Music

Twelve-tone composition

Posted on by John

Atonal music is difficult to compose because it defies human instincts. It takes discipline to write something so unpleasant to listen to.

One technique that composers use to keep their music from falling into tonal patterns is the twelve-tone row. The composer creates some permutation of the 12 notes in a chromatic scale and then uses these notes strictly in order. The durations of the notes may vary, and the notes may move up or down octaves, but the pitch classes are recycled over and over in order.

There are variations on this technique that allow a small amount of variety, such as allowing the the tone row to be reversed, inverted, or both. The retrograde version of the sequence is the original (prime) sequence of notes in the opposite order. The inverted form of the tone row inverts each of the intervals in the original sequence, going up by k half steps when the original when down by k half steps and vice versa. The retrograde inverted sequence is the inverted sequence in the opposite order.

Here is an example, taken from Arnold Schoenberg’s Suite for Piano, Op. 25.

Some math

Since a tone row is a permutation of 12 notes, there are 12! possible tone rows. However, since the notes of a tone row are played in a cycle, the same sequence of notes starting at a different point shouldn’t count as a distinct tone row. With that way of counting, there are 11! possible tone rows.

The operations of creating the retrograde and inverted forms of a tone row are the generators of an Abelian group. Let P (for prime) be the null operation, leaving a sequence alone. Then the elements of the group are P, R, I, and RI (= IR). The two generators have order 2, i.e. R² = I² = P. Therefore the group is isomorphic to ℤ2 × ℤ2.

Although I enjoy music and math, I do not enjoy most attempts to use math to compose music. I do not like atonal music, though I do like some “math rock.” It seems that math applied to rhythm results in more palatable music than math applied to melody.

A concert story

When I was in college I often walked from my dorm over the music building for concerts. I probably heard more concerts in college than I have heard ever since.

One night I went to an organ concert. At the end of the concert the organist took melodies on strips of paper that he had not seen before and improvised a fugue on each. After the concert I ran into a friend in the music building who had not been to the concert. I enthusiastically told him how impressed I was by the improvised fugues, especially the last one that sounded like Schoenberg tone row.

The organist overhead my conversation and walked up to me and said that he was impressed that I recognized the Schoenberg tone row. To be fair, I did not recognize the music per se. The music sounded random, and I came up with the only example of random music I could think of, and said it sounded like a Schoenberg tone row. I certainly did not say “Ah, yes. That was Schoenberg’s tone row from ….” It was a lucky guess.

Typesetting sheet music with AI

Posted on by John

Lilypond is a TeX-like typesetting language for sheet music. I’ve had good results asking AI to generate Lilypond code, which is surprising given the obscurity of the language. There can’t be that much publicly available Lilypond code to train on.

I’ve mostly generated Lilypond code for posts related to music theory, such as the post on the James Bond chord. I was curious how well AI would work if I uploaded an image of sheet music and asked it to produce corresponding Lilypond code.

In a nutshell, the results were hilariously bad as far as the sheet music produced. But Grok did a good job of recognizing the source of the clips.

Test images

Here are the two images I used, one of classical music

and one of jazz.

I used the same prompt for both images with Grok and ChatGPT: Write Lilypond code corresponding to the attached sheet music image.

Classical results

Grok

Here’s what I got when I compiled the code Grok generated for the first image.

This bears no resemblance to the original, turning one measure into eight. However, Grok correctly inferred that the excerpt was by Bach, and the music it composed (!) is in the style of Bach, though it is not at all what I asked for.

ChatGPT

Here’s the corresponding output from ChatGPT.

Not only did ChatGPT hallucinate, it hallucinated in two-part harmony!

Jazz results

One reason I wanted to try a jazz example was to see what would happen with the chord symbols.

Grok

Here’s what Grok did with the second sheet music image.

The notes are almost unrelated to the original, though the chords are correct. The only difference is that Grok uses the notation Δ for a major 7th chord; both notations are common. And Grok correctly inferred the title of the song.

I edited the image above. I didn’t change any notes, but I moved the title to center it over the music. I also cut out the music and lyrics credits to make the image fit on the page easier. Grok correctly credited Johnny Burke and Jimmy Van Heusen for the lyrics and music.

ChatGPT

Here’s what I got when I compiled the Lilypond code that ChatGPT produced. The chords are correct, as above. The notes bear some similarity to the original, though ChatGPT took the liberty of changing the key and the time signature, and the last measure has seven and a half beats.

ChatGPT did not speculate on the origin of the clip, but when I asked “What song is this music from?” it responded with “The fragment appears to be from the jazz standard ‘Misty.'”

Giant Steps

Posted on by John

John Coltrane’s song Giant Steps is known for its unusual and difficult chord changes. Although the chord progressions are complicated, there aren’t that many unique chords, only nine. And there is a simple pattern to the chords; the difficulty comes from the giant steps between the chords.

