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KARL JOHNSON: Good evening. Welcome and thank you for joining us for the 2018 Beimfohr-Neuss Lecture. My name is Karl Johnson. I'm the executive director of Chesterton House, which is a center for Christian studies here at Cornell and an affiliate member of Cornell University Religious Work.
The Beimfohr-Neuss Lecture is designed to address questions related to faith and public life. Past lecturers have included historians, a sociologist, a literary scholar, a medical doctor, a farmer, the leader of one of the largest nonprofit organizations in the world, and this year, a pianist. The Beimfohr-Neuss Lecture is made possible by Carl and Elaine Neuss as well as Al and Linda Beimfohr. The late Al Beinfohr was a member of the Cornell class of 1966.
And we are very privileged this evening to have Carl and Elaine Neuss with us here this evening. And I would like to introduce them so that we can give them our appropriate thanks and also ask Elaine to say a couple of words about the lectureship.
[APPLAUSE]
ELAINE NEUSS: Thank you. It's great to be here. I realize this is your spring, but for us from Orange County it's freezing cold. And I layered every bit of clothing I had on today, walking around.
And a real highlight for me today was to have lunch with several students at the Beta Residence House, along with Dr. Chung. So I thank you for your lovely hospitality. Al, Linda, Carl, and myself were very good friends. We traveled a bit.
Al especially loved classical music. He was a real fan, and he and I really connected in that way. I am a pianist-violinist, and I compose. So we had a lot to talk about.
We were also members of the Philharmonic Society of Orange County, which was a presenting group that brings in the Vienna Philharmonic and a lot of large names, Joshua Bell. And so he was instrumental in making that happen in our community. So I know if Al were here, he would be so pleased to be seeing what Dr. Chung is going to be presenting and to see all of you here.
And I thank you for that support. And I know Carl and I will continue to look forward to being a support to the Chesterton House lecture series. Thank you. And I know Carl has a few words to say. Are you sure?
CARL NEUSS: I was born in Orange County, New York. She means Orange County, California.
[LAUGHTER]
[APPLAUSE]
KARL JOHNSON: Thank you again, and a special welcome also to the many folks who have tuned in remotely via live stream. College is at least in part preparation for the world of work. We speak during the college years of career exploration, or sometimes in language with slightly more religious overtones of vocational discernment.
In the Christian tradition, vocation is said to be discerned because it's discovered, rather than merely constructed. And it's discovered rather than constructed because it's inextricably linked to the world as it really is. And the world, it turns out, is ripe with potential for further development.
The architect who designs a building and the engineer who builds it do not start with a blank paper. They start with the way the world as it has been given to us-- so, too, with the animal scientist seeking to, for example, keep pets healthy, the economist aiming to mitigate poverty, and the viticulturist, seeking to bring forth from the earth the finest of wines. But what about music?
If vocation consists largely of bringing to fruition the latent potential of the world in which we live, music just might be the archetypal example. And we find a fine illustration of this in an enigmatic poem entitled "The Strange Music" by the British man of letters GK Chesterton, in which the poet likens himself to a harpist whose only problem is that he doesn't know how to play the harp.
"Other loves may sink and settle. Other loves may loose and slack. But I wander like a minstrel with a harp upon his back. Though the harp be on my bosom, though I finger and I fret, still my hope is all before me, for I cannot play it yet."
Then in the very lovely second stanza, he turns to his harp and addresses it in the second person. "In your strings is hid a music that no hand hath ere let fall. In your soul is sealed a pleasure that you have not known at all, pleasure subtle as your spirit, strange and slender as your frame, fiercer than the pain that folds you, softer than your sorrow's name."
In Chesterton's view, we are all hapless but hopeful harpists. And the world is waiting for us to make known the fullness of its hidden beauty. Fortunately for us, we have with us this evening a special guest who is an expert at bringing beauty out of stringed instruments.
Dr. Mia Chung is among the most accomplished of American pianists. When she was just 16, the famous Russian cellist and conductor, Rostropovich, selected her for a televised collaboration. She attended Harvard, where she graduated magna cum laude, while at the same time continuing her training in performance independently from her academic studies.
She went on to complete a master's degree from Yale University and a doctorate from the Juilliard School. She has studied under some of the finest pianists in the world, including Peter Serkin, and performed as a concert soloist with symphonies all over the world under several noted conductors, including Leonard Slatkin. She is the recipient of awards too numerous to mention, but including the Concert Artists' Guild Competition in 1993, and in 1997 an Avery Fisher career grant, which is the highest recognition for young concert artists in the United States.
Her performances have received all sorts of accolades. The New York Times, for example, referred to a recital she provided at Carnegie Hall's Weill Recital Hall as, quote, "uncommonly insightful, individualistic, and lively." Her playing, the reviewer wrote, was dazzling.
But Dr. Chung is not only a performer. She is also a scholar and an educator, often providing musically illustrated lectures, as she is doing for us this evening. She served as the first young artist in residence for NPR's Performance Today, where her work earned her a nomination as debut artist of the year.
She went on to positions as artist in radio at public radio stations in Boston and Washington, DC, and has appeared on public radio throughout the United States. For over 20 years, Dr. Chung served as artist in residence and professor of music at Gordon College, an institution that Chesterton House partners with to provide courses in biblical studies, as noted on the back of your program. And in 2012, she joined the faculty of the Curtis Institute of Music in Philadelphia. It's our great privilege to have Dr. Chung with us this evening. Please join me in extending her a warm welcome.
[APPLAUSE]
MIA CHUNG: Thank you very much, Karl. That is an introduction I do not deserve-- very kind, warm, and caring. I have to say that in hearing how the Beimfohr-Neuss Lecture Series was birthed, I was deeply moved by the caring, friendship, and love that exists between the Beimfohr and Neuss family. And so this is really a great privilege for me to be speaking on this lectureship series tonight.
