Wikipedia:Reference desk/Archives/Science/2021 May 29

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May 29

Where Can I Read the Dissertation of America's 1st Black PhD Physicist?

How can I get access to an electronic form of the doctoral dissertation of Edward Bouchet, Yale University? The title is "Measuring refractive indices." He finished it in 1876. He was the first Black man to earn a PhD in physics in the US. If there is no electronic copy, does Yale still have a physical copy? If so, can a member of the general public check it out or access it?

Perhaps the National Society of Black Physicists (email: headquarters@​nsbp.org) can help; see this post. There is also a biography, Edward Bouchet: The First African-American Doctorate, ISBN 981-02-4909-8.  --Lambiam 05:32, 29 May 2021 (UTC)

Finding copies of old dissertations is usually hopeless, but if you want to try, I recommend that you ask on Wikipedia:WikiProject Resource Exchange/Resource Request. – b_jonas 14:10, 30 May 2021 (UTC)

If I was looking for it, I'd start with the Yale University library. --184.145.50.201 (talk) 21:55, 30 May 2021 (UTC)

This search of the Yale archives[1] has several hits but I don't see a PhD thesis. I'm sure you could ask directly via their published email address, which is mssa.assist@yale.edu. Mike Turnbull (talk) 10:15, 1 June 2021 (UTC)

Which mammal has the most bones

In trying to research the answer to this question I have learned how similar different mammal skeletons can be - but I can't find anything about variations in the numbers of ribs and so on to tell me which mammal has the most bones. Hayttom (talk) 05:26, 29 May 2021 (UTC)

Googling "Which mammal has the most bones" brings-up this discussion, which says "It could maybe be elephants or blue whales – apparently they both have around 350 bones".

I haven't been able to pin down an entirely reliable source, but How Many Bones Does a Whale Have? by "a staff writer" says: "The number of bones varies from one species of whale to another and is dependent, in part, on the length of the whale's spine. Sperm whales contain 164 bones, while the blue whale, the largest animal on earth, has 356 bones".

Our Elephant article says: "The skeleton of the elephant is made up of 326–351 bones" and that "African elephants have 21 pairs of ribs, while Asian elephants have 19 or 20 pairs". Alansplodge (talk) 14:37, 29 May 2021 (UTC)

Bear in mind that the number of bones in any given mammal tends to change during their lifetime. The cranial bones fuse after infancy and I think the bones of the sternum also fuse in later life, for example. At least, in humans they do. I'm guessing that the same occurs in most, if not all, mammals. nagualdesign 21:35, 29 May 2021 (UTC)

No, bears are not in the running for this prize. (That was a joke.) I get the point, but I'd say that humans have the number of bones they start off with, not a reduced number after fusing. Thanks, though, nagualdesign.Hayttom (talk) 18:59, 31 May 2021 (UTC)

@Hayttom: Thanks for making me laugh at my silly mistake. The number of bones in the human body is generally given as 206, but children have as many as 300, depending upon how you count them. I've never read anywhere that "humans have 300 bones", so I think that the general consensus is to count fused bones as a single bone. See List of bones of the human skeleton. nagualdesign 01:07, 1 June 2021 (UTC)

While not explicitly resolved, I'm marking this...

Resolved

Hayttom (talk) 18:59, 31 May 2021 (UTC)

Chemistry questions.

1. Why can't the O in CO2 be a Lewis base? It has 2 lone pairs. The O in water can act as a Lewis base, why not for CO2?

2. Can hydrogen peroxide act as a Lewis acid and a Lewis base? What about hydrazine?

3. Compounds that absorb IR tend to also emit IR right after, right, are there compounds that absorb IR without emitting?

4. But not with UV absorption, right, most things that absorb UV only mildly re-admit or don't admit at all. Can there be something that emits UV as it absorbs? Thanks. 67.165.185.178 (talk) 17:17, 29 May 2021 (UTC).

