WEBVTT 00:00:00.000 --> 00:00:02.419 Welcome to today's Deep Dive. We are so glad 00:00:02.419 --> 00:00:05.000 to have you with us. Today we're looking at something 00:00:05.000 --> 00:00:07.719 that, honestly, it acts less like a traditional 00:00:07.719 --> 00:00:11.300 piece of music and more like... Like a mathematical 00:00:11.300 --> 00:00:14.279 virus. Yes, exactly. A mathematical virus. It's 00:00:14.279 --> 00:00:17.739 this highly specific grouping of notes that somehow 00:00:17.739 --> 00:00:20.379 just managed to infect the minds of, well... 00:00:20.600 --> 00:00:22.620 the greatest composers of the 20th century. It 00:00:22.620 --> 00:00:24.679 really did. It served as this hidden architectural 00:00:24.679 --> 00:00:27.960 blueprint for decades of avant -garde music. 00:00:28.079 --> 00:00:30.480 Right. So our mission for today's deep dive is 00:00:30.480 --> 00:00:33.740 to explore a single fascinating Wikipedia article 00:00:33.740 --> 00:00:36.039 that is entirely dedicated to a musical concept 00:00:36.039 --> 00:00:39.359 that sounds honestly more like a magical incantation 00:00:39.359 --> 00:00:42.020 or a secret code. Oh, for sure. It is called 00:00:42.020 --> 00:00:44.750 the Ode to Napoleon Hexachord. Which is quite 00:00:44.750 --> 00:00:47.210 the name. It is. So, okay, let's unpack this. 00:00:47.270 --> 00:00:49.689 At its most basic fundamental level, a hexachord 00:00:49.689 --> 00:00:51.750 is just a collection of six musical pitch classes. 00:00:52.009 --> 00:00:54.270 And in this specific case, for the Ode to Napoleon 00:00:54.270 --> 00:00:56.670 hexachord, we are talking about the notes C, 00:00:56.909 --> 00:01:02.130 D -flat, E, F, G -sharp, and A. Or, if we step 00:01:02.130 --> 00:01:04.569 out of traditional letter names and look at it 00:01:04.569 --> 00:01:07.090 through the lens of musical set theory, it's 00:01:07.090 --> 00:01:10.670 represented by the numbers 0, 1, 4, 5, 8, 9. 00:01:11.340 --> 00:01:13.260 Yeah, and to give you the foundational context 00:01:13.260 --> 00:01:15.939 for those six notes, this specific collection 00:01:15.939 --> 00:01:18.700 wasn't just some abstract theoretical concept 00:01:18.700 --> 00:01:20.859 pulled out of thin air by a mathematician. No, 00:01:20.920 --> 00:01:23.340 not at all. It derives its rather dramatic name 00:01:23.340 --> 00:01:27.099 from a very specific piece of music. Arnold Schoenberg's 00:01:27.099 --> 00:01:31.239 1942 12 -tone composition, Ode to Napoleon Bonaparte. 00:01:31.579 --> 00:01:35.359 Opus 41. Opus 41. Right. That work is a setting 00:01:35.359 --> 00:01:38.500 of a text by the poet Lord Byron, and the entire 00:01:38.500 --> 00:01:41.239 structural foundation of the piece relies heavily 00:01:41.239 --> 00:01:44.099 on this exact hexachord. Which is wild to think 00:01:44.099 --> 00:01:46.900 about. It is. But what is truly remarkable when 00:01:46.900 --> 00:01:48.900 looking through our source material is how this 00:01:48.900 --> 00:01:51.280 configuration of notes took on a life of its 00:01:51.280 --> 00:01:54.019 own entirely beyond Schoenberg. Yeah, you see 00:01:54.019 --> 00:01:56.540 it popping up everywhere. Exactly. You'll find 00:01:56.540 --> 00:01:58.560 it going by several aliases in the theoretical 00:01:58.560 --> 00:02:01.519 literature. It's often referred to as the hexatonic 00:02:01.519 --> 00:02:04.459 collection or the hexatonic set class. Some theorists 00:02:04.459 --> 00:02:08.080 even