I also want to extend a word of thanks for all those, the hands and feet, who make this presentation possible-- Dr. Karl Johnson and his team, Julie Johnson, Billy Wiley, Nicole Wiley, many, many other names, so et al., OK, in the academic terminology. Thank you so much for investing your energy in making this happen.
The title of my talk tonight, Music, Math, and Mortality, I hope you enjoy the alliteration. I worked very hard at it to bring these very disparate areas together. Music and math, maybe not so disparate, but certainly music, math, and mortality is the case, a bit of a distance, a jump.
But I hope that I am able to present a compelling argument for why these things intersect, why they come together, and why they are deeply meaningful and inform one another. OK, so there's nothing like music to sort of pull you in. So I'm going to start off right off the bat by playing the opening theme of Beethoven's "Fifth Symphony in C Minor," which was composed 1804 to 1808.
I'm using this example because it's one of the most famous classical works. And you're going to go, I remember hearing that in a cartoon on television. So even if you're not a classical musicphile, you will know this piece.
[MUSIC - BEETHOVEN, "FIFTH SYMPHONY IN C MINOR"]
Now I'm bringing this piece into the conversation first because I want to drive home one concept right off the bat, and that is the idea of resolution in music. All tonal works, which are all works in the history of Western music leading up into the 20th century, use a tonal vocabulary. That means it's situated in a key, C major, C minor, F major, F minor.
Most of you have sort of that understanding of tonality. It's pleasing to the ear. It's consonant. And the bedrock of tonal definition is something called the 5-1 cadence, dominant to tonic cadence. Sometimes, it's dominant seventh, OK. It's still the dominant entity. And this is what it sounds like.
[MUSIC - BEETHOVEN, "FIFTH SYMPHONY IN C MINOR"]
Which is the very end of the first movement of the "Symphony Number Five." Now as soon as you hear that, what's your impulse? You say, closure, resolution, 5-1, ah, delicious.
And for some of you, you may have made the mistake of applauding, right? And unfortunately, the performers go on into the second movement. And you think, oh, holy smokes. That was totally intended, OK, the moment of embarrassment.
So we've all done that at some point. The reason you did that is because you feel it inside your bones. There was a resolute conclusion to the movement.
But the question is why? Why do you feel it as such? So that's resolution. But now I'm going to move to another example, to draw your attention to something called tension, harmonic tension.
And for this case study, I'm going to go to the prelude of Wagner's "Tristan and Isolde." The prelude is the orchestral portion that's played, and Wagner introduces all of the musical leitmotifs, the themes, that he's going to be expounding upon in the context of the opera, even before the singers come in. Now Wagner was a 19th-century German opera composer.
In 1859, he penned "Tristan and Isolde." And it was a game-changing opera, game changing for the principal reason that he averts resolution with every fiber and cell of his body. It was meant to depict something, and I'll explain in just a second what that was.
So this is tension that's avoiding resolution at all costs, and this is how it sounds. This is the prelude. And I'm going to play it a little faster than orchestras play it because we don't have enough hours in the day to listen to, you know, the slowness with which it would be performed. We're not going to live long enough. OK, so here's the prelude.
[MUSIC - PRELUDE TO "TRISTAN AND ISOLDE"]
Now this opera, "Tristan and Isolde," is about two lead characters, Tristan and Isolde, a man and a woman who are in love with each other. However, their love is never meant to be. It was never destined to be.
So it's this extraordinary, long, three-act opera that expounds upon a love that will not come to fulfillment. And so the music-- and this was Wagner's extraordinary contribution to composition-- was that the music should tell the story. The music is going to depict this unresolved tension. So let me show you how he does this harmonically. In the very beginning, did you hear this incredible dissonance?
[DISSONANT MUSIC]
You heard that, right? That's the Tristan chord, controversial guy. And Wagner then moves to the five chord, the dominant. But he doesn't resolve it. What comes after the dominant? Do you remember?
The tonic, the one chord, right? He leaves it on the five chord. And then this is what he does. That's once. Twice, he transposes it up a whole step, and he goes--
[DISSONANT MUSIC]
Another Tristan chord, and another dominant seventh chord, and no tonic. Big rests, right? Silences, and we're like, what's going on? Then a third time.
[DISSONANT MUSIC]
Another Tristan chord, another dominant seventh chord, then he repeats it, another dominant seventh chord, no tonic. He repeats that little motif. And then he gives us a five chord, and he goes to something called a deceptive cadence. In other words, he averts the tonic.
Do you see what I mean by avoiding closure or resolution at all costs? The music is depicting or telling you the story through these themes what the emotional content and narrative of the opera is going to be. Now, Billy, if you would kindly pull up the first slide.
Here's the score. See, for my students, I love them to engage through ears only first, because if I put that up there, you would all be going like this, right? Truthfully.
So what would happen if the five chord that I was playing or referencing in each example was followed by a tonic chord? It would close it off, right? It gives you a sense of resolution. And then I did it again. Beautiful, huh? Third time.
And then I went like this. And you'd stand up, and you'd applaud, right? OK, end of story and end of opera. Do you see how banal that sounds?
It's banal because it in actuality is contradicting the very spirit of this chromatic music. Wagner wanted to avoid the tonic, and he does so brilliantly for many hours, in fact, in this opera. So this is the quintessence of tension, harmonic tension, that Tristan chord.
Let me tell you this. This work was so game changing in music for this very reason. Composers had never encountered a music that had so defiantly objected against resolution and the move to one, the tonic, which makes us feel closure, satisfying conclusion, resoluteness.
That certainty was defied front on by this opera. And so generations of composers to come had to grapple with this game-changing opera. And, in fact, it was so impactful, it influenced composers going way into the 20th century. And folks like Arnold Schoenberg, the man who emancipated dissonance, right, tension in music, he freed it. And his students, like Anton Rayburn, and those who followed, Milton Babbitt, all these great 20th-century composers can draw their lineage all the way back to this seminal work.
Amazing, isn't it? But it was the perfect depiction of this unfulfilled love and longing, right? OK. Now we yearn for resolution. And the question is, why?
