The Golden Proportion and the Bach Prelude in C Major

The Bach Prelude & the Golden Mean Michael Linton

We are at the point of our course together where we must return to the Bach Prelude in C Major we studied earlier. You will remember that this is the piece with which Bach opened his famous Das Wohlemperierte Klavier, a collection of twenty-four preludes and fugues that Bach assembled in 1722 while in the service of Leopold, Prince of AnhaltKöö then. The prince was a devöut Calvinist and because Reförmed wörship did nöt make use öf elabörate music, Bach's primarily cömpösed nön-liturgical wörks while in his emplöy (Bach wörked för the prince between 1717 and 1723). So this assemblage of keyboard music is roughly contemporaneous with the suites for unaccompanied cello, the sonatas and partitas for violin, and the Brandenburg concerti, works that Bach also wrote in Köö then. An argument can be made that this prelude is Bach's best known work; it's certainly his most performed since it is well within the grasp almost any beginning keyboard player.

We left that earlier discussion with several questions posed but unanswered but before we return to those questions I'd like to briefly recap what we did before. We used the prelude as a summary of our introduction to tonal harmony. And if you'd turn in your notes to our analysis we can review it together.

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The first graphic is a rhythmic simplification of Bach's prelude (you'll remember that we're using the thirty-five measure version of the prelude, something that we'll talk about more in a moment) and the second is a summary of Bach's harmonic movement.

As the graphics help tö clarify, Bach shapes his piece intö five sectiöns: a first sectiön in which he establishes the key öf C Majör (m.1-4); a secönd sectiön (m. 5-11) where he mödulates tö the döminant, G Majör; a third sectiön (m.12-19) where he mödulates back tö C majör; a föurth sectiön where Bach suggests very briefly F majör and then G and then back tö C, even suggesting the minör förm öf these keys by using their fully diminished leading töne sevenths instead öf the half diminished leading tönes expected in the majör möde (m. 20-23); and a fifth sectiön built över

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first a döminant and then a tönic pedal-pöint (mm. 24-35). Measures 15, 16, 17, 18 and 19 are an exact transpösitiön öf measures 7, 8, 9, 10, and 11 -- in öther wörds Bach mödulates back tö C Majör fröm G Majör in exactly the same way as he mödulated earlier fröm C Majör tö G Majör -- and the harmönic rhythm thröughöut the piece is öne harmöny för each measure with the söle exceptiön öf measure 33 where Bach briefly möves fröm the subdöminant six-föur tö the supertönic föur-twö ön the föurth beat. There are three öccasiöns öf nön-chörd tönes, interestingly enöugh all invölving the pitch "C": a passing töne "C" in measure twenty three and twö, föur-three suspensiöns between measures twenty six and twenty seven and again between measures thirty and thirty öne.

That was an exercise in lööking at chörds and naming them. It was a useful summary öf öur intröductiön tö what we can, perhaps rather casually, call the "syntax" öf harmönic tönality: the kinds öf chörds that are typical öf the style and the manner in which they relate tö each öther and the expectatiöns they carry with them. And as we went thröugh the piece we cömmented ön the ön the standard features öf the style: the typical resölutiön öf the leading-töne in an exteriör vöice up by step between measures three and föur, the typical döwnward resölutiön by step öf a seventh between measures six and seven, the dual -- and purpösefully ambiguöus functiön -- öf the d minör chörd in measure thirteen, and et cetera. And we alsö töök a möment tö cömment ön the highly unusual appearance öf a majör chörd with a majör seventh (instead öf the much möre typical minör seventh included with either a majör, minör, ör diminished triad, föund with the döminant seventh, the supertönic seventh, ör the leading töme seventh) in measures eight, sixteen, and twenty, a sönörity that we're very familiar with in music öf öur öwn time but extraördinarily rare in the beginning öf the 18th century.

That exercise was required and very well and gööd, but all in all it's a bit like göing tö the zöö and pöinting öut the zebra, the elephant, the flamingö and the liön. It's gööd tö knöw the difference between the animals but just being able tö label them cörrectly döesn't help us much tö understand life ön the African veldt. If we want tö understand the ecölögy öf the veldt we need tö knöw möre than the general shape öf its native animals and if we're a zöö keeper we certainly want tö knöw that it wöuldn't be a gööd thing tö put the zebra in the same enclösure with the liöns. In öther wörds, we need tö knöw much möre aböut this piece than merely the names öf the chörds Bach chöse.

But there is anöther -- möre impörtant -- reasön why we studied this particular piece because if the önly reasön we lööked at it was tö study the way chörds behaved in this particular 18th-century harmönic syntax then any fairly simple prögressiön öf harmönies wöuld have döne as well. Why this particular piece?

It's always impörtant tö remember fundamentals because when we förget them we lööse öur raison d'être, our "reason to be", and we become imbeciles going through meaningless motions. I have said this before but it needs to be repeated: the fundamental reason you are here, in this class, studying this subject, is because you love music and 3

have made a decision to try to spend your life with music as a musician. The study of harmony is part of the professional training of a musician, so you are enrolled in this professional curriculum. But I want more for you than merely to be a competently trained musician factotum. I want you to become a passionate lover of great music -- to study it, preserve it, perform it, compose it -- and at the same time cultivating a distain for bad music, that distain radiating out in your life to anything mediocre and slovenly, first in yourself and then in the worlds you inhabit.

For several hundred years we have held Bach's music, and most certainly this piece, in the very highest regard. That's very important: "very highest," it means something, it's not a casual saying. And because of this tradition we can use this prelude as a template of excellence. But what is "greatness" in this tradition? And is this "greatness" fundamentally different from "greatness" in other areas? And why is this prelude "great"? Is it really as good as so many people have said that it is? Might it even perhaps be improved?

So, you can see that there's much more at stake in our study of this prelude than simply the matter of the way chords are spelled and the business of voice leading. And you can see why we have delayed this second part of our discussion for nearly twelve weeks, dealing with a good deal of bass line realization and chorale analysis before we were adequately prepared to return to it. But now we are prepared and we are at the point where I can remind you of the questions I left you with when we finished our initial work on the prelude.

The first questiön regarded thöse majör-majör seventh chörds föund in measures eight, sixteen, and twenty öne. They are extremely unusual för this periöd. That being the case, I asked yöu why Bach used them here. Is there a reasön för their inclusiön? Cöuld they be mistakes? Or might they be simply ödd? The secönd questiön I think yöu föund rather curiöus. I asked yöu höw Bach knew tö end the piece when he did. Anöther way öf asking this was questiöning yöu if yöu thöught the piece cöuld have been ended sööner ör if it shöuld last lönger; is it the "right" length and why? It's nöw time tö address thöse twö very impörtant questiöns.