The basis of #1 is false. [HOCO]⁺ is known, and that's the conjugate acid of CO₂ using an O as the basic atom. Some Metal carbon dioxide complexes are known that involve binding solely via O as Lewis base. Both answers to #2 are "yes". You might want to clarify in your questions whether you are including all types of Lewis acid/base interactions, or only those that are not also considered Brønsted–Lowry modes. DMacks (talk) 17:21, 30 May 2021 (UTC)

Ah, guess we need users like, if I recall, Wyzant, who I think is inactive now. I also had a 5th question but I already have my own answer for, but could use better ones. The question being, what are some compounds that can absorb both IR and UV? Pretty much anything polar will absorb IR. And things that absorb UV have to do with band gaps, and so metal oxides, and conjugated organics (such as avobenzones), are good UV absorbers. Since the latter are also polar, then conjugated organics and avobenzones can absorb both IR and UV. But to what extent can they absorb IR and UV and emit at the same time, does either interfere the other? I am curious to know. 67.165.185.178 (talk) 18:14, 30 May 2021 (UTC).

IR absorption is largely dependent upon a change in the dipole moment with respect to a change in nuclear coordinates (i.e. a molecular vibration). The result is that most compounds will have an absorption somewhere in the IR spectrum, with the main things that do not being homonuclear diatomic molecules, like O2 and N2. They have no permanent dipole moment, and they only have one vibrational mode, and it doesn't create a transient dipole moment. It's incorrect to say that being polar really has much to do with being an IR absorber. Polar molecules almost always do absorb in the IR, but that is because their dipole moment almost always changes with respect to nuclear motion. Many/most non-polar molecules will also absorb, however. Take carbon dioxide, which is non-polar, but its bending and asymmetric C=O stretching modes result in transient dipole moments, which makes it a rather strong IR absorber. Methane (and all alkanes, for that matter) are also non-polar, but excellent IR absorbers. You are generally correct that UV absorption, especially near-UV absorption (and visual absorption, for that matter) are often compounds with conjugated pi systems. Far-UV, however, doesn't necessarily need these conjugated systems. As for emission, I suggest reading our article on fluorescence. It's not at all uncommon for UV absorbers to also have emission, always at a longer wavelength (this might still be in the UV, or it might be in the visual spectrum. Emission from IR absorption is more rare, since fluorescence is generally a property of electronic excitation first, and then a small non-radiative relaxation between vibrational energy levels in the excited state, followed by emission as the system returns either to the ground state, or to a vibrational level above the electronic ground state. IR absorption isn't a good candidate for this as it rarely involves electronic excitation. Usually, IR absorption is from the ground state to the first vibrational level within the ground electronic state. Outside of rotational levels, there isn't anything for that type of absorption to have a non-radiative relaxation to be followed by an emission of different energy. I suppose an IR absorption to a second or third harmonic could have an emission with different energy, but the absorption will already be so weak that the weaker emission may not be readily detectable. I hope that helps. --OuroborosCobra (talk) 20:00, 30 May 2021 (UTC)

Oh, almost everything that's a good UV absorber will also absorb in the IR, but that's also because almost anything with more than one atom will absorb in the IR, outside of exception I previously noted. --OuroborosCobra (talk) 20:06, 30 May 2021 (UTC)

You know I forgot so much about fluorescence, and phosphorescence. Did you just put emitting IR and emitting UV as both fluorescence category? I thought those were about emitting radiation at a different level than received. Not re-emitting what you just absorbed. 67.165.185.178 (talk) 00:34, 1 June 2021 (UTC).