Why do we want this tension to be resolved? It's because of the mathematics that undergird this music. In order to go down that line of inquiry, I'm going to have to go all the way back to the sixth century before Christ. Now the next slide, please.
A gentleman by the name of Pythagoras, for the humanities majors in here, they're like, oh, yuck, right? This is the man we associate in geometry with, you know, right triangles. A squared plus B squared equals C squared, right?
He was the first one to experiment with frequency. And as controversial legend has it, one day, he was walking down the street. And he heard blacksmiths pounding their hammers against an anvil simultaneously.
And he was intrigued, because he realized that depending on the hammers that were being used, different intervals were sounding simultaneously. So intervals are sort of a collection of two pitches. So let me play you, like, the perfect fifth.
One, two, three, four, five, that's an example of an interval. Perfect fourth, one, two, three, four. An octave, one, two, three, four, five, six, seven, eight, OK? So he goes back to his lab. Forgive me, technical troubles here.
He goes back to his lab, and he decides to experiment with string lengths on his monochord. This monochord was a single-string instrument, over a box that was hollowed out. And the monochord had a movable bridge, so he could shorten or reach the full length of the string and figure out, test out how the proportions of the string can create different pitches and how these pitches mathematically relate to one another.
And this is what he discovered. For a vibrating string, that full length of the string, OK, let's just call that one. He could actually create by simple integer ratios pitch relationships.
So to create an octave, he would divide the string in half, which gives you two halves, generating the ratio of one to two. So here's the bass pitch, the one. And then if he were to move the bridge halfway along this string, for this-- you see that there-- and he plucked just half of that string, he would get the octave pitch above it. OK, two to one, or one to two ratio.
Wow. He said, let's go with this further. Then he said, hmm. Let's divide the string in three and see what happens.
So then plucking this full string here, the top string, he notices if he places the bridge, the movable bridge, right there, using the length of 2/3 of the string, he gets the perfect fifth, OK, two to three ratio, or three to two ratio. And then if he divides the string into fourths like this, and he plucks the full string, and takes 3/4 of the length, he puts the bridge right there and allows this part of the string to resonate, he gets the perfect fourth. Hmm, he experiments with consonance even further. Next slide, please.
And this is what he finds. These gorgeous, simple integer relationships generate these other very consonant-sounding intervals. You have the major sixth, three to five. You have the minor sixth, five to eight. You have the major third, right, four to five, and the minor third, five to six.
Now why is it that intervals like the perfect fifth sound so consonant? We're going to turn to the waveforms of this to talk about that. The perfect fifth, three to two ratio, is like the interaction of 200 hertz and 300 hertz wave cycles.
Now just know that in frequencies, even though we use these sinusoidal graphs, in actuality, when sound moves through the medium of air or music, the sound wave is like this contraction or compression and rarefaction that goes on three-dimensionally speaking. And it reaches our ears, and we hear it as a pitch. But we transfer it to this two-dimensional depiction of this interaction, compression and rarefaction.
And what we find is that in a 300 to 200 hertz interaction, you will have a waveform. You see the red form there. It goes through two cycles. And the cycle, of course, is defined by its intersection with the x-axis, which measures time.
So in a matter of just roughly 0.0067 seconds, which is that other y-axis, from 0 to that next increment in time, the waveform is going through two cycles, 200 hertz, OK? And then you have a blue waveform. It goes through three cycles. See that interaction?
Now the beauty of consonance is that these waveforms align. They line up frequently in any given interval of time. So do you see the intersection on the x-axis between the red and blue wave right in the center there? You see that? And then they actually align at the next sort of time interval y-axis there, right?
This is why it's consonant. Our ears hear it as pleasing. It registers in the center of our brains as a pleasing sound.
But what happens with dissonance-- the next slide, please-- is that these waves, for example, in the most dissonant interval, which is the minor second, they're just a half step apart. I'm playing C and C sharp, OK? What happens here is that the waveforms actually do not intersect, rarely or if ever. And so in our brains, it registers as unpleasing, dissonant.
See the interaction? Consonance, dissonance, tension, resolution, OK, these are all concepts that are at play when you listen to music. If I fast forward 2,000 years-- think about that-- from Pythagoras all the way to the 16th century, there happened to be two mathematician-physicists, one in the East and one in the West. We can turn to the next slide.
Listen to this. This is fascinating. Zhu Zaiyu, brilliant mathematician, physicist, music theorist of the Ming dynasty, Ming dynasty, guys, all right-- this guy's not in Germany; he's in China-- discovers how to divide the octave into 12 half steps or semitones. That's best depicted by the keys on the keyboard, to understand that.
Can you imagine? He finds a mathematically satisfying way to divide up the frequencies such that the octave is constructed by half steps. So this is when it's C, to C sharp, to D, to D sharp, to E, to F, F sharp, or G flat. Those are enharmonic spellings, G, G sharp or A flat, A, A sharp or B flat, B, and C.
That was in 1584. Zhu Zaiyu comes on that discovery. Simultaneously, in 1585, Simon Steven makes the same discovery, but with a few more mathematical errors. So he's not viewed in the scholarship as credibly as Zhu Zaiyu in the discovery of this math.
This is a game changer. To divide the octave into 12 half steps means now that you can get these pitches to interact in chord forms. We think of triads, major and minor triads.
So if I play, for example, a C major triad, C, E, G, it's very pleasing. This is the most consonant-sounding assemblage of pitches you can imagine. Now what happens if I lower the third, and I go C, E flat, G? I get a minor-sounding triad, which is deeply sad.
OK, so you see the interaction? All right, now, that's just one triad. Now imagine building those triads on every single key. Now you have a vocabulary of harmonies that can now interact to communicate something deeply powerful and emotive.
The question is, why does this math move us so powerfully? We actually don't know. We do not know enough about brain science. It's just starting, actually.
Since about 2000, I think there's been much more research about the neuroscience behind music, listening, impactfulness, and psychology, why we interpret it as being this mood or that mood. It's a very recent phenomenon. But we can ask, what is going on in the math?