Giant Steps chords

If you wrap the chromatic scale around a circle like a clock, there is a three-fold symmetry. There is only one type of chord for each root, and the three notes not represented are evenly spaced. And the pattern of the chord types going around the circle is

minor 7th, dominant 7th, major 7th, skip
minor 7th, dominant 7th, major 7th, skip
minor 7th, dominant 7th, major 7th, skip

To be clear, this is not the order of the chords in Giant Steps. It’s the order of the sorted list of chords.

For more details see the video The simplest song that nobody can play.

Related posts

Tritone substitution

Posted on by John

Big moves in roots can correspond to small moves in chords.

Imagine the 12 notes of a chromatic scale arranged around the hours of a clock: C at 12:00, C♯ at 1:00, D at 2:00, etc. The furthest apart two notes can be is 6 half steps, just as the furthest apart two times can be is 6 hours.

Musical clock

An interval of 6 half steps is called a tritone. That’s a common term in jazz. In classical music you’d likely say augmented fourth or diminished fifth. Same thing.

The largest possible movement in roots corresponds to almost the smallest possible movement between chords. Specifically, to go from a dominant seventh chord to another dominant seventh chord whose roots are a tritone apart only requires moving two notes of the chord a half step each.

For example, C and F♯ are a tritone apart, but a C7 chord and a F♯7 chord are very close together. To move from the former to the latter you only need to move two notes a half step.

Musical clock

Replacing a dominant seventh chord with one a tritone away is called a tritone substitution, or just tritone sub. It’s called this for two reasons. The root moves a tritone, but also the tritone inside the chord does not move. In the example above, the third and the seventh of the C7 chord become the seventh and third of the F♯7 chord. On the diagram, the dots at 4:00 and 10:00 don’t move.

Tritone substitutions are a common technique for making basic chord progressions more sophisticated. A common tritone sub is to replace the V of a ii-V-I chord progression, giving a nice chromatic progression in the bass line. For example, in the key of C, a D min – G7– C progression becomes D min – D♭7 – C.

Related posts

Recamán’s sequence

Posted on by John

I recently ran into Recamán’s sequence. N. J. A. Sloane, the founder of the Online Encyclopedia of Integer Sequences calls Recamán’s sequence one of his favorites. It’s sequence A005132 in the OEIS.

This sequence starts at 0 and the nth number in the sequence is the result of moving forward or backward n steps from the previous number. You are allowed to move backward if the result is positive and a number you haven’t already visited. Otherwise you move forward.

Here’s Python code to generate the first N elements of the sequence.

    def recaman(N):
        a = [0]*N
        for n in range(1, N):
            proposal = a[n-1] - n
            if proposal > 0 and proposal not in set(a):
                a[n] = proposal
            else:
                a[n] = a[n-1] + n
        return a

For example, recaman(10) returns

    [0, 1, 3, 6, 2, 7, 13, 20, 12, 21]

There’s a Numberphile video that does two interesting things with the sequence: it visualizes the sequence and plays a representation of the sequence as music.

The rules for the visualization are that you connect consecutive elements of the sequence with circular arcs, alternating arcs above and below the number line.

Here’s code to reproduce an image from the video.

    import numpy as np
    import matplotlib.pyplot as plt
    
    def draw_arc(a, b, n):
        c = (a + b)/2
        r = max(a, b) - c
        t = np.linspace(0, np.pi) * (-1)**(n+1)
        plt.plot(c + r*np.cos(t), r*np.sin(t), 'b')
           
    N = 62
    a = recaman(N)
    for n in range(N-1):
        draw_arc(a[n], a[n+1], n)
    plt.gca().set_aspect("equal")
    plt.show()

Here’s the output:

To create a music sequence, associate the number n with the nth note of the chromatic scale. You can listen to the music in the video; here’s sheet music made with Lilypond.

Update

Here is another Recamán visualization going further out in the sequence. I made a few aesthetic changes at the same time.

I’ve also made higher resolution versions of the images in this post. Here are SVG and PDF versions of the first visualization.

Here is a PDF of the sheet music.

And here are SVG and PDF versions of the new visualization.

Overtones and Barbershop Quartets

Posted on by John

I’ve heard that barbershop quartets often sing the 7th in a dominant 7th a little flat in order to bring the note closer in tune with the overtone series. This post will quantify that assertion.

The overtones of a frequency f are 2f, 3f, 4f, 5f, etc. The overtone series is a Fourier series.

Here’s a rendering of the C below middle C and its overtones.

\score { \new Staff { \clef treble <c c' g' c'' e'' bes''>1 } \layout {} \midi {} }

We perceive sound on a logarithmic scale. So although the overtone frequencies are evenly spaced, they sound like they’re getting closer together.