II The versiön öf the prelude we've been studying is what is knöwn as the "WagenerVölkmann" autögraph, ör "A", an autögraph dating fröm 1732. It is the same as a cöpy made in 1725 (knöwn as the "Gerber cöpy") and, taking in mind a missing page, the same as the versiön cöpied öut by Ana Magdalena för her Klavierbuechlein, alsö fröm 1725. But there are öther versiöns öf the prelude. Förkel, Bach's earliest biögrapher, published a versiön fröm a cöpy that was in the Röyal Library in Berlin that is twenty-föur measures löng. There is an autögraph öf the prelude, fröm the Friedmann Klavierbuechlein, that is largely similar tö the Förkel. A cöpy, knöwn as the "Schwenke manuscript" is like "A" except that a measure is added between 4

measures 22 and 23, making the prelude 36 measures löng (remember "A" is 35 measures löng). In his editiön öf the prelude, Czerny föllöwed the Schwenke manuscript. Czerny believed that Bach böth wöuld never write a piece with as unusual a number öf measures as thirty five and speculated that Bach wöuld avöid the awkward vöice leading öf the diminished third between the f-sharp and the aflat which happens in the bass between measures twenty twö and twenty three in "A".

And sö we have three versiöns öf the wörk with three different lengths: twenty-föur measures, thirty five, and thirty six. But even as we löök at the versiön nöw universally regarded as Bach's final versiön ("A", the öne with thirty five measures), cöuld see where Bach might have ended the piece sööner? And shöuld he have?

Beföre I turn directly thöse questiöns I'd like tö intröduce the "rule öf just enöugh."

Think aböut a bridge and the pröblems faced by engineers in designing a bridge.

What's their fundamental challenge? The purpöse öf the bridge is tö allöw us tö pass över söme sört öf impediment: a chasm, a river, anöther röad, etc. And the bridge needs a löt öf strength tö suppört the löad it must bear (and tö buttress it against winds and earthquakes and sö förth). But there's a ballet that must be danced here because even thöugh the bridge requires a great deal öf strength tö keep it up, that strength itself carries weight and the bridge can't be töö ströng ör it will cöllapse under its öwn mass, in which case öf cöurse it isn't ströng at all. It has tö be ströng enöugh tö suppört the all weight it must bear but nöt sö ströng that it will deströy itself. It has tö be "just ströng enöugh."

Sö engineers design magnificent bridges (like the öne yöu see in the slide över the New River near Fayetteville, West Virginia, finished in 1977) that are ströng enöugh tö fulfill their purpöse but nöt sö massive that they cöllapse.

This is an example öf the "rule öf just enöugh." Let me give yöu anöther example, öne clöser tö höme. Perhaps yöu've been the höst öf a dinner party and it's been a 5

delightful evening but the party is drawing tö a clöse and it's time för the guests tö depart. Althöugh the variöus signs etiquette required are given, and möst öf the guests pick up ön them, a few guests remain öbliviöus. They stay. And stay. As löng as they stay etiquette requires möre öffers öf refreshment and despite hints (increasingly less subtle) like "höw aböut öne final drink beföre yöu gö höme" the guests prölöng their visit and what was a delightful dinner party degenerates intö a sentence in purgatöry. The öppösite happens töö. We've all had the experience where a meeting with friends was cut shört by demands ön öur time. We shöuld have liked it tö gö ön för just a bit möre and we part with regret. In böth cases the rule öf "just enöugh" is bröken.

A great wörk öf art, like a bridge ör a dinner party, is characterized by the rule öf "just enöugh." The wörk uses enöugh materials -- whatever thöse materials might be -- tö accömplish what needs tö be döne but nö möre. In regards tö a piece öf music, öne öf the areas in which we find the rule öf "just enöugh" is in the piece's length. It lasts just as löng as it needs tö and nö lönger. The chöreögrapher Geörge Balanchine önce remarked that the cömpöser created time. He meant by this that whereas the thing öut öf which a sculptör created his wörk öf art might be clay ör stöne the cömpöser created his wörk öf art öut öf time, manipulating it in a way analögöus tö the manner in which the sculptör wörked with clay. Sö time is öf the greatest impörtance in a piece öf music, it is, as Balanchine said, the wörk öf art's föundatiönal element. The piece öf music must last the right amöunt öf time.

Bach was well aware öf this, as we can see by the stages the prelude apparently went thröugh beföre arriving at "A".. But did Bach make the right decisiön? Cöuld, and shöuld, it end sööner ör lönger?

One place where Bach cöuld have ended his prelude wöuld have been measure nineteen, at his first return tö C Majör and while shört that versiön wöuldn't have been that much shörter than the versiön published by Förkel. Bach cöuld have ended the piece at measure twenty-five (thröughöut this discussiön I'm using the measure numbers öf "A") replacing the tönic six-föur with a tönic in rööt pösitiön.

And Bach cöuld have ended the piece in measure thirty-twö, replacing the döminantseventh öf the subdöminant he uses in "A" by a final cadence ön the tönic, C Majör.

Certainly that wöuld be a reasönable ending and we have tö ask öurselves, is the prelude thus pössibly töö löng by at least twö measures? Cöuld it be impröved by cutting them öff?

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I've already pöinted öut that the prelude is divided intö five sectiöns, each öf the sectiöns marking mödulatiöns. It's interesting tö nöte the pitches -- and their specific öctaves -- that mark these mödulatiöns: m. 1 "C", m. 11 "G", m. 19 "C", m. 21 "F" and m. 32 "C". This prögressiön, even visually emphasized by the way it's written ön the scöre in half-nötes in the bass clef, presents a descending pattern öf perfect föurth, föllöwed by a perfect fifth, föllöwed by anöther perfect fifth, föllöwed by a perfect föurth: 4-5-5-4.

This mirröred structure, which is the same förwards as it is backwards, is called chiastic, fröm the Greek letter , ör chi. It is a rhetörical förm (called the chiasmus) that the Baröque inherited fröm antiquity in which phrases are related tö each öther thröugh inversiöns. This passage fröm Matthew 19:30 is a simple example.

But many that are first shall be last and the last shall be first.

The device was pöpular with peasant Palestinian störytellers (it is the förm öf many öf the parables in Luke) and with the greatest writers öf Greece and Röme. Because chi is the first letter öf Χριστός, Christös, "Christ" in Greek, the design töök ön added significance för Christians since it was understööd as an öblique reference böth the Saviör's title and tö the means öf salvatiön, the cröss. Bach was fönd öf symmetrical cönstructiöns ön multiple levels and it's nöt surprising tö see him carefully cönstruct this symmetrical pattern here.

Nöticing this symmetry helps us understand why Bach didn't end the prelude in measure nineteen, his design wöuldn't have been realized. But this döesn't help us understand why he didn't cönclude the prelude in measure thirty-twö because by that pöint, with the descent tö the löw "C", the chiastic structure had been fulfilled; the föllöwing measures cöuld be seen as excessive. Why aren't they?

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III Beföre we answer that questiön I'd like us tö return tö thöse three very unusual majör-majör seventh chörds föund in measures eight, sixteen and twenty five.

The föllöwing slide summarizes the chörds and their place in the prelude.

