WEBVTT 00:01.610 --> 00:07.290 >>MAJOR FUNDING FOR THIS PROGRAM WAS PROVIDED BY: SHELBY CULLOM DAVIS CHARITABLE FUND INC, 00:07.290 --> 00:13.640 BY ANDREW DAVIS. ADDITIONAL FUNDING WAS PROVIDED BY ACEVES-LONDON 00:13.640 --> 00:21.750 FAMILY FUND AT THE SANTA FE COMMUNITY FOUNDATION AND BARBARA ERDMAN FOUNDATION. 00:21.750 --> 00:29.820 AND BY THE KIND WORLD FOUNDATION, PAT AND RUTH CONNERY, CATHERINE WEEB, HAROLD FOLLEY 00:29.820 --> 00:37.570 AND JENNIE NEGIN, THE ESTATE OF DAVID ELLIOTT YOUNG BY MARY McCACHREN… AND NEW MEXICO 00:37.570 --> 00:41.640 PBS VIEWERS LIKE YOU. THANK YOU. 00:41.640 --> 00:47.380 (birds chirping) 00:47.380 --> 00:59.600 (sipping coffee) 00:59.600 --> 02:27.020 (music begins-“Also sprach Zarathustra”-Richard Strauss) 02:27.020 --> 02:35.020 (phone rings) 02:35.020 --> 02:36.160 >>Rosemary Moore: Hi Dad! 02:36.160 --> 02:38.620 >>Cris Moore: Hey! >>Rosemary: We’re gonna be late for the 02:38.620 --> 02:40.680 concert. Meet Mom and me at the plaza. Come on! 02:40.680 --> 02:44.160 >>Cris: O.K. I’m on my way. 02:44.160 --> 02:46.640 My name is Cris Moore. I’m a mathematician 02:46.640 --> 02:52.880 here at the Santa Fe Institute. I love math, but some of my friends tell me that their 02:52.880 --> 02:55.620 experience with it was a bit more like this. 02:55.620 --> 03:01.660 >>”C squared equals A square plus B squared.” (yawning) 03:01.660 --> 03:04.030 >>Cris: But for me, math is more like this. 03:04.030 --> 03:06.360 Listening to music on the plaza. 03:06.360 --> 03:12.020 (Singing and playing) 03:12.020 --> 03:16.840 >>Cris: It all starts with a vibration. The 03:16.840 --> 03:20.850 drummer strikes the drum, and makes it vibrate. (Drum beat in slow motions)… which makes 03:20.850 --> 03:30.690 the air vibrate… which makes your ear drums vibrate (pulses). Your mind does the rest. 03:30.690 --> 03:42.720 (music continues). Translating those vibrations into beats and tones (singing). 03:42.720 --> 03:49.840 Whether these vibrations come from a drum, a reed or a string, we hear them as musical notes. 03:49.840 --> 03:53.540 (Clarinet and guitar-jazzy music). 03:53.540 --> 03:55.760 >>Cris: each instrument virbates in a unique 03:55.760 --> 04:01.440 way, with a unique mix of frequencies that comes from its shape and materials. From the 04:01.440 --> 04:03.240 math that tells it how to move. 04:03.240 --> 04:06.860 (music) 04:06.860 --> 04:10.280 >>When we put these sounds together, we get 04:10.280 --> 04:16.280 harmonies, chords, melodies, songs… And now math is making music. And whether that 04:16.280 --> 04:21.590 music makes you get up and dance, or sing the blues, or feel truly transcendent, that’s 04:21.590 --> 04:27.620 no coincidence. It’s your intuitive connection to the notes, and the relationships between them. 04:27.620 --> 04:30.180 (Fiddle and bass music) 04:30.180 --> 04:33.010 >>Cris: The ratios and rhythms and patterns 04:33.010 --> 04:39.550 that excite our minds and move our hearts are the same harmonies we find in subatomic 04:39.550 --> 04:49.580 particles, and the motions of planets and stars. Music and math let us hear and see 04:49.580 --> 04:55.500 the patterns all around us. The elegance and wonder of the world. When you come up with 04:55.500 --> 05:01.050 a new melody, or I find a new pattern in math, we’re using the same parts of our brains, 05:01.050 --> 05:06.210 and feeling the same feelings. And I have a feeling it’s getting a bit late. My family! 05:06.210 --> 05:13.730 <Come on, we’ve got to go!> We’re headed to Santa Fe’s historic Lensic Theater. The 05:13.730 --> 05:18.920 seats are filled with music lovers. And, after tonight’s special performance by the Santa 05:18.920 --> 05:23.010 Fe Symphony Orchestra, I hope math lovers as well. 05:23.010 --> 05:27.760 Join me on a journey to discover the Majesty of Music and Math. 05:27.760 --> 05:32.100 (Orchestra performs) 05:32.100 --> 05:48.540 (Applause) 05:48.540 --> 05:52.680 >>Guillermo Figueora: Ladies and gentlemen, the Santa Fe Symphony welcomes you. My name 05:52.680 --> 05:58.150 is Guillermo Figueroa and I have the great honor of being the conductor of this wonderful 05:58.150 --> 06:03.920 orchestra. Joining me on stage tonight is a mathematician and music fan. 06:03.920 --> 06:08.020 Please welcome Cris Moore. 