The last chapter of George Box’s book Improving Almost Anything contains the lyrics to “I Am the Very Model of a Professor Statistical,” to be sung to the tune of “I Am the Very Model of a Modern Major General” by Gilbert & Sullivan.

Here’s the original:

The original song has a few funny math-related lines.

I’m very well acquainted, too, with matters mathematical,
I understand equations, both the simple and quadratical,
About binomial theorem I’m teeming with a lot o’ news,
With many cheerful facts about the square of the hypotenuse.

I’m very good at integral and differential calculus;
I know the scientific names of beings animalculous:
In short, in matters vegetable, animal, and mineral,
I am the very model of a modern Major-General.

Here are a few lines from George Box’s version.

I relentlessly uncover any aberrant contingency
I strangle it with rigor and stifle it with stringency
I understand the different symbols be they Roman, Greek, or cuneiform
And every distribution from the Cauchy to the uniform.

With derivation rigorous each lemma I can justify
My every estimator I am careful to robustify
In short in matters logical, mathematical, idealistical
I am the very model of a professor statistical.

Gilbert & Sullivan have come up on this blog a couple other times:

George Box has come up too, but only once. (I’m surprised he hasn’t come up more; I should rectify that.) This post has a great quote from Box: “To find out what happens to a system when you interfere with it, you have to interfere with it (and not just passively observe it).”

The song YYZ by Rush opens with a theme based on the rhythm of “YYZ” in Morse code:

-.-- -.-- --..

YYZ is the designation for the Toronto Pearson International Airport, the main airport serving Toronto. The idea for the song came from hearing the airport identifier in Morse code.

However, the song puts no spaces between rhythm corresponding to each letter. Here’s what the opening riff would look like in sheet music:

Each dash is a middle C and each dot is an F# a tritone below middle C.

When I listen to the song, I don’t hear YYZ. My mind splits up the rhythm with each sequence of long notes starting a group:

-. ---. ----..

So I hear the 20/8 time signature as (3 + 7 + 10)/8.

In terms of Morse code, -. is N. Interpreting the other groupings depends on what you mean by Morse code. The American amateur radio community defines Morse code as 40 characters: the 26 letters of the Latin alphabet, 10 digits, and 4 more symbols: / = , . Using that definition of Morse code, there are no symbols corresponding to ---. or ----... There is no symbol corresponding to ---- either. More on unused sequences here.

However, sometimes ---. is used for Ö and ---- for Š. So the way I hear “YYX” would be more like “NÖŠI”.

There are many other ways to parse -.---.----.. into Morse code symbols. For example, NO1I

-. --- .---- ..

Enumeration

How many ways could you split -.---.----.. into valid Morse code?

Here’s an outline of a recursive algorithm to enumerate the possibilities.

Start at the beginning and list the possible symbols formed by consecutive dots and dashes. In our case the possible symbols are T, N, K, and Y. So the possibilities are

T (-) added to the front of all sequences that start with .---.----..

N (-.) added to the front of all sequences that start with ---.----..

K (-.-) added to the front of all sequences that start with --.----..

Y (-.--) added to the front of all sequences that start with -.----..

So for the first bullet point, for example, how would we find all sequences that start with .---.----..? Use the same idea.

E (.) added to the front of all sequences that start with ---.----..

A (.-) added to the front of all sequences that start with --.----..

W (.--) added to the front of all sequences that start with -.----..

J (.---) added to the front of all sequences that start with .----..

So pull off all the symbols you can from the beginning of the list of dots and dashes and in each case recurse on the rest of the list.

We finished a bottle of wine this evening, and I blew across the top as I often do. (Don’t worry: I only do this at home. If we’re ever in a restaurant together, I won’t embarrass you by blowing across the neck of an empty bottle.)

The pitch sounded lower than I expected, so I revisited some calculations I did last year.

As I wrote about here, a wine bottle is approximately a Helmhotz resonator. The geometric approximation is not very good, but the pitch prediction usually is. An ideal Helmholtz resonator is a cylinder attached to a sphere, and a typical wine bottle is more like a cylinder attached to a larger cylinder. But the formula predicting pitch is robust to departures from ideal assumptions.

As noted before, the formula for the fundamental frequency of a Helmholtz resonator is

where the variables are as follows:

f, frequency in Hz

v, velocity of sound

A, area of the opening

L, length of the neck

V, volume

The opening diameter was 2 cm, the neck length 9 cm, and the volume 750 cm³. All these are typical. The predicted frequency is f = 118 Hz. The measured frequency was 106 Hz, measured by the Sonic Tools phone app.

The actual frequency was about 10% lower than predicted. This is about a whole step lower in musical terms. I could certainly hear an interval that large if I heard the two pitches sequentially. But I don’t have perfect pitch, and so I’m skeptical whether I could actually notice a pitch difference of that size from memory.