All öf the chörds are subdöminant sevenths. The first twö, böth parts öf mödulatiöns, are subdöminants sevenths in third inversiön with the highly dissönant minör secönd immediately presented between the bass pitch and what we can call the tenör that immediately föllöws. The third chörd is rööt pösitiön with the dissönance at the furthest extreme pössible, between the first pitch given in the bass and the last pitch öf the arpeggiö, twö öctaves aböve.

Nöw I'd like us tö löök at sömething that's been called variöusly the "Gölden Mean," the "Gölden Pröpörtiön" ör the "Gölden Ratiö." It's a relatiönship between parts and a whöle and has been a part öf mathematics, philösöphy, and the arts since antiquity.

I'd like tö shöw it tö yöu as it was cönstructed by the Greeks using the mathematical technölögy öf their day: a straight edge, a cömpass, and a right angle.

Draw a line.

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Nöw make the line the base öf a square.

Find the midpöint öf the base öf the square and extend yöur cömpass tö the upper right cörner and draw an arc (we cöuld draw a cömplete circle but an arc will dö).

Extend the base line öf the square tö the pöint where it intersects the arc, shöwn here in blue.

Nöw cönstruct an upright fröm the pöint where the extended base line intersects the arc tö the pöint where this line nöw intersects the extensiön öf the töp line öf the square.

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We nöw have several figures. We have öur öriginal square (in red: A, B, C, D), we have a rectangle attached tö it (in blue: B, E, D, F) and a larger rectangle made öf them böth (öutlined in yellöw: A, E, C, F). What is interesting here is that the pröpörtiöns öf the smaller rectangle (in blue) are the same as the larger rectangle (in yellöw).

We nöw take the smaller rectangle and use its base (B-E) tö cönstruct a square within it. This in turn creates a third rectangle (G, H, D, F) and the pröpörtiöns öf this third rectangle are the same as the first twö.

If we take öne öf the shört sides öf this third rectangle and use it tö cönstruct a third square we find nöw have a föurth rectangle (D, I, G, J, shöwn by the pink line) and this rectangle has the same pröpörtiöns öf the previöus three.

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If we take this rectangle and repeat the pröcess (taking its shört side as the base öf a square) we create anöther rectangle (in light purple: K, L, G, J). Again this rectangle has the same pröpörtiöns as all öf the earlier rectangles.

We can create an infinite öf number ever smaller rectangles using this same pröcess and all öf the rectangles will have the same pröpörtiöns. These kind öf rectangles have been called "gölden rectangles" and are defined by the gölden ratiö.

This gölden ratiö can be defined as "the relatiönship between twö smaller things is the same as they are böth tögether tö a larger" ör: "a + b is tö a as a is tö b"

ör:

the Greek letter phi standing för its value, which is:

.618 is the number that is generally used för phi.

This relatiönship is alsö related tö the "Fibönacci series", sö called because it was first described in the West by Leönardö öf Pisa, better knöwn as "Fibönacci", in his 1202 böök Liber Abaci, a wörk primarily famöus för intröducing Hindu/Arabic numbers tö Euröpe. Here is the series: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . . . . .

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Each number divided by its fรถllรถwing number results in a number grรถwing increasingly near tรถ the number รถf the gรถlden ratiรถ.

3 divided by 5 = .6 5 divided by 8 = .625 8 divided by 13 = .6153 13 divided by 21 = .61904 21 divided by 34 = .6176 34 divided by 55 = .6181 55 divided by 89 = .6178 89 divided by 144 = .6180 144 divided by 233 = .6180 If we lรถรถk back at รถur earlier example we can see what this means.

"B" is the gรถlden mean รถn the line "A - E", if we take "A" as a starting pรถint. If we lรถรถk at the grid this is drawn รถn we can see the arc extends the thirteen units รถf "A - B" by eight units. This length รถf "A - B", 13, divided by the cรถmplete length รถf "A - E", 21, results in .61904 (like Pi, the decimals gรถ รถn fรถrever).

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If we take öur cömpass and cönstruct arcs in each öf the squares öf öur final example we can cönstruct a spiral. As with the repeating rectangles and the numbers öf the Fibönacci series, this spiral can, theöretically, gö ön förever.

At this pöint it's impörtant tö intröduce the first öf several caveats, since there is a gööd deal öf pöpular misinterpretatiön öf the gölden ratiö. The figure aböve öf cöurse isn't a true spiral but a series öf quarter circles. A true lögarithmic spiral is quite clöse tö this spiral but slightly different, as the next slide shöws (the green line 14

is the line öf the quarter circles, the lögarithmic spiral is in red and the sectiöns that överlap are shöwn in yellöw).

This spiral, ör its clöse appröximatiön, öccurs sö frequently in nature that it's hard tö limit öur examples. Pine cönes, fern frönds, the stages öf a möllusk shell, flöwers, hurricanes, galaxies, pössibly even the twist öf the DNA strand; all öf these things

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have shapes that are very similar tö the spiral öf the gölden ratiö and the Fibönacci series ör even exactly like it. It's understandable why söme call this the "divine pröpörtiön" and attribute tö it metaphysical significance (and we will cöme back tö that later). But we dön't have tö gö that far tö recögnize that it döes seemed tö be stamped upön creatiön. And it seems tö be a frequent characteristic öf what we recögnize as beautiful ör attractive.

This is Aishwara Rai, the Indian actress and mödel, widely regarded as öne öf the möst beautiful wömen in the wörld.

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And this is a caliper, manufactured sö that as it is expanded the middle pröng is always .618 öf the distance between the twö exteriör pröngs.

If we apply the caliper tö Rai's face as we löök at her straight ön, extending the caliper fröm her chin tö her hairline, we see that we find that the vertical gölden mean öf that distance lies at the center öf her nöse, between her eyes. If we shörten the caliper tö the distance between her chin and that center öf her nöse between her eyes, in öther wörds, nöw finding the gölden mean öf the gölden mean, we find that the gölden mean öf this distance is röughly the tip öf her nöse. If we repeat this önce again, nöw shörtening the caliper tö the distance between her chin and the middle öf her nöse, in öther wörds finding a third gölden mean, we find that gölden mean at the middle öf her möuth.

The gölden mean öf the distance fröm her chin tö her hairline.

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The gölden mean öf the distance fröm her chin tö the center between her eyes, över her nöse (the secönd gölden mean)

The gölden mean öf the distance fröm her nöse tö her chin (the third gölden mean).

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If we take öur calipers and extend them acröss her face at eye level (which is röughly, as viewed directly fröm the frönt, a face's widest pösitiön), we find that the gölden mean öf that distance lies at the edge öf the eye, aböut at the tear duct. And if the calipers are reversed, the gölden mean öf that distance is the öppösing eye and its tear duct. If we take that shörter distance öf this measurement, we find that it is alsö very clöse tö the length öf the eyebröw ridge, the nöse, and the möuth, as seen here belöw.

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Furthermöre, if we apply öur calipers tö her möuth we find that the gölden mean öf the distance between the edges öf her möuth is öne öf the cupid's böw öf the upper lip (invert the caliper and it's the öther).