refer to it as the magic hexachord. And 00:02:08.080 --> 00:02:11.120 for those who utilize Alan Forte's rigorous taxonomic 00:02:11.120 --> 00:02:13.800 system for classifying pitch class sets, it is 00:02:13.800 --> 00:02:16.740 officially designated as Forte number 620. Forte 00:02:16.740 --> 00:02:19.120 number 620. I mean, it literally sounds like 00:02:19.120 --> 00:02:22.259 a catalog number for a rare cursed artifact or 00:02:22.259 --> 00:02:24.580 something. It really does. But here's where it 00:02:24.580 --> 00:02:27.620 gets really interesting. when you start to dissect 00:02:27.620 --> 00:02:31.400 the structural makeup of 620 you realize it is 00:02:31.400 --> 00:02:33.879 a complete mathematical anomaly oh absolutely 00:02:33.879 --> 00:02:35.919 our source breaks down the component intervals 00:02:35.919 --> 00:02:38.500 measured from the root note so the set contains 00:02:38.500 --> 00:02:41.810 a minor second a major third a perfect fourth, 00:02:42.009 --> 00:02:44.810 an augmented fifth, and a major sixth, along 00:02:44.810 --> 00:02:47.370 with the root itself. Right. It's this incredibly 00:02:47.370 --> 00:02:50.110 specific, almost artificially restricted set 00:02:50.110 --> 00:02:52.689 of distances between the notes. And what's fascinating 00:02:52.689 --> 00:02:55.930 here is how those distances interact when you 00:02:55.930 --> 00:02:59.090 analyze the collection as a whole entity. In 00:02:59.090 --> 00:03:01.229 post -tonal music theory, there's a tool called 00:03:01.229 --> 00:03:03.289 an interval vector. Right, the interval vector. 00:03:03.330 --> 00:03:05.639 Essentially, it's a six -digit code. It tallies 00:03:05.639 --> 00:03:07.939 up every single possible distance, every interval 00:03:07.939 --> 00:03:10.659 class between every possible pair of notes within 00:03:10.659 --> 00:03:13.240 a given set. Okay. The interval vector for the 00:03:13.240 --> 00:03:23.900 620 hexachord is 303630. 303630. Exactly. Now, 00:03:23.960 --> 00:03:26.139 if we decode what those numbers actually represent 00:03:26.139 --> 00:03:29.120 based strictly on the text, it reveals a structural 00:03:29.120 --> 00:03:32.060 extreme. The six digits correspond to the six 00:03:32.060 --> 00:03:34.740 basic interval classes, from the smallest to 00:03:34.740 --> 00:03:36.620 the largest. Right. The second digit represents 00:03:36.620 --> 00:03:39.180 interval class two, which is the major second, 00:03:39.319 --> 00:03:43.139 or a whole step. The sixth digit represents interval 00:03:43.139 --> 00:03:45.740 class six, which is the tritone. And both of 00:03:45.740 --> 00:03:48.610 those digits are zero. Precisely. Because there 00:03:48.610 --> 00:03:51.610 are zeros in those positions, it means this hexachord 00:03:51.610 --> 00:03:54.250 completely lacks any whole steps, and it completely 00:03:54.250 --> 00:03:56.830 lacks any tritones. And if we translate that 00:03:56.830 --> 00:03:58.930 from math into what our ears actually perceive, 00:03:59.270 --> 00:04:02.189 the absence of those specific intervals is staggering. 