06:08.020 --> 06:15.640 (applause) 06:15.640 --> 06:17.760 >>Moore: You may think of music and math as 06:17.760 --> 06:22.070 separate, but it hasn’t always been this way. In Ancient Greece, music was combined 06:22.070 --> 06:28.360 with arithmetic, geometry, and astronomy in a liberal arts package later called the quadrivium, 06:28.360 --> 06:34.240 the “four-fold way”. Plato believed these four disciplines allowed us to study numbers 06:34.240 --> 06:40.810 and patterns in different ways. Just as geometry lets us see patterns in space, music lets 06:40.810 --> 06:47.800 us hear patterns in time. The most basic connection between music and math is rhythm. If we start 06:47.800 --> 06:51.570 with a steady beat (drum beating: 1:4 time) 06:51.570 --> 06:56.640 >>And add another twice as fast (drum beating: 1:4, 2:4 time) 06:56.640 --> 07:01.910 >>And another twice as fast as that (drum beating: 1:4, 2:4, 4:4 time) 07:01.910 --> 07:07.830 >>We get 4 beats per measure. On the other hand, 3 beats per measure gives 07:07.830 --> 07:16.880 the familiar rhythm of a waltz: (Violin plays waltz beat) 07:16.880 --> 07:21.960 >>Let’s hear the Santa Fe Symphony playing “The Blue Danube.” 07:21.960 --> 08:01.280 (Orchestra plays “Blue Danube” excerpt (Strauss) 08:01.280 --> 08:07.960 (applause) 08:07.960 --> 08:10.490 >>Cris: Some composers alternate these rhythms, 08:10.490 --> 08:14.170 to create interesting combinations. >>Guillermo: Listen to the Santa Fe Symphony, 08:14.170 --> 08:21.070 as we play a selection from Leonard Bernstein's West Side Story, where he alternates two beats 08:21.070 --> 08:28.090 and three beats in his classic celebration of immigrant life in America. Notice how your 08:28.090 --> 08:34.600 ear picks up the alternate rhythms: one-ta-ta, 2-ta-ta and 1-2-3. 08:34.600 --> 09:09.440 (Orchestra plays a selection from “America” (West Side Story Selections-L. Bernstein) 09:09.440 --> 09:15.340 (applause) 09:15.340 --> 09:17.400 >>Cris: Leonard Bernstein created that rhythm 09:17.400 --> 09:23.280 by taking a unit of time and dividing it sometimes into two pieces and sometimes into three. 09:23.280 --> 09:28.930 As we’ll see, ratios like these play a role not just in rhythms, but in notes, chords 09:28.930 --> 09:34.620 and melodies. But, what is a musical note anyway? When the first violinist plays a note 09:34.620 --> 09:46.890 on his violin (note plays), what’s happening in his violin, and in our ears? Well, sound 09:46.890 --> 09:52.800 is vibration. Every time a string wiggles or a hummingbird flaps its wings, a wave of 09:52.800 --> 09:59.240 pressure travels through the air. When a vibration is fast enough, we hear this series of waves 09:59.240 --> 10:04.440 as a continuous tone instead of as individual beats. The number of vibrations per second 10:04.440 --> 10:09.390 is called the frequency, and different notes have different frequencies. For instance, 10:09.390 --> 10:17.740 a middle C, which we’ll now hear from the Symphony’s first trumpet (note plays) vibrates 10:17.740 --> 10:24.060 at a frequency of 261 times a second. On the other hand that low note from the contrabassoon 10:24.060 --> 10:30.800 that you heard at the very beginning of “Also Sprach Zarathustra” (note plays) vibrates 10:30.800 --> 10:36.340 only 33 times per second. That’s such a low frequency that you can almost hear the 10:36.340 --> 10:46.270 individual beats of it, the way we would in a drumroll (drumroll plays) or a cat’s purr 10:46.270 --> 10:53.960 (cat purring, laughter). At the other end of the scale, a high note from a piccolo, 10:53.960 --> 11:00.430 (note plays), is vibrating more than 4,000 times per second. Now, it turns out each time 11:00.430 --> 11:05.820 we go up an octave, for instance from low C, to middle C, to high C, this corresponds 11:05.820 --> 11:11.780 to doubling the frequency, vibrating twice as fast. That high note from the piccolo is 11:11.780 --> 11:16.640 seven octaves above the low note from the contrabassoon, which means it’s vibrating 11:16.640 --> 11:24.960 2, times 2, times 2, times 2 times 2 times 2 times 2, or 128 times faster. 