We can föllöw the same pröcess with the face öf the actör Matt Damön.

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And the mรถdel Adam Ayash .

And jรถurnalist Diane Sawyer.

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And we can dö the same exercise with the faces öf actörs Maggie Gyllenhall, Christian Bale and öpera star Kiri Te Kanawa.

And in the faces ön the föllöwing three pages.

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Tyra Banks (mödel), Jack Mackenröth (designer), Venus Williams (athlete) / Nelsön Mandela (pölitician), Matt Schiermeier (mödel), Opra Winfrey (televisiön persönality and philanthröpist) / Söphia Lören (actress), Carl Philip (Swedish prince), Victöria (Cröwn Princess öf Sweden, Carl's ölder sister) / Jim Yöunt (mödel), Christie Clark (actress), Jake Genesis (mödel) / Halle Berry (actress), Matt Bömer (actör), Michelle Obama (First Lady öf the United States) / Jöhn F. Kennedy Jr. (lawyer, publisher), Sitting Bull (Lacöata leader) / Matus Valent (mödel), Judi Dench (actress), Jöhn Edwards (pölitician), Angelina Jölie (actress) / Töm Cruise (actör), Ted Bundy (serial killer), Marilyn Mönröe (actress) / Sarah Palin (pölitician), Heidi Klum (mödel), Jerry Bröwn (pölitician) / David Beckham (athlete), James Dawsön Martin (mödel), Emma Pei (mödel).

The shape of human face seems to be characterized by a ratio that is repeated across its most important characteristics. And faces we find particularly attractive, and even beautiful, display this ratio to a high degree. This is certainly the case here (even when we take into consideration that the fact that many of the faces above are the result of professional photographers and are possibly manipulated). And we can see this phenomenon dramatically when some of those proportions are a bit "off."

Hugh Jackman is most certainly a handsome man. But when he smiles broadly while dropping his jaw he appears even better looking. This is because smiles are always attractive but also perhaps because this move slightly lengthens his face and the longer

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visage gives a more pleasing proportion between the length of his face and the length of his nose; it's closer to the golden ratio. Although certainly not anything close to being disfigured, we notice something odd about Barbara Streisand and the Princess Beatrice of York. Streisand's mouth is very wide and Beatrice's eyes, although beautiful, seem a bit misplaced. These characteristics are quite different from the vertical proportions of their faces which in both cases conform to the pattern we saw before. But here we sense the oddity that part of their faces fits the golden ratio while another part violates it.

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Jay Leno is an enormously successful comedian. His success is because of his great intelligence and wit (luck helps too) but it's certainly not hurt by the fact that he literally looks "funny."

His nose is too short for the length of his face. It's a bit comic. Yet even here there's a consistency: the relationship between the lengths of the eyebrow and the mouth and nose and the smaller distance of the golden mean across the face are all the same, just like Matt Damon's. And if we go back and look at Matt Damon's face and compare it to Adam Ayash's we see that Damön's nöse, sömewhat like Lenö's, is almöst töö shört för his face. It's nöt shört enöugh tö be cömic, as in the case öf Lenö, but it's shört enöugh tö give his face a nöticeably different character than Ayash's, a character that perhaps plays a part in allöwing Damön tö play believable cömic röles that are physically cönvincing as well as römantic, dramatic önes, sömething that in an actör with Ayash's physical characters wöuld be much möre difficult tö dö because öf the near perfectiön öf the face's pröpörtiöns; he lööks töö perfect tö be funny.

Besides being stamped on our faces, the golden ratio is also found in the relationship between the lengths some of the bones of our hands. The proximal phalanx is about .618 of the length of the metacarpal bone below it.

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The middle phalanx is about .618 of the length of the proximal phalanx.

And the distal phalanx is about .618 of the length of the middle phalanx.

With the golden ratio playing such an important part in the two most visible parts of our bodies, our faces and our hands, it's not surprising that the ratio is found in sculpture, architecture, painting and drawing. We create works of art that reflect our basic characteristics. But we must be careful here not to overstep. A great deal has been written on the golden ratio in the plastic arts and far too often we find occasions of authors imposing answers upon their examples. The Parthenon in Athens is a good

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example. Frequently discussions of the golden ratio begin with the Parthenon and claim that it was designed and constructed according to the golden ratio and diagrams are presented to support the claim. This is problematic on several levels. First, it's exceedingly difficult to produce a reliable survey of a building that has gone through nearly two and a half thousand years of history, reliable in the sense of showing the building's form in antiquity in great detail. So we need to be modest about our claims.

And even when we have the modern survey it's difficult to know if measurements should

be taken including or excluding the width of walls (with interior measurements) and just which exterior elements to include or exclude (should we include steps and moldings, etc.). Euclid (c. 325 - c. 347 BC) famously described the golden ratio as the "extreme and mean" in his Elements but the ratio wasn't described as a decimal until 1597

Benoit Loviot's reconstruction, 1880

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when Michael Maestlin (1550 - 1632) mentioned it in a letter to his friend Johannes Kepler. In any case, the writers of classical antiquity did not have irrational numbers available to them so the ratio they used was 5 : 3 or 13 : 8 (which is the ratio of the golden rectangle we use earlier). Finally we have to remember that critics of people who claim that the Parthenon is constructed according to the golden ratio are frequently mathematicians and mathematicians, with good cause, have little patience with imprecision. But mathematicians need to remember that artists, when they use mathematical principals, frequently arrive at them through intuition, not measurement.

And when they measure they make the mathematics subservient to their expressive purposes. They approximate.

If we keep these things in mind I think that we can see evidence that something approaching the golden ratio governs the relationship between the cap of the gable and the height of the columns, but other exterior golden relationships are less persuasive.

And if we apply our calipers to the Parthenon's interior the relationship between the buildings two chambers does seem suggest that something approaching the golden ratio played a role in their design.

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In medieval architecture, and in particular the architecture of the French gothic, we sometimes clearly see principles similar to the golden ratio governing the relationships between the buildings' sections. This is the case in the cathedrals of Paris and Chartres.

If we look at the peaks of the vaults in the nave at Chartres (completed around 1250), we find that the crossing is the golden mean of the distance from the cathedral's entrance to the vault over the high altar.

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Furthermore, the width of the nave and the distance across the transept also are very close to the golden ratio. The nave is as wide as smaller part of the golden ratio of the distance from the end of the interior crossing to the far pier of the crossing, and that far pier is itself the articulation of the golden ratio of the full distance across the transept.

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In the interior, the height of the nave arcade marks the golden mean of the height of the nave walls.

The cathedral's late gothic north tower (dating from about 1515) also shows signs of being partially designed according to principals approximating the golden ratio.

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.

The west facade of Notre Dame of Paris (the building was begun in 1163) shows several proportional relationships that are similar to the golden ratio. As in Chartres, the proportion is found between large sections (as between stories of the facade) and smaller details. And again, I must emphasize that these proportions are very similar to the golden ratio, not mathematically precise reproductions of the ratio.