00:04:02.550 --> 00:04:04.469 It really is. I mean, the major second, the whole 00:04:04.469 --> 00:04:06.650 step is the fundamental building block of almost 00:04:06.650 --> 00:04:09.490 every traditional Western scale. It's what gives 00:04:09.490 --> 00:04:11.569 melodies their forward motion, you know. Oh, 00:04:11.569 --> 00:04:14.610 absolutely. It's the staircase. Yes. And the 00:04:14.610 --> 00:04:17.910 tritone is the primary engine of harmonic tension 00:04:17.910 --> 00:04:20.430 in classical music. It's that crunchy interval 00:04:20.430 --> 00:04:23.310 that desperately wants to resolve. But because 00:04:23.310 --> 00:04:26.569 Forte 620 completely lacks both of them, any 00:04:26.569 --> 00:04:29.430 composer using this set is forced to build melodies 00:04:29.430 --> 00:04:32.589 and harmonies without the very tools that traditionally 00:04:32.589 --> 00:04:35.110 give music its sense of scale and resolution. 00:04:35.490 --> 00:04:37.850 Right. But to compensate, look at the other numbers 00:04:37.850 --> 00:04:40.649 in that vector. The threes, the fours, and the 00:04:40.649 --> 00:04:43.050 sixes. Wait, the vector was three, zero, three, 00:04:43.089 --> 00:04:46.629 six, three, zero. So there's a six in the fourth 00:04:46.629 --> 00:04:49.230 position. Exactly. The vector indicates that 00:04:49.230 --> 00:04:51.410 this hexachord has the highest possible number 00:04:51.410 --> 00:04:54.329 of interval classes, three and four, of any six 00:04:54.329 --> 00:04:56.750 -member set. Okay, wow. Interval class three 00:04:56.750 --> 00:04:58.689 is the minor third, and interval class four is 00:04:58.689 --> 00:05:01.850 the major third. So you have an absolute overabundance 00:05:01.850 --> 00:05:04.910 of thirds, but a total vacuum of whole steps 00:05:04.910 --> 00:05:07.589 and tritones. It's so weird. It's like a jigsaw 00:05:07.589 --> 00:05:09.329 puzzle that is... completely missing certain 00:05:09.329 --> 00:05:11.389 shapes of pieces. Like you open the box and there 00:05:11.389 --> 00:05:13.509 are absolutely no edge pieces. Right. But you 00:05:13.509 --> 00:05:16.370 have a massive, massive overabundance of those 00:05:16.370 --> 00:05:18.790 weird middle pieces with three knobs. That is 00:05:18.790 --> 00:05:21.019 a great analogy. It shouldn't be able to form 00:05:21.019 --> 00:05:24.120 a coherent picture with such a skewed distribution 00:05:24.120 --> 00:05:27.519 of materials. Yet the geometry of it interlocks 00:05:27.519 --> 00:05:31.220 flawlessly. And it interlocks flawlessly because 00:05:31.220 --> 00:05:34.160 of a mind -bending level of mathematical symmetry. 00:05:34.480 --> 00:05:36.199 Let's get into the symmetry because this blew 00:05:36.199 --> 00:05:38.600 my mind. It's wild. According to our source, 00:05:38.779 --> 00:05:43.019 the 620 hexachord maps onto itself exactly three 00:05:43.019 --> 00:05:46.800 times under transposition. specifically at transposition 00:05:46.800 --> 00:05:51.139 levels 0, 4, and 8. Okay, so to ground that conceptually 00:05:51.139 --> 00:05:53.920 for you listening, transposition just means shifting 00:05:53.920 --> 00:05:56.680 the entire collection of notes up or down the 00:05:56.680 --> 00:05:58.800 keyboard by a certain distance. Right. So if 00:05:58.800 --> 00:06:00.959 you take this hexachord and shift every single 00:06:00.959 --> 00:06:04.839 note up by four semitones, which is a major third 00:06:04.839 --> 00:06:07.339 take, you don't get a new set of notes. You land 00:06:07.339 --> 00:06:09.759 on the exact same six pitch classes you started 00:06:09.759 --> 00:06:12.060 with. It maps perfectly back onto itself. It's 00:06:12.060 --> 00:06:14.639 like rotating a perfect circle. Exactly. Furthermore, 00:06:14.759 --> 00:06:16.579 the source notes it maxed onto itself exactly 00:06:16.579 --> 00:06:19.680 three times