11:24.960 --> 11:30.160 In ancient Greece, Pythagoras discovered the math behind these notes, by doing experiments 11:30.160 --> 11:35.580 on simple instruments. Let’s follow in his footsteps. This is a monochord. It’s just 11:35.580 --> 11:41.240 a string, stretched across a movable bridge, attached to a sound box. We’ve tuned it 11:41.240 --> 11:46.980 so that right now, more or less, it’s a middle C. (plays note). Now let’s see what 11:46.980 --> 11:53.980 happens when we make the string shorter. If we make the string half as long (plays note, 11:53.980 --> 12:01.860 moves bridge), we hear the same note an octave higher, another C. What’s going on? Well, 12:01.860 --> 12:07.060 if the string is half as long, the vibrations have half as far to go along it, so the string 12:07.060 --> 12:14.060 vibrates twice as fast. And that 2:1 ratio corresponds to the octave. Let’s make the 12:14.060 --> 12:19.920 string one-third as long, tripling the frequency from what we had originally. This takes us 12:19.920 --> 12:30.130 to a new note, from C to the G above it (plays note). Here’s a middle C on the piano (plays 12:30.130 --> 12:36.930 note). Each time we go up an octave (plays notes) the frequency gets twice as fast. Each 12:36.930 --> 12:43.190 time we go down by an octave (plays notes) it gets twice as slow. We hear these doublings 12:43.190 --> 12:50.010 without realizing it, but our ears can only take us so far. A grand piano has seven ocatves, 12:50.010 --> 12:56.100 but mathematically these frequencies go on forever, both above and below our hearing. 12:56.100 --> 13:02.870 To help you visualize this, we’ve made you a much grander piano. An endless piano. Below 13:02.870 --> 13:08.360 the notes we can hear, elephants communicate using sub-sonic rumbles (rumbling noise). 13:08.360 --> 13:16.510 And even the earth and sun ring like bells at 16 octaves below middle C. Far above the 13:16.510 --> 13:24.930 notes we can hear, dolphins use high frequency chirps 9 octaves above middle C. Molecules 13:24.930 --> 13:31.850 vibrate trillions of times per second, gamma rays bring us bursts from distant supernovas, 13:31.850 --> 13:38.750 making the electromagnetic field oscillate a billion billion times per second, 60 octaves 13:38.750 --> 13:44.060 above middle C. This range of frequencies, from low to high, 13:44.060 --> 13:48.510 lets the orchestra tell us some amazing stories. Let’s hear the Santa Fe Symphony play the 13:48.510 --> 13:53.000 Grand March from Prokofiev’s Peter and the Wolf… 13:53.000 --> 14:27.140 (Orchestra plays Peter and the Wolf-Prokofiev) 14:27.140 --> 14:34.600 (applause) 14:34.600 --> 14:41.690 >>Cris: When one object vibrates, it can “resonate”, or cause sympathetic vibrations, in another. 14:41.690 --> 14:46.890 Every physical object, such as this wine glass, has natural frequencies at which it likes 14:46.890 --> 14:52.330 to vibrate. (clinks wine glass). If you provide a vibration which is powerful enough, at just 14:52.330 --> 14:58.230 the right frequency, you can create some exciting effects. Famously, if an opera singer sings 14:58.230 --> 15:02.830 loud enough, at just the right tone, she can cause a wine glass to vibrate so much that 15:02.830 --> 15:04.570 it breaks. 15:04.570 --> 15:08.720 (Violinist sings note operatically) 15:08.720 --> 15:13.100 (laughter) 15:13.100 --> 15:18.230 >>Resonances don’t just make things fall apart. They make each musical instrument vibrate 15:18.230 --> 15:24.490 in a unique way. This chladni plate, named after its inventor Ernst Chladni lets us see 15:24.490 --> 15:29.860 the shape of these vibrations. It has a speaker underneath which makes it vibrate and lets 15:29.860 --> 15:36.810 us control the frequency. I'm going to sprinkle some sand on top and once it gets going the 15:36.810 --> 15:40.810 sand will get bounced off the parts of the plate that are vibrating the most, and it 15:40.810 --> 15:45.740 will collect at the still points between them. The shapes and patterns the sand makes will 15:45.740 --> 15:51.640 show us how the plate is responding or resonating to different frequencies. Let's start with 15:51.640 --> 16:00.940 a low frequency. If you remember the monochord, the single stringed instrument from before, 16:00.940 --> 16:06.550 the lowest frequency consists of the entire string going up or down together, but at higher 16:06.550 --> 16:10.000 frequency, one part of the string is wiggling up at the same moment another is wiggling 16:10.000 --> 16:15.460 down. The same thing is happening here and the sand is just collecting at the point in 16:15.460 --> 16:22.960 between them. Let's see what we can do at an even higher frequency. And as the frequency 16:22.960 --> 16:29.060 gets this high the shapes of the vibrations get very complicated. It does look like a 16:29.060 --> 16:34.430 raccoon in a kaleidoscope doesn't it? This violin-shaped plate will give us