To demonstrate that, I'm actually going to move to a piece of music because we can all say with absolute certainty that we may not understand the mathematics behind pitch interplay, right? Music is this complex assemblage of pitch, rhythm, harmonies, right? But we do know this.
Music is governed by natural law, the natural law of the universe. The frequencies interact with one another in a way that's governed by something beyond musical imagination. It's governed by universal principles-- symmetry, simple integer ratios.
Did you know that this half step has a ratio of 15 to 16? Who likes that? Right? It's not easy guys, all right, for us to swallow that mathematically, nor is it easy for our brains to understand it. Why is the big question. We don't really know.
Now let's move from this incredibly important question to a piece by Ludwig van Beethoven. And let's demonstrate how this half step idea, this idea of dissonance, is going to be manipulated by one of the greatest composers in Western music toward the aim of tremendous emotional impact. OK, shall we do that together?
All right, remember, "The Fifth Symphony" was composed 1804 to 1808. The "Impassionata Sonata," which means with passion, composed 1804 to 1805, so very much contemporaneous with "The Fifth Symphony." So for this exercise, Billy, maybe we do project the next slide.
All right, I'll give you the score. OK, so this piece is in F minor. F minor, key of F minor, all tonal music is built on scales. So it's built on the F minor scale. F, G, A flat, B flat, C, D flat, E natural, F.
[PLAYING SCALE]
And all the harmonies that lie within that scale, that can be built on those pitches, the triads and seventh chords, et cetera, et cetera, will interact with each other in this domain, this home territory of F minor. So if F is the tonic, my friends, anyone want to hazard a guess where the dominant is built? Which pitch? F, G, A flat, B flat, C, one, two, three, four, five, right?
OK, so you're going to be expecting that for resolution. And then you're going to stand up and applaud, right? You say, yay. It's over. Now I can get my Starbucks, right?
But let me tell you this. That half-step tension that I referenced before is not going to come in the form of a Tristan chord, sort of a simultaneous sounding of discord. It's not a Tristan chord or anything like it.
The dissonance here happens to be between two harmonic territories. So this is a different definition of dissonance. The two harmonic territories happen to be a half step apart. Mm, now you might say, so what?
A half step is close. They're close in pitches. So it must be easy to resolve that tension.
Actually not, because in F minor, the half step above that is G flat. Guess what? G flat ain't in the scale of F minor.
OK, so he's brought in a foreign body, an intruder, a harmonic intruder into this narrative. And now he's got to resolve this controversy. OK, enough talk. Let me play the exposition of this piece, OK?
[EXPOSITION]
So that's the exposition of the "Impassionata Sonata." That's where Beethoven lays out all the important thematic materials. Now I don't know if you noticed this, but right off the bat, he presents this ultimate controversy between two harmonic areas, F minor and G flat major.
Did you hear it, by the way, in the beginning? How many of you heard it, actually? I'd love to know.
OK, if you didn't hear it, you'll definitely hear it by the time I'm done. So here's the F minor. Here's the theme.
[THEME IN F MINOR]
Then it goes to G flat major.
[THEME IN G FLAT MAJOR]
Up a half step, right? OK, look it, guys. Because of Lady Gaga and Madonna-- now I'm actually revealing my age-- we are no longer shocked by these things, because we're like, so what, because we've seen so many more radical things in music happen since.
In Beethoven's day, that would have been eyebrow-raising. People would be like, what, heavens, right, as they heard that. OK, but look it. He does this right off the bat. No composer in the classical era does that, unless they intend to weave a narrative, a story, if you will, that is all about bringing this fundamental controversy to resolution.
Do you understand? That's the brilliance. And Beethoven was the king of drama. He is one of the greatest Western composers for a reason, because he knew how to maximize musical anticipation, energy, excitement, and impact by drawing out harmonic controversy in the form of these two neighboring, disputing harmonic areas for as long as he possibly could.
And it so turns out, he's in 238 of the 261 measures of this movement, talking about this controversy. Now let me point out for you all the little references to this fundamental half-step controversy. It's everywhere.
And if you look at the score, it's kind of like, where's Waldo? You know what I mean? You're looking for it, and you find him all over the place. Let me show you where. So I just played that opening theme.
[OPENING THEME]
Now F minor to G flat major. Whoa. Holy smokes, what is that? That sounds like the fate motif of Beethoven's "Fifth."
[FATE MOTIF]
You hear that rhythmic reference? But really what it is, it's almost like this motivic reference to the fundamental controversy. You understand? This half step, right? And then he harps on it.
Almost like a Greek chorus, right? OK, let me show you a few more examples of this half step.
Hear it? Hear the half step? Half step. Half step. Half, going down, half step.
Now we come to something called the secondary theme, and it is sort of the antithesis of this controversy. It's peaceful, placid, totally major, and wonderful, a respite.
[SECONDARY THEME]
I love the nobility of this theme. Enough of the niceness. Half step. Half step. Half step.
Half step. Half step. Half step, half step, half step, half step. Do you get my point? How many of you found, like, 15 Waldos? I hope that you did.
Now that's just the exposition. When he gets to the development, which is the middle section of the movement, where he's going to really expound on this fundamental controversy, this is what he does. And every time you hear a half step, I want you to raise your hand, OK?
[MIDDLE OF MOVEMENT]
Good. Got it? Good. Now watch.
In case you missed it, OK, A flat, A, A flat, A, over and over. He's like, come on, just in case you didn't get it the first 50 times.
That's the development. That's a portion of the development. Now what happens in the recap?
He brings back all the materials of the exposition, and then you reach something called a cadenza. And the cadenza is the moment where he launches into radical virtuosity, because he's somehow got to up energy even more to bring it to this grand climax. And this is what the cadenza sounds like.
[CADENZA]
Now I'm going to stop there. If you heard carefully, it was loaded with half steps. Half step, half step, half step, it was like he was throwing oranges and apples at us to really impact us. Listen to this.