under inversion at levels 1, 4, and 00:06:19.680 --> 00:06:21.740 9. And inversion means turning the intervals 00:06:21.740 --> 00:06:24.819 upside down. Right. So between the transpositions 00:06:24.819 --> 00:06:26.899 and the inversions, this single collection of 00:06:26.899 --> 00:06:30.220 notes possesses six degrees of symmetry. Six 00:06:30.220 --> 00:06:32.839 degrees of symmetry. Because it is so highly 00:06:32.839 --> 00:06:35.199 symmetrical, so perfectly balanced upon itself, 00:06:35.480 --> 00:06:38.920 there are only four distinct forms of this hexachord 00:06:38.920 --> 00:06:41.360 in existence. Which is incredibly rare. Yeah. 00:06:41.819 --> 00:06:45.279 Most standard collections of notes can be transposed 00:06:45.279 --> 00:06:48.480 or inverted into 12 or even 24 distinct forms. 00:06:48.720 --> 00:06:52.519 But 620 is locked into a closed loop of just 00:06:52.519 --> 00:06:55.019 four. And the way these four forms interact with 00:06:55.019 --> 00:06:57.439 each other is mathematically beautiful. It really 00:06:57.439 --> 00:07:00.000 is binary. When you compare them, you find that 00:07:00.000 --> 00:07:02.939 any two... forms either overlap perfectly by 00:07:02.939 --> 00:07:05.459 way of an augmented triad Or they don't overlap 00:07:05.459 --> 00:07:07.459 at all. We should probably clarify the concept 00:07:07.459 --> 00:07:09.199 of the augmented triad for a moment, just because 00:07:09.199 --> 00:07:11.279 it's central to why this hexachord sounds the 00:07:11.279 --> 00:07:13.860 way it does. Oh, good call. So an augmented triad 00:07:13.860 --> 00:07:16.279 is essentially a chord built by stacking two 00:07:16.279 --> 00:07:18.740 major thirds on top of each other. Think of the 00:07:18.740 --> 00:07:21.800 notes C, E, and G sharp. Unlike a traditional 00:07:21.800 --> 00:07:25.319 major or minor chord, an augmented triad divides 00:07:25.319 --> 00:07:27.740 the octave into three perfectly equal parts. 00:07:28.300 --> 00:07:31.180 Because of that perfect symmetry, your ear cannot 00:07:31.180 --> 00:07:33.939 easily pinpoint a definitive root note. It just... 00:07:33.959 --> 00:07:37.180 floats yes it sounds suspended floaty and inherently 00:07:37.180 --> 00:07:40.240 uneasy it totally lacks the grounding of a traditional 00:07:40.240 --> 00:07:43.180 chord and our source points out that the entire 00:07:43.180 --> 00:07:46.939 620 hexachord can be completely exhausted meaning 00:07:46.939 --> 00:07:49.540 every single one of its six notes is accounted 00:07:49.540 --> 00:07:52.660 for by just combining two augmented triads two 00:07:52.660 --> 00:07:55.339 augmented triads that's it alternatively the 00:07:55.339 --> 00:07:58.199 text states you can exhaust the entire set using 00:07:58.199 --> 00:08:00.560 six minor and major triads that have their roots 00:08:00.560 --> 00:08:03.720 located along the notes of an augmented It's 00:08:03.720 --> 00:08:06.000 an architecture of endless structural redundancy. 