Intervals of a fourth, such as the interval from C to F, are common in western music, but consecutive intervals of this size are not. Quartal harmony is based on intervals of fourths, and quartal melodies use a lot of fourths, particularly consecutive fourths.

Maybe the most famous quartal melody is the opening fanfare to Star Trek (original series). Here’s a transcription of the opening line:

And here is the same music with the intervals of a fourth circled.

The theme opens with two consecutive fourths, there’s an augmented fourth in the middle, then two more consecutive fourths. There are two major thirds in the phrase above, which you could call diminished fourths.

Incidentally, there are four bell tones before the melody above begins, and the interval between the first two tones is a fourth.

Making the sheet music

Here’s the Lilypond source code I used to create the images above.

The lowest C on a piano is called C1 in scientific pitch notation. The C one octave up is C2 and so forth. Middle C is C4.

The frequency of Cn is approximately 2^{n+4} Hz. This would be exact if C0 were 16 Hz, but it’s a little flat. In order to make A4 have frequency 440 Hz, CO must have frequency 16.3516.

Notes other than C take their number from the nearest C below. So A4 is the A above C4, middle C. The lowest note on a standard piano, the A below C1, is A0.

At one point in time C0 was defined to be exactly 16 Hz. The frequencies of notes have been defined slightly differently across time and location.

Mathematically perfect octaves, however, don’t sound quite right. The highest notes on a piano would sound flat if every octave were exactly twice the frequency of the previous octave. So we tune the lowest notes a little lower than the math would say to, and the high notes higher.

In the original thread I said that C0 was the lowest C on a piano when I should have said C1.

No discussion of mathematics and piano tuning would be complete without mentioning Fermi problems. As I discuss here,

These problems are named after Enrico Fermi, someone who was known for being able to make rough estimates with little or no data.

A famous example of a Fermi problem is “How many piano tuners are there in New York?” I don’t know whether this goes back to Fermi himself, but it’s the kind of question he would ask. Of course nobody knows exactly how many piano tuners there are in New York, but you could guess about how many piano owners there are, how often a piano needs to be tuned, and how many tuners it would take to service this demand.

A few days ago I wrote about Frequency Shift Keying (FSK), a way to encode digital data in an analog signal using two frequencies. The extension to multiple frequencies is called, unsurprisingly, Multiple Frequency Shift Keying (MFSK). What is surprising is how MFSK sounds.

When I first heard MFSK I immediately recognized it as an old science fiction sound effect. I believe it was used in the original Star Trek series and other shows. The sound is in once sense very unusual, which is why it was chosen as a sound effect. But in another sense it’s familiar, precisely because it has been used as a sound effect.

Each FSK pulse has two possible states and so carries one bit of information. Each MFSK pulse has 2^{n} possible states and so carries n bits of information. In practice n is often 3, 4, or 5.

Why does it sound strange?

An MFSK signal will jump between the possible frequencies in no apparent; if the data is compressed before encoding, the sequence of frequencies will sound random. But random notes on a piano don’t sound like science fiction sound effects. The frequencies account for most of the strangeness.

MFSK divides its allowed bandwidth into uniform frequency intervals. For example, a 500 Hz bandwidth might be divided into 32 frequencies, each 500/32 Hz apart. The tones sound strange because they are uniformly on a linear scale, whereas we’re used to hearing notes uniformly spaced on a logarithmic scale. (More on that here.)

In a standard 12-note chromatic scale, the ratios between consecutive frequencies is constant, each frequency being about 6% larger than the previous one. More precisely, the ratio between consecutive frequencies equals the 12th root of 2. So if you take the logarithm in base 2, the distance between each of the notes is 1/12.

In MFSK the difference between consecutive frequencies is constant, not the ratio. This means the higher frequencies will sound closer together because their ratios are closer together.

Pulse shaping

As I discussed in the post on FSK, abrupt frequency changes would cause a signal to use an awful lot of bandwidth. The same is true of MFSK, and as before the solution is to taper the pulses to zero on the ends by multiplying each pulse by a windowing function. The FSK post shows how much bandwidth this saves.

When I created the audio files below, at first I didn’t apply pulse shaping. I knew it was important to signal processing, but I didn’t realize how important it is to the sound: you can hear the difference, especially when two consecutive frequencies are the same.

Audio files

The following files are a 5-bit encoding. They encode random numbers k from 0 to 31 as frequencies of 1000 + 1000k/32 Hz.

Here’s what a random sample sounds like at 32 baud (32 frequency changes per second) with pulse shaping.

If you know of examples of MFSK used as a sound effect, please email me or leave a comment below.

Here’s one example I found: “Sequence 2” from this page of sound effects sounds like a combination of teletype noise and MFSK. The G7 computer sounds like MFSK too.