[CADENZA]
Hear that in the bass? Now watch. You got it?
And then you think, OK, how is he going to bring this to closure? He's absolutely going nuts, ballistic with these half steps. And listen to what he does. You guys are just going to be mesmerized by the brilliance of this.
[CONCLUSION]
What's that, guys? That motif, right, I call it the angst motif, right, because it's the ultimate tension.
[ANGST MOTIF]
And we've had it with this half step. Jesus bring me home, OK? OK. Five, seven, to one, did you hear the dominant tonic?
Did you feel the resolution, the conclusion to this? Now it's brilliant for you music theory-ophiles. D flat is a half step above C.
D flat is the sixth degree in the scale. The sixth degree moves out, pushes out towards the C, which is the five. And then he could go five, one. Isn't that brilliant?
The proximity, he makes you feel like it's some half-step idea out there. But actually there's this little trap door that enables him to go right to the five and then to the one. Wow.
Now you asked yourself, why on earth is this music so impactful? Why is it that audiences come back time and time again to hear these pieces? What is it that's going on?
It's the math. It's the interaction of those frequency waves. In fact, you understand that the bass line of a five-to-one resolution is built on the perfect fifth, right, the most consonant of intervals.
[INTERVALS]
Right, that three-to-two ratio, and that's just the baseline. Imagine the interaction of all the other voices in the chords. If I had the ability, I would graph for you all the frequency interactions in different colors. And then you would see. And your brain is processing all of these waveforms in its consonant and dissonant forms and making sense out of it emotionally.
Is that by chance, or is there something else going on here? What we find compelling about this music, fundamentally, if you strip it down to its absolute core, it has to do with dissonance and consonance, tension and resolution. That's how Beethoven was able to weave this formidable narrative.
And then you ask yourself, well, if this is mathematics speaking to us, how did this work for Beethoven? Beethoven was not a mathematician, my friends. He knew very little about math.
In fact, according to historical record, he wasn't really extraordinary at anything else other than composing, writing music. But he had this most extraordinary musical imagination. And, in fact, think of this. He went deaf, even more mind-blowing.
So there was a point in his composition where he could hear nothing, and he was still writing this glorious music. The reason this music is so impactful, the foundation of tension and resolution, is because that little coupling, this yin and yang dynamic between dissonance, consonance, tension, and resolution, is what defines every moment of our lives. You do see this, right?
Think about this. So let me give you a day in the life of a Cornellian, OK? You wake up in the morning, starving because you just studied so hard.
You studied so hard, OK? You're starving. And then you go to the Beta House, and you have breakfast. And you're satiated. You're happy, right, content.
Then you go through your day. And you have a battle with your roommate over the silliest thing, clothes all over the floor. But ultimately, you find resolution by solving your problems amicably-- war, peace.
You lose an object, your ID swipe card. And you're looking frantically all over the room to find it, and you locate it under the couch. You're writing a paper. It just kills you because it's so difficult. Or you have a problem set, or you're writing your dissertation.
And that period of angst and tension finally comes to resolution with this gloriously convincing defense and an A on the paper, A-plus on the problem set. Even better, right? Do you get what I'm saying?
Tension, resolution marks every moment and minute of our lives. The ultimate point of tension and resolution for Beethoven came in 1802, when he was 32 years old. Beethoven was living in Vienna, which was the capital center of Europe at the time, in the 19th century.
And he used to frequent an area called Heiligenstadt, which is a forested area that he would retreat to, to get away from the urban stress, the hustle and bustle of life. He loved nature, because it was then that he felt the power of the divine. Next slide, please.
It was during this time that he penned something called the Heiligenstadt Testament, in 1802. And he meant for this testament to be read by his brothers after he died. So he hid this letter. It wasn't discovered until after he died, in 1827. And this is what he said.
"But what humiliation when anyone beside me heard a flute in the far distance, while I heard nothing, or when others heard a shepherd singing, and I still heard nothing. Such things brought me to the verge of desperation and well-nigh caused me to put an end to my life." So this was a letter of goodbye to his brothers.
Thankfully, he did not end his life. In fact, the "Appassionata Sonata," "The Fifth Symphony" come two years after this crisis moment in his life. And it was during this time that he felt great ambition and drive to put down on paper all the glorious statements in his mind that were gestating. Do you understand?
He felt an urgency, a pressing need to bring and display his brilliance by writing. So he wrote frantically. And the "Appassionata Sonata" and "The Fifth Symphony" are products of this sort of resurgence of hope and energy in 1804.
So what is the fundamental dissonance that harangues most of us? For Beethoven, it was deafness. It was the prospect of never being able to bring to fruition the greatness of his talent through the compositions that he knew he could write.
In point of fact, losing hearing was one of the best things that happened because then he plumbed the depths, reached levels of profundity and depth in his musical statements that otherwise, I believe, would not have happened had he not become deaf. But the amazing thing is he knew nothing about the underlying mathematics at play that would impact you and me.
What is the ultimate dissonance in our lives? I would say it's mortality. This is a dissonance that begs resolution. And this dissonance runs through our psyche almost like a river.
It informs us, whether consciously or unconsciously, all the time. Think about the ways that we try to extend our lives. For those of you who are vegan or vegetarian, I know why, why you exercise three times a day.
You're fighting your mortality, for those of us who seek beauty treatments or want to look younger, for those of us who want to create heaven on earth through luxury and beauty, to escape the reality of dissonance that we experience every day. Or how about this? We work to achieve something that will be remembered for generations.
Do you see? It's around us, in us, all the time. It motivates us to do things.
Perhaps the math in Beethoven's music is pointing us towards some greater truth, one that offers a victorious resolution to this very heavy reality that we live with every moment of our lives. Now for centuries, scholars, mathematicians viewed the math of the universe, whether it's in music, or in the rotation of the planets, or in the arrangement of pedals in a flower, or the proportions of a nautilus. They saw these as manifestations of God's creative genius. In the last two centuries, however, the beauty and power of music has been made anthropological, as if it is merely the expression of emotion, feeling, completely divorced from the mathematical principles that drive the musical expression.