00:08:06.019 --> 00:08:08.180 It really is. And I want to add one more mathematical 00:08:08.180 --> 00:08:10.500 nugget regarding this redundancy to round out 00:08:10.500 --> 00:08:13.220 this section. Lay it on me. In set theory, the 00:08:13.220 --> 00:08:15.800 complement of a pitch class set is the collection 00:08:15.800 --> 00:08:17.860 of all the notes from the 12 -tone chromatic 00:08:17.860 --> 00:08:20.079 scale that are not included in your original 00:08:20.079 --> 00:08:23.279 set. Since 620 has six notes, its complement 00:08:23.279 --> 00:08:26.120 must naturally be the remaining six notes. But 00:08:26.120 --> 00:08:28.319 the complement of the 620 hexachord is actually 00:08:28.319 --> 00:08:31.399 itself. It is another instance of 620. Wait, 00:08:31.439 --> 00:08:34.059 so which means if you play the six notes of the 00:08:34.059 --> 00:08:36.379 Ode to Napoleon hexachord, and then you play 00:08:36.379 --> 00:08:38.120 the remaining six notes on the piano that you 00:08:38.120 --> 00:08:40.360 haven't touched yet, those leftover notes form 00:08:40.360 --> 00:08:43.259 the exact same intervallic structure. Precisely. 00:08:43.259 --> 00:08:45.899 It is its own complement. That is insane. Furthermore, 00:08:46.120 --> 00:08:48.759 the source details its subset and superset relationships. 00:08:49.379 --> 00:08:52.740 The only five -note subset of 620 is designated 00:08:52.740 --> 00:08:56.090 as 521. If you look for the complement of 521, 00:08:56.230 --> 00:08:58.090 which would be a seven note set, you get 721. 00:08:58.350 --> 00:09:01.690 And it just so happens that 721 is the only superset 00:09:01.690 --> 00:09:04.389 of 620. It's a completely closed mathematical 00:09:04.389 --> 00:09:07.250 system. It is. In fact, the text notes that the 00:09:07.250 --> 00:09:09.590 only hexachord in existence that is mathematically 00:09:09.590 --> 00:09:12.230 more redundant than this one is the whole tone 00:09:12.230 --> 00:09:15.389 scale, which is forte number 635. Wow. Okay. 00:09:15.529 --> 00:09:19.250 So what does this all mean? We've mapped out 00:09:19.250 --> 00:09:21.809 the transpositions, the inversions, all the vectors, 00:09:21.909 --> 00:09:24.350 the subsets but we have to bring this back to 00:09:24.350 --> 00:09:26.529 the ear. We have to bring it back to the actual 00:09:26.529 --> 00:09:28.929 music and why you should care about these overlapping 00:09:28.929 --> 00:09:31.870 notes. Right. Arnold Schoenberg was not just 00:09:31.870 --> 00:09:34.389 doing geometry on a chalkboard. He was writing 00:09:34.389 --> 00:09:38.590 music, particularly in 1942 when he wrote the 00:09:38.590 --> 00:09:41.409 Ode to Napoleon Bonaparte. Right. Why would a 00:09:41.409 --> 00:09:43.490 composer intentionally restrict themselves to 00:09:43.490 --> 00:09:46.450 such a redundant, symmetrical, closed loop set 00:09:46.450 --> 00:09:49.090 of notes? That brings us to the brilliant paradox 00:09:49.090 --> 00:09:51.330 of Schoenberg's use of this hexachord in his 00:09:51.330 --> 00:09:54.500 ode. As you likely know, Schoenberg is the primary 00:09:54.500 --> 00:09:57.700 pioneer of the 12 -tone technique. This is an 00:09:57.700 --> 00:09:59.659 intonal system designed deliberately to ensure 00:09:59.659 --> 00:10:01.720 that all 12 notes of the chromatic scale are 00:10:01.720 --> 00:10:03.679 sounded as often as one another. To completely 00:10:03.679 --> 00:10:06.600 avoid any traditional tonal centers. Exactly. 