Mars, magic squares, and music

About a year ago I wrote about Jupiter’s magic square. Then yesterday I was listening to the New Sounds podcast that mentioned a magic square associated with Mars. I hadn’t heard of this, so I looked into it and found there were magic squares associated with each of solar system bodies known to antiquity (i.e. Sun, Mercury, Venus, Moon, Mars, Jupiter, and Saturn).

Here is the magic square of Mars:

The podcast featured Secret Pulse by Zack Browning. From the liner notes:

Magic squares provide structure to the music. Structure provides direction to the composer. Direction provides restrictions for the focused inspiration and interpretation of musical materials. The effect of this process? Freedom to compose.

The compositions on this CD use the 5×5 Magic Square of Mars (Secret Pulse), the 9×9 Magic Square of the Moon (Moon Thrust), and the ancient Chinese 3×3 Lo Shu Square found in the Flying Star System of Feng Shui (Hakka Fusion, String Quartet, Flying Tones, and Moon Thrust) as compositional models.  The musical structure created from these magic squares is dramatically articulated by the collision of different musical worlds …

I don’t know how the composer used these magic squares, but you can listen to the title track (Secret Pulse) on the podcast.

Related posts:

All day long I’d bidi-bidi-bum

This evening I watched my daughter in Fiddler on the Roof. I thought I knew the play pretty well, but I learned something tonight.

Before the play started, someone told me that the phrase “bidi-bidi-bum” in “If I Were a Rich Man” is a Yiddish term for prayer. I thought “All day long I’d bidi-bidi-bum” was a way of saying “All day long I’d piddle around.” That completely changes the meaning of that part of the song.

When I got home I did a quick search to see whether what I’d heard was correct. According to Wikipedia,

A repeated phrase throughout the song, “all day long I’d bidi-bidi-bum,” is often misunderstood to refer to Tevye’s desire not to have to work. However, the phrase “bidi-bidi-bum” is a reference to the practice of Jewish prayer, in particular davening.

Unfortunately, Wikipedia adds a footnote saying “citation needed,” so I still have some doubt whether this explanation is correct. I searched a little more, but haven’t found anything more authoritative.

Now I wonder whether there’s any significance to other parts of the song that I thought were just a form of Klezmer scat singing, e.g. “yubba dibby dibby dibby dibby dibby dibby dum.” I assumed those were nonsense syllables, but is there some significance to them?

Update: At Jason Fruit’s suggestion in the comments, I asked about this on Isaac Moses replied that the answer is somewhere in between. The specific syllables are not meaningful, but they are intended to be reminiscent of the kind of improvisation a cantor might do in singing a prayer.

John Coltrane versus Kenny G

My previous post began with a story about a performance by John Coltrane. Douglas Groothuis left a comment saying that he used the same story in his book Truth Decay. Before telling the Coltrane story, Groothuis compares the philosophies of Kenny G and John Coltrane.

Kenny G’s philosophy is as shallow as his music.

I just play for myself, the way I want to play, and it comes out sounding like me.

Coltrane’s philosophy, like his music, is more ambitious.

Overall, I think the main thing a musician would like to do is give a picture to the listener of the many wonderful things he knows and senses in the universe. That’s what music is to me — it’s just another way of saying this is a big, wonderful universe we live in, that’s been given to us, and here’s an example of just how magnificent and encompassing it is. That’s what I would like to do. I think that’s one of the greatest things you can do in life, and we all try to do it in some way. The musician’s is through his music.

As Groothuis comments, Kenny G only spoke of expressing himself, while Coltrane “expressed a yearning to represent objective realities musically.”

Nunc dimittis

In his book The Call, Os Guinness tells the following story of John Coltrane.

After one utterly extraordinary rendition of “A Love Supreme,” Coltrane stepped off the stage, put down his saxophone, and said simply “Nunc dimittis.” … Coltrane felt he could never play the piece more perfectly. If his whole life had been lived for that passionate thirty-two minute jazz prayer, it would have been worth it. He was ready to go.