Materials will emit infrared radiation because they are warm, and the energy of the vibrations is in the same range as infrared photons. See black body radiation. To avoid this thermal radiation in the infrared you will have to cool your substance to well below 1K, millikelvin will do. Then it should still be able to absorb in infrared. In a liquid or solid it should then down-convert the energy to lower energy vibrations (phonons). For a gas it may fluoresce at other infrared wavelengths. To have emission in the ultraviolet through thermal radiation, the material will have to be thousands of degrees.Graeme Bartlett (talk) 22:16, 30 May 2021 (UTC)

• Also, I'm not sure I follow what is meant by numbers 3 and 4; anything capable of absorbing some photon of light of any wavelength can also emit photons of light; sometimes those photons are of a different wavelength than absorbed, this occurs in fluorescence and phosphorescence; these types of radiation are associated with energy transitions in vibrational modes of the molecules in question; the reason why the wavelength is different in emission than absorption is due to vibrational energy relaxation. IR, UV, and visible light can also all be absorbed and emitted through Molecular electronic transition as well, this usually happens at the same wavelength as it is absorbed. --Jayron32 13:35, 1 June 2021 (UTC)

So if a compound does not release the radiation as much as it absorbed, it gets warmer, right? I thought things who receive UV, generally do not emit it as much as they receive, compared to IR. Plants are good UV absorbers but do not emit as much, compared to clouds who are good IR absorbers but also emit them back a lot. 67.165.185.178 (talk) 13:59, 1 June 2021 (UTC).

There are WAY too many factors here to make broad general statements. Absorbed light energy can do any number of really complicated things, and making a statement about what will or will not happen in vague terms like "If anything absorbs UV light, what will it do" depends on the specifics of the situation. The devil is in the details. --Jayron32 14:20, 1 June 2021 (UTC)

Okay, earlier I was under the generalization that things that absorb IR will emit them back at a high % it absorbed, compared to UV. But OurosboroCobra brought up a different factor of emitting a different wavelength than received. I suspect this is the case for absorbing UV? Things that absorb UV, if they do emit, will emit at a different wavelength instead? So clouds that absorb IR will just emit them back at almost the same wavelengths and %. I was just looking for examples of UV-absorbers that behave like IR-absorbers, readmitting at the same wavelengths. 67.165.185.178 (talk) 14:58, 1 June 2021 (UTC).

Most substances (possibly all) have UV spectral series in their electron transition analogous to the Lyman series of hydrogen, and they will absorb and emit at the exact same wavelengths. --Jayron32 15:30, 1 June 2021 (UTC)

Okay, I have a few related questions now, on criss-cross. Are there classic examples of something absorbing IR, but emitting UV, or absorbing UV, but emitting IR? Or more importantly, something that emits light from absorbing anything radiation other than light? But I realize, I may have a pre-question for that. Looking at fluorescence and phosphorescence, both of those cases are for emitting radiation at a longer wavelength and absorbed. (So absorbing UV and emitting IR, falls under this category). What is the other-way-around phenomenon called? Or should I ask beforehand, is there even a other-way-around phenomenom, found in nature? Emitting shorter wavelengths than absorbed? Thanks, and sorry for the hassles. 67.165.185.178 (talk) 16:42, 1 June 2021 (UTC).

So one of the things here that we probably need to clear up is that there are two different ways that a particle of a substance can absorb and emit light energy, either through electron transitions, or through changes in kinetic energy (translation, vibration, rotation of the molecule itself). When something absorbs and/or emits light it can happen through any number of mechanisms. Most of what you are talking about is kinetic energy effects, and in a solid we're talking mostly about vibrational modes. But that doesn't mean that electronic modes are not also happening at the same time. That's why it's difficult to parse a straightforward answer. Thermal radiation is continuous across a wide spectrum, while things like molecular and atomic electron transitions and molecular bond vibrational modes are not and will occur at discrete wavelengths. There really isn't a simple "Will something absorb X and emit Y" blanket answer. All of that having been said, no, we generally don't see things absorb IR and emit UV, because there are usually only energy losses rather than energy gained in processes such as phosphorescence and fluorescence; UV photons are higher energy than IR photons. This was explained above in the already linked articles. --Jayron32 17:06, 1 June 2021 (UTC)

So when something absorbs and emits wavelengths at essentially the same wavelength, the 2 possibilities are 0 electronic transitions and 0 kinetic energy, or a big electronic transition, but a big counter-kinetic energy in the opposite direction, thus neutralizing it to 0? Can you think of classic examples for either of them? 67.165.185.178 (talk) 17:20, 1 June 2021 (UTC).

