November 29, 1947: The Story of a Vote


Seventy years ago to this day, the United Nations voted on Resolution 181, the partitioning of the British Mandate into Jewish and Arab states as recommended by the UN’s special investigative commission (UNSCOP).

The story behind the scenes is told in this short movie, which combines historical footage with recent interviews of people who lived the event. The woman in the thumbnail is Suzy Eban, wife of Abba Eban.

A two-thirds majority was needed. In the end, thirty-three countries voted in favor:

• Latin American and Caribbean Group: Bolivia, Brazil, Costa Rica, Dominican Republic, Ecuador, Guatemala, Haiti, Nicaragua, Panama, Paraguay, Peru, Uruguay, Venezuela

• Western Europe and Others: Belgium, Denmark, France, Iceland, Luxemburg, Netherlands, Norway, Sweden

• Eastern Europe: Byelorussian SSR (Belarus), Ukrainian SSR (Ukraine), USSR, Czechoslovakia, Poland

• African: Liberia and South Africa

• Asia-Pacific: Australia, New Zealand, Philippines

• North America: USA and Canada

Ten countries abstained:

• Latin American and Caribbean Group: Argentina, Chile, Colombia, El Salvador, Honduras, Mexico

• Four other countries: UK (the outgoing mandate holder), China, Ethiopia, and Yugoslavia

Thailand was absent from the vote.

Thirteen countries voted against, ten of them Muslim:

•  Arab or Islamic countries: Afghanistan, Iran, Iraq, Lebanon, Pakistan, Saudi Arabia, Syria, Yemen, Turkey, and Egypt

• Others: India, Cuba, and Greece. It should be noted that Greece then had a large diaspora in countries like Egypt, and was thus vulnerable to threats.

Voting happened by voice vote, alphabetically. The vote that put the resolution over the top was cast by the Philippines.

The day of the vote is remembered in Israel to this day as kaf-tet be-November  (from the Hebrew notation of the number 29, כ׳׳ט). The British Mandate was to end at midnight between May 14-15, 1948. On the afternoon of May 14, around 4pm, a hastily convened assembly gathered at a museum building in Tel-Aviv, and with a minimum of pomp and circumstance, David Ben-Gurion proclaimed the independent State of Israel.


Our hope is not yet lost

The hope of two thousand years

To be a free people in our land

The land of Zion, Jerusalem





Saturday delight: Bach’s “Chaconne” on 11-string guitar

I accidentally stumbled on Moran Wasser’s amazing performance of Bach’s Chaconne in D minor, BWV 1004, on an 11-string guitar, embedded below:

What’s the deal with an… 11-string guitar?! Sounds pretty scary, no? Actually, 11-string and 13-string guitars are similar to the Baroque lute in conception:  The top six or seven strings are played like a standard guitar, while the additional bass strings are typically tuned ad hoc to cover the bass notes of the piece, and plucked as open strings with the thumb as a harmonic foundation. I am sure that sympathetic vibration also adds a lot to the body of the sound when these strings are not explicitly struck.

But let’s talk about the piece now. Many instrumental jazz and rock improvisations are based on a repeated “riff” or bass line that forms the foundation. This is, however, not something invented in the modern era. Early Western classical music had a form called a “ground” where exactly the same was done: take, for example, William Byrd’s virginal/harpsichord piece “The Bells” (1580). During the early baroque period, two forms of Western art music evolved with a repeated-riff structure: the Passacaglia and the Chaconne. Significantly, both were originally slow, stately dances in 3/2 rhythm.

It seems nobody is quite sure what is the difference between the two: I remembered it as “in a chaconne, the repeated riff is always in the bass, while in a passacaglia, it can move through all voices” — but it appears this definition was too narrowly based on J. S. Bach’s monumental examples, the Passacaglia and Fugue for Organ in C minor, BWV 582, (about which I have blogged previously), and the Chaconne from the Partita for Solo Violin in D minor, BWV 1004.

This piece, which stretches the capabilities of violin and violinist to the very limit, has numerous times been arranged for other instruments: for piano (by Ferruccio Busoni and by Alexander Siloti), for piano left hand (by Brahms), for organ, and indeed for orchestra (by Leopold Stokowski). It is particularly often performed on guitar (either in Andres Segovia’s arrangement or directly from the original score).

Moran Wasser’s arrangement is transposed one half-step down from the original, i.e., to C# minor: this sounds equivalent to playing it in “baroque tuning” (A=415 Hz) in its original key. Note that he places a capo on the 2nd fret over the seven top strings.

For those who prefer a violin original, here is Hilary Hahn’s performance:

The piano arrangements for two hands both tend toward the flashy, but Siloti’s is to my ears the more musical of the two. Here is a surprisingly powerful recent performance by a young pianist named Tanya Gabrielian:



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!