The acoustics of kettledrums

typmani

Kettledrums (a.k.a. tympani) produce a definite pitch, but in theory they should not. At least the simplest mathematical model of a kettledrum would not have a definite pitch. Of course there are more accurate theories that align with reality.

Unlike many things that work in theory but not in practice, kettledrums work in practice but not in theory.

A musical sound has a definite pitch when the first several Fourier components are small integer multiples of the lowest component, the fundamental. A pitch we hear at 100 Hz would have a first overtone at 200 Hz, the second at 300 Hz, etc. It’s the relative strengths of these components give each instrument its characteristic sound.

An ideal string would make a definite pitch when you pluck it. The features of a real string discarded for the theoretical simplicity, such as stiffness, don’t make a huge difference to the tonality of the string.

An ideal circular membrane would vibrate at frequencies that are much closer together than consecutive integer multiples of the fundamental. The first few frequencies would be at 1.594, 2.136, 2.296, 2.653, and 2.918 times the fundamental. Here’s what that would sound like:

(download)

I chose amplitudes of 1, 1/2, 1/3, 1/4, 1/5, and 1/6. This was somewhat arbitrary, but not unrealistic. Including more than the first six Fourier components would make the sound even more muddled.

By comparison, here’s what it would sound like with the components at 2x up to 6x the fundamental, using the same amplitudes.

(download)

This isn’t an accurate simulation of tympani sounds, just something simple but more realistic than the vibrations of an idea membrane.

The real world complications of a kettledrum spread out its Fourier components to make it have a more definite pitch. These include the weight of air on top of the drum, the stiffness of the drum head, the air trapped in the body of the drum, etc.

If you’d like to read more about how kettle drums work, you might start with The Physics of Kettledrums by Thomas Rossing in Scientific American, November 1982.

Octave holes on a saxophone

I’ve played saxophone since I was in high school, and I thought I knew how saxophones work, but I learned something new this evening. I was listening to a podcast [1] on musical acoustics and much of it was old hat. Then the host said that a saxophone has two octave holes.  Really?! I only thought there was only one.

When you press the octave key on the back of a saxophone with your left thumb, the pitch goes up an octave. Sometimes this causes a key on the neck to open up and sometimes it doesn’t [2]. I knew that much.

Saxophone with octave key not open on a high note

Saxophone with octave key open on a high note

 

I thought that when this key didn’t open, the octaves work like they do on a flute: no mechanical change to the instrument, but a change in the way you play. And to some extent this is right: You can make the pitch go up an octave without using the octave key. However, when the octave key is pressed there is a second hole that opens up when the more visible one on the neck closes.

Octave hole for low notes on a saxophone

According to the podcast, the first saxophones had two octave keys to operate with your thumb. You had to choose the correct octave key for the note you’re playing. Modern saxophones work the same as early saxophones except there is only one octave key controlling two octave holes.

* * *

[1] Musical Acoustics from The University of Edinburgh, iTunes U.

[2] On the notes written middle C up to A flat, the octave key raises the little hole I wasn’t aware of. For higher notes the octave key raises the octave hole on the neck.

 

Musical pitch notation

How can you convert the frequency of a sound to musical notation? I wrote in an earlier post how to calculate how many half steps a frequency is above or below middle C, but it would be useful go further have code to output musical pitch notation.

In scientific pitch notation, the C near the threshold of hearing, around 16 Hz, is called C0. The C an octave higher is C1, the next C2, etc. Octaves begin with C; other notes use the octave number of the closest C below.

C4, middle C

The lowest note on a piano is A0, a major sixth up from C0. Middle C is C4 because it’s 4 octaves above C0. The highest note on a piano is C8.

Math

A4, the A above middle C, has a frequency of 440 Hz. This is nine half steps above C4, so the pitch of C4 is 440*2-9/12. C0 is four octaves lower, so it’s 2-4 = 1/16 of the pitch of C4. (Details for this calculation and the one below are given in here.)

For a pitch P, the number of half steps from C0 to P is

h = 12 log2(P / C0).

Software

Here is a page that will let you convert back and forth between frequency and music notation: Music, Hertz, Barks.

If you’d like code rather than just to do one calculation, see the Python code below. It calculates the number of half steps h from C0 up to a pitch, then computes the corresponding pitch notation.

from math import log2, pow

A4 = 440
C0 = A4*pow(2, -4.75)
name = ["C", "C#", "D", "D#", "E", "F", "F#", "G", "G#", "A", "A#", "B"]
    
def pitch(freq):
    h = round(12*log2(freq/C0))
    octave = h // 12
    n = h % 12
    return name[n] + str(octave)

The pitch for A4 is its own variable in case you’d like to modify the code for a different tuning. While 440 is common, it used to be lower in the past, and you’ll sometimes see higher values like 444 today.

If you’d like to port this code to a language that doesn’t have a log2 function, you can use log(x)/log(2) for log2(x).

Powers of 2

When scientific pitch notation was first introduced, C0 was defined to be exactly 16 Hz, whereas now it works out to around 16.35. The advantage of the original system is that all C’s have frequency a power of 2, i.e. Cn has frequency 2n+4 Hz. The formula above for the number of half steps a pitch is above C0 simplifies to

h = 12 log2P – 48.

If C0 has frequency 16 Hz, the A above middle C has frequency 28.75 = 430.54, a little flat compared to A 440. But using the A 440 standard, C0 = 16 Hz is a convenient and fairly accurate approximation.

