DSP_fact is for DSP, digital signal processing: filters, Fourier analysis, convolution, sampling, wavelets, etc.

MusicTheoryTip is for basic music theory with a little bias toward jazz. It’ll tweet about harmony, scales, tuning, notation, etc.

Here’s a full list of my 15 daily tip twitter accounts.

If you’re interested in one of these accounts but don’t use Twitter, you can subscribe to a Twitter account via RSS just as you’d subscribe to a blog.

If you’re using Google Reader to subscribe to RSS feeds, you’ll need to switch to something else by July 1. Here are 18 alternatives.

Source: Radiolab

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 judaism.stackexchange.com. 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.

2012 is also prime as a base-five number.

Update: Here’s some Mathematica code to find other bases where 2012 is prime.

f[n_] := 2 n^3 + n + 2
For[n = 3, n < 100, n++, If[PrimeQ[f[n]], Print[n]]]

Related posts:

Odd numbers in odd bases
Prime words
Prime telephone numbers
Sonnet primes

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.”

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)

See also March in 7/4 time and Blue Rondo à la Turk.

Related posts

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:

Just-in-case versus just-in-time
Learners versus the learned
Feed the stars, milk the cows, and shoot the dogs
Evaluate people at their best or at their worst?

Odd meters

Music in 5/4 time
Blue Rondo à la Turk
March in 7/4 time

Music and computers

Typesetting music in LaTeX with LilyPond
Windows XP and Ubuntu start-up music

Music and math

Opening chord of “A Hard Days Night”
Circle of fifths and number theory
Circle of fifths and roots of two
Logarithms, music, and arsenic
Calendars, Connections, and Cats

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:

I miss quiet libraries
Audio book narrators
Listening to music with your whole body

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.

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.

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 http://www.ams.org/rss/mathmoments.rss.

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.

Related post:

Circle of fifths and roots of two

An octave is a ratio of 2 to 1. Suppose a string of a certain tension and length produces an A when plucked. If you make the string twice as tight, or keep the same tension and cut the string in half, the string will sound the A an octave higher. The new sound will vibrate the air twice as many times per second.

A fifth is a ratio of 3 to 2 in the same way that an octave is a ratio of 2 to 1. So if we start with an A 440 (a pitch that vibrates at 440 Hz, 440 vibrations per second) then the E a fifth above the A vibrates at 660 Hz.

We can go up by fifths and down by octaves to produce every note in the chromatic scale. For example, if we go up another fifth from the E 660 we get a B 990. Then if we go down an octave to B 495 we have the B one step above the A 440. This says that a “second,” such as the interval from A to B, is a ratio of 9 to 8. Next we could produce the F# by going up a fifth from B, etc. This progression of notes is called the circle of fifths.

Next we take a different approach. Every time we go up by a half-step in the chromatic scale, we increase the pitch by a ratio r.  When we do this 12 times we go up an octave, so r¹² must be 2. This says r is the 12th root of 2. If we start with an A 440, the pitch n half steps higher must be 2^n/12 times 440.

Now we have two ways of going up a fifth. The first approach says a fifth is a ratio of 3 to 2. Since a fifth is seven half-steps, the second approach says that a fifth is a ratio of 2^7/12 to 1. If these are equal, then we’ve proven that 2^7/12 equals 3/2. Unfortunately, that’s not exactly true, though it is a good approximation because 2^7/12 = 1.498. The ratio of 3/2 is called a “perfect” fifth to distinguish it from the ratio 1.498. The difference between perfect fifths and ordinary fifths is small, but it compounds when you use perfect fifths to construct every pitch.

The approach making every note via perfect fifths and octaves is known as Pythagorean tuning. The approach using the 12th root of 2 is known as equal temperament. Since 1.498 is not the same as 1.5, the two approaches produce different tuning systems. There are various compromises that try to preserve aspects of both systems. Each set of compromises produces a different tuning system. And in fact, the Pythagorean tuning system is a little more complicated than described above because it too involves some compromise.

Related post: Circle of fifths and number theory

I was totally broke. I was living on beans and I know thousands of bean recipes. It’s scary at any age, but it’s particularly scary in your fifties when all my friends are retiring and my goal is to save up for an avocado.

The quote comes from the BrightSideBroadcast podcast featuring her music.

Listen to the podcast.

video

JC: Your .NET Rocks podcast goes back further than podcasting. Did the show start on radio or was it always online?

CF: It was always online. Although I was inspired by public radio programs like Car Talk and Whad’Ya Know, I always thought the audience was too narrow for general radio. That, and I had web resources readily available.

JC: So the show was a set of downloadable MP3 files before RSS feeds came along to organize the files into a podcast?

CF: Exactly. We had a site more or less like it is now, with links to and info about the current show on the front page, and an archives page. We also had a newsletter we used to notify people of new shows.

