For many years, rivals University of Texas and Texas A&M University played each other in football on Thanksgiving. In 1999, the game fell one week after the collapse of the Aggie Bonfire killed 12 A&M students and injured 27.
The University of Texas band’s half time show that year was a beautiful tribute to the fallen A&M students.
Amplitude modulated signals sound rough to the human ear. The perceived roughness increases with modulation frequency, then decreases, and eventually disappears. The point where roughness reaches is maximum depends on the carrier signal, but for a 1 kHz tone roughness reaches a maximum for modulation at 70 Hz. Roughness also increases as a function of modulation depth.
Amplitude modulation multiplies a carrier signal by
1 + d sin(2π f t)
where d is the modulation depth, f is the modulation frequency, and t is time.
Here are some examples you can listen to. We use a pure 1000 Hz tone and Gaussian white noise as carriers, and vary modulation depth and frequency continuously over 10 seconds. he modulation depth example varies depth from 0 to 1. Modulation frequency varies from 0 to 120 Hz.
First, here’s a pure tone with increasing modulation depth.
Next we vary the modulation frequency.
Now we switch over to Gaussian white noise, first varying depth.
And finally white noise with varying modulation frequency. This one sounds like a prop-driven airplane taking off.
Related: Psychoacoustics consulting
I’ve posted an online calculator to convert between two commonly used units of loudness, phon and sone. The page describes the purpose of both units and explains how to convert between them.
Fluctuation strength is similar to roughness, though at much lower modulation frequencies. Fluctuation strength is measured in vacils (from vacilare in Latin or vacillate in English). Police sirens are a good example of sounds with high fluctuation strength.
Fluctuation strength reaches its maximum at a modulation frequency of around 4 Hz. For much higher modulation frequencies, one perceives roughness rather than fluctuation. The reference value for one vacil is a 1 kHz tone, fully modulated at 4 Hz, at a sound pressure level of 60 decibels. In other words
(1 + sin(8πt)) sin(2000πt)
where t is time in seconds.
Since the carrier frequency is 250 times greater than the modulation frequency, you can’t see both in the same graph. In this plot, the carrier is solid blue compared to the modulation.
Here’s what the reference for one vacil would sound like:
See also: What is an asper?
Acoustic roughness is measured in aspers (from the Latin word for rough). An asper is the roughness of a 1 kHz tone, at 60 dB, 100% modulated at 70 Hz. That is, the signal
(1 + sin(140πt)) sin(2000πt)
where t is time in seconds.
Here’s what that sounds like (if you play this at 60 dB, about the loudness of a typical conversation at one meter):
And here’s the Python code that made the file:
from scipy.io.wavfile import write
from numpy import arange, pi, sin, int16
def f(t, f_c, f_m):
# t = time
# f_c = carrier frequency
# f_m = modulation frequency
return (1 + sin(2*pi*f_m*t))*sin(2*f_c*pi*t)
def to_integer(signal):
# Take samples in [-1, 1] and scale to 16-bit integers,
# values between -2^15 and 2^15 - 1.
return int16(signal*(2**15 - 1))
N = 48000 # samples per second
x = arange(3*N) # three seconds of audio
# 1 asper corresponds to a 1 kHz tone, 100% modulated at 70 Hz, at 60 dB
data = f(x/N, 1000, 70)
write("one_asper.wav", N, to_integer(data))
See also: What is a vacil?
This afternoon I was working on a project involving tonal prominence. I stepped away from the computer to think and was interrupted by the sound of a leaf blower. I was annoyed for a second, then I thought “Hey, a leaf blower!” and went out to record it. A leaf blower is a great example of a broad spectrum noise with strong tonal components. Lawn maintenance men think you’re kinda crazy when you say you want to record the noise of their equipment.
The tuner app on my phone identified the sound as an A3, the A below middle C, or 220 Hz. Apparently leaf blowers are tenors.
Here’s a short audio clip:
And here’s what the spectrum looks like. The dashed grey vertical lines are at multiples of 55 Hz.
The peaks are perfectly spaced at multiples of the fundamental frequency of 55 Hz, A1 in scientific pitch notation. This even spacing of peaks is the fingerprint of a definite tone. There’s also a lot of random fluctuation between peaks. That’s the finger print of noise. So together we hear a pitch and noise.
