Explaining the Sine Formula to Acoustics Students

[DaydreamNation]

I'm (a music academic) teaching a musical acoustics course to a very mixed group (music students, science students, and neither). Today, I covered the basic concept of simple harmonic motion and how this produces a sine wave pattern of motion over time. In the extra time we had left over, I (perhaps foolishly) tried to expand by connecting this to how the sine function arises from trigonometry, the unit circle, and ultimately giving them the formula:
y(t) = Asin[360o(t)(f) + θ]

I stated what each variable there refers to but I think I went too fast. I feel like I was going right over a lot of their heads and will probably need to review this. What are some strategies that you have used to explain this material to non-science students and make it make sense (ideally connecting it to music)? (I did play an electronic sine tone with oscilloscope visualization.)

I found these, which might help: http://www.businessinsider.com/7-gifs-trigonometry-sine-cosine-2013-5
I also like this video:

It might help if I can get them into the computer lab next time and create a little program in Max/MSP where they can control the frequency and phase shift of a sine wave (with both audio and oscilloscope visual). Maybe tough to put all that together in a day but not impossible...

 

[Simon Bridge]

make your reference circle so the circumference is 360 units... so the radius is ##360/2\pi## units.
measure angles in the usual way - the size of the angle is, therefore, the distance around the circumference inside the corner.
drop a chord in the usual way for the unit circle thing you know about ... the sine of the angle is half the length of this chord using the same ruler you used for the circumference. (Rolling a circular protractor along the line should also work.)

But, this sort of thing requires some prior knowledge of the geometry - similar triangles etc. You may be better making the link to a practical demonstration instead.

 

[olivermsun]

Visualizations of the wave pattern of various instruments with harmonics could be very helpful to students with a music background. There have been a lot of materials like this developed over the years. (There may even have been a segment about this somewhere in Disney's Fantasia!) Letting the students play with a synthesizer and oscilloscope display is a great idea. Another idea might be having students actually generate sounds on their own musical instruments and watching the waveforms on a display! (Bonus points if you can relate the note you just played to the waveform!)

One connection I suggest making early is the relationship between sine waves in time and the physical oscillations that arise in instruments (usually functions in space). So, for example, standing waves in strings and Pythagoras' identification of harmonics created by dividing the string; oscillating air columns in wind instruments, etc.

 

[DaydreamNation]

Hi, thanks for the replies. I did use degrees rather than radians. The students have not learned about harmonics/overtones yet but they will. However, I did point out that we study SHM because it is the simplest version of the sorts of vibrations that we find in any instrument, whether a vibrating guitar string, a column of air, a reed, etc.

 

[DaydreamNation]

I'm thinking that maybe next class I could have them try graphing different sine functions with this: https://www.desmos.com/calculator/nqfu5lxaij
before moving on to actually playing with an audio sine tone generator with oscilloscope?

 

[Andy Resnick]

Hi, thanks for the replies. I did use degrees rather than radians. The students have not learned about harmonics/overtones yet but they will. However, I did point out that we study SHM because it is the simplest version of the sorts of vibrations that we find in any instrument, whether a vibrating guitar string, a column of air, a reed, etc.

A colleague and I are keen to re-awaken our 'Physics of Music' course (she has a BS in Oboe from Peabody in addition to her Physics degrees), and we are wrestling with some of these same issues- how to incorporate quantification into a course (hopefully) populated by many non-science folks. We've done a few 1-hour demos that were well received, so perhaps you and I can learn from each other.

We bring in various instruments- she brings her oboe, I bring in a few cymbals and bells as well as a tone generator. She found a great app that generates a real-time spectrogram, we leave it playing in the background the whole time:

https://itunes.apple.com/us/app/spectrumview/id472662922?mt=8
https://techfortheclassicalsinger.w...-for-visualizing-the-overtones-in-your-voice/

Some of the students brought instruments as well- saxophones, guitars, trumpets. All told, we have the noise-maker, a microphone connected to an oscilloscope, the app, and everyone's ears. Thus, when we make a jazz noise here, people can see it and hear it there.

She and I have different approaches; she begins with sines and tones, while I begin with noise and broad-spectrum sounds. Conceptually, it's a lot like that bit in 'Fantasia' with the talking soundtrack (I'm dating myself).

Overall, my question for you is: why do you want your students to have a detailed mathematical understanding of sine waves in the context of music? You partially answered the question above, but I'm challenging you to justify the time and effort since SHM is largely useless for understanding timbre. I can justify the math for people interested in "engineering" sounds through synthetic tones and/or extended technique or for people interested in designing acoustic speakers and spaces, but as part of a broad class on music, I'm not convinced that it should take up more than 1 session.