00:10:06.940 --> 00:10:09.639 The explicit goal was to move beyond the familiar 00:10:09.639 --> 00:10:13.159 major and minor keys of classical music. Yet 00:10:13.159 --> 00:10:15.399 despite operating within this uncompromising, 00:10:15.460 --> 00:10:18.399 mathematically strict avant -garde system, the 00:10:18.399 --> 00:10:20.820 primary form of the tone row used in the ode 00:10:20.820 --> 00:10:23.220 possesses a very unique property. And this is 00:10:23.220 --> 00:10:26.039 the magic trick. It is. The source highlights 00:10:26.039 --> 00:10:28.360 that because of the internal structure of the 00:10:28.360 --> 00:10:31.519 620 hexachord, traditional familiar sounds, specifically 00:10:31.519 --> 00:10:35.200 the triads of G minor, E flat minor, and B minor, 00:10:35.320 --> 00:10:38.269 are able to easily appear. Which is just a massive 00:10:38.269 --> 00:10:40.090 contradiction. I mean, think about it. If the 00:10:40.090 --> 00:10:42.629 entire premise of his 12 -tone system was to 00:10:42.629 --> 00:10:45.549 abandon traditional keys and tonal centers, why 00:10:45.549 --> 00:10:48.470 intentionally construct a row that builds a backdoor 00:10:48.470 --> 00:10:50.830 right back to traditional minor chords? Right. 00:10:50.909 --> 00:10:53.350 Inside a strict, mathematically rigorous avant 00:10:53.350 --> 00:10:55.669 -garde system, Schoenberg essentially built a 00:10:55.669 --> 00:10:57.889 backdoor for traditional music to emerge. It 00:10:57.889 --> 00:11:00.070 is less a compromise and more a masterful act 00:11:00.070 --> 00:11:02.769 of subversion. Oh, I like that. He engineered 00:11:02.769 --> 00:11:06.789 this specific row using this specific magic hexachord 00:11:06.789 --> 00:11:09.250 so that he could adhere strictly to the rules 00:11:09.250 --> 00:11:11.730 of the new atonal world while simultaneously 00:11:11.730 --> 00:11:14.809 summoning the ghost of the old tonal world whenever 00:11:14.809 --> 00:11:17.950 he deemed it dramatically necessary. Wow. He 00:11:17.950 --> 00:11:19.769 is using the extreme mathematical constraints 00:11:19.769 --> 00:11:23.730 of 620 not to limit his expression but to breed 00:11:23.730 --> 00:11:27.269 a new kind of creativity. It's an illusion. To 00:11:27.269 --> 00:11:28.909 the analyst looking at the score, the music is 00:11:28.909 --> 00:11:31.480 perfectly atonal and mathematically sound. But 00:11:31.480 --> 00:11:34.399 to the listener, a traditional minor triad suddenly 00:11:34.399 --> 00:11:36.980 emerges from the chaos. He used a radically modern 00:11:36.980 --> 00:11:39.460 system to smuggle traditional elements into the 00:11:39.460 --> 00:11:42.240 composition. It's genius. And because that concept 00:11:42.240 --> 00:11:44.419 is so powerful, you can begin to see why this 00:11:44.419 --> 00:11:47.179 specific hexachord became a foundational tool 00:11:47.179 --> 00:11:49.580 for other legendary composers. If we connect 00:11:49.580 --> 00:11:51.539 this to the bigger picture, the proliferation 00:11:51.539 --> 00:11:54.299 of the 620 hexachord demonstrates the shared 00:11:54.299 --> 00:11:56.720 vocabulary of the 20th century avant -garde. 00:11:56.860 --> 00:11:59.159 It wasn't just Schoenberg's secret weapon. No, 00:11:59.159 --> 00:12:01.059 not at all. Our source... lists an incredible 00:12:01.059 --> 00:12:03.399 lineup of heavy hitters who utilize this exact 00:12:03.399 --> 00:12:06.159 collection of notes. We have Alexander Scrab 00:12:06.159 --> 00:12:08.299 in step who is famously obsessed with building 00:12:08.299 --> 00:12:10.960 his own mystic chords and esoteric harmonies. 00:12:10.960 --> 00:12:14.679 Exactly. We have Bela Bartok, an absolute mastermind. 