a taste of 16:34.430 --> 16:39.900 how the soundboard of a violin responds to different musical notes. Of course, real violins 16:39.900 --> 16:48.140 are much more complicated. When you hear a violin being played you should imagine the 16:48.140 --> 16:54.610 body of the violin vibrating with these beautiful intricate patterns, responding to different 16:54.610 --> 17:05.030 notes. It's an amazing degree of complexity it also shows you how a high note on a violin 17:05.030 --> 17:11.190 isn't just a low note played faster. The violin responds in completely different ways to different 17:11.190 --> 17:17.900 tones. This is partly why the sound of the violin is so rich and so complex and sounds 17:17.900 --> 17:26.340 so wonderful in our ears. In fact, even a single note on a violin has many different 17:26.340 --> 17:35.770 frequencies inside it… not just the basic or fundamental one we hear. These layers of 17:35.770 --> 17:41.750 sound give each instrument a unique flavor. We call these hidden frequencies harmonics. 17:41.750 --> 17:48.090 They're kind of like backup singers, singing at frequencies twice as high, three times 17:48.090 --> 17:54.840 as high, and so on above the fundamental. But where are they hiding? Let's look inside 17:54.840 --> 18:00.690 a violin note and see how these harmonics combine to make its unique sound. This audio 18:00.690 --> 18:07.730 analyzer breaks down the violin’s sound wave and shows us the frequencies inside it. 18:07.730 --> 18:13.710 These Peaks here show how strong the different harmonics are and how much each one contributes 18:13.710 --> 18:20.940 to the sound. The fundamental frequency and the hidden harmonics two, three, four times 18:20.940 --> 18:27.240 as high and so on. Now let's try to reconstruct the violin by putting these harmonics back 18:27.240 --> 18:34.080 together. If we play just the fundamental… it doesn't sound like a violin at all. Let's 18:34.080 --> 18:40.690 add the second harmonic… and the third. As we add more harmonics including more and 18:40.690 --> 18:50.720 more of these hidden frequencies the richness and color of the real violin appear. 18:50.720 --> 18:55.330 Let's hear how the same note sounds on different instruments, each with its own unique mix 18:55.330 --> 19:01.980 of harmonics. Here's a middle C on the oboe… 19:01.980 --> 19:08.480 the trombone… 19:08.480 --> 19:11.980 the piano… 19:11.980 --> 19:17.330 and a kazoo. 19:17.330 --> 19:22.590 We love how the orchestra uses these instruments to represent different characters. In our 19:22.590 --> 19:25.630 next piece, We’ll hear the finale from Prokofiev's “Peter 19:25.630 --> 19:30.980 and the Wolf.” See if you can hear all the different characters in it including the flute 19:30.980 --> 19:41.520 as the bird… 19:41.520 --> 19:56.380 the oboe as the duck… 19:56.380 --> 20:13.650 and the clarinet as the cat, and the bassoon as the grandfather have now joined the parade… 20:13.650 --> 20:19.000 Here's the Santa Fe Symphony playing the finale from Prokofiev's "Peter and the Wolf". 20:19.000 --> 21:33.180 (Orchestra plays “Peter and the Wolf”-Prokofiev) 21:33.180 --> 21:40.140 (Applause) 21:40.140 --> 21:42.110 >>Cris: We've learned that the unique sound 21:42.110 --> 21:48.250 of each instrument comes from its shape and how it resonates, but when you add electrons, 21:48.250 --> 21:55.120 vacuum tubes, and antennas to the mix things get even more interesting. Let's visit a remarkable 21:55.120 --> 22:02.630 workshop in Santa Fe to learn more about this. It's a place filled with rare antique clocks 22:02.630 --> 22:08.410 and vintage electronics too. The man at the center of this collection is Andrew Barron. 22:08.410 --> 22:24.900 You can hear him now playing one of his prized possessions. (theremin in the background) 22:24.900 --> 22:29.160 “Hey Andrew!” >>Andrew Barron: Chris it's good to see you. 22:29.160 --> 22:30.790 >>Cris: Thanks so much for showing me your workshop. 22:30.790 --> 22:34.090 >>Andrew: It's my pleasure. >>Cris: This is fantastic. This is an original 22:34.090 --> 22:36.470 Theremin? >>Andrew: It is. it's one of the 120 or so 22:36.470 --> 22:39.810 that survived from the original 1929 production by RCA. 22:39.810 --> 22:44.390 >>Cris: So one of these antennas controls the frequency and the other controls the loudness? 