See optical frequency multiplier, second-harmonic generation, high harmonic generation, nonlinear optics. --Amble (talk) 17:45, 1 June 2021 (UTC)

Okay intresting. Going on to my last question, about a net balance between electronic transitions and kinetic energy cancelling out, what would be examples where there is a positive electronic transition, and negative kinetic energy, cancelling out, vs. negative electronic transition, and positive kinetic energy, still cancelling out to 0 or near 0? For when compounds absorb radiation? Thanks. 67.165.185.178 (talk) 03:07, 2 June 2021 (UTC).

BMI / Body Mass Index for Males

I looked online and can't find what I want. Does anyone know where I can find a BMI / body mass index chart or table (for males) that includes heights measured to the half-inch (not the full inch)? Specifically, for a height of 5 feet 7.5 inches (i.e., 67.5 inches). I want a chart or a table, not just a calculator. Something similar to this: [2] ... but measured in half-inches. (Also, a chart that uses American measurements of feet, inches, and pounds ... not European measurements of meters, centimeters, and kilograms.) Thank you in advance. 32.209.55.38 (talk) 20:16, 29 May 2021 (UTC)

The chart you linked to shows everything you need. Measurements are shown in both metric and imperial. Just read between the lines for 5'7" and 5'8". If one shows a BMI of 27 for a particular weight and the other shows a BMI of 26 then it'll be 26~27 for someone whose height is in the middle. It's hardly an exact science. If you want a precise calculation just use a BMI calculator. nagualdesign 21:05, 29 May 2021 (UTC)

Thanks. The chart that I linked does not have, specifically, what I am seeking. That is, a chart that measures height by half-inch increments. I understand that I can do mathematical calculations myself, or via an online calculator. I am looking for a chart that already has these calculations all laid out, in a table ... similar to the one that I linked. Thanks. 32.209.55.38 (talk) 23:02, 29 May 2021 (UTC)

Is it not an issue that the weight goes up in increments of 5 lbs? That has about the same effect as the whole-inch increments, so going by half-inch increments, while leaving the weight increments as 5 lbs, will result in only a marginal improvement on the overall accuracy, bringing the RMS deviation of 0.41 down to 0.37.  --Lambiam 01:21, 30 May 2021 (UTC)

Thanks. I am 67.5 inches. So, I am only interested in a chart that contains that exact value. I am not interested in any of the other height values, as they are not applicable to me. I am not "too worried" about the actual weight values or increments, as my weight is a constantly fluctuating number. Whereas the height is stable / constant. I understand that I can do the mathematical calculations myself, or via an online calculator. But, I was looking for a chart that already has these calculations all laid out for me. Thanks. 32.209.55.38 (talk) 15:58, 31 May 2021 (UTC)

Your height changes more often than your weight. You're at your tallest in the morning and then you shrink throughout the day as your spine compresses. Matt Deres (talk) 15:49, 1 June 2021 (UTC)

Height variation is about 1.5ish cm over the course of a day, about 3/5 of an inch. Weight variation can be about 5-6 pounds per day. Assuming 67.5 inches, .6/67.5 is about a 0.89% difference. Assuming a weight of 150 pounds, a 5-pound variation would be a difference of around 3.3%. So your height does NOT change more than your weight does. That being said, the OP is asking for a level of false precision in seeking a table which measures height to more precise than the whole inch, given that his claimed height of 67.5 inches will actually vary between 67-68 inches and probably a little on either side, throughout the day. The OP asking for such a table does not mean one exists, also. Requesting to be provided one does not make it so. --Jayron32 16:32, 1 June 2021 (UTC)