When using the tone-to-noise ratio algorithm from the ECMA-74, only the spike at 110 Hz is prominent. A limitation of that approach is that it only considers single tones, not how well multiple tones line up in a harmonic sequence.
This post looks at loudness and sharpness, two important psychoacoustic metrics. Because they have to do with human perception, these factors are by definition subjective. And yet they’re not entirely subjective. People tend to agree on when, for example, one sound is twice as loud as another, or when one sound is sharper than another.
Loudness is the psychological counterpart to sound pressure level. Sound pressure level is a physical quantity, but loudness is a psychoacoustic quantity. The former has to do with how a microphone perceives sound, the latter how a human perceives sound. Sound pressure level in dB and loudness in phon are roughly the same for a pure tone of 1 kHz. But loudness depends on the power spectrum of a sound and not just it’s sound pressure level. For example, if a sound’s frequency is too high or too low to hear, it’s not loud at all! See my previous post on loudness for more background.
Let’s take the four guitar sounds from the previous post and scale them so that each has a sound pressure level of 65 dB, about the sound level of an office conversation, then rescale so the sound pressure is 90 dB, fairly loud though not as loud as a rock concert. [Because sound perception is so nonlinear, amplifying a sound does not increase the loudness or sharpness of every component equally.]
Here are the audio files from the previous post:
Clean note:
Clean chord:
Distorted note:
Distorted chord:
Here’s the loudness, measured in phons, at both sound pressure levels.
|-----------------------+-------+-------| | Sound | 65 dB | 90 dB | |-----------------------+-------+-------| | Clean note | 70.9 | 94.4 | | Clean chord | 71.8 | 95.3 | | Note with distortion | 81.2 | 103.7 | | Chord with distortion | 77.0 | 99.6 | |-----------------------+-------+-------|
While all four sounds have the same sound pressure level, the undistorted sounds have the lowest loudness. The distorted sounds are louder, especially the single note. Increasing the sound pressure level from 65 dB to 90 dB increases the loudness of each sound by roughly the same amount. This will not be true of sharpness.
Sharpness is related how much a sound’s spectrum is in the high end. You can compute sharpness as a particular weighted sum of the specific loudness levels in various bands, typically 1/3-octave bands. This weight function that increases rapidly toward the highest frequency bands. For more details, see Psychoacoustics: Facts and Models.
The table below gives sharpness, measured in acum, for the four guitar sounds at 65 dB and 90 dB.
|-----------------------+-------+-------| | Sound | 65 dB | 90 dB | |-----------------------+-------+-------| | Clean note | 0.846 | 0.963 | | Clean chord | 0.759 | 0.914 | | Note with distortion | 1.855 | 2.000 | | Chord with distortion | 1.281 | 1.307 | |-----------------------+-------+-------|
Although a clean chord sounds a little louder than a single note, the former is a little sharper. Distortion increases sharpness as it does loudness. The single note with distortion is a little louder than the other sounds, but much sharper than the others.
Notice that increasing the sound pressure level increases the sharpness of the sounds by different amounts. The sharpness of the last sound hardly changes.
The other day I asked on Google+ if someone could make an audio clip for me and Dave Jacoby graciously volunteered. I wanted a simple chord on an electric guitar played with varying levels of distortion. Dave describes the process of making the recording as
Fender Telecaster -> EHX LPB clean boost -> Washburn Soloist Distortion (when engaged) -> Fender Frontman 25R amplifier -> iPhone
Let’s look at the Fourier spectrum at four places in the recording: single note and chord, clean and distorted. These are a 0:02, 0:08, 0:39, and 0:43.
The first note, without distortion, has most of its spectrum concentrated at 220 Hz, the A below middle C.
The same note with distortion has a power spectrum that decays much slow, i.e. the sound has more high frequency components.
Here’s the A major chord without distortion. Note that since the threshold of hearing is around 20 dB, most of the noise components are inaudible.
Here’s the same chord with distortion. Notice there’s much more noise in the audible range.
Update: See the next post an analysis of the loudness and sharpness of the audio samples in this post.
Photo via Brian Roberts CC
I’ve written before about how to convert between frequency and pitch and scientific pitch notation. I’ve also written about the Bark scale.
Here’s a little online calculator to convert between Hz, Bark, and music notation. You can enter one of the three and it will compute the other two.
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.