22:44.390 --> 22:48.940 >>Andrew: Exactly right. Yes. When you bring your hand close to this antenna the note goes 22:48.940 --> 22:57.260 higher and if you want to make it louder you can 22:57.260 --> 23:03.500 bring your left hand up over this loop on the left side of the cabinet. 23:03.500 --> 23:09.310 >>Cris: So, this is an electronic instrument. You might think it would make a boring simple 23:09.310 --> 23:11.670 beep, but the sound is so much richer than that. 23:11.670 --> 23:15.380 >>Andrew: Leon Theremin wanted something that was harmonically rich, that would emulate 23:15.380 --> 23:21.410 the sound of orchestra instruments. And, he did it by having two frequency generators 23:21.410 --> 23:26.410 in the instrument, we call them oscillators. Theremin connected one of those oscillators 23:26.410 --> 23:32.250 to this antenna that controls the pitch and so when you bring your hand near it, it shifts 23:32.250 --> 23:38.190 the frequency of that one oscillator and it tugs on the other, and it actually pulls that 23:38.190 --> 23:43.690 pure symmetrical wave into a different shape. >>Cris: It creates some more complex <yes> 23:43.690 --> 23:45.360 sound wave. >>Andrew: Exactly right. 23:45.360 --> 23:46.470 >>Cris: Can I try it out? >>Sure. 23:46.470 --> 23:49.740 >>Cris: I promise I won't break it. >>Andrew: That's all right. Come over here 23:49.740 --> 23:53.050 and let me lead this for the moment, okay. 23:53.050 --> 24:00.300 (playing the theremin-“Auld Lang Syne) 24:00.300 --> 24:03.860 >>Cris: People associate this now with bad 24:03.860 --> 24:08.510 sci-fi and horror movies, but back in the 20s and 30s this was an instrument… it was 24:08.510 --> 24:11.820 part of the orchestra, people wrote music for it. There were great players. 24:11.820 --> 24:17.660 >>Andrew: The original repertoire was actually all classical, and in the first decade or 24:17.660 --> 24:24.620 so of the instrument’s existence. It wasn't until 1945 Alfred Hitchcock, using theremin 24:24.620 --> 24:27.300 in the soundtrack of a movie called “Spellbound.” 24:27.300 --> 24:38.640 (Clip from the movie with theremin playing in the background.) 24:38.640 --> 24:43.540 >>Cris: And who were the great players, the famous virtuoso of the theremin back in the 24:43.540 --> 24:46.720 day? >>Andrew: The really superstar of her day 24:46.720 --> 24:51.830 was Clara Rockmore and she performed with the Philadelphia Orchestra. 24:51.830 --> 24:56.280 >>Cris: Let's relive the glory days of the theremin and listen to Clara Rockmore. 24:56.280 --> 25:22.200 (Clara Rockmore Theremin Performance) 25:22.200 --> 25:26.590 >>Cris: Harmonics are part of the math inside individual notes, but they also let us combine 25:26.590 --> 25:31.450 notes into melodies and harmonies that are beautiful to the ear. Remember how going up 25:31.450 --> 25:36.620 an octave corresponded to doubling the frequency or ratio of two to one? About as simple as 25:36.620 --> 25:41.630 a ratio could be. It turns out that many of the most beautiful chords and harmonies come 25:41.630 --> 25:46.240 from ratios that are almost as simple. >>Guillermo: For instance, if you take middle 25:46.240 --> 25:52.610 C and play the G right above that, we get a ratio of three to two, what we musicians 25:52.610 --> 26:01.780 call a fifth, and it sounds like this (notes played on trumpet). 26:01.780 --> 26:08.280 If we take that same G, and play the C right above that we get a ratio of four to three… 26:08.280 --> 26:17.060 what we call a Fourth. And it sounds like this… (notes played on trumpet) 26:17.060 --> 26:24.170 A ratio of 5 to 4 gives us the ingredients of major chords such as C major, which is 26:24.170 --> 26:37.740 made up of C, E & G, and we’ll add another C in the middle just for fun. So, we get a 26:37.740 --> 26:47.440 C major chord (chord played). And, a ratio of six to five gives us a minor 26:47.440 --> 26:55.940 chord, as in C minor (chord played). Now, in case you've recognized all this little 26:55.940 --> 27:03.700 excerpts we just played, it's because that's the iconic beginning of "Also sprach Zarathustra" 27:03.700 --> 27:10.720 of Ricard Strauss, with which this program opened. Let's hear it again. 27:10.720 --> 27:23.500 (Orchestra plays excerpt from “Also sprach Zarathustra”-Strauss) 27:23.500 --> 27:30.690 >>Cris: Thank you. These ratios have been resonating with us long before Ricard Strauss 27:30.690 --> 27:35.450 used them in his compositions, and to help us understand this ancient connection between 27:35.450 --> 27:40.690 music and math, my friend Penelope Penland has organized this party at the Hotel Santa 27:40.690 --> 27:45.280 Fe. Hey Penelope. Nice to see you. >>Penelope: How are you? 27:45.280 --> 27:52.430 >>Cris: Everybody, your attention please. Thank you very much for being here tonight. 