JC: Could you say something about your podcasts, ones you host, produce, etc.?

CF: Well, .NET Rocks is a twice-weekly interview show for .NET devs. I am the host and Richard Campbell is the co-host. It’s an hour long, more or less. Topics range from low-level techie stuff to new technologies and methodologies to speculation about the future.

We also produce a weekly video screencast/interview show also about an hour long called dnrTV. Topics are hands-on practical. It’s recorded at 1024×768 so it will fit most projectors.

Hanselminutes is a 30-50 minute podcast with Scott Hanselman covering a wide variety of developer and technology topics. Also weekly.

RunAs Radio is a 30-50 minute weekly interview show on Microsoft-centric IT topics with Richard Campbell and Greg Hughes.

We also do an adult comedy podcast called Mondays. Richard Campbell and I basically spend an hour or so laughing at the stories and wit of Mark Miller and Karen Mangiacotti. NSFW but hilarious.

JC: On .NET Rocks, you’re the alpha geek programmer, but sometimes you mention your life as a musician and entrepreneur.Were you a musician first?

CF: Yes. I was singing in the Westerly Chorus from age 8. Piano since age 4. Guitar since age 10. Trumpet since age 10. Bass and drums came later. Programming didn’t come around till I was 17. I went to Berklee School of Music in 85-86 and Full Sail School of Recording Arts 86-87. Learned computers on my own. I was lucky to have many smart programmer friends who were willing to share their knowledge. That experience has shaped everything I have done since.

JC: How did you get started as a programmer?

CF: My dad bought a TRS-80 model 4 when I was a kid to do taxes and bills. I think VisiCalc was the only program he used. It had a guide to BASIC programming that I started reading. Between that and the TRSDOS manual I started writing some cool programs. Then I got a modem and was introduced to the BBS world. That was it. I was hooked on writing serial communications programs.

JC: You’ve mentioned Franklins.Net and Pwop productions on .NET Rocks. Could you describe these businesses and how you got started?

CF: Franklins.Net was started in 1999 as a training company. I taught VB6 and then VB.NET for several years. Pwop was started as a media production company to support the podcasts. Now Franklins.Net is the .NET education brand and Pwop is all about audio/video/music production.

JC: Do you have any other businesses?

CF: No.

JC: Let’s go back to your music. Who are some musicians that influenced you? Who do you like to listen to now?

CF: I was brought up on good old classic rock. On acoustic guitar I was influenced by John Fahey, Leo Kottke, Jorma Kaukonen, and the like. On electric guitar: Jeff Beck, Brian May, Peter Frampton, Eagles, Skynyrd, Duane Allman, Jerry Garcia, and more recently John Scofield, John Pisano, Lee Rittenour, and Pat Martino. Nowadays I’m on a New Orleans kick, hanging out with The Meters and Professor Longhair.

JC: Sounds like you’re active as a performer and a producer.

CF: Yes. I’ve produced music for a handful of artists and I play in local venues regularly.

JC: Your web site says recorded a CD with your brother Jay a few years ago. Where can we find it?

CF: We will announce a website soon with our new album, and free links to our old album.

JC: Tell me about the new CD you’re working on.

CF: It’s all original but you’ll be able to hear and identify our influences easily.

JC: Anything else you want to talk about?

CF: Sounds good to me! Thanks!!!

Related post:

Best podcast intro music (Includes a couple links to Carl’s music.)

Apparently he only recently started recording with his own group, the Brian Lopes Trio. According to the podcast, Brian Lopes has played with Chick Corea, Frank Sinatra, Aretha Franklin, Ray Charles, and other well known musicians. Finding his music is difficult, but you can buy his first CD at Blue Canoe Records. (Apparently you can’t actually buy a physical CD, but you can buy the MP3 files, sans DRM, that make up the CD.)

Image credit: Eclectic Mix podcast

Related posts:

Micheal Brecker
Talent alone won’t pay the bills

.NET Rocks by Carl Franklin and Richard Campbell.

Carl Franklin composed the intro theme, Toy Boy, and recorded the song with his brother Jay. The tune is catchy, the words are clever, and Carl’s a great musician. Richard and Carl talk over the intro, but you can hear these odd phrases poking out, such as “got a transmitter banned by the FCC.” After listening to the podcast for a while, I decided I had to find the theme song and listen to Toy Boy without the voice overs. Here’s more music by Carl Franklin.

Hanselminutes by Scott Hanselman.

The theme song is just a short loop, but it’s fun music. I wrote Scott a note asking him about the intro. I was hoping the loop taken from a longer song I could buy somewhere and thought I’d like to find more music by the same composer. Scott said that his theme song was written for his podcast by Carl Franklin. I was surprised that Carl came up again, but this isn’t totally unexpected since Carl’s company Pwop Productions produces Hanselminutes.

Accidental Creative by Todd Henry.

