The “Magical Mystery Chord” finally revealed?

The classic Beatles song, “A Hard Day’s Night”, opens with a complex ringing chord that has had songbooks (and musicians) arguing among themselves for decades. Complicating the answer is that even Paul McCartney can’t exactly remember what was done.

Full disclosure: I relate to the Beatles much the way I relate to Mozart: I recognize their musical genius but much of their most popular music does not ‘move’ me either intellectually or emotionally. But I love a good musical puzzle as much as can be.

In principle, given modern computer technology, the problem of transcribing a piece of music should be simple: digitize the audio, carry out a Fourier analysis, and convert the resulting frequencies to note names. Right?

Well… Feed in unaccompanied flute and this will work fine. (As anybody who’s owned an analog synth knows, a triangle wave is a pretty decent starting point for a flute sounds — and while a triangle does have some harmonics, the fundamental is very strong and there are only odd harmonics so you can tell apart the fundamentals pretty easily from the rest in the Fourier spectrum.) Feed in a Hammond organ with just a single drawbar open: ditto. Feed in a more complex sound but with restricted harmony (e.g., a violin playing only single notes), no problem. Feed in a complex chord played by multiple instruments on top of each other, and things get hairier. Have some of the multiple instruments not be quite in tune, or let some be in equal temperament and others in just intonation, and things gets even worse.

An applied mathematician at Dalhousie University did a Fourier analysis on the opening chord some time ago and turned that into a paper.  Does this sound like an academic with too much time on his hands, “partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada,” no less? Well, to me it sounds like a good “torture test” for the robustness of a musical transcription code. And where it comes to science popularization, this definitely hits the spot with the musically minded: only yesterday I saw another popular article about the now a decade old analysis being linked on Instapundit.

Just retaining all frequencies with relative amplitudes above 0.02 still gave him 48 frequencies, from which he squeezed a solution that looks good in theory but just doesn’t sound “quite right”.

A musical transcription site run by somebody with the delightful pseudonym “Waynus of Uranus” points out a fly in the ointment that people who grew up with digital recording wouldn’t even have thought of. Back in the day, loud bass tones meant pushing against the limitations of vinyl singles and lo-fi audio equipment alike, so the deep end of the bass (about 80 Hz and lower) was routinely rolled off with an equalizer or a highpass filter during mixing or mastering. What this means, for example: if Paul were to strike an open D string on his bass guitar (or an A string at the fifth fret) his fundamental would be below the filter cutoff, and the Fourier spectrum would instead have the second harmonic much stronger — leading to claims like “Paul played a D3 and a soft D2 at the same time”. I know bass players like Geddy Lee or Rush or Steve Harris of Iron Maiden play lots of double-stops, but this really is a progressive rock or metal thing to do, not a pop thing.

Applied mathematician Kevin Houston takes it from there and digs further in a very geekish way. While the original record was mono, it turns out there is a stereo mix made for the movie—and in the early days of stereo, it was not unusual for recording engineers to just put some instruments all left and others all right, with the vocal in the center. (This is, pretty much, how I used to jam along with Deep Purple records: Jon Lord’s organ and Ritchie Blackmore’s guitar were usually at opposite end of the stereo image, so you could single out their parts by listening to one stereo channel at a time.) In the stereo

In the stereo mix of AHDN, Paul (bass) and George (12-string guitar) are off to one side, and John (acoustic guitar) off to the other, together with producer George Martin on piano. Better still: after subtracting the left channel from the right (i.e., “phase-inverting”), it becomes clear that the acoustic is playing an Fadd9 chord. (That means: an F major chord with an added ninth, a.k.a. a “Steely Dan chord“. It differs from a major ninth chord F9 in that the seventh is omitted.)

To cut a very long story short (some mathematicians can get quite verbose ;)), this is the solution (which relies on a good dose of Occam’s razor/the Law of Parsimony as well):

  • Paul just plays a low D2, but because of EQing off the deep end, the D3 overtone/second harmonic comes through louder than the fundamental, hence the acoustic illusion that the bass note played is D3
  • John plays F2 A2 F3 A3 C4 G4 (in standard tuning, frets 1-0-3-2-1-3)
  • George plays the same chord, but on a 12-string in standard tuning—where the bottom four “courses” have the second string one octave higher. Hence aside from the slight tuning discrepancy with John, he adds F4 A4 as new pitches
  • Finally, George Martin on the piano, with the sustain pedal down, plays D2 G2 D3 G3 C4, which one could call a Gsus4/D chord. Sympathetic resonance from the undamped piano strings adds the wash of low-level extra pitches that befuddles the Fourier analysis.