00:12:14.720 --> 00:12:17.940 And we have Anton Webern, who used it extensively 00:12:17.940 --> 00:12:21.549 in his concerto Opus 24. And it became deeply 00:12:21.549 --> 00:12:24.309 embedded in post -tonal music theory across different 00:12:24.309 --> 00:12:27.330 theoretical camps. The Hungarian musicologist 00:12:27.330 --> 00:12:30.909 Erno Lenvai, for instance, analyzed this exact 00:12:30.909 --> 00:12:33.190 pitch collection. But he referred to it under 00:12:33.190 --> 00:12:36.090 a different framework as the 1 to 3 model scale. 00:12:36.559 --> 00:12:39.259 The 1 .3 model. And then we have Milton Babbitt. 00:12:39.419 --> 00:12:41.799 Right. The text mentions that Babbitt called 00:12:41.799 --> 00:12:45.179 the 620 hexachord one of his six all -combinatorial 00:12:45.179 --> 00:12:47.179 hexachord source sets. We should probably define 00:12:47.179 --> 00:12:50.000 combinatoriality really quickly as it is crucial 00:12:50.000 --> 00:12:51.840 to understanding Babbitt's interest in the set. 00:12:52.039 --> 00:12:54.480 Go for it. Combinatoriality is a structural property 00:12:54.480 --> 00:12:56.960 where a composer can take a specific hexachord 00:12:56.960 --> 00:12:59.539 and pair it with a transformed version of itself. 00:13:00.110 --> 00:13:02.570 Perhaps a transposed or inverted version. Okay. 00:13:02.669 --> 00:13:04.970 If the hexachord is combinatorial, those two 00:13:04.970 --> 00:13:07.110 halves will lock together to form a complete 00:13:07.110 --> 00:13:09.850 12 -tone row without a single note overlapping 00:13:09.850 --> 00:13:12.389 or repeating. It's like finding two puzzle pieces 00:13:12.389 --> 00:13:15.169 that perfectly complete the whole picture. Exactly. 00:13:15.210 --> 00:13:18.210 It is a highly efficient way to generate full 00:13:18.210 --> 00:13:21.549 chromatic harmony while maintaining strict structural 00:13:21.549 --> 00:13:25.149 control. Because 620 is so heavily symmetrical, 00:13:25.570 --> 00:13:29.039 it is all combinatorial. meaning it possesses 00:13:29.039 --> 00:13:31.799 this property across multiple different transformations. 00:13:32.159 --> 00:13:34.940 It's the ultimate modular building block, and 00:13:34.940 --> 00:13:37.720 Babbitt put it to rigorous use. The source notes 00:13:37.720 --> 00:13:39.960 he used it in the third and fourth movements 00:13:39.960 --> 00:13:42.279 of his composition for four instruments, written 00:13:42.279 --> 00:13:45.700 in 1948, as well as in his composition for 12 00:13:45.700 --> 00:13:47.779 instruments. And it's actually worth noting that 00:13:47.779 --> 00:13:50.320 Schoenberg himself didn't just abandon the hexachord 00:13:50.320 --> 00:13:53.080 after the ode, he also used it combinatorially 00:13:53.080 --> 00:13:56.169 in his suite, Op. 29. The structural potential 00:13:56.169 --> 00:13:58.690 of the set was so profound that it literally 00:13:58.690 --> 00:14:01.950 inspired other composers to write direct homages 00:14:01.950 --> 00:14:05.549 to its mathematical properties. Yes. Luigi Nono 00:14:05.549 --> 00:14:08.590 was so captivated by it that in 1950, he literally 00:14:08.590 --> 00:14:11.289 wrote a piece called Variazione Canonica sulla 00:14:11.289 --> 00:14:14.940 Serie dell 'Op, 41 Dion Schoenberg. He is directly 00:14:14.940 --> 00:14:17.320 referencing, right there in the title, the very 00:14:17.320 --> 00:14:20.120 opus 41 piece that gives the Haute -Napoleon 00:14:20.120 --> 00:14:23.220 hexachord its name, and Bruno Moderna is another 00:14:23.220 --> 00:14:26.159 major figure mentioned in the text who utilized 00:14:26.159 --> 00:14:29.200 this specific set. They recognized that Forte 00:14:29.200 --> 00:14:31.580 620 wasn't just a random