Overtones and equal temperament

Let’s look at the notes in the chord above and compare the frequencies between the overtone series and equal temperament tuning.

Let f be the frequency of the lowest note. The top four notes in this overtone series have frequencies 4f, 5f, 6f, and 7f. They form a C7 chord [1].

In equal temperament, these four notes have frequencies 224/12 f, 228/12 f, 231/12 f, and 234/12 f. This works out to 4, 5.0397, 5.9932, and 7.1272 times the fundamental frequency f

The the highest note, the B♭, is the furthest from its overtone counterpart. The frequency is higher than that of the corresponding overtone, so you’d need to perform it a little flatter to bring it in line with the overtone series. This is consistent with the claim at the top of the post.

Differences in cents

How far apart are 7f and 7.1272f in terms of cents, 100ths of a semitone?

The difference between two frequencies, f1 and f2, measured in cents is

1200 log2(f1 / f2).

To verify this, note that this says an octave equals 1200 cents, because log2 2 = 1.

So the difference between the B♭ in equal temperament and in the 7th note of the overtone series is 31 cents.

The difference between the E and the 5th overtone is 14 cents, and the difference between the G and the 6th overtone is only 2 cents.

More music posts

[1] The dominant 7th chord gets its name from two thing. First, it’s called “dominant” because it’s often found on the dominant (i.e. V) chord of a scale, and it’s made from the 1st, 3rd, 5th, and 7th notes of the scale. It’s a coincidence in the example above that the 7th of the chord is also the 7th overtone. These are two different uses of the number 7 that happen to coincide.

Equipentatonic scale

Posted on by John

I ran across a video that played around with the equipentatonic scale [1]. Instead of dividing the octave into 12 equal parts, as is most common in Western music, you divide the octave into 5 equal parts. Each note in the equipentatonic scale has a frequency 21/5 times its predecessor.

The equipentatonic scale is used in several non-Western music systems. For example, the Javanese slendro tuning system is equipentatonic, as is Babanda music from Uganda.

In the key of C, the nearest approximants of the notes in the equipentatonic scale are C, D, F, G, A#. Approximate equipentatonic scale

In the image above [2], the D is denoted as half sharp, 50 cents higher than D. (A cent is 1/100 of a half step.) The actual pitch is a D plus 40 cents, so the half sharp is closer, but still not exactly right.

The F should be 20 cents lower, the G should be 20 cents higher, and the A# should be 10 cents higher.

Notation

The conventional chromatic scale is denoted 12-TET, an abbreviation for 12 tone equal temperament. In general n-TET denotes the scale that results from dividing the octave into n parts. The previous discussion was looking at how 5-TET aligns with 12-TET.

A step in 5-TET corresponds to 2.4 steps in 12-TET. This is approximately 5 steps in 24-TET, the scale we’d get by adding quarter tones between all the notes in the chromatic scale.

Math

When we talk of dividing the octave evenly into n parts, we mean evenly on a logarithmic scale because we perceive music on a log scale.

The notation will be a little cleaner if we start counting from 0. Then the kth note the n-TET scale is proportional to 2k/n.

The proportionality constant is the starting pitch. So if we start on middle C, the frequencies are 262 × 2k/n Hz. The nth frequency is twice the starting frequency, i.e. an octave higher.

We can measure how well an m-TET scale can be approximated by notes from an n-TET scale with the following function:

d(m, n) = \max_j \min_k \left|\frac{j}{m} - \frac{k}{n} \right|

Note that this function is asymmetric: d(m, n) does not necessarily equal d(n, m). For example, d(12, 24) = 0 because a quarter tone scale contains exact matches for every note in a semitone scale. But d(24, 12) = 1/24 because the quarter tone scale contains notes in between the notes of the semitone scale.

The equipentatonic scale lines up better with the standard chromatic scale than would a 7-note scale or an 11-note scale. That is, d(5, 12) is smaller than d(7, 12) or d(11, 12). Something similar holds if we replace 12 with 24: d(5, 24) is smaller than d(m, 24) for any m > 1 that is relatively prime to 24.

Related posts

[1] The video first presents 10-TET and then defines 5-TET as taking every other note from this scale.

[2] The image was created with the following Lilypond code.

\score {
  \new Staff {
    \relative c' {
      c1 dih f g aih c \bar "|."
    }
  }
  \layout {
    \context {
      \Staff
      \remove "Bar_engraver"
      \remove "Time_signature_engraver"
    }
  }
}

Grokking the James Bond chord

Posted on by John

The last chord of the James Bond theme is known as the James Bond chord. If you’ve ever seen a Bond movie, you’ll recognize it. Here’s a quick video showing how to play it on guitar.

I tried asking Grok 3 to generate some LilyPond code for sheet music a few days ago. LilyPond is pretty obscure as programming languages go—it’s not like asking for Python code—so I was skeptical, but Grok did well.