If you think about the medieval academy, the quadrivium, the university training-- music, geometry, arithmetic, astronomy-- these were seen. These were grouped together because they were multiple lenses, pointing all toward some essential and central truth. That was divine order-- proportion, symmetry, balance, all these things that exist in its myriad forms, through the sciences, through nature, et cetera.
So are we, in fact, depriving ourselves of a much greater realization because we've made music so anthropologically oriented, so human focused? Next slide, please. Julian Johnson, the British musicologist, once said this.
"If it now strikes us as amusing the music was once linked to astronomy or natural science, that is only because we fail to recognize ourselves there and the historical development of our own attempts to understand the world. If we no longer take music seriously as a way of defining our relationship to the external world, perhaps we have become not more sophisticated, but simply more self-absorbed."
The Bible says that man was created in the image of God. And nowhere is this image more evident than in our capacity to create, to invent, to compose. But if we are created in the image of God, the creator of the universe, is it possible that the creator of the universe is trying to communicate to us through all these different means which converge into one central truth, this relationship between God and man?
Beethoven most certainly thought so. The math in music speaks to the heart of a creative god who wants to communicate to us. He does so not only through natural laws, but through our minds, through our emotions, our spirit. He aims to communicate through multiple means.
Beethoven's music speaks of tension, which we've all experienced in life, and the resolution that we long for. In the Bible, the God of our universe addresses this ultimate tension and resolution. But his resolution is a definitive one.
Our peace and our solace rests not simply in the cadence of a five-to-one chord progression, but in the one who created this math, and music, our rationality, our emotions, our spirit, and the deepest yearnings of our heart. If you're grappling with your mortality, the ultimate dissonance of human existence, your resolution can be found in him. Thank you.
[APPLAUSE]
Thank you. Thank you, everybody.
KARL JOHNSON: Well, what a wonderful talk. There's an awful lot I'm sure we can continue talking about. We are going to take a little bit of time for some questions and interaction with Dr. Chung.
We do not have microphones to pass around, and this is a cavernous space. So [INAUDIBLE] Dr. Chung, everybody, we would invite you to stand up. Project your voice.
And, Dr. Chung, if you would be able to sort of quickly re--
MIA CHUNG: I'll reiterate the question, or I can hand this over to you.
KARL JOHNSON: That would be very helpful for those who are watching.
MIA CHUNG: OK, great, sounds good. Yes?
AUDIENCE: OK, I'll start. That was fantastic, by the way, [INAUDIBLE] My question is, I was thinking about cultures that have, like, atonal music, like Middle Eastern, Indian, where you have this very distinct thing played for hours at a time.
Is there something different about that? Is that a cultural thing? Is it a math thing? I don't get it after hearing all this.
MIA CHUNG: Fantastic question, so his question was other cultures that use perhaps different forms of dissonance that extend for prolonged periods, he doesn't sense that there's tonality. Let me tell you. There's way more universal principles at play that bring together world music with Western music than anyone cares to admit. The fact of the matter is, we are all bound by the octave.
[OCTAVE]
We all have the challenge of dividing this octave. In the West, we've divided it into 12 semitones. In the Far East, we may have divided it into the pentatonic scale, penta meaning five.
[PENTATONIC SCALE]
OK, and it's a scale form, in that example, that avoids tension. The question is really, do you feel the placid sort of nature? You do. You still feel that emotion because of the consonance that exists in that pentatonic scale.
Regarding cultures that use sort of microtone or quarter-tone inflections, what you will find is a universal principle, no matter what the scale form is, is this desire for home territory, home pitch. And by the way, many other cultures did not define or elaborate their musical language through harmony in the way that Western music did. It's not about complex harmonic relationships.
It may be instead rhythm or elaborate melody. But what we see in melodies, even those that are quarter tonal, they always lean toward a central pitch. There's a sense of tension and release, right? If you listen to the tunes of Fanta Damba, from Africa, for example, she will circle around a central pitch, maybe move around it, even these quarter tone slides away, and return back to the pitch.
So this sense of tension-resolution is lived out in the context of melody, not in the context of harmonic interaction. And maybe in Central Asia, in the Middle East, in Asia, you have a drone, a drone that's going on. Tell me. That's tonally conceived.
You're based in a key. And you may move around and away from that central pitch, and you may even use quarter tones. But guess what? You always return back home.
So journeying away, returning home, tension, release, it's manifesting itself no matter what the musical culture is. And by the way, the reason why the semi-tone is so dominant, it's one of the smallest perceptible, perceivable, musical increments. That's why music tends not to go into intervals that are smaller than that.
You do have quarter tone, microtonal, right? And that is less perceptible because the distance between two pitches is not a half step anymore. It's a quarter. OK, it's half of that distance, much harder to perceive.
But often, those quarter tones are meant to elaborate on a pitch that it leans against. Does that make sense to you? So there are many more universals.
You know what? As a follow up to that question, let me tell you about this. This blew my mind, OK? There is a gentleman named Dr. Thomas Fritz, who's from the Max Planck Institute in Berlin. He studies cognitive science.
What really intrigued me about this cultural conversation dialogue is whether or not a person who has never been exposed to tonal Western music can understand the emotional meaning of that music. So you know what Thomas Fritz did? He was intrigued by this question.
He had to find a people group that had never heard anything from Western music, nothing tonal or anything, nothing with the harmonic language that we're talking about. So you couldn't have listened to a radio. You couldn't have stepped into a church, nothing.
And guess what he found. He found the Mafa tribe, in the outer reaches of Cameroon, a very rural area. And he found a gentleman in this tribe who was 110 years old.
And he presented three images-- the image of a smiling face, the image of a sad face, and an image of anxiety, of fear. He said, now I'm going to play for you 30, 40 excerpts of music. And I want you to point to the picture that best illustrates or correlates to the musical feeling, the emotion that you're getting from this patch of music.