27:52.430 --> 27:57.000 Humans have been making music for as long as we've been human. Whether you write music, 27:57.000 --> 28:01.890 play it, or just listen to it, you're part of that history. And beautiful music has beautiful 28:01.890 --> 28:09.590 math behind it; patterns that we sense intuitively, that make music fun, moving and sacred. Hildegard 28:09.590 --> 28:14.160 von Bingen is one of the earliest composers whose name has come down to us and one of 28:14.160 --> 28:19.580 the earliest we know of who wrote complex melodies. Here is the University of New Mexico's 28:19.580 --> 28:25.200 Hildegard Schola performing one of her pieces. 28:25.200 --> 29:29.220 (singing) 29:29.220 --> 29:35.880 (applause) 29:35.880 --> 29:40.020 >>Cris: The music you just heard is almost a thousand years old, but these patterns go 29:40.020 --> 29:45.430 back even farther. Thousands of years ago, Native American flute makers here in New Mexico 29:45.430 --> 29:50.360 created beautiful instruments, and these ratios still ring in their music today. 29:50.360 --> 29:56.970 >>Penelope: I'm delighted to have you all meet Marlon Magdalena from Jemez Pueblo, and 29:56.970 --> 30:01.340 I'm so interested in your flutes. >>Marlon: Most of these flutes I've handmade. 30:01.340 --> 30:05.700 This particular one, I've hand-carved with a small knife, to get the shapes that you 30:05.700 --> 30:11.030 see on here. So, as a person from Jemez Pueblo, we have traditional ways of life, and a lot 30:11.030 --> 30:17.500 of those ways are very religious. So we view everything that we do as a part of that religion, 30:17.500 --> 30:22.860 as a part of our way of life. So this song that I'm gonna play, it's called “Welcoming the Buffaloes." 30:22.860 --> 31:25.580 (flute performance) 31:25.580 --> 31:32.580 (applause) 31:32.580 --> 31:37.260 >>Cris: before humans ever walked the earth, these ratios and resonances were at work in 31:37.260 --> 31:42.010 the universe. Ganymede, Europa and IO, three of the moons 31:42.010 --> 31:46.710 of Jupiter that Galileo saw through his telescope, which you can see too, through a backyard 31:46.710 --> 31:53.220 telescope or even a decent pair of binoculars, are in a 1 to 2 to 4 resonance. In the time 31:53.220 --> 31:58.880 it takes Ganymede to go around Jupiter once, Europa goes around twice and IO goes around 31:58.880 --> 32:04.740 four times. It's as if they're singing a low C, a middle C, and a high C. The music of 32:04.740 --> 32:09.820 the spheres has always inspired music down here on earth. The composer Gustav Holst may 32:09.820 --> 32:14.060 have been thinking more of astrology then astronomy when he wrote his suite, “The 32:14.060 --> 32:19.390 Planets,” but it’s full of these heavenly ratios. If you listen carefully to his movement, 32:19.390 --> 32:25.110 “Mars the Bringer of War,” you'll hear that he uses a rhythm of five beats per measure 32:25.110 --> 32:29.990 to create a menacing march toward battle, with a mixture of pleasant and unpleasant 32:29.990 --> 32:36.580 harmonies. Here's the Santa Fe Symphony playing Mars from Holst's, “The Planets” 32:36.580 --> 35:21.160 (Orchestra performs an excerpt from “The Planets”-Holst) 35:21.160 --> 35:27.640 (applause) 35:27.640 --> 35:30.820 >>Cris: One of the things which makes “Mars's 35:30.820 --> 35:36.190 March to War” so menacing is a chord called the tritone, or the devil's interval. Remember 35:36.190 --> 35:43.250 the pleasing sounds of the fifth? (chord played) The tritone is just one 35:43.250 --> 35:52.410 semitone lower, but that makes all the difference (Chord played). Sounds strange doesn't it? 35:52.410 --> 35:58.180 Dissonant, and unsettling. See if you can hear it in Ricard Wagner's classic chord from 35:58.180 --> 36:05.800 “Tristan and Isolde,” where he uses it to convey the tragedy of those two lovers. 