Not only does this not require attributing instrumental acrobatics to the Beatles that are out of character for them, but actually playing those notes on the respective instruments does produce a sound quite like the record. (Listen at 7:17 in the video below.)

Kevin and his collaborators could not readily find an electric 12-string, so they simulated that by layering two six-string electric chords: once fretted 1-0-3-2-1-3, the second time 13-12-15-14-1-2 with an extra hand. “Fake Nashville Tuning“, if you like.)

If this isn’t  the solution, it sounds much closer than anything else I’ve heard. Enjoy the above video!


Who first said: “We must keep an open mind, but not so open that our brains fall out”?

I have heard the quote in the title attributed to all sorts of people, ranging from mathematician Alan Ross Anderson to Mark Twain to Prince Charles [OK, the sophomoric jokes write themselves]. But who really said this?

Quoteinvestigator did the legwork and also cites another article researching the origin of the quote.

Let us keep our minds open, by all means, as long as that means keeping our sense of perspective and seeking an understanding of the forces which mould the world. But don’t keep your minds so open that your brains fall out! There are still things in this world which are true and things which are false; acts which are right and acts which are wrong, even if there are statesmen who hide their designs under the cloak of high-sounding phrases.

— Walter Kotschnig November 8, 1939

Now, who is Walter Kotschnig? This American academic and diplomat of Austrian-Jewish origin has a fairly detailed bio in the German-language Wikipedia, but none in the English version. A brief summary:

He was born in the historical town Judenburg in Steiermark/Styria, Austria as the son of a school principal. The town name is first documented in 1074: it was an important commercial center at the time and, as the name suggests, had a significant Jewish community (which was expelled in 1496). During the Third Reich, there were attempts to change the “embarrassing” name, but a decision was postponed until after the “Endsieg” (final victory), which thank G-d never came.

Kotschnig started his university studies in nearby Graz. As he became ill with tuberculosis, he was briefly cared for by an American relief organization based in the Netherlands: the experience made him passionate about international collaboration. Upon obtaining his doctorate in political science at the U. of Kiel, Germany in 1924 and marrying (to psychologist Elinid Prys), he took a position with the International Student Service in Geneva, and from 1927 until 1934 served as secretary-general of the organization. Subsequently, he worked for the League of Nations (the interbellum predecessor of the UN) as director of the High Commission for German Refugees. In 1936 he emigrated with his family to the USA, where he took up teaching positions at two of the “Seven Sisters” women’s colleges, Smith College and Mount Holyoke. In addition, he published scholarly papers on education policy planning. He became a US citizen in 1942, published a book with proposals for democratic education reforms in formerly fascist countries, and in 1944 was involved in the planning of the Dumbarton Oaks conference, which was the cradle of the UN. In 1947 he became the head of the International Organizations desk at the US State Department, to eventually rise to the position of Assistant Secretary of State (1965-1971).

At any rate, on November 8, 1939, he gave a speech at Smith College in honor of the upcoming Armistice day, where he made the above remark. The manuscript of his speech has been found in his collected papers at SUNY Albany.

The speech was later reported on in an article in the Smith Alumnae Quarterly [“Chapel and Assembly Notes”, Vol. 31(2), p. 153 (1940)] where the quote first appears in print in that form.

Tim Farley in his article does, however, note an earlier quote in a Yale Law Journal article by law professor Max Radin, “On Legal Scholarship,”  ) that may have been a direct inspiration.

[Practical gentlemen] have a number of bitterly sarcastical comments on persons whose minds are so open that their brains fall out.

Radin may have borrowed it in turn from somebody else, but Kotschnig is clearly the first documented person to use the quote in substantially its present form.


If it keeps on raining…



The other day I heard a strange and wonderful cover of a blues classic, performed by Tool vocalist Maynard James Keenan’s second band.