The theme song is My City In Healing  from A Slave Left Dreaming by Joshua Seurkamp. The song is a blend of Eastern and Western music, appropriate for a podcast that emphasizes creatively combining ideas.

I also wanted to mention the theme from the Science Magazine podcast. It’s not music I particularly enjoy listening to, but it is written in 5/4 time, something that has come up for discussion on this blog.

Related post:

Interview with Carl Franklin

But I wonder how many know about his album “To Hope! A Celebration.”

The album is a Roman Catholic mass containing beautiful mixture of classical and jazz music. It features the Cathedral Choral Society Chorus & Orchestra as well as the Dave Brubeck Quartet. It was recorded live at Washington National Cathedral on June 12, 1995.

According to Wikipedia, Brubeck was not a Catholic when the mass was commissioned but joined the Catholic church shortly after the piece was finished.

Related posts:

Blue Rhondo a la Turk
Music in 5/4 time

Full size sheet music

audio file

Thanks, Paul, for making the audio file.

The sheet music was created using LilyPond.

I posted a note on StackOverflow and got some good responses. Nikhil Chelliah suggested I look at LilyPond. I had looked at LilyPond before, and @jleedev explained how to integrate LaTeX and LilyPond.

Here’s some sheet music I included in my previous post, March in 7/4 time.

Here’s a full-sized PDF file version of the music above. And here’s the LilyPond source code used to create the music.

\relative c' {
\time 7/4
\key f \major
\clef treble
f g f \times 2/3{ c8 c c} f4 g a
g a8. bes16 a4 g f g c,
f g f \times 2/3{ c8 c c} f4 g a
g a8. bes16 a4 g f e f
}

The notation looks cryptic at first, but it makes sense after a few minutes. The command relative c' means that the following pitches will be relative to middle C. For example, the first note, F, is the F closest to middle C. Each note is the same length as the previous note by default, and the first note is a quarter note by default. The notation c8 means that the C is an eighth note, except it’s in the context of a triplet (times 2/3) and so it’s an eighth note triplet. The next F is denoted f4 to indicate that we’re back to quarter notes.

The notation a8. says that the A is a dotted eighth note. For the next note, bes16 means a B-flat sixteenth note. The suffix “es” stands for “flat” and “is” stands for “sharp.” (The documentation says it’s Dutch. I’ve never seen it before.) I don’t understand why I had to tell it that the B was flat. The code specified earlier that the key was F major, which implies B’s are flat. I suppose the code for individual notes is decoupled from the code to draw the key signature. That would make entering music painful in keys that have lots of sharps or flats. Maybe there’s a way to specify default sharps or flats.

The comma in c, gives the absolute pitch of the C. In relative mode, LilyPond assumes by default that each pitch name refers to the pitch closest to its predecessor. The C closest to the previous note, F, would have been the C up one fourth rather than down one fifth, so the comma was necessary to tell LilyPond to go down.

If I were to do a lot of music processing, I’d probably look at a commercial package such as Sibelius. But for now I’m just interested in producing small excerpts like that above, and it looks like LilyPond may be fine.

Update: I double checked the rules about flats etc. Yes, I do have to specify explicitly that the B in this example is B-flat. If I just say b rather than bes, LilyPond will add a natural sign in front of the B! It’s strange. It is aware of the key signature: when I tell it the B is flat, it says “OK, then I don’t have to mark that specially since it’s implicit in the key signature.” And if I don’t tell it the B is flat, it says “Oh, that’s an exception to the key signature. Better mark it with a natural sign.”

In case the music above is too hard to read, here’s a full-sized PDF file version.

Marches are always in even meters: your left foot has to come down on the first beat of every measure when you’re marching. And yet this odd meter tune comes across as a convincing march. (It was a concert march. Actually marching to it would have been odd, pun intended.)

This march had a 4/4 + 3/4 feel, emphasis on the first and fifth beats of each 7/4 measure.

I’ve just started blogging about music recently, and I’ve got a lot to learn. I’m not set up to record audio clips. My next post will describe the software I used to post the sheet music above.

Update: Many thanks to Nikhil Chelliah for identifying the march. It’s the first movement from Third Suite by Robert Jager. The sheet music and a sound clip are available here.

Related post: Blue Rondo à la Turk

And here’s a video of Pattillo playing Peter and the Wolf.

Now I finally get what “à la Turk” means. It must be a reference to the Turkish rhythm of the 9/8 theme. You can hear a short excerpt of Blue Rondo à la Turk at here.

Update: The article on Blue Rondo in Wikipedia says that it was based on Mozart’s Rondo alla Turca. I listened to Mozart’s rondo. It’s a famous tune — you’d probably recognize it — but I didn’t know it by name. I would never have drawn a connection between the Mozart rondo and Brubeck’s rondo. Maybe the Wikipedia article is wrong, or maybe Brubeck’s imagination moved pretty far from his inspiration.