assortment of avant 00:14:31.580 --> 00:14:34.820 -garde pitches, it was a highly refined structural 00:14:34.820 --> 00:14:38.220 mechanism. It offered extreme mathematical control 00:14:38.220 --> 00:14:40.879 while simultaneously offering startling musical 00:14:40.879 --> 00:14:43.279 flexibility. As we start to wrap up this deep 00:14:43.279 --> 00:14:45.340 dive, I want to trace the path we've walked today 00:14:45.340 --> 00:14:47.700 for you. We started with a list of six simple 00:14:47.700 --> 00:14:52.639 notes, C, D flat, E, F, G sharp, and A. Right. 00:14:52.759 --> 00:14:55.379 And we uncovered a highly symmetrical, heavily 00:14:55.379 --> 00:14:58.240 redundant mathematical marvel. We looked at its 00:14:58.240 --> 00:15:00.779 highly unusual interval vector and learned how 00:15:00.779 --> 00:15:03.220 the total absence of major seconds and tritones 00:15:03.220 --> 00:15:06.120 forces a composer into a highly constrained sonic 00:15:06.120 --> 00:15:08.539 space. And we explored its staggering symmetry. 00:15:08.970 --> 00:15:10.929 leading to the fact that only four distinct forms 00:15:10.929 --> 00:15:13.549 of this hexachord even exist. Exactly. We saw 00:15:13.549 --> 00:15:15.809 how this redundant mathematical anomaly served 00:15:15.809 --> 00:15:18.389 as a secret architectural blueprint for the 20th 00:15:18.389 --> 00:15:20.850 century's most important composers. Which I think 00:15:20.850 --> 00:15:23.070 reinforces the value of this knowledge for you 00:15:23.070 --> 00:15:25.490 as a listener. It proves that even in modern 00:15:25.490 --> 00:15:28.389 art that might seem chaotic or unstructured to 00:15:28.389 --> 00:15:30.909 the untrained ear, there is often a profound, 00:15:31.090 --> 00:15:34.980 rigorous, and almost magic hidden order. holding 00:15:34.980 --> 00:15:37.259 everything together. It's not just random noise. 00:15:37.519 --> 00:15:39.779 Not at all. And this raises an important question, 00:15:39.860 --> 00:15:41.919 something for you to mull over after we finish 00:15:41.919 --> 00:15:45.580 today. I love these. Consider the concept of 00:15:45.580 --> 00:15:48.159 hiding traditional elements inside radical new 00:15:48.159 --> 00:15:51.320 systems. If Schoenberg could use strict mathematical 00:15:51.320 --> 00:15:54.399 symmetry to secretly smuggle traditional minor 00:15:54.399 --> 00:15:57.679 triads into a 12 -tone piece, effectively hiding 00:15:57.679 --> 00:16:00.649 the old world inside the new, What other seemingly 00:16:00.649 --> 00:16:03.009 rigid or chaotic systems in our modern world 00:16:03.009 --> 00:16:05.669 might be secretly harboring familiar patterns? 00:16:05.889 --> 00:16:07.929 Oh, wow. Just waiting to be decoded by someone 00:16:07.929 --> 00:16:10.169 who knows the magic key. That is a phenomenal 00:16:10.169 --> 00:16:13.110 thought to leave on. The chaos we perceive might 00:16:13.110 --> 00:16:15.529 just be a highly symmetrical puzzle we haven't 00:16:15.529 --> 00:16:17.590 yet learned to read. Thank you so much for joining 00:16:17.590 --> 00:16:19.950 us on this deep dive into the ode to Napoleon 00:16:19.950 --> 00:16:22.990 hexachord. We've loved dissecting this hidden 00:16:22.990 --> 00:16:25.490 architecture with you. Keep questioning the structures 00:16:25.490 --> 00:16:27.990 around you. Keep looking for the underlying order 00:16:27.990 --> 00:16:29.970 in the noise, and we will catch you on the next 00:16:29.970 --> 00:16:30.210 one.