And what he found was stunning. This man who had never heard tonal music in his entire life was dead on every time. He heard major-sounding music. He pointed to the happy face. He heard minor-sounding music. He pointed to the sad face.
Think about that. Isn't that amazing? It's astounding. There is something going on in our brains. And the amazing thing about what Thomas Fritz has done, he goes back to his lab at the Max Planck Institute.
And he does brain scans of people, because he wants to know what's going on in the brain. Why is it that consonance registers? Why is it that we can remember tonal tunes, consonant tunes? "Twinkle, Twinkle, Little Star," can you imagine if I did, you know--
["TWINKLE, TWINKLE, LITTLE STAR" IN MINOR KEY"]
You're not going to remember that.
[LAUGHTER]
Well, at least, I hope you don't. But in any case, there's a reason. It turns out that the brain absorbs consonance. It registers with your brain's processing.
Dissonance is actually not absorbed. It's cast out. That's why you have a hard time remembering dissonant melodies. So Arnold Schoenberg in the 1920s, he had this firm belief that if he wrote atonal music, this keyless music, dissonant, that people will come out because it's so ordered, highly ordered, mathematically ordered, but with no pleasing consonance, he thought people would come out of the concert singing the tune, you know, of a melody that goes--
[DISSONANT MUSIC]
And do you think anyone was humming his tunes? No, because actually it wasn't registering in our brains the way that we are designed to. Now don't get me wrong. I love Schoenberg's music, perhaps for different reasons.
I don't sing it or whistle it when I'm cooking. I can tell you that much, OK? But I love the ideas that inspired his music and the revolutionary manner, the experimental matter that he treated it.
So have I answered your question? Sorry. That was a bit roundabout. That was my "Apassionata Sonata" on that question. OK, any other questions? yes?
AUDIENCE: [INAUDIBLE]
MIA CHUNG: So true, so she said, our own voices, the production of sound through our mouths, informs so much of how we perceive or think of music. Is that a fair articulation of your question? It's absolutely right.
And, actually, I'll drop back even further. Music, if you go back to the earliest point in time, is all informed by speech. So if you look at different musical traditions, even the rhythms of speech inform the musical rhythms that are written. You look at Hungarian music and the sort of first syllable emphasis.
[HUMMING]
It's all because of the speech accents and inflections that happen in Hungarian language. So you're right. The voice, what we're capable of producing, what our voices can sing, even just the muscles and the interaction of the physical element of sound production would absolutely inform the way we hear music and produce it. So true. Yes? Karl?
KARL JOHNSON: So in listening to you, it makes me think about [INAUDIBLE] We can hear the world, sounds, and reactions in the physical world. We hear each other speak. And then we hear music. These things happen almost every day.
So the sounds of the world alert us [INAUDIBLE] We speak to each other in a direct way so that we can communicate with each other. So what would you say is God's purpose for music, as opposed to [INAUDIBLE]
MIA CHUNG: So he's saying, all different forms of communication, speech, you hear of all these stimuli, aural stimuli, that come through your brains, music as well. But what is the distinctive of music? Is that fair, Karl? Is that right?
So what's the distinctive of music? What's God's intended purpose for music? For me, I believe it's to pull us away from the everyday and the mundane, to transport us, to energize us emotionally, but also to enable us to experience the transcendent.
You know, there's nothing transcendent about going down the grocery aisle with your cart and realizing that you have five minutes before you have to pick up your son from soccer practice. Do you know what I'm saying? So, you know, the everyday sort of discipline of life, we lift up our countenance. That's what scripture does for our souls.
That's also what music does for our souls. And music holds a very, very special place in scripture. God does give a power to that medium other than communication. So I think it's, yes, to lift us up, our souls, towards something more heavenward, to lift our countenance upward.
Any other questions? Yes?
AUDIENCE: [INAUDIBLE]
MIA CHUNG: Absolutely, and, you know, it's funny. She said, is music a kind of offering, just like tithing? I do think there are those similarities.
You know, actually, one student said to me one day, Dr. Chung, how can you justify practicing six hours a day when there are people starving, you know, in the outer reaches of our neighborhood? I said, you know what, John? I do this-- trust me. Practicing six hours is not like that I want to do this all the time.
But it is a form of worship for me. It's a sacrifice that I give every day joyfully because that's what it calls me to do. And so I do think worship in services, and the quality of that worship, and the kind of music that you use, it's all expressions of your soul, of thanksgiving and praise. So, yes, it is a gift given and a gift returned. Way back.
AUDIENCE: [INAUDIBLE] Then I was wondering about [INAUDIBLE]
MIA CHUNG: Yeah, yeah, so we experience these things a little bit differently. So he's talking about the tuning systems, the sort of equal temperament tuning system which I explained to you. You have the Pythagorean tuning system. You have means sort of tuning, you know, where you're actually dickering with the thirds and the triads to make them a little bit sharper, a little bit flatter. And there are different emotional sort of responses to that.
But I will say this. Those, to me, we still resonate with those. But they fall within pretty narrow parameters. In other words, the distance of emotion is not fundamentally redesigned.
Does that make sense? So even if you tweak the third by a third comma or a quarter comma, you're not going to be like, oh, my gosh. Now we're moving to anxiety. We went from happiness to total anxiety and anarchy.
It doesn't happen. But we may have a subtle sort of response to that. So that's my answer to you.
But it's interesting that we still resonate. We resonate with those intervals, even when they're tweaked a little bit. It's an imperfect system. Anybody else? Yes?
AUDIENCE: [INAUDIBLE] I just wanted to know. Some people say that [INAUDIBLE]
MIA CHUNG: Right, right, right, so how does music relate to how much of the brain we use? Is that your question?
AUDIENCE: [INAUDIBLE]
MIA CHUNG: Because it resonates with the consonance, resonates with us, do we end up using more of that portion of the brain that resonates with that consonance? Is that a fair reiteration of your question?
AUDIENCE: Sort of.