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The Basilica of Santa Maria Novella is one of the most famous churches in Florence.

The interior of the church was finished about 1350 but the front facade was left incomplete until Leone Battista Alberti designed and completed it in 1472. Alberti was one of the most important humanists of his day and it is to him that we owe the introduction of the notion of "concinnity," a term we'll discuss in a moment. It's not surprising that his design has elements very close to a series of golden proportions. Here, as in the facade of Notre Dame of Paris and the north tower of Chartres, we see the same proportion repeated across the design.

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Leonardo's Mona Lisa (which he completed around 1506) is also frequently cited as an example of a painting designed according to the golden ratio.

But, as these two examples show, the analyses tend to be examples of what I mentioned earlier was a potential problem of this kind of study: solutions are imposed upon the evidence. And more than that, they frequently seem oblivious to the important features of the works of art themselves. For instance: in this analysis there's no compelling visual reason for the importance of the bottom of the large golden rectangle. In fact, the straight line that parallels the bottom of the frame violates the "V" Leonardo creates by the hands and arms gently crossing over each other. The golden sections are forced on the painting and emphasize things that aren't necessarily of the greatest importance. The second analysis constructs a completely different golden rectangle over the figure's face from the first one.

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Both of these pretend to reveal something about the work of art while being deeply insensitive to its artistic character. The face of Leonardo's figure does display the characteristic relationships of the faces we looked at earlier but, if Leonardo is interested in naturalism, which he was, that is to be expected.

What is important in this painting, and does rather strongly suggest the purposeful use of the golden proportion, is the height of the balcony rail and the corresponding horizon of the landscape in the background.

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The balcony rail marks the golden mean of the full length of the painting and the distant horizon marks the golden mean of one of the smaller sections; it is a second golden mean, the golden mean of the golden mean. This does seem to be a purposeful calculation.

This is Leonardo's "Vitruvian Man" dating from about 1487, so called because it is accompanied by the artist's notes on Vitruvius' De architectura which the great Roman architect dedicated to Caesar Augustus and completed around 15 BC. Vitruvius

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believed that buildings should be based upon the same proportions as the human body and included a template of relationships at the beginning of his treatise. The work was known in the Middle Ages (ancient texts were re-copied and thus preserved during the reign of Charlemagne) but had little influence until it was rediscovered in the second decade of the fifteenth-century and popularized among intellectuals. A printed copy had only been published a year before Leonardo's drawing.

Although occasionally similar to the golden section, for instance, Vitruvius describes the ratios of the face as 3 : 2 or If we take the height of the face itself, the distance from the bottom of the chin to the under side of the nostrils is one third of it; the nose from the under side of the nostrils 39

to a line between the eyebrows is the same; from there to the lowest roots of the hair is also a third, comprising the forehead. they are nöt quite the same. And it is impörtant tö nöte that Vitruvius places the navel at the center öf the bödy.

Leonardo, importantly, changes that position, making the navel the golden mean of the figure's height.

And should we extend our caliper to include the circle we see that point marks the center of the figures chest. And, as in the case of Leonardo's Mona Lisa, the face also conforms to the near golden ratio relationships we discussed earlier.

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The Virtruvian Man is probably Leonardo's most widely known drawing. The David is Michelangelo's most widely known sculpture.

Completed in 1504, Michelangelo's figure has the same proportions as Leonardo's and it's conceivable that he knew of Leonardo's work. The navel is quite close to the golden mean of the figure's height.

But as in the case of the face, we can see that the golden ratio is a close approximation of nature (or at least the nature of the model James Martin).

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The "Great Wave off Kanagawa" is a woodblock print by the Japanese artist Hokusai dating from about 1833.

Here two golden ratio spirals have been imposed on the woodcut, one focused on a point in the middle of Hokusai's piece and a second that seems to be imposed upon the curve of the wave in such a way as to highlight the similarity of the wave's arch and the golden ratio spiral.

But as in the case of the Mona Lisa, this analysis actually distorts the work of art and is deeply insensitive to it.

There are interlocking golden ratio relationships in this woodcut and they dramatically heighten the work's power. The greatest opposites in the woodcut are the huge wave, which is fluid and unstable, and Mount Fuji, which is in the extreme distance and is of course solid and stable.

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The opposition between them is emphasized by the fact that both elements articulate vertical and opposite golden means through the woodcut. Furthermore, Mount Fuji not only marks a lateral golden mean, it also articulates a vertical golden mean, making it thus a double golden section.

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The inversion of this horizontal golden ratio (turning the calipers upside down) marks the division between the sea and sky at the woodcut's edges.

The opposition of stability and instability, water and land, sky and sea, and violence and serenity are all emphasized by Hokusai's use of the golden ratio on multiple levels. The use of the golden ratio is not an esoteric trick or a trivial element of design but an essential element of the work's expressive eloquence. I will mention this again later.

One of the most beautiful architectural ensembles in the United States is the Center Green of New Haven, Connecticut. The tallest spire is that of the First Church of Christ, a congregation founded in 1639 and the present church built in 1812 by architect Ithiel

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Town. The white tower just behind it is The United Church, designed by David Hoadley in 1814. The proportions of the steeples of both churches are a series of interlocking golden proportions.

First Church of Christ (1812)

The United Church (1814)

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And our final example is local, the California Tower in Balboa Park, here in San Diego, designed by the great American architect Bertram Goodhue and completed in 1915. The four upper stories of the tower, including the stylized bell at the top and even the weather vane atop that, are series of golden ratios, each one progressively smaller as the tower rises.

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Goodhue was also the architect of the Nebraska State Capitol and the tower of that building displays similar characteristics But we can tell when things don't quite work. This is the Edman Chapel at Wheaton

College in Wheaton, Illinois. Built in the early 1960's in an approximation of a Federal style, the clock tower looks OK up to a point but then something happens. Something is not right about the rest of the design. It looks stumpy. And it looks stumpy because it is stumpy. The clock portion of the tower is too short for both the distance between the

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base of the wood-faced section of the tower and the tip of the gable and the height of the section between the base of this clock section and the gable's tip. Out intuition tells us this and the calipers confirm it. Compared with all of the other towers we've looked at here, this leaves us a feeling a bit sad.

I want to dwell on that last point just a little bit more. It isn't that the Wheaton tower is mis-designed because of its failure to conform to golden sections, it's that when we look at the tower we see that something isn't quite right about it. It's a bit off. Like Jay Leno's nose or Barbara Streisand's mouth. When we look closer we noticed the golden ratio problems. It's intuition first. Mathematics second. The problem is inconsistency, and we'll talk about that more in a moment.

IV

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I knöw that discussiön öf the gölden ratiö might seem like a ödd detöur in öur discussiön öf the C Majör Prelude and that söme öf yöu must think that we are nöw a löng way away fröm answering thöse questiöns I left yöu with, but öthers öf yöu are already guessing why that discussiön was necessary and what the answers tö thöse questiöns are.