 

[DaydreamNation]

Thanks for the tips. For the first lab they submit, they will analyse the waveforms of various instrumental recordings in Audacity. I've also demonstrated the waveforms in a jazz recording I've mixed in class: they can see that e.g. a kick drum's waves is aperiodic while a sax and bass both produce periodic waveforms with different shapes. I like the real-time live spectrogram idea, though, and I might use that.

The professor who taught this class previously leaves out the mathematical analysis of sine waves, as does our textbook, although most textbooks on the subject do at least give the equation. I might leave it out next time. I guess I was thinking that it's useful to have a basic understanding of how a sine wave works as a springboard for dealing with more complex waves, especially when some of them are taking Computers and Music (where they do synthesize tones) concurrently or have taken it previously. (They are going to combine sine waves in a subsequent lab in this class also.) I spent half of a 75m session on it yesterday and was thinking of spending another 35-45m or so on it tomorrow, so that's not too much more than what you're recommending, I think? I wasn't planning on having them solve a lot of problems with that formula but I thought it might be useful for them to e.g. be able to graph a pure tone at A4=440Hz?

 

[Andy Resnick]

I spent half of a 75m session on it yesterday and was thinking of spending another 35-45m or so on it tomorrow, so that's not too much more than what you're recommending, I think? I wasn't planning on having them solve a lot of problems with that formula but I thought it might be useful for them to e.g. be able to graph a pure tone at A4=440Hz?

Seems like a reasonable amount of time, but again: how does being able to graph a sine function fit in with your learning objective? Especially since Wolfram alpha provides much of this already:

https://www.wolframalpha.com/examples/MusicTheory.html

I'm not judging your choices, I'm simply making suggestions :) - we use the above module as part of a pre-calculus module on trig functions.

 

[DaydreamNation]

It enables people to visualize the concepts of frequency, period, amplitude, and phase shift with the most simple type of wave before they go on to apply these concepts with more complex waves. They will then actually listen to what pure sine tones sound like before they start adding and combining them.

 

[DaydreamNation]

Maybe it is unnecessary. I'm not really sure yet.

 

[olivermsun]

Overall, my question for you is: why do you want your students to have a detailed mathematical understanding of sine waves in the context of music? You partially answered the question above, but I'm challenging you to justify the time and effort since SHM is largely useless for understanding timbre. I can justify the math for people interested in "engineering" sounds through synthetic tones and/or extended technique or for people interested in designing acoustic speakers and spaces, but as part of a broad class on music, I'm not convinced that it should take up more than 1 session.

I guess I'm a little surprised by your question. The OP stated that the class was one on "musical acoustics." How does one go through a course in that and not deal with the mathematics of sine waves, even at a basic level?

 

[Andy Resnick]

I guess I'm a little surprised by your question. The OP stated that the class was one on "musical acoustics." How does one go through a course in that and not deal with the mathematics of sine waves, even at a basic level?

It's crucially important to match course content to the students. The course described by the OP (and our nascent course) are aimed at a "mixed group", ours would be primarily taken by music majors. Given that those students may not have seen trig functions since 10th grade (call it 5 years before taking this course), it's not realistic to expect those students to want to put in any time or effort in mastering trig functions. One of the reasons our class went dormant (about 10 years ago) is because music majors stopped enrolling: there was "too much hard math"- trig and algebra.

If this were an upper division course primarily for electrical engineers, architects, civil engineers, etc., then obviously there's a different set of expectations regarding mathematical fluency.

One thing my colleague and I want to do is find out how the music department teaches music theory- how much mathematical detail do they provide?

 

[DaydreamNation]

I've taught music theory at a number of universities, have been published in a theory journal, and have presented at music theory conferences. It is definitely a rigorous and challenging subject but it is not really a mathematics-intensive discipline. I'd sooner compare it to learning a language. Numbers are used but you could go up to graduate-level work without needing more mathematical knowledge than basic arithmetic/modular arithmetic. (Basic algebra could help when it comes to pcset theory. When you go deeper into this, statistical analysis helps. We use terms such as "set theory" and "interval vectors" but the operations that one needs to apply are not particularly advanced.) You could apply higher-level mathematics to music theory and composition but it's not a requirement to be successful in the field.

 

[olivermsun]

It's crucially important to match course content to the students. The course described by the OP (and our nascent course) are aimed at a "mixed group", ours would be primarily taken by music majors. Given that those students may not have seen trig functions since 10th grade (call it 5 years before taking this course), it's not realistic to expect those students to want to put in any time or effort in mastering trig functions. One of the reasons our class went dormant (about 10 years ago) is because music majors stopped enrolling: there was "too much hard math"- trig and algebra.