36:05.800 --> 36:24.920 (Orchestra performs excerpt from “Tristan and Isolde”) 36:24.920 --> 36:33.800 >>Cris: A tritone is exactly half of an octave. For instance, it takes us from C up to F sharp 36:33.800 --> 36:40.460 and another tritone will take us from there up to the next C. But remember that going 36:40.460 --> 36:47.610 up by an octave is equivalent to multiplying by 2, so that means the tritone must multiply 36:47.610 --> 36:55.750 by something, which when we multiply it by itself again, we get 2. We call that number 36:55.750 --> 37:01.870 the square root of 2, but it isn't a simple fraction. It isn't the ratio of any pair of 37:01.870 --> 37:09.950 whole numbers. Lots of ratios come close, but nothing quite works. Written in decimal, 37:09.950 --> 37:17.060 the square root of 2 never ends or repeats. It goes on forever like pi. We call it an 37:17.060 --> 37:23.810 irrational number. The followers of Pythagoras loved whole numbers, so this came as quite 37:23.810 --> 37:29.750 a shock. Legend tells that they threw its discoverer overboard to punish him. Perhaps 37:29.750 --> 37:35.590 this irrationality is why the tritone sounds so strange and why composers like Wagner use 37:35.590 --> 37:41.560 it to create dissonance and tension. Chords and melodies that step outside the simple 37:41.560 --> 37:46.890 Pythagorean intervals are called chromatic from the Greek chroma for color, and they 37:46.890 --> 37:52.790 add a lot of color to a piece. Composers like John Williams use them to express the eerie 37:52.790 --> 37:58.530 and the unfamiliar, like a school of teenage wizards fighting the forces of darkness. 37:58.530 --> 38:03.980 Next, the Santa Fe Symphony goes to Hogwarts. 38:03.980 --> 40:34.620 (Orchestra performs an excerpt from “Harry Potter”- John Williams) 40:34.620 --> 40:43.580 (applause) 40:43.580 --> 40:46.940 >>Cris: We've seen and heard that humans love 40:46.940 --> 40:52.190 patterns. We’re tuned to enjoy them without even realizing it. We like making patterns 40:52.190 --> 40:55.960 where there were none before, and we like finding them when they were hidden. That's 40:55.960 --> 41:01.590 why we like puzzles so much, and at our house puzzles are a big deal. Hey guys. 41:01.590 --> 41:04.430 >>Rosemary: Hi. >>Cris: What did you just build there? 41:04.430 --> 41:10.230 >>Rosemary: This is a soccer ball. >>Cris: In both mathematics and music, a classic 41:10.230 --> 41:19.740 kind of pattern comes from symmetry. We look about the same if you flip our image in a 41:19.740 --> 41:25.500 mirror. Some of our more distant relatives look the same if you rotate them. These intricate 41:25.500 --> 41:31.840 moorish mosaics from the Alhambra in Spain look the same if we shift them over, or rotate 41:31.840 --> 41:43.120 them. Another kind of symmetry, both in mathematics and the natural world, is a fractal. A fractal 41:43.120 --> 41:47.410 is something which, when you zoom in on it, it looks the same, because each part is a 41:47.410 --> 41:50.830 smaller copy of the whole. >>Rosemary: Hey Dad, we have a fractal in 41:50.830 --> 41:53.840 the fridge. >>Cris: A fractal in the fridge? <Yeah> A 41:53.840 --> 41:57.420 fractal in the fridge? >>Rosemary: We got it the other day while 41:57.420 --> 42:02.070 we were at the farmers market >>Cris: Nice. It's a broccoli. <Yeah> But, 42:02.070 --> 42:05.370 what makes it a fractal? >>Rosemary: See how there's one big cone <uh-huh>. 42:05.370 --> 42:09.770 This cone is made up of lots of little cones which are made up of lots of little cones 42:09.770 --> 42:13.370 which are made of lots of little cones, which just keeps getting smaller and smaller. 42:13.370 --> 42:15.920 >>Cris: Can we eat it later? >>Rosemary: Yeah! 42:15.920 --> 42:22.650 >>Cris: The most famous fractal is the Mandelbrot set. Zooming in on different areas reveals 42:22.650 --> 42:30.850 endless variety, but always with a smaller copy. The entire set hiding deep inside it. 42:30.850 --> 42:36.570 A musical fractal would be a piece where the theme harmonizes with a slowed down version 42:36.570 --> 42:42.400 of itself. That way, the small-scale structure of the piece is echoed on a larger, grander 42:42.400 --> 42:45.070 scale. >>Guillermo: Johann Sebastian Bach loved to 42:45.070 --> 42:51.940 play with symmetry. In many of his fugues, the themes are heard upside down or even backwards. 42:51.940 --> 42:57.070 He also liked to play with fractal symmetry, where the themes are given in different scales 42:57.070 --> 43:03.040 or sped up or slowed down. We're gonna play a fugue by Bach and you'll hear first the 43:03.040 --> 43:13.020 theme in the French horn… 43:13.020 --> 43:26.260 And then an upside down version of it in the second trumpet... 