This afternoon I asked Grok to create sheet music for the James Bond chord. Specifically, I asked

  1. What is the “James Bond chord”?
  2. Create sheet music for the James Bond chord using Lilypond.
  3. Move the chord up an octave.

Here’s the output of each step.

What is the chord?

According to Grok,

The “James Bond chord” is a distinctive musical chord famously associated with the James Bond film franchise. It’s not just a single chord but rather a specific voicing and progression that evokes a sense of mystery, danger, and sophistication—perfectly matching the suave spy’s persona. Officially, it’s often identified as an E minor major 9 chord (EmMaj9), though its exact interpretation can vary slightly depending on the context.

The chord was first introduced by composer Monty Norman in the “James Bond Theme” for Dr. No (1962), arranged by John Barry, whose orchestration gave it its iconic swagger. …

First output

Grok produced the following LilyPond code

\version "2.22.0"

\score {
  \new Staff {
    \clef treble
    \key c \major % Default key, no sharps/flats; we'll specify accidentals manually
    \time 4/4
     <e g b dis' fis'>1 % EmMaj9 chord: E G B D# F#
  }
  \layout {}
  \midi {}
}

When compiled this gave the following music.

Grok also produce a MIDI file.

Second output

I wanted to move the chord up an octave for aesthetic reasons, putting the notes inside the staff. Grok complied, changing one line in the code, essentially adding an extra prime mark after each note.

<e' g' b' dis'' fis''>1 % EmMaj9 chord moved up an octave: E' G' B' D#'' F#''

This compiled to the following music.

Problems and prospects

Grok’s not perfect. In another experiment it produced code that wouldn’t compile. But when I told Grok that the code didn’t compile and asked it to try again, it worked.

I tried to remove the time signature, the C symbol. I asked Grok to remove it, and it did not. I asked Grok “How do you get LilyPond to produce music without a time signature?” and it told me two ways, neither of which worked.

I’ve used LilyPond occasionally for years, not to produce full sheets of music but to produce little fragments for blog posts. I’ve always found it a bit mysterious, in part because I jumped in and used it as needed without studying it systematically. There have been times when I thought about including some music notation in a blog post and didn’t want to go to the effort of using LilyPond (or rather the effort of debugging LilyPond if what I tried didn’t work). I may go to the effort more often now that I have a fairly reliable code generator.

Posts using LilyPond

The Clausen function

Posted on by John

I ran across the Clausen function the other day, and when I saw a plot of the function my first thought was that it looks sorta like a sawtooth wave.

Plot of Clausen function Cl_2

I wondered whether it also sounds like a sawtooth wave. More on that shortly.

The Clausen function can be defined in terms of its Fourier series:

\text{Cl}_2(x) = \sum_{k=1}^\infty \frac{\sin(kx)}{k^2}

The function commonly known as the Clausen function is one of a family of functions, hence the subscript 2. The Clausen functions for all non-negative integers n are defined by replacing 2 with n on both sides of the defining equation.

The Fourier coefficients decay quadratically, as do those of a triangle wave or sawtooth wave, as discussed here. This implies the function Cl2(x) cannot have a continuous derivative. In fact, the derivative of Cl2(x) is infinite at 0. This follows quickly from the integral representation of the function.

\text{Cl}_2(x)=-\int_0^x\log \left|2\sin\frac{t}{2} \right|\, dt

The fundamental theorem of calculus shows that the derivative

\text{Cl}'_2(x)=-\log \left|2\sin\frac{x}{2} \right|

blows up at 0.

How does it sound?

What does it sound like if we create music with Clausen waves rather than sine waves? I initially thought it sounded harsh, but that turned out to be an artifact of how I’d make the audio file. A reader emailed me a better recording using the first few notes of a famous hymn. It’s a much more pleasant sound than I had expected.

ein_feste_burg.wav

 

Related posts

The Real Book

Posted on by John

I listened to the 99% Invisible podcast about The Real Book this morning and thought back to my first copy.

My first year in college I had a jazz class, and I needed to get a copy of The Real Book, a book of sheet music for jazz standards. The book that was illegal at the time, but there was no legal alternative, and I had no scruples about copyright back then.

When a legal version came out later I replaced my original book with the one in the photo below.

The New Real Book Legal

The podcast refers to “When Hal Leonard finally published the legal version of the Real Book in 2004 …” but my book says “Copyright 1988 Sher Music Co.” Maybe Hal Leonard published a version in 2004, but there was a version that came out years earlier.

The podcast also says “Hal Leonard actually hired a copyist to mimic the old Real Book’s iconic script and turn it into a digital font.” But my 1988 version looks not unlike the original. Maybe my version used a kind of typesetting common in jazz, but the Hal Leonard version looks even more like the original handwritten sheet music.