With all due respect, I think there's a big separation between asking the students to "master" trig functions and asking them to re-familiarize themselves with the basic notation for a sine wave. How else do you talk about the "acoustics" of musical sounds if amplitude, frequency, and phase in one line are "too much hard math"?

If this were an upper division course primarily for electrical engineers, architects, civil engineers, etc., then obviously there's a different set of expectations regarding mathematical fluency.

One thing my colleague and I want to do is find out how the music department teaches music theory- how much mathematical detail do they provide?

Not sure what you would be asking about there. You mean the mathematical details of chord progressions and polyphony and stuff like that?

 

[Andy Resnick]

With all due respect, I think there's a big separation between asking the students to "master" trig functions and asking them to re-familiarize themselves with the basic notation for a sine wave. How else do you talk about the "acoustics" of musical sounds if amplitude, frequency, and phase in one line are "too much hard math"?

Let's not argue over too fine a point- the issue is not just what do I want my students to learn, but also what do the students who enroll in this elective class *want* to learn? Our class was a failure because students stopped enrolling. Why did they stop enrolling? They didn't want to learn any math. Period. Deplorable? Perhaps. That's doesn't change the facts.

Not sure what you would be asking about there. You mean the mathematical details of chord progressions and polyphony and stuff like that?

Sure, but also the most basic stuff- how do they teach the concept of an octave? Time signatures and irrational meters? How do they talk about timbre? How do they incorporate unpitched instruments? How do they discuss dissonance and resolution? If I can improve my communication with music majors, I can better sculpt the content.

Edit: Actually, on further reflection, the topic I'd *actually* want to talk about with music faculty is how they would discuss effects pedals: distortion, flangers, chorus, wah=wah, etc...

 

[DaydreamNation]

I used the video and a GIF from the above links, then had them try graphing sine waves with the desmos link, find their periods, etc., and then had them play with sine wave oscillators so they could view and listen to them. It worked! They had a great time and seemed to really click with the concept.

 

[Andy Resnick]

I used the video and a GIF from the above links, then had them try graphing sine waves with the desmos link, find their periods, etc., and then had them play with sine wave oscillators so they could view and listen to them. It worked! They had a great time and seemed to really click with the concept.

Awesome!

 

[DaydreamNation]

Let's not argue over too fine a point- the issue is not just what do I want my students to learn, but also what do the students who enroll in this elective class *want* to learn? Our class was a failure because students stopped enrolling. Why did they stop enrolling? They didn't want to learn any math. Period. Deplorable? Perhaps. That's doesn't change the facts.

I think my class might be different from yours. First, I am at a small liberal arts college. The course is offered by the music department and taught by a composer/music technologist but counts as a science elective. It fills up, with a long waiting list, when it is offered, in no small part because students want to use it for their science elective (but also this is the sort of school that gets a lot of people interested in media and music/arts tech).

Sure, but also the most basic stuff- how do they teach the concept of an octave? Time signatures and irrational meters? How do they talk about timbre? How do they incorporate unpitched instruments? How do they discuss dissonance and resolution? If I can improve my communication with music majors, I can better sculpt the content.

Have you looked at standard music theory textbooks (Clendinning/Marvin, Benward/Saker, Kostka/Payne)? You are asking very basic questions about a core subject in the field. Perhaps your oboist colleague could help with some of these questions?

Edit: Actually, on further reflection, the topic I'd *actually* want to talk about with music faculty is how they would discuss effects pedals: distortion, flangers, chorus, wah=wah, etc...

While it is possible to go through three degrees in music without ever having to study about these, I do cover some of these (digital versions) in my Computers and Music class. (Signal processing for the electric guitar was a major part of my dissertation so this is right in my wheelhouse personally!) Feel free to PM if you want to talk about this.

 

[olivermsun]

(Signal processing for the electric guitar was a major part of my dissertation so this is right in my wheelhouse personally!)

Wow, cool topic to study!

 

[Andy Resnick]

I think my class might be different from yours.

That's entirely possible- our (defunct) class was listed as an upper-division PHY course- that's intimidating enough!

Have you looked at standard music theory textbooks (Clendinning/Marvin, Benward/Saker, Kostka/Payne)? You are asking very basic questions about a core subject in the field. Perhaps your oboist colleague could help with some of these questions?

Exactly- I defer to her about most of that. My point is that she and I are going to completely redesign the course, and part of that effort is me having a dialog with the music department to get a better sense of their perspective. She and I are thinking about textbooks, I really like Benade's "fundamentals of musical acoustics", she's found one that is more up-to-date: digital methods, mostly.