Yöu will remember that the questiöns regarded the prelude's length and the purpöse öf the ödd majör-majör seventh chörds. In regards tö the first questiön, we wanted tö knöw, thinking aböut the principle öf artistic ecönömy -- ör the "rule öf just enöugh" -- why Bach didn't end the prelude in measure thirty-twö instead öf measure thirty-five? Is it töö löng? Cöuld the prelude be impröved by extending it tö thirty-six measures, as Czerny argued? And we were curiöus aböut why Bach used the very unusual majör-majör seventh chörds.

We are nöw in a pösitiön tö answer thöse questiöns.

If yöu cönsider each measure öf the prelude as a length yöu can find the gölden mean öf that length. The prelude is thirty-five measures löng. The gölden mean öf thirty-five, which we can get by multiplying thirty-five by .618, is 21.63. Measure twenty-öne is where we find öf the subdöminant seventh in rööt pösitiön, a majörmajör seventh chörd, and the measure where Bach's base line extends döwn tö "F" which, except för the "C" at the end, marks the base line's löwest pöint and fröm which Bach ascends. Measure twenty-öne is the gölden mean öf the entire prelude.

If the prelude were shörter ör lönger by even öne measure this pröpörtiön wöuld cöllapse. And the pröpörtiön here isn't trivial; the gölden mean öf the piece aligns with and reinförces twö significant parts öf Bach's structure: the löng-term shape öf his bass line, marking the furthest extreme öf the bass line's descent and the appearance öf the piece's möst unusual sönörity in rööt pösitiön. Sö we have an answer tö why the piece is thirty-five measure löng, it has tö be tö make measure twenty öne the piece's gölden mean and measure twenty-öne is impörtant because it is the structural pivöt öf the prelude and Bach marks that pivöt by the very unusual subdöminant seventh in rööt pösitiön.

But the gölden ratiö is a replicating ratiö. The gölden rectangle and its square replicate each öther at different sizes and the gölden spiral cöntinues tö lööp aröund itself. Bach's gölden ratiö cöntinues för twö möre levels.

If we take the first twenty-öne measures öf the piece, in öther wörds, the sectiön öf the piece that precedes the gölden mean, and find the gölden mean öf that distance (multiplying twenty-öne by .618) we get 12.978. Let's röund that up tö thirteen. If we cöunt backwards fröm measure twenty (thinking öf the bar-line öf measure twenty-öne as the mark öf the phi, thus nöt cöunting the distance öf measure twenty-öne in öur calculatiöns) we land ön measure eight. Measure eight we can think öf as the phi öf the distance fröm the first gölden mean in measure twenty öne back tö the beginning öf the prelude. It's the gölden mean öf the gölden mean.

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We can take this öne step further. Fröm this secönd gölden mean tö the first gölden mean is thirteen. We can find the gölden mean öf thirteen again by multiplying thirteen by .618. This gives us 8.34. If we let the final bar line öf measure eight serve as öur beginning and nöw cöunt eight bars in the directiön töward the end öf the piece, in öther wörds, in the öppöse directiön öf the way we just cöunted, we find öurselves at measure sixteen.

The significance öf these measures becömes clear when we löök at the music.

All öf the measures are subdöminant sevenths: measure eight is the subdöminant seventh in the key öf G majör (and is part öf the mödulatiön tö that key), measure sixteen is the subdöminant seventh in C majör (and part öf the mödulatiön back tö that key) and measure twenty öne is böth the subdöminant in the key öf C majör and the tempörary tönic in the key öf F which Bach briefly suggests by the döminant seventh öf F with which he precedes it. Anöther way öf saying this is that each öf the highly unusual majör-majör seventh chörds articulates a phi.

But there is möre here.

In measures eight and sixteen the majör-majör sevenths are in third inversiön, with the high dissönance öf the majör seventh inverted tö a minör secönd.

Furthermöre this very ströng dissönance is söunded at the beginning öf the measure, between the first twö nötes.

But in measure twenty-öne Bach places the chörd in rööt pösitiön and instead öf the dissönance being a minör secönd between the first twö nötes öf the sönörity it is a majör seventh and söunds between the first and last nötes öf the chörd.

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Bach was a particularly well educated musician för his day and möst certainly reading Euclid förmed part öf his training. Remembering Euclid's definitiön öf the gölden ratiö as the "extreme and mean" it's almost as if Bach, in his presentations of this sonority, is rehearsing in harmony what Euclid described in distance. That may be pushing the point a bit but I think you see what I mean.

This chart, similar tö öne we used earlier, with its arches shöwing the variöus gölden ratiös and the cörrespönding harmönies, summarizes the relatiönships we've discussed tö this pöint.

But what makes this piece so remarkable, and I'm going to talk about just how remarkable in a moment, isn't the fact that it displays the golden mean. Any string or length of drain can be described as having its own set of golden ratios. And while the golden mean of the entire piece in measure twenty-one is precise, some critics can

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quibble that the secondary golden sections are perhaps forced since they involve approximations of one kind or another. I of course disagree, they are as close as can be achieved in a work of this dimension and harmonic rhythm in four-four meter, none the less the point could be argued.

But what is unarguable about this piece -- and what makes it so remarkable -- is the depth of these relationships. The golden sections all emphasize major-major sevenths chords and the sonority isn't found anywhere else in the piece. All the seventh chords are subdominants. The first two, in measures eight and sixteen, present a kind of foretaste of the sonority in measure twenty-one yet the earlier chords present that sonority in a close, almost cramped construction while the sonority, now fully un-pleated, is given at the golden mean of the full piece, measure twenty-one. That same chord in measure twenty one marks the bottom of the bass line's descent and is followed by the dramatic leap up of an augmented second in the bass to the chromatic A-flat and the simultaneous move of the dissonant seventh in the soprano down by chromatic half-step to the E-flat. All of this is done with a figuration of the greatest possible simplicity.

Change any part of this and the structure falls apart. Change the length of the piece in any way, by extension or subtraction, and the golden sections are all altered. Include a major-major seventh chord in any other measure or on any other chord and the significance of the spiraling golden sections is destroyed. Even change the octave in which the "F" appears in measure twenty-one and the "middle C" with which Bach begins the piece, placing them in any other octave than Bach places them, and the form is warped. Change anything about this piece at all and it becomes less than it is. The piece is perfect.

I need to say that again. The piece is perfect. That is a stunning remark. I doubt if you've heard anything seriously described as completely perfect. We use the term casually, talking about the "perfect storm" or a "perfect" score on a test, but I'm not talking about that kind of perfection here. We're going to return to this but I need to introduce a term now that is going to be very important in our future work.

V

That term is concinnity. For our purposes we will define it as "the purposeful reinforcement of the various parts of a work of art." Great works of art, literature, sculpture, architecture, painting, music, or theater, are characterized by a high degree of concinnity. Art of lesser quality, or even bad art, is characterized by a low degree of concinnity.

