John D. CookModify

2026-03-18 10:50:12

The alternating series test is part of the standard calculus curriculum. It says that if you truncate an alternating series, the remainder is bounded by the first term that was left out. This fact goes by in a blur for most students, but it becomes useful later if you need to do numerical computing.

To be more precise, assume we have a series of the form

where the a_i are positive and monotonically converge to zero. Then the tail of the series is bounded by its first term:

The more we can say about the behavior of the a_i the more we can say about the remainder. So far we’ve assumed that these terms go monotonically to zero. If their differences

also go monotonically to zero, then we have an upper and lower bound on the truncation error:

If the differences of the differences,

also converge monotonically to zero, we can get a larger lower bound and a smaller upper bound on the remainder. In general, if the differences up to order k of the a_i go to zero monotonically, then the remainder term can be bounded as follows.

Source: Mark B. Villarino. The Error in an Alternating Series. American Mathematical Monthly, April 2018, pp. 360–364.

Related posts

• Euler’s method for accelerating an alternating series
• Cohen’s method for accelerating an alternating series

2026-03-17 21:26:13

If a number has a finite but nonzero fractional part, so do all its powers. I recently ran across a proof in [1] that is shorter than I expected.

Theorem: Suppose r is a real number that is not an integer, and the decimal part of r terminates. Then r^k is not an integer for any positive integer k.

Proof: The number r can be written as a reduced fraction a / 10^m for some positive m. If s = r^k were an integer, then

10^mks = a^k.

Now the left side of this equation is divisible by 10 but the right side is not, and so s = r^k must not be an integer. QED.

The only thing special about base 10 is that we most easily think in terms of base 10, but you could replace 10 with any other base.

What about repeating decimals, like 1/7 = 0.142857142857…? They’re only repeating decimals in our chosen base. Pick the right base, i.e. 7 in this case, and they terminate. So the theorem above extends to repeating decimals.

[1] Eli Leher. √2 is Not 1.41421356237 or Anything of the Sort. The American Mathematical Monthly, Vol. 125, No. 4 (APRIL 2018), page 346.

2026-03-17 21:22:39

The previous post introduced the idea of a twelve-tone row, a permutation of the twelve pitch classes of a chromatic scale.

The earlier post also introduced a group of operations on a tone row with elements P, R, I, and RI. Here P, which stands for “prime”, is the identity operator: it leaves the tone row alone. R stands for retrograde, which means to reverse the sequence. I stands for inversion, inverting each of the intervals in the row. If you apply R then I, you get the same result as if you first applied I then R. See the previous post for an example of each.

Do these operations always return a new tone row? In other words, are P(t), R(t), I(t), and RI(t) always distinct for all tone rows t?

It’s easy to see that P(t) and R(t) are always different. The first and last elements of t are different, and so the first element of P(t) does not equal the first element of R(t).

There is one interval that remains the same when inverted, namely the tritone. It’s possible to create a two-tone row that does not change under inversion, such as the sequence [B, F]. A tone row with more than two notes must have an interval that is not a tritone, and so inverting the tone row changes it.

If a tone row t has at least three notes, then P(t), R(t), I(t), and RI(t) are all distinct.

The pitch classes in a tone row are repeated in a cycle, so we might consider all rotations of a tone row to be in a sense the same. In mathematical terms, we can mod out by cyclic permutations.

If we apply R, I, and RI to equivalence classes of tone rows, the results are not always unique. For example, let t be the chromatic scale.

C C♯ D D♯ E F F♯ G G♯ A A♯ B

Then R(t) is

B A♯ A G♯ G F♯ F E D♯ D C♯ C

and I(t) is

C B A♯ A G♯ G F♯ F E D♯ D C♯

which is a rotation of R(t).

The chromatic scale is a boring tone row. For a random tone row t, all the variations P(t), R(t), I(t), and RI(t) will very likely be distinct, including rotations. The following Python code will illustrate the points above and estimate the probability that manipulations of a tone row are distinct even when equating rotations.

import random
import itertools

def inversion(x, m = 12):
    return [(2*x[0] - x[i]) % m for i in range(m)]

def rotate_left(x, n):
    return x[n:] + x[:n]

def rotdiff(x, y):
    for i in range(len(x)):
        if rotate_left(x, i) == y:
            return False
    return True

c = 0
for _ in range(1_000_000):
    P = list(range(12))
    random.shuffle(P) # shuffle in place
    R = list(reversed(P))
    I = list(inversion(P))
    RI = list(reversed(I))
    for pair in combinations([P, R, I, RI], 2):
        assert(pair[0] != pair[1])
        if not rotdiff(pair[0], pair[1]):
            c += 1
print(c)

This generates and tests a million random tone rows. When I ran it, it found 226 tone rows were one manipulation of the tone row was equal to a rotation of another. So when I said the variations are “very likely” to be distinct, you could quantify that as having over a 99.9% chance.

2026-03-16 07:39:20

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.

Update: More math in the next post. Do the applications of R, I, and RI always produce different sequences of notes? What if you consider two rotations of a tone row to be equivalent?

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.

2026-03-16 03:49:13

Notice anything special about the following sequence?

8 6 10 3 1 11 1 3 6 8 12 9 7 10 4 2 5 11 2 4 7 9 5 12

Each of the numbers 1 through 12 appear twice. Between the two 1s there is one number. Between the two 2s there are two numbers. Between the two 3s there are three numbers, etc.

Langford’s problem of order n is to arrange two copies of the integers 1 through n so that there are k numbers between the two ks. This problem has a solution if and only if n is congruent to 0 or 3 mod 4.

You can find much more on Langford’s problem here.

2026-03-13 23:15:50

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.'”