A delta blues purist might get an even bigger stroke than they would from Led Zeppelin’s famous version. But precisely because of the change of context, and Maynard’s emotional yet understated delivery, the song hit me like a hammer.

The original was written about the 1927 Great Mississippi Flood, the most destructive river flood in the history of the USA, which made hundreds of thousands homeless. Many of those were black, and joined the Great Migration from the agricultural South to the industrial cities of the Northeast and Midwest.

But the metaphor of a levee about to break speaks to me on a number of levels.

There is the general sense that insanity and inanity in the political system and the popular culture have reached a level where the rest of us feel like we are drowning in derp and d-baggery. Exhibit, well, T: My Beautiful but Evil Space Mistress’s article on the left’s long post-election tantrum.

At another level, the Harvey Weinstein scandal (and another shoe about to drop) show that the depravity of some beacons of popular culture has risen to such levels that even with the help of a fawning, compliant press it can no longer be contained. “When the levee breaks, you’ll have no place to stay.” Not that it came as a great surprise to anyone familiar with the inner workings of certain industries.

My friend “masgramondou” comments here on the peculiar “bootlegger and Baptist coalition” (or is that a CAT coalition: cads and Tumblristas?) that has arisen in an attempt to change the subject. (Mayim Bialik learned the hard way what happens when you deviate from the party line.)

At a third level, one sees something more hopeful. The ever-increasing shrillness of the would-be opinion makers and virtue signalers in politics, media (but I repeat myself), academia, and popular culture are causing ever more of us to “cut the cord” and tune them out entirely. Too many alternatives are available nowadays, and if none are to our liking, the entry barriers to creating our own have never been lower. (The flip side, of course, is the ever greater challenge to stand out from the crowd of creators.)

Are we at a tipping point, and is a return to sanity near? “And grace and good sense will be found in the eyes of G-d and man” (ומצא חן ושכל טוב בעיני א׳ ואדם), as it says in the Grace After Meals. May it happen speedily and in our days.

Post-Yom Kippur reflection on intergenerational guilt

During the Yom Kippur service, we repeat many times the “Vidui” (confession) prayer. While reading the commentary in the Artscroll Machzor (AM below), I was struck by the gloss on the line

‘But we and our ancestors have sinned’ (אבל אנחנו ואבותינו חטאנו)
The gloss asks: why are the sins of ancestors mentioned, which we did not commit? And indeed, Leviticus 26:39-40 reads (KJV translation):
And also in the iniquities of their fathers shall they pine away. [But] if they shall confess the iniquity and the iniquity of their fathers…
Now does that mean that the people of Israel today are on the hook in perpetuity for, say, the sin of the Golden Calf? The implications of a “yes” answer for contemporary political reparations debates in the US speak for themselves: are present-day nonblack, non-aboriginal Americans on the hook for slavery abolished in 1865, or for the tragedy of the American Indians? (Actually,  the “reparations” advocates go one step further and expand the “blood guilt” to people whose ancestors weren’t even in the US in those times!)
Closer to home: what does Lev. 26:39-40 imply for the responsibility of present-day Germans for the Shoah and other genocidal and democidal campaigns that happened before the defeat of National Socialism? That is, do people who were born or came of age after these crimes against humanity were committed bear some sort of blood guilt?
As pointed out in AM, the Talmud (TB Sanhedrin 27b) explains that we are punished for our ancestors’ sins only if we approve of their way of life, and especially if we adopt it.
The Moroccan Jewish Torah commentator Rabbi Chaim Ibn-Attar, in his commentary Or haChaim, notes ad loc. Lev. 26:40 that a proper understanding of our ancestors’ sins is often a prerequisite of repentance. Paraphrasing AM, sometimes we accept family or community traditions as a proper way of life because ‘it’s always been done this way and no-one was ever punished.’ Thus we are to ‘confess’ — i.e., acknowledge — such sins of the past.
If true teshuva is achieved (repentance, but literally: “return” [to G-d] or “backtracking” from the evil ways) then the guilt has been washed away.  The Torah describes Amalek as what amounts to the first terrorist (Deut. 25:17-18): the Amalekites avoided combat with the Israelite warriors but lay in ambush and attacked the women, the children, and the elderly. Indeed, Amalek becomes a symbol, or a synecdoche if you like, for mortal enemies of the Jewish people that arise in every generation.
The villain in the book of Esther, Haman, is identified as a descendant of Amalek. But the sages also refer to Shimon and Levi has “having the seed of Amalek in them”, making it clear this is not a matter of biological descent.
And the Talmud (TB Gittin 57b) indeed gives a long list of descendants of evildoers who have now embraced the Torah, including “the descendants of Haman [and hence of Amalek] are now students of Torah in Bnei Brak” (see also TB Sanhedrin 96b). They have made full teshuvah, have fully rejected the poisonous doctrine and practices of Amalek, and hence are cleansed of his guilt.
Summing up: Judaism rejects the concept of intergenerational guilt in the narrow sense. In a broader sense, it does extend the guilt to descendants who continue walking in their ancestors’ ways, but once the descendants make a clean break with those behaviors and attitudes, they also make a clean break with the guilt.