Nunc dimittis is Latin for “Now dismiss.” These are the opening words of the Vulgate translation of the Song of Simeon, Luke 2:29–32. Simeon says he is ready to die because he has seen what he was waiting for, the promised Messiah.

Lord, now lettest thou thy servant depart in peace, according to thy word:
For mine eyes have seen thy salvation,
Which thou hast prepared before the face of all people;
A light to lighten the Gentiles, and the glory of thy people Israel.

Coltrane’s story brings several things to mind. First, it is awe-inspiring to imagine an accomplishment so fulfilling that you would say “That was it. I’m ready to die.”

Next, it’s interesting to ponder Coltrane’s eclectic spirituality. I knew Christianity was part of his spiritual gumbo, but I was surprised to hear that he made a spontaneous reference to Latin liturgy.

Coltrane was canonized by the African Orthodox Church in 1982. Truth is stranger than fiction.

Finally, I was interested in the name Nunc dimittis itself. I hadn’t heard it before. (I’ve only been part of non-liturgical churches.) I thought the name might only be familiar to Catholics, being a Latin term. But an Episcopalian friend informed me that the Anglican mass preserves many Latin titles even though the liturgy itself is in English. I suppose Coltrane encountered this Anglican name via the Episcopalian influence on the African Methodist Episcopalian Zion Church.

Closely related post:

John Coltrane versus Kenny G

Less related posts:

Software sins of omission (Software and the Book of Common Prayer)
Doing good work with bad tools (Charlie Parker story)
Dave Brubeck mass (Mass composed by a jazz icon)

Accelerated learning

Derek Sivers tells how a mentor was able to teach him a semester’s worth of music theory in three hours. His mentor also prepared him to place out of four more classes in four sessions. He gives the details in his blog post There’s no speed limit. It’s an inspiring story.

However, Sivers didn’t go through his entire education this way. He finished his degree in 2.5 years, but at the rate he started he could have finished in under a semester. Obviously he wasn’t able to blow through everything as fast as music theory.

Some classes compress better than others. Theoretical classes condense better than others. A highly motivated student could learn a semester of music theory or physics in a short amount of time. But it would take longer to learn a semester of French or biology no matter how motivated you are because these courses can’t be summarized by a small number of general principles. And while Sivers learned basic music theory in three hours, he says it took him 15 years to learn how to sing.

Did Sivers’ mentor expose him to everything students taking music theory classes are exposed to? Probably not. But apparently Sivers did learn the most important material, both in the opinion of his mentor and in the opinion of the people who created the placement exams. His mentor not only taught him a lot of ideas in a short amount of time, he also told him when it was time to move on to something else.

It’s hard to say when you’ve learned something. Any subject can be explored in infinite detail. But there comes a point when you’ve learned a subject well enough. Maybe you’ve learned it to your personal satisfaction or you’ve learned it well enough for an exam. Maybe you’ve reached diminishing return on your efforts or you’ve learned as much as you need to for now.

One way to greatly speed up learning is to realize when you’ve learned enough. A mentor can say something like “You don’t know everything, but you’ve learned about as much as you’re going to until you get more experience.”

Occasionally I’ll go from feeling I don’t understand something to feeling I do understand it in a moment, and not because I’ve learned anything new. I just realize that maybe I do understand it after all. It’s a feeling like eating a meal quickly and stopping before you feel full. A few minutes later you feel full, not because you’ve eaten any more, but only because your body realizes you’re full.

Related posts:

Casablanca and Einstein

Humphrey Bogart and Ingrid Bergman in Casablanca

If you’ve ever seen Casablanca, you’ve heard the song As Time Goes By, but only the chorus.

You must remember this
A kiss is just a kiss, a sigh is just a sigh.
The fundamental things apply
As time goes by.

Did you know the song includes references to relativity and four-dimensional geometry?

Here’s the first verse.