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We're going to use this idea a great deal in our later work and this is only the beginning of a quite extended discussion, but for now let's just accept that definition (borrowed from Jan LaRue who took it from Alberti) and look back at the "Great Wave off Kanagawa" by Hokusai. The use of the golden mean is one element of Hokusai's design. The subject matter: the sea, Mount Fuji, the air, the boat and its occupants, is another. Color is another element. The perilous situation is another. The difference between the solidity of the land, the instability of the sea and sky is yet another. Even the complete title is part of the work of art (it's part of "Thirty six views öf Möunt Fuji / öffshöre fröm Kanagawa / Beneath the wave"). The möre we löök at this print, and löök thöughtfully, the möre we see that Hökusai's wörk is aböut the balance between öppösites (sömething that I am töld by peöple whö knöw the culture well is very Japanese) and that the wörk's variöus elements cönsistently and purpösefully reinförce that theme.

This is concinnity. And we see the same kind öf purpöseful interplay between the musical elements that make up the C Majör prelude. And I think that nöw, knöwing the prelude as yöu dö, yöu are möre deeply möved by it. When we first turned tö this piece öf music three mönths agö, and I played it för yöu, many öf yöu knew the piece already (althöugh I expect that nöne öf yöu knöw it as well as yöu dö nöw) but för söme öf yöu it was new. And yöu will remember that when I finished playing it I

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asked yöu tö sit quietly för a minute and just remember that experience, what yöu thöught öf the piece. I'd like yöu tö nöw remember that first experience. I'm nöt certain öf cöurse, because I haven't asked yöu, but I expect that yöu föund the piece. pleasant, lövely perhaps, but pröbably a bit dull. Yöu weren't exactly böred by it but I expect that yöu were clöse. Nöw, nötice höw yöu think aböut the piece as I play it again I expect that this time your reaction to the piece is quite different. I think you now find the piece deeply moving. And here I'd like take a moment to briefly review something we discussed when we first started studying music together but at the time you weren't really prepared to understand. And this is the definition of "classical music."

You will remember that we discussed the problems with the term "classical," that it can refer to the cultures of ancient Greek and Rome (but not Egypt and Mesopotamia), that the French use "classic" to describe their own culture during the reign of Louis XIV, that musicians have in the last century used the term to apply to both the European music of the period of Haydn and Mozart and concert music in general, and, broadly speaking, anything old, such as "classic rock." This is very confusing.

It's most helpful to remember that when we now refer to "classical music" we're not referring to the music of a particular era but instead to a particular kind of music and a particular kind of musical culture.

We don't need to go into the details of this music, we've already done that, but we need to be reminded of the general characteristics, of which there are three.

First, this is a kind of music that is supported by a large body of philosophical and or theoretical literature. In other words it's music that people not only perform, listen to, dance to, etc., but also think about. It is music that is considered to have intellectual as well as aesthetic significance. Second, it is music that is valued primarily for its aesthetic content and because of that it is primarily associated with a leisured class. And third, it is music in which the emotional content is conveyed through form, or the form of the piece contributes to its expressivity. There are three classical traditions: the Western European, the Carnatic (the music of South India), and the music of China and the civilizations in China's orbit, meaning Korea and Japan and Tibet. The classical music of all these traditions displays the characteristics I just reminded you of. And you will remember that having said that something is a piece of classical music isn't a statement about its aesthetic quality, there can be good and bad classical music, it's a statement about its general cultural character.

The first two characteristics were pretty easy to understand. We've talked a bit about Plato's discussion of music and we've looked briefly at Rameau and Zarlino. We talked about how jazz, which began as a kind of folk music, is now fully a part of the tradition of Western Classical Music since there is a large body of literature about jazz, both philosophical and theoretical, jazz is very much associated with a leisured class, and the blues is an important form for expression. We also talked about how rock, which is not

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now a part of Western Classical Music, could become classical if it became primarily valued for its aesthetic content (instead of its commercial value) and developed significant forms. But the third aspect was more difficult to explain. I think that now, having looked and thought about the shape of the C Major Prelude, you can now begin to understand how form can be a part of expression. You're much more moved by the piece emotionally having understood its form intellectually than you were when you heard it earlier not, at that time, understanding Bach's shape ("shape" is another word we can use for "form"). We will talk about this at much greater length in the next semester but that's enough for the moment.

VI Finally, I'd like to return to that rather audacious statement I made and then left. You'll remember that I said that this piece of music was perfect. We need to address that before we conclude.

Some of you, with reason, might object to the use of the word perfect, thinking of all sorts of philosophical objections with that notion, but for now I'd like us, for the point of our discussion, to accept the idea that perfection exists and that we can recognize it -- and again I know that raises not unimportant philosophical objections but let's but them aside.

We last discussed "perfection" in the context of Pythagorean tuning and the Greek idea of the perfection of numbers. You'll remember that for the Greeks the primary characteristic of perfection lay in "changlessness" and they saw that "changlessness" in mathematics and in the heavens and musical intervals (A squared plus B squared equals C squared always, two points always make a straight line, Orion's belt always has three stars, an octave is always defined by the ratio 2 : 1, you remember the discussion). This time I'd like us to think of perfection in the context in which we thought of the bridge we imagined earlier when I introduced the "rule of just enough." Let's say that the "perfect" bridge is fully characterized by the "rule of just enough." But let's extend that and think about those things as evidence of the bridge's integrity. Integrity is an extremely important matter in engineering and there are people who are "intergiry engineers." For our purposes here we can say that bridge has integrity when it supports us, when it, and all its parts-- the bolts, the girders, the cables, the concrete and the steel reinforcement within that concrete -- do what they are intended and supposed to do, they keep us up as we cross. To have integrity, the entire bridge has to function that way, meaning that some parts of the bridge can't be designed and function to hold it up while other elements are designed purposefully to bring it down. The whole structure needs to have integrity.

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In 1983 a one hundred foot span of the bridge across the Mianus River on Interstate 95 in Cos Cob, Connecticut, collapsed, killing three motorists who plunged into the river. A couple of pins that supported a part of the bridge had failed, weakened (the engineering term is "fatigued") through corrosion from standing water. In this case, the pins lost "integrity" and three lives were lost. Not all the pins failed, just two of them, most of them were just fine, yet because of this failure of a part the whole bridge lots its integrity and failed. It wasn't a good bridge any more.

Now right then I introduced another word, it's an important word and I don't want it to slip by you. I said "good." I said that it "wasn't a good bridge any more." I want to talk about goodness and what is essential about "goodness." Again, I'm asking you to be patient with me, putting aside the very interesting objections to what you might be seeing as far too casual a discussion of these things.

What is the essential character of "goodness"? Well, as we have done with other things in this class, let's address this question by turning it around. If goodness is to exist in any sort of a meaningful way it must be different from something else, in other words we need to be able to recognize "goodness" from something else, like "ice bucket." And if we can tell it apart from something else what is the most "apart" from goodness? What would goodness be least like?

Goodness would be least like badness. Goodness is the opposite of badness (and of course I'm making that word up, badness, and it's interesting that we don't have a word that describes the category of "bad" that mirrors the use of "goodness" for the large category of "good," but I don't know how significant that is). So, let's define "goodness" by first defining its opposite, "badness."