Rush, “Between the Wheels” – A song for uncertain times


You know how that rabbit feels
Going under your speeding wheels
Bright images flashing by
Like windshields towards a fly
Frozen in that fatal climb
But the wheels of time just pass you by…

Wheels can take you around.
Wheels can cut you down

We can go from boom to bust
From dreams to a bowl of dust
We can fall from “rockets’ red glare”
Down to “brother, can you spare?”
Another war
Another wasteland
And another lost generation

German Elections II: Up is down, down is up

Browsing through German election updates in Die Welt over lunch, I got a feeling it isn’t just the US that has entered Robert Heinlein’s “Crazy Years”. (see my previous post)

  • Various virtue-signalers are of course hand-wringing about the “extreme right” AfD. The actual NS-nostalgists of the NPD, however, polled less than 0.5%, worse than the satirical Die Partei. (As explained by the article in Die Welt: under German law,  a party must poll at least 0.5% nationally to be eligible for certain subsidies.)
  • It is, however, clearly true that the AfD is riven by a power struggle between two camps: a “right-liberal” one around Frauke Petry that is pro-free-market, Euroskeptic, and populist, and a “national-conservative” camp currently led by former CDU politician Alexander Gauland. The latter camp appears to include some truly unsavory elements, presumably “entryists” from the extreme right.
  • Now Frauke Petry and her husband Marcus Pretzell [sic] are leaving AfD, citing its “radicalization”, and just bought a new internet domain Die Blauen (the blue ones – in Europe blue has traditionally been the color of classical liberalism). She herself was elected to the Bundestag directly (about half the seats are constituency seats) so she doesn’t have to vacate her seat.
  • This leaves Alice Weidel as the co-chair representing the party’s “right-liberal” wing.  Weidel, a Ph.D. economist who speaks fluent Mandarin Chinese, works as a management consultant to startup companies and… is an overt lesbian who lives in a domestic partnership with a Swiss filmmaker of Sri Lankan origins. The couple have a second residence in Biel, Switzerland and raise her partner’s two biological children together. [How many leftie heads have exploded yet?] Weidel is opposed to same-sex marriage and adoption, as well as what she calls “pushing gender idiocy on prepubescent children”, but says she “supports lifestyles other than the traditional family” as well as, naturally, domestic partnerships.
  • Lest you think that the AfD (which may well implode) is the only party riven by internal contradictions: the Greens have pretty much for all their existence been divided between a “Fundi” (fundamentalist Green) extremist wing and a pragmatic “Realo” wing. Coalition negotiators for the Greens are having to placate both camps.
  • And as if this weren’t enough headaches for Merkel (at this point I feel almost sorry for her): her Bavarian sister party the CSU is now signaling that their presence in her coalition is not to be taken for granted. They are quite nervous about regional elections next year, and are in particular demanding an upper limit on refugee admissions. The Greens, for their part, have indicated that this is a nonstarter for them.

As the SPD had earlier announced it was taking the opposition cure following its historical nadir this election, Merkel’s options are basically reduced to a “Jamaica Coalition” of CDU/CSU (black), FDP (yellow), and Greens. But the latter is increasingly looking like an exercise in squaring the circle.

And I would not rule out an internal coup against Merkel by the right wing of the CDU.