This day and age we’re living in
Gives cause for apprehension
With speed and new invention
And things like fourth dimension.

Yet we get a trifle weary
With Mr. Einstein’s theory.
So we must get down to earth at times
Relax relieve the tension.

And no matter what the progress
Or what may yet be proved
The simple facts of life are such
They cannot be removed.

Here are the full lyrics.

Via Math Mutation podcast #134.

Screwtape on music and silence

From Screwtape, the senior demon of The Screwtape Letters:

Music and silence — how I detest them both! … no square inch of infernal space and no moment of infernal time has been surrendered to either of those abominable forces, but all has been occupied by Noise — Noise, the grand dynamism, the audible expression of all that is exultant, ruthless, and virile … We will make the whole universe a noise in the end. We have already made great strides in that direction as regards the Earth. The melodies and silences of Heaven will be shouted down in the end. But I admit we are not yet loud enough, or anything like it. Research is in progress.

Related posts:

Windows XP and Ubuntu start-up music

I just realized that the start-up music for Ubuntu is a variation on the start-up music for Windows XP. (You can hear the Ubuntu theme in this video at around 0:10 [Update: video has been removed]. The Windows XP theme is in this video at around 1:16.) If you don’t hear the similarity, concentrate on the rhythm rather than the melody.

The Ubuntu music style is African, like the word ubuntu. It was influenced by Windows, like the Ubuntu user interface, but it’s a new composition.

I’ve been running Ubuntu on a virtual machine. The sound quality was so bad that I never clearly heard Ubuntu start up. But I recently installed Ubuntu on a physical machine and heard the start-up music clearly for the first time.

Eclectic mix podcast

If you’re looking for a way to discover some new music, check out Eclectic Mix. The show lives up to its name, featuring all kinds of music. For example, here’s a show with Latin Giants of Jazz and here’s one with The Monks and Choirs of Kiev Pechersk Lavra.

Calendars, Connections, and Cats

James Burke had a television series Connections in which he would create a connection between two very different things. For example, in one episode he starts with the discovery of the touchstone for testing precious metals and tells a winding tale of how the touchstone led centuries later to the development of nuclear weapons.

I had a Connections-like moment when a calendar led to some physics, which then lead to Andrew Lloyd Webber’s musical Cats.

A few days ago I stumbled on Ron Doerfler’s graphical computing calendar and commented on the calendar here. When I discovered Ron Doerfler’s blog, I bookmarked his article on Oliver Heaviside to read later. (Heaviside was a pioneer in what was later called distribution theory, a way of justifying such mathematical mischief as differentiating non-differentiable functions.) As I was reading the article on Heaviside, I came to this line:

At one time the ionosphere was called the Heaviside layer …

Immediately the lyrics “Up, up, up to the Heaviside layer …” started going through my head. These words come from the song “The Journey to the Heaviside Layer” from Cats. I had never thought about “Heaviside” in that song as being related to Mr. Heaviside. I’ve never seen the lyrics in print, so I thought the words were “heavy side” and didn’t stop to think what they meant.

Andrew Lloyd Webber based Cats on Old Possum’s Book of Practical Cats by T. S. Eliot. The song “The Journey to the Heaviside Layer” in particular is based on the poem Old Deuteronomy from Eliot’s book. Webber used the Heaviside layer as a symbol for heaven, based on an allusion in one of T. S. Eliot’s letters. The symbolism is obvious in the musical, but I hadn’t thought about “Heaviside layer” as meaning “the heavens” (i.e. the upper atmosphere) as well as heaven in the theological sense.

The opening chord of "A Hard Day’s Night"

The opening chord of the Beatles song “A Hard Day’s Night” has been something of a mystery. Guitarists have tried to reproduce the chord with limited success. Turns out there’s a good reason why they haven’t figured it out: the chord cannot be played on a guitar alone.

Jason Brown has digitally analyzed the chord using Fourier analysis and determined that there must have been a piano in the recording studio playing along with the guitars. Brown has determined what notes each member of the Beatles were playing.