MIA CHUNG: Yeah, yeah, yeah, listen. You're frightening, man. I can't imagine what you're going to be like when you're 18. So [INAUDIBLE] but tell me. Refine your question a little bit more.
AUDIENCE: [INAUDIBLE]
MIA CHUNG: Yes, but other things?
AUDIENCE: [INAUDIBLE]
MIA CHUNG: Oh, bravo, man, OK, let's do this. So his question was-- [SIGH]
[LAUGHTER]
He's asking about the way our brains are designed. And because we had to hunt and gather and all this, we developed a portion of our brains that was larger, to service that need. If that's the case, can you translate that principle to music making?
You know, what does it do for our brains? If we end up using music even more, how will that enhance our brain capacity? Is that fair? Awesome.
So let me tell you this. OK, remember, I mentioned the neuroscience that's coming out since 2000? Now because of brain scans and our ability to measure things scientifically, we are discovering-- first of all, let me say this.
Music, playing music, taxes the brain more than any other activity in life. And improvisation in music taxes the brain about 75%. It uses about 75% of the capacity.
So you think, my goodness. In daily life, I'm probably only using about 30% of my mental capacity. But what we have found is that musicians-- think about what's involved in playing music.
You have to read notes, translate visual symbols, move, physically move your hands to where those pitches are, respond by hearing the sound. Sometimes, if you play violin or other instruments, you have to tune yourself. You listen to your stand partner. You may have to watch a conductor going like this.
We find musicians have incredibly sharp executive function, which happens in this prefrontal area of the brain, because of these things. They have a keener sense of language definition and understanding and comprehension in the auditory portion of the brain because of the music they play. They have a better ability to learn language, acquire new languages, because of the music that they play.
They have incredible facility neuromuscular response. I mean, you think about it. There's, like, 2,000 signals going between the brain and the hands every second that a musician is performing, a piano piece, for example. So that neural musculature develops.
Memory, I mean, I could go on and on. So because music draws upon all these different compartments of the brain, you are finding that their lives are enhanced in many other ways. Memory, you think about dementia, staving off dementia. You think about coordination, gross motor, fine motor.
The list is endless. So your question is so central to what my message is in the public at large. Play music. Play an instrument.
Sing. It has power that you have no idea existed. When people sing together, guess what. Their heart rates synchronize.
Is there any wonder when you sing in ensemble, you feel this incredible sense of unity and togetherness, oneness? There's a physiological reason why. OK, have I answered your question?
KARL JOHNSON: I don't know if anybody is brave enough to follow that question.
[LAUGHTER]
We have reached the end of our talk. Perhaps there's time for just one more question.
MIA CHUNG: Yes?
AUDIENCE: So I was thinking as you were talking about the [INAUDIBLE] and just thinking about, say, Beethoven's "Appasionata" [INAUDIBLE] just repeating [INAUDIBLE] of tension, of resolution, of mortality and life, and the way we write music?
MIA CHUNG: Now?
AUDIENCE: Now.
MIA CHUNG: Versus back then? Yeah, so his question was, if you compare Beethoven's music to, say, something that was written now, say, in the contemporary popular form, which tends to only draw upon a few chords, many of them one and four and five-- aha, right, remember [INAUDIBLE] ratios-- what does it say about how we engage music? What is it saying about us maybe? Is that your question?
AUDIENCE: Yeah, and how we--
MIA CHUNG: Oh, and how we approach mortality.
AUDIENCE: [INAUDIBLE]
MIA CHUNG: That's right. So we're not engaging the tension that's fundamental in the music of Beethoven. And by the way, just a side note before I address your question, we don't have the capacity anymore to listen to a musical narrative that works its way through 23 minutes of time.
How many of us actually have the attention span anymore to follow or listen to that kind of narrative? So our modern world has shrunk our capacity for engaging these sorts of issues like tension and resolution, life and death, these larger existential questions. Never mind those, man, right?
OK, so when it comes to the musical language of today, it is very simple. And it is very sensory. It's very gratifying because the harmonies are simple, and the rhythm has been enlivened.
Maybe there are words that are very compelling. But it does avoid us dealing with or experiencing the mathematics on the level that we could. Isn't that true? It's a lost opportunity.
But you know what? Music is not the only form that shields us from this harsh reality. If you look at the Academy, I mean, how we live our lives now, and the constant stimulation, and the inability to pay attention or to attend to one another in relationships, or to suffer, we're dealing with mental health and emotional issues on a gargantuan scale.
Is it potentially because of the avoidance of these fundamental interactions with things like music that take us through that journey of dissonance and consonance, tension and resolution? Maybe we have not given our souls window to interact with those things because we've surrounded ourselves with communication and stimulation that avoids this fundamental question that enriches our lives.
KARL JOHNSON: Thank you. This was wonderful. Thank you.
MIA CHUNG: Thank you, Karl.
[APPLAUSE]
Thank you. A pleasure, thank you.
KARL JOHNSON: [INAUDIBLE]
MIA CHUNG: Thank you.
Music is math—it is a sophisticated compilation of harmonies, pitches, and rhythms built upon the physical laws of sound that render emotional impact. Music moves us powerfully. Can math do the same? Might you come out of your next math lecture with tears in your eyes?
In order to examine these questions more truthfully, a new way of viewing these disciplines is essential. Math isn't just an objective understanding of the universe, where answers are correct and incorrect, and music isn't simply a subjective expression of deep emotion. The two are inextricably linked. The language of music uses mathematics to communicate powerful emotional messages about the questions that matter most to us in life and in death.
Pianist Mia Chung delivered the annual Beimfohr-Neuss Lecture on April 26, 2018. Chung was the first-prize winner of the 1993 Concert Artists Guild Competition and a recipient of the 1997 Avery Fisher Career Grant. She has appeared with the Baltimore Symphony, National Symphony, Alabama Symphony, New Haven Symphony, Harrisburg Symphony, Boston Pops, and the Seoul Philharmonic, among others, under the direction of such conductors as Leonard Slatkin, John Nelson, Andrew Litton, and Richard Westerfield.