43:26.260 --> 43:46.700 And a slowed down version of it in the first trumpet... 43:46.700 --> 44:18.700 And finally, a slowed down version of the theme on the tuba... 44:18.700 --> 44:22.870 Now see if you can hear all of these themes 44:22.870 --> 44:27.730 as members of the brass section of the Santa Fe Symphony put it all together when they 44:27.730 --> 44:44.240 play Contrapunctus Number Seven from the "Art of the Fugue" by Johann Sebastian Bach. 44:44.240 --> 47:46.580 (Music) 47:46.580 --> 47:55.240 (applause) 47:55.240 --> 48:00.320 >>Cris: Our brains love music like that Bach Fugue, because we love patterns. We love to 48:00.320 --> 48:05.610 recognize a melody and hear it again in a new form. But, we also love to be surprised. 48:05.610 --> 48:11.040 We love it when patterns go sideways and turn out to be more complicated than we thought. 48:11.040 --> 48:16.260 Take the prime numbers for example. A prime is a number which cannot be broken down into 48:16.260 --> 48:22.520 smaller factors. Five and seven are prime, but six isn't because it's two times three. 48:22.520 --> 48:28.620 We can find the primes, if we're willing to do a little work. Start with all the numbers. 48:28.620 --> 48:36.950 One doesn't count. Circle two, and now cross out all the even ones. Next, circle three 48:36.950 --> 48:43.540 and cross out all the multiples of three. Six, nine, 12… and so on. The next one left 48:43.540 --> 48:49.460 over is five. Circle it, and cross out the multiples of five. Circle seven, and cross 48:49.460 --> 48:57.780 out its multiples. And so on. The ones that are left are the primes. That wasn't so hard, 48:57.780 --> 49:02.490 but the primes have many secrets. You'll notice that there are some pairs of primes that are 49:02.490 --> 49:10.430 just two apart. 5 and 7, 11 and 13 and so on. But what's the pattern here? Are there 49:10.430 --> 49:16.330 an infinite number of these pairs? Do they go on forever? No one knows. It's these patterns 49:16.330 --> 49:21.490 and surprises that fascinate me as a mathematician. They're what makes math so intriguing and exciting. 49:21.490 --> 49:25.680 >>Guillermo: Prime numbers can add suspense 49:25.680 --> 49:32.850 to music. Here's a piece in five beats, a prime number, which creates a sense of excitement 49:32.850 --> 49:39.220 and international intrigue. 1, 2, 3, 4, 5. 1… 2… 3… 4… 5… 49:39.220 --> 50:16.000 (Music-"Mission Impossible Theme.") 50:16.000 --> 50:22.740 (applause) 50:22.740 --> 50:29.320 >>Cris: We've heard the math inside notes and instruments, the ratios that make up rhythms, 50:29.320 --> 50:39.210 chords and harmonies and the symmetries and surprises that make music so fascinating, 50:39.210 --> 50:46.340 from ancient times to today. Music and math both reveal something fundamental about our 50:46.340 --> 50:55.560 world… a tension between simplicity and complexity, between order and chaos. They 50:55.560 --> 51:07.070 speak to our brains and our hearts in endlessly surprising ways. And they're both very much 51:07.070 --> 51:16.840 a part of what makes us human. In our finale tonight, the composer John Adams starts out 51:16.840 --> 51:21.550 with a simple rhythm on the wood block, but he quickly overlays this with more and more 51:21.550 --> 51:26.670 complex rhythms, making it harder and harder for us to hold onto it all. If you can feel 51:26.670 --> 51:32.220 this joyful confusion, the sense that a pattern is always just a little too complicated to 51:32.220 --> 51:36.750 grasp, so that you're always just on the brink of understanding it, you'll know what it's 51:36.750 --> 51:43.330 like to explore the frontiers of mathematics. Ladies and gentlemen the Santa Fe Symphony 51:43.330 --> 51:46.880 playing John Adams's "Short ride in a Fast Machine." 51:46.880 --> 56:12.820 (music) 56:12.820 --> 57:28.500 (Applause) 57:28.500 --> 57:31.340 >>MAJOR FUNDING FOR THIS PROGRAM WAS PROVIDED BY: 57:31.340 --> 57:37.360 SHELBY CULLOM DAVIS CHARITABLE FUND INC, BY ANDREW DAVIS. 57:37.360 --> 57:41.580 ADDITIONAL FUNDING WAS PROVIDED BY ACEVES-LONDON FAMILY FUND AT THE SANTA FE COMMUNITY FOUNDATION. 57:41.580 --> 57:47.360 AND BARBARA ERDMAN FOUNDATION. 57:47.360 --> 57:56.180 AND BY THE KIND WORLD FOUNDATION, PAT AND RUTH CONNERY, CATHERINE WEBB, HAROLD FOLLEY 57:56.180 --> 58:03.670 AND JENNIE NEGIN, THE ESTATE OF DAVID ELLIOTT YOUNG BY MARY McCACHREN… AND NEW MEXICO 58:03.670 --> 58:08.670 PBS VIEWERS LIKE YOU. THANK YOU.