I heard Jason Brown’s story on the Mathematical Moments podcast. In addition to the chord discussed above, Brown talks about other things he has discovered about the Beatles and about the relationship between music and math in general. Unfortunately, Mathematical Moments does not make it easy to link to individual episodes. Here is a link to a PDF file of show notes with the audio embedded. The file is slow to download, and your PDF viewer may not support it. Here’s a link directly to just the MP3 audio file.

The Mathematical Moments podcast also does not make it obvious that you can subscribe to the podcast; they only provide links to individual episodes with fat PDF files. However, you can subscribe by using the URL

Click to learn more about consulting help with signal processing


Circle of fifths and number theory

Why does music have a circle of fifths but no circle of thirds or circle of sixths?

If you start at on any note and go up by fifths, you’ll cycle through the entire chromatic scale. For example: C, G, D, A, E, B, F#, C#, G#, D#, A#, F, C. If you go up by fourths, you’ll get the same sequences of notes but in the reverse order. So there’s a cycle of fifths and a cycle of fourths, but there are no other ways to cycle through the chromatic scale other than the chromatic scale itself.

If you start at C and go up by minor thirds, for example, you’ll only hit four distinct notes before returning to where you started: C, D#, F#, A, C. You don’t cycle through all 12 notes, only four of them. Instead of filling out a chromatic scale, you fill out a diminished chord. You could fill out two other diminished chords by starting on C# or on D. Going up by major sixths produces the same sequence of notes as going down by minor thirds.

What’s special about fourths and fifths that their cycles cover the chromatic scale while cycles of other intervals partition the chromatic scale into smaller groups of notes? A fourth is 5 chromatic steps and a fifth is 7 chromatic steps. The numbers 5 and 7 are relatively prime to 12, that is, they share no factors with 12 (other than 1, which doesn’t count).

The numbers less than 12 and relatively prime to 12 are 1, 5, 7, and 11. These intervals correspond to the ascending chromatic scale, the circle of fourths, the circle of fifths, and the descending chromatic scale.

The numbers less than 12 and not relatively prime to 12 are 2, 3, 4, 6, 8, 9, and 10. Going up by 2 chromatic steps produces a whole-tone scale. Going up by 10 steps produces the same sequence of notes but in the opposite order. Going up by 3  or 9 steps produces a diminished chord. Going up by 4 or 8 steps produces an augmented chord. Going up by 6 steps produces a tritone pair. (I’m used to jazz terminology which uses the term “tritone.” Classical musicians would more likely say “augmented fourth” or “diminished fifth.”)

Now imagine a non-traditional scale that divided the octave into some number of parts other than 12. Suppose this new scale has n notes. Cycling in steps of size m will cover all n notes if and only if m and n are relatively prime. For example, if we divide the scale into 15 parts, we could cover all 15 pitches if we went up 4 steps at a time. We could play notes 1, 5, 9, 13, 2, 6, 10, 14, 3, 7, 11, 15.

If m and n are not relatively prime, let d be their greatest common divisor, the largest number that divides both m and n. Then going up d parts at a time will cycle through m/d notes and there will be d distinct cycles. For example, if there were 15 notes in our scale and we went up in intervals of 10 notes, we would cover 3 distinct notes, and we could make 5 different such three-note chords. For example, one such chord would be notes 1, 11, and 6, and another would be notes 2, 12, and 7.

If a scale had a prime number of notes, then every interval (other than an octave) would cycle through all notes.

Why is the 12-note scale so common? There have been other systems, but these are mostly subsets (at least approximately) of the 12-note scale. The answer seems to have something to do with the fact that intervals in the 12-tone scale have simple frequency ratios. For example, a fifth is a ratio 3:2 and a forth is a ratio 3:4. (More on that here.) These intervals are pleasant to our ears. There was a prehistoric flute in the news a few weeks ago and it appears to have been based on the same musical intervals common in modern music.