in several senses) the numbers composing the harmonies, Or, a Pythagorean might say, the number 10 is composed of those harmonies. On this thought at any rate a music-constiiuted cosmology could be constructed. The pyramid that is 10, interlocking with other pyramid-10s, can be imagined to construct physical reality. The number 10, referred to by the Pythagoreans as the mystic “tetractys,” was the perfect number, the number which “contains the fount and root of ever-flowing nature.” lis method of “containment” is explained somewhat obscurely (by Sextus) history of strations are apparently being felt and explcred for the first time in very rich. are tions implica its For s. emphasi ar particul s western culture, deserve number of it could hardly be denied that a perception of the constitutive character Repubthe , itself plays a profound role in Plato's thought, and that, for example with paired y, harmon ike lic’s final definition of justice in the soul as a music-l the Phaedo’s earlier proposed definition of the soul in terms of harmony, is testimony to Pythagersan influence of some interesting Sort.” oo IL in a universe whose comprehensibility is thoroughly predicated upon, and is rea 8 But more broadly, the musical/mathematicai universe of ine Pythago believes ist rational the For gy. cosmolo ist rational truly any to displavs kinship explanaguaranteed by, its accessibility to mathematical (or mathematical like) ol a desired is what be to system. a tion, Deductive certainty appears, in such y, certaint matic paradig possess results tical satisfactory epistemology. Mathema tions. explana ic scientif and phical philoso for ideal an as n and therefore functio This constellation of ideas is therefore of undeniable importance, not only phy. in Greek philosophy courses, but also in courses on carly modern Philoso science, in humanities courses dealing with Greek Culture or the rise of modern weli. as courses theory music and in historicaliv-oriented om The Pythagorean intervals are susceptible of a vivid and lucid classro [T]he whole universe is arranged according to attunement [kara harmonian], and the attunement is the system of three concords, the fourth, the fifth, and used this deraonsiration, which 1 describe in the remainder of this paper. I have is handout à don. demonstration with great success at several levels of instruc forget and ration useful, since the students will become absorbed in che demonst the octave, and of three concords the propertions are found in the four numbers to take notes. as follows: | just mentioned, in one, two, three, and icur.? The Pythagorcans did not themselves discover these basic harmonic inter :, They were already ubiquitously present in the demain of music practice. TI val Garde inravin original contribution was intellectteed nce, As Walter Burkert writes, i yo |, measuring

single-stringed x runs Uisecvered, probably by experimenting ona menochord. instrument), can also bedemonstratedon a single string of any modern siringed instrument. One simply calculates the musical intervals by: measuring the string length (the part of the string that is free to vibrate). On a violin, viola, etc., this is the string length between the nut (raisedportion between: scroll and fingerboard) and the bridge (thin arched slice of wood which rests on the belly of the instrument). This length (x inches)is then divided into two parts (to create the interval of the octave), or three parts (to create the fifth), or four parts (to create the fourth).’ For each interval, one shouid sound alternately the open string and the stopped string. The distance between the two sounds is the musical interval in question. he ratio of the octave, the Py thagoreans saw, is 2:/. On a violin (for xample; of average size, the part of the string that is free te vibrate will be is a ratio of 13:6.5 . So, if we about 13 inches long. Thus, the octave interval measure 6.5 inches from the scrollof a violin, and mark this spot, we should ooctave-point, or the note that is eight notes above the note have located the soundedby ihe open string. (Measuringand marking can be done wich a ruler: and chalk, and should be done in full view of the students ) The ratio of the fifth Give notes above the open siring’s note) is, according to the Pythagoreans, 3:2. Thus. staying wiih the example of the vioiin, we divide the 13 inch length into three parts, each of which will measure 4.33 inches. We 3 the bravest instructor will burst into song atBis point. It is in addition the ntervalbetween any two open strings of vic!in, viola, or celle, se that the less intrepid may simply play any two veighboring sstrings in succession to show that the chalk-marked, Pythagoreanly-derived fifthreally is a fifth. The ratio of the fourth (four notes above the ope we are using a violin, we wil divid 3 ® The tetractys replicates the musica! between the top two layers corresponds to intervals perfectly, since me ratio that of the octave; me ratio be tween ol tom t the fifth: and the relation ° tbe the second two mirrors the ratio of We a eg : into e h. This “frequency har layers mirrors the ratio ofthe fourt basic the for date candi ct perfe ctys a kind of sonar coherence. made> the tetra ealllv. world is in a very real sense mm erin to the Pythagoreans, the do not heer, and therefore We ons. rti propo osed of musicai/r umerical comp cosmic U cannot appreciate, this symphony, pretiselv because We tune it out, in the same way that it is always there. essent we tune out the faint buzz ofthe Fuor mere myStical adherents a ese VIEWS light fixtures in some classrooms. Some and strict mental discipline, we an have maintained that, through meditatior music emanating from th€ near ° develop the ability to hear the myster:ious ity. Others have maintained that its 7beauty is too reali oy great a would roy gee d|thai the Pythagore ac Xnewledee Lrs t natural law to be the ratios represent us. However this may be. it is generally musical intervals in their numerical formulated mathematically.’ then measure 4.33 inchesfrom the scroll-end of the string and make our mark. This is the location cf the interval of the fifth. The fifthis also the interval between the first and second notes of the song “Twinkle, Twinkle. Little Star”: 3 Notes Mi Jane eM Prof. Janc 10 Prof. ted to . ! am indebted nstration For considerable helpin work ing out this demo tment. An anony: Snyder, of Ohic State University’ss Classics Depar journal provided helpful criticisms of an earlier draft. Raven, and Schofieldin The |. Sextus, Ad. Math. Vii. 94-5, as translated by Kirk, 233 a.2. p. idge, Cambr 1982, ed, . 2nd Pre-Sucretie Philosophers 2, Ibid., pp. 233-4. , . > Dn En Je ard 107% 3, Walter Bu i Science in Ancien: Pythagoreanism, Harvard, 1978. Rivers Meta; : 3-7 of the generation of the world à \SMIVETSIEY . Dlate Padi vtr sroduses init

ne syinbols in the symbols written : are tr 2 CUT EVER ra the languaz This bookis written in the maathematical s, circles, and other geometrical figures, without ic is impossibie to comprehend a singie word of it wiihout which cre ugh a dark labyrinth,” Guthrie's quctation irom them physicist Weizir wanders sacker, on pn. 225-6 of History ef Greek Philecepky, vol. L is also illuminating. 8. The idea for this demonstration came originally from a diagram in Robinson's book. Unfertunately there is an error in Robinson’s description of the ratios. The ratios he refers to are all (in terms of bis diagram) “AB to AC”; thus the line of text immed lately above the diagram is mistaken. Also. the diagram itself misrepresents the lengths of the fifthand fourth intervals. i] perfect integers themselves, and sv we are using 7. The lengths are thus ac irrationals or surds to demonst e harmony among rational teal) elements, but it is the ratios represented that areen al lo theFPythagorean insight. Apurist might insist that s of varving length sothat their le pgthe would correspond we use string harmonics, s 2 cneral acknowledg ersiry of here can later be pointed out inc de UT FINS super ip ea es ur. 8.2 3er vearforindivduas ner ‘Sers VOLUME XIX:2-3 385 35 87.50 Geest Co Hutor: Av. Mand: Sditor- Marx Wartofsky | | | Minnesota, Duluth Minnesota3582-2496 USA though he himself iQ sly formulating n forthe Pythagoreans. to perfect integers: 211, 3:2,„5. But sing harmonics are much too difficult for the ssion results. ja fact, the use ofrationals nateur te achieve reliable«class: m