Music

Amazing presentation from Evelyn Glennie, a deaf percussionist.

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

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

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A chromatic scale in Western music divides an octave into 12 parts. There are slightly different ways of partitioning the octave into 12 parts, and the various approaches have long and subtle histories. This post will look at the root of the differences.

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

Ellen Finn describes how she quit her job and exhausted her retirement savings to become a musician when she was around 50 years old.

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

Brian Lopes is amazing. I’d never heard of him until he was featured on the Eclectic Mix podcast a few days ago. The podcast describes his music “a high energy expedition crossing from jazz to R&B to funk and back again.” On his web site, Brian Lopes lists as his influences John Coltrane, Michael Brecker, Wayne Shorter, David Sanborn, and Cannonball Adderly. These are some of my favorite musicians, and listening to Lopes is like listening to all of these at once.

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

I’ve started a new page to list CDs for music mentioned here.

Here are three of my favorite podcast intro themes.

.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

Judging from the comments on previous posts, it seems a good number of Dave Brubeck fans read this blog. Everyone familiar with Dave Brubeck knows about Take Five from his album Time Out.

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

There’s a tune I heard at a concert when I was growing up that has stuck in my head ever since. If you know the name of the tune, please let me know.

Full size sheet music

audio file

Thanks, Paul, for making the audio file.

The sheet music was created using LilyPond.

Apparently there’s no HTML entity for the flat symbol, ♭. In my previous post, I just spelled out B-flat because I thought that was safer; it’s possible not everyone would have the fonts installed to display B♭ correctly.

So how do you display music symbols for flat, sharp, and natural in HTML? You can insert any symbol if you know its Unicode value, though you run the risk that someone viewing the page may not have the necessary fonts installed to view the symbol. Here are the Unicode values for flat, natural, and sharp.

Since the flat sign has Unicode value U+266D, you could enter ♭ into HTML to display that symbol.

The sharp sign raises an interesting question. I’m sure most web pages referring to G-sharp would use the number sign # (U+0023) rather than the sharp sign ♯ (U+266F). And why not? The number sign is conveniently located on a standard keyboard and the sharp sign isn’t. It would be nice if people used sharp symbols rather than number signs. It would make it easier to search on specifically musical terms. But it’s not going to happen.

Related posts:

Entering Unicode characters in Linux
Three ways to enter Unicode characters in Windows
Greek letters and math symbols in (X)HTML

I tried typesetting music in LaTeX some time ago and gave up. The packages I found were hard to install, the examples didn’t work, etc. This weekend I decided to try again. I tried plowing through the MusiXTeX documentation and got no further than I did last time.

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.”

After writing my post on music in 5/4 time, I remembered a march in 7/4 time that I played in band many years ago. Here’s an excerpt, about all I can remember.

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