Related posts

Structure in jazz and math

Last night I went to a concert by the Branford Marsalis Quartet. One of the things that impressed me about the quartet was how creative they are while also being squarely within a tradition. People who are not familiar with jazz may not realize how structured it is and how much it respects tradition. The spontaneous and creative aspects of jazz are more obvious than the structure. In some ways jazz is more tightly structured than classical music. To use Francis Schaeffer’s phrase, there is form and freedom, freedom within form.

Every field has its own structure, its tropes, its traditions. Someone unfamiliar with the field can be overwhelmed, not having the framework that an insider has to understand things. They may think something is completely original when in fact the original portion is small.

In college I used to browse the journals in the math library and be completely overwhelmed. I didn’t learn until later that usually very little in a journal article is original, and even the original part isn’t that original. There’s a typical structure for a paper in PDEs, for example, just as there are typical structures for romantic comedies, symphonies, or space operas. A paper in partial differential equations might look like this:

  1. Motivation / previous work
  2. Weak formulation of PDE
  3. Craft function spaces and PDE as operator
  4. A priori estimates imply operator properties
  5. Well posedness results
  6. Regularity

An expert knows these structures. They know what’s boilerplate, what’s new, and just how new the new part is. When I wrote something up for my PhD advisor I remember him saying “You know what I find most interesting?” and pointing to one inequality. The part he found interesting, the only part he found interesting, was not that special from my perspective. It was all hard work for me, but only one part of it stood out as slightly original to him. An expert in partial differential equations sees a PDE paper the way a professional musician listens to another or the way a chess master sees a chess board.

While a math journal article may look totally incomprehensible, an expert in that specialization might see 10% of it as somewhat new. An interesting contrast to this is the “abc conjecture.” Three and a half years ago Shinichi Mochizuki proposed a proof of this conjecture. But his approach is so entirely idiosyncratic that nobody has been able to understand it. Even after a recent conference held for the sole purpose of penetrating this proof, nobody but Mochizuki really understands it. So even though most original research is not that original, once in a while something really new comes out.

Related:

Electrical hum

If you hear electrical equipment humming, it’s probably at a pitch of about 60 Hz since that’s the frequency of AC power, at least in North America. In Europe and most of Asia it’s a little lower at 50 Hz. Here’s an audio clip in a couple formats: wav, mp3.

The screen shot above comes from a tuner app taken when I was around some electrical equipment. The pitch sometimes registered at A# and sometimes as B, and for good reason. In a previous post I derived the formula for converting frequencies to musical pitches:

h = 12 log(P / C) / log 2.

Here C is the pitch of middle C, 261.626 Hz, P is the frequency of your tone, and h is the number of half steps your tone is above middle C. When we stick P = 60 Hz into this formula, we get h = -25.49, so our electrical hum is half way between 25 and 26 half-steps below middle C. So that’s between a A# and a B two octaves below middle C.

For 50 Hz hum, h = -28.65. That would be between a G and a G#, a little closer to G.

Update: So why would the frequency of the sound match the frequency of the electricity? The magnetic fields generated by the current would push and pull parts, driving mechanical vibrations at the same frequency.

Related: Accoustics consulting

What key has 30 sharps?

Musical keys typically have 0 to 7 sharps or flats, but we can imagine adding any number of sharps or flats.

When you go up a fifth (seven half steps) you add a sharp. For example, the key of C has no sharps or flats, G has one sharp, D has two, etc. Starting from C and adding 30 sharps means going up 30*7 half-steps. Musical notes operate modulo 12 since there are 12 half-steps in an octave. 30*7 is congruent to 6 modulo 12, and six half-steps up from C is F#. So the key with 30 sharps would be the same pitches as F#.

But the key wouldn’t be called F#. It would be D quadruple sharp! I’ll explain below.

Sharps are added in the order F, C, G, D, A, E, B, and the name of key is a half step higher than the last sharp. For example, the key with three sharps is A, and the notes that are sharp are F#, C#, and G#.

In the key of C#, all seven notes are sharp. Now what happens if we add one more sharp? We start over and start adding more sharps in the same order. F was already sharp, and now it would be double sharp. So the key with eight sharps is G#. Everything is sharp except F, which is double sharp.

In a key with 28 sharps, we’ve cycled through F, C, G, D, A, E, and B four times. Everything is quadruple sharp. To add two more sharps, we sharpen F and C one more time, making them quintuple sharp. The note one half-step higher than C quintuple sharp is D quadruple sharp, which is enharmonic with F#.

You could repeat this exercise with flats. Going up a forth (five half-steps) adds a flat. Or you could think of a flat as a negative sharp.

Related posts:

How to convert frequency to pitch

I saw somewhere that James Earl Jones’ speaking voice is around 85 Hz. What musical pitch is that?

Let P be the frequency of some pitch you’re interested in and let C = 261.626 be the frequency of middle C. If h is the number of half steps from C to P then

P / C = 2h/12.

Taking logs,

h = 12 log(P / C) / log 2.

If P = 85, then h = -19.46. That is, James Earl Jones’ voice is about 19 half-steps below middle C, around the F an octave and a half below middle C.

More details on the derivation above here.

There’s a page to do this calculation for you here. You can type in frequency or pitch and get the other back.

(The page also gives pitch on a Bark scale, something you may not care about that is important in psychoacoustics.)

 

Photo credit Wikipedia
Music image created using Lilypod.

New Twitter accounts for DSP and music theory

I’ve started two new Twitter accounts this week: @DSP_fact and @MusicTheoryTip.

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.