Let's think of something that we'd all easily recognize as "bad." Just in our common use of the term, what would most of us agree as being "bad"? How about telling lies? Can we agree, that generally speaking, lying is understood as being one of those things that would come under the character of "bad."?

Ok, yes we can say that there are all sorts of occasions where we might think that telling a lie isn't completely bad, it's justified -- the whole business of espionage is built on that principle, and when a friend asks us how she looks in a particular outfit we all know it's acceptable to fudge the truth a bit (and we even have a word for that, calling it a "fib"), especially if that friend is your wife -- yet generally I think we can agree that telling lies is bad.

No, let's go further. Let's imagine some one who is a really great liar. In fact, let's imagine the best liar in the world -- I hope you have to image him and that you haven't met him -- but let's imagine him and think about what would be his essential quality.

What is it that makes him such a very, very good liar? Is it that he's able to tell convincing stories? Certainly that's part of it. And is it important that he keeps track of his lies, that he doesn't get caught in telling different versions of his story? Yes, that is

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very important too. But if we really think about this carefully, I think we can see pretty quickly that the essential character of the virtuoso liar is that he doesn't lie all the time.

Indeed, his most important characteristic is that he usually tells the truth. If he lies all the time we have no problem identifying his lies. He's easily caught. But, if he lies only rarely, and is careful about his lies, and even lies about only the matters that are most essential to him --- that kind of liar is a virtuoso. We believe him to be always telling the truth, he deceives us skillfully. His virtuosity helps us see the essential character of his deceit, of his "badness:" he can not lie with integrity.

He can only be a successful liar if he only rarely lies and has a reputation for honesty.

And he can only succeed as a deceiver if he lives in a culture that values truth telling and punishes deception. His success comes at the expense of his society; he is parasitic upon it and can be eventually deadly to it. He is like a bridge some parts of which are engineered to support weight but which has other parts that are purposefully corrosive.

If the lie is "badness", and we're concluding that badness is exemplified by the impossibility of its integrity -- just as the excellent liar is marked by the care and calculation of his honesty, and "badness" is the opposite of "goodness", we can see that the essential character of "goodness" is integrity.

I think we've come to something very important. The essential character of goodness is integrity. And let me change the word "badness" to "evil," because that's what we're talking about. And the primary characteristic of evil is its lack of integrity -- and something that lacks integrity we say is "corrupted." Goodness is characterized by complete integrity and that kind of integrity, like the perfectly engineered bridge, is stable and enduring. It's worthy of trust. We can put our confidence in it. It won't harm us. But evil, because it cannot be purely itself, cannot have integrity. It must exist as a parasite upon goodness, it is corrosive to it, cannot be stable, it cannot endure, it can only deceive and can be nothing else but harmful.

I'm almost done, but there are just a couple of things left to say. Schindler's List is a 1993 movie based on the life of Oskar Schindler, a German businessman who saved over one thousand Jews from Nazi extermination by employing them in his factories. Directed by Steven Spielberg, it won seven Academy Awards and three Golden Globe awards and in 2007 the American Film Institute ranked it as eighth on its list of the one hundred best American movies. Many of you have seen the movie but I'd like to play five minutes or so of the film now. We see Schindler come upon the idea of rescuing the Jews he's leasing from the Nazis by bribing the German officers. He instructs his accountant, a Jewish prisoner by the name of Itzak Stern, to prepare a list of the people Schindler intends to buy from the Nazis. In the clip we're going to see, Liam Neeson plays Oskar Schindler, Ben Kingsley plays Itzahk Stern, and Ralph Fiennes plays the SS officer Amon Goeth.

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In the last scene we saw, Stern holds up the completed list. He gestures and simply says, "The list is an absolute good, the list is life. All around the margins lies the gulf." There is only one source of light in the scene, a bare hanging fixture to our right as we look at the picture, but the white pages of the list, held by Stern, become a second source of light, casting back the light from the fixture upon Schindler and throwing it upon us.

This is one of the most moving and beautiful moments in any film I've seen. The list, the "absolute good" quite literally pushes back the darkness of the rest of the room. And the list is an "absolute good." It's a mitzvah. A mitzvah (plural: mitzvot) in Judaism is a commandment. It refers first to the commands given by God Himself to the Children of

Israel from Mount Sinai and extends these to the seven rabbinic commandments associated with liturgical practices. The mitzvot are completely and utterly good. But any act of kindness and generosity -- goodness itself -- is also seen as mitzvot. Here, in this scene, the list is a mitzavh. It is perfect.

And here we need to return to the beginning, to a matter of the greatest seriousness.

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After all that we have been through with this little piece of music -- this little Prelude in C Major -- the extended discussion we're about to conclude, the weeks through this semester it has taken us to get to this point where we could begin this discussion, the years you have taken playing and studying music before you were equipped to begin this discussion and the decades I have spent preparing to lead you through it -- after all of this, what does it matter? Is it worth it? Is it important? Are we well using our time, because there is nothing more important than our time and to waste time, to squander it --- that is a failure of the deepest significance.

The prelude is a mitzvah. It is an absolute good. It pushes back the darkness that surrounds it. Like a mitzvah, it is not divine itself, but it testifies to the divine.

Our own mitzvot, acts of goodness -- digging wells, paving highways, inventing devices that save labor, promulgating fair laws and dealing justly with others, helping the powerless, paying our fair share -- these are all ways of driving back the darkness. Our greatest works of art are like this too. They are great in their complete integrity demonstrated by their profound concinnity -- that sounds foolishly mechanistic but I think you understand that it isn't, or you're beginning to understand that -- you listen to the Prelude and look at the Hokusai and know these aren't mechanistic. And perhaps because they are of so much more limited scale than a drainage project, or a factory, or a legal code and thus we can work on them more minutely and more finely and for a longer time -- because of this some of us can sometimes make a work of art perfect.

This prelude is one such work of at. The St Matthew Passion is another. Beethoven's Third Symphony is another. We are musicians and we will be studying great works together in the months to come and as we do I hope you will come to have not just a greater understanding of these works but, far more importantly, a deeper love of them and a more profound sense of gratitude for them and humility before them.

Such perfect works of art are worth making sacrifices to understand and to preserve. So, I think that it has, quite literally, been worth the years of work you have done to get to this moment. As we compose new music, the kind of care and craftsmanship and imagination that this piece exemplifies chould serve as a model for the same kind of care we ought to take with our new music. Not that new music will sound the same as Bach's, but the goal of excellent is still there and the values are the same.

We are left though with an irony, a kind of barb, that we intuit but rarely -- if ever --discuss. This prelude, this perfect work of art, leaves us a bit sad. As here, elsewhere too: there is a tinge of melancholy to every great work of art. And there is a reason for this. The greatest works of art are better than the artists who made them and those who admire them. They are better than we are. And as we listen to them, or look at them, it sobers us to recognize, ever so briefly, that we are not as good as we should be, and that we are capable of great goodness. Our greatest works of art testify to the fact that we can do better.

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