According to this page, a dial tone in DTMF [1] is a chord of two sine waves at 350 Hz and 440 Hz. In musical notation:

According to the same page, a busy signal is a combination of 480 Hz and 620 Hz with pulses of 1/2 second.

Note that the bottom note is an B half flat, i.e. midway between a B and a B flat, denoted by the backward flat sign. The previous post on DTMF tones also used quarter tone notation because the frequencies don’t align well with a conventional chromatic scale. The frequencies were chosen to be easy to demodulate rather than to be musically in tune.

Audio files

Here are audio files corresponding to the notation above.

Keenan Pepper left a comment on my previous post saying that the DTMF tones used by touch tone phones “are actually quite close to 14 equal divisions of the octave (rather than the usual 12).”

Let’s show that this is right using a little Python.

By dividing the octave into 14 points the DFMT system largely avoids overlap between the set of tones and the set of their harmonics. However, the first overtone of the first tone (1394 Hz) is kinda close to the fundamental of the 6th tone (1336 Hz).

The sounds produced by a telephone keypad are a combination of two tones: one for the column and one for the row. This system is known as DTMF (dual tone multiple frequency).

I’ve long wondered what these tones would be in musical terms and I finally went to the effort to figure it out when I ran across DTMF in [1].

The three column frequencies are 1209, 1336, and 1477 Hz. These do not correspond exactly to standard musical pitches. The first frequency, 1209 Hz, is exactly between a D and a D#, two octaves above middle C. The second frequency, 1336 Hz, is 23 cents [2] higher than an E. The third frequency, 1477 Hz, lands on an F#.

In approximate musical notation, these pitches are two octaves above the ones written below.

Notice that the symbol in front of the D is a half sharp, one half of the symbol in front of the F.

Similarly, the four row frequencies, starting from the top, are 697, 770, 852, and 941 Hz. In musical terms, these notes are F, G (31 cents flat), A (54 cents flat), and B flat (16 cents sharp).

The backward flat symbol in front of the A is a half flat. As with the column frequencies, the row frequencies are two octaves higher than written.

These tones are deliberately not in traditional harmony because harmonic notes (in the musical sense) are harmonically related (in the Fourier analysis sense). The phone company wants tones that are easy to pull apart analytically.

Finally, here are the chords that correspond to each button on the phone keypad.

Maybe this has been done before, but I haven’t seen it: Morse code in musical notation.

Here’s the Morse code alphabet, one letter per measure; in practice there would be less space between letters [1]. A dash is supposed to be three times as long as a dot, so a dot is a sixteenth note and a dash is a dotted eighth note.

Morse code is often at a frequency between 600 and 800 Hz. I picked the E above middle C (660 Hz) because it’s in that range.

Rhythm

Officially a dash is three times as long as a dot. But there’s also a space equal to the length of a dot between parts of a letter. So the sheet music above would be more accurate if you imagined all the sixteenth notes are staccato and the dotted eighth notes are really eighth notes followed by a sixteenth rest.

This doesn’t make much difference because individual operators have varying “fists,” styles of sending Morse code, and won’t exactly follow the official length and spacing rules.

You could rewrite the music above as follows, but it’s all an approximation.

Tempo

According to Wikipedia, “the dit length at 20 words per minute is 50 milliseconds.” So if a sixteenth note has a duration of 50 milliseconds, this would mean five quarter notes per second, or 300 beats per minute. But according to this video, the shortest duration people can distinguish is about 50 milliseconds.

That would imply that copying Morse code at 20 wpm is pushing the limits of human hearing. But copying at 20 wpm is common. Some people can copy Morse code at more than 50 words per minute or more, but at that speed they’re not hearing individual dits and dahs. An H, for example, four dits in a row, sounds like a single rough sound. In fact, they’re not really hearing letters at all but recognizing the shape of words.

How the image was made

I made the image above with LaTeX and Lilypond.

Adding the letters above each measure was kind of a hack. I used rehearsal markings to label the measures, but there was one problem: the software skips from letter H to letter J. That meant that the labels I and all subsequent letters were one ahead of what they should be, and the final letter Z was labeled AA. I tried several tricks, and Lilypond steadfastly refused to label a measure with ‘I’ even though I’ve seen such a label in the documentation.

My way around this was to make it label two consecutive measures with H, then in image editing software I turned the second H into an I. No doubt there’s a better way, but this worked.

I may play around with this and try to improve it a bit. If you have any suggestions, particularly related to Lilypond, please let me know.

[1] You could think of the musical score above as a sort of transcription of the Farnsworth method of teaching Morse code. Students learn the letters at full speed, but with extra space between the letters at first. The faster speed discourages consciously counting the dits and dahs, forcing the student to listen to the overall rhythm of the letters.