@Jayron32: (1) First, thanks for the input. (2) I think my point about height and weight was that I can do a lot of things (i.e., diet / exercise / etc.) that can make my weight vary considerably; there is essentially nothing that I can do to vary my height. (3) I don't think that measuring one's height at 67.5 inches is any different than measuring one's height at 67 or 68 inches. They "all" have the same "margin of error" ... plus or minus that 3/5 inch that you cite. To measure yourself in a "half-inch" increment is no more "falsely precise" than using a full-inch increment. (4) Asking for a certain table does not mean that it exists; that was the whole point of my question. I was looking for something; could not find it; came here to the Wikipedia Help Desk to get some input / feedback on what I was looking to find. (5) I don't see anywhere above where I "demanded" anything. Sounded more like a simple and polite request; no "demands". Thanks. 32.209.55.38 (talk) 18:09, 2 June 2021 (UTC)

So corrected. --Jayron32 18:12, 2 June 2021 (UTC)

Thank you for the correction; I was being hyperbolic, but should not have been. Matt Deres (talk) 16:53, 1 June 2021 (UTC)

@Jayron32: I think you're probably right that such a table does not exist. However, the graph used at the BMI article is a good start. I'm in the process of creating a new version where the BMI for any given height and weight is explicitly stated. Instead of a grid where each cell contains a rounded integer it will have curved bands, so you can read off your BMI to high precision. Pointless, I know. nagualdesign 17:35, 1 June 2021 (UTC)

@Nagualdesign: Thanks. I would not say it's "pointless". Where exactly is that new chart that you created? Did you add it into the body mass index article? Thanks. 32.209.55.38 (talk) 18:15, 2 June 2021 (UTC)

I haven't finished it yet. I have an ongoing issue with my laptop that I'm still battling. I'll be sure to leave a message here once I've uploaded it.

Perhaps pointless was an overstatement. It will certainly be an improvement over any of the tables or graphs that are currently available, since you'll be able to read it to arbitrary precision. The reason I called it pointless is because such precision is effectively useless. BMI charts generally provide figures to the nearest integer for good reason. The only really important thing is to see whether someone is underweight or overweight. BMI is effectively a rule of thumb. nagualdesign 19:04, 2 June 2021 (UTC)

@Nagualdesign: Well, yes, I agree with all you say. But -- generally speaking -- one goes into reading a BMI chart, knowing its limitations / definitions / meanings / intents / purposes / etc. Or, at least, should. Thanks. 32.209.55.38 (talk) 19:59, 3 June 2021 (UTC)

...Okay, I've uploaded a first draft (see below). I haven't added it to the article yet as I think it could use some more work but I've had enough for today. Any feedback is welcome. nagualdesign 00:15, 3 June 2021 (UTC)

Graph showing body mass index (BMI) for a range of heights and weights, both metric and imperial. Colours indicate BMI categories; underweight, normal weight, overweight, moderately obese, severely obese and very severely obese.

Wow, impressive. Can I ask, what program you used? 67.165.185.178 (talk) 00:35, 3 June 2021 (UTC).

It's hand drawn in Photoshop. nagualdesign 00:57, 3 June 2021 (UTC)

 Done I've made a few changes to the image; toning down the colours, aligning the BMI values diagonally, and adding incremental scales along the edges. If anyone has any more suggestions I'm all ears, but I'm pretty happy to call that done. nagualdesign 18:54, 3 June 2021 (UTC)

@Nagualdesign: Wow, the chart looks perfect. Thanks for the hard work! Did you / will you add it to the BMI article? Thanks again! 32.209.55.38 (talk) 19:54, 3 June 2021 (UTC)

Here is a table for, specifically, a body height of 67.5 inches:

                       101–106 : 16
                       107–113 : 17
                       114–119 : 18
                       120–126 : 19
                       127–132 : 20
                       133–139 : 21
                       140–145 : 22
                       146–152 : 23
                       153–158 : 24
                       159–165 : 25
                       166–171 : 26
                       172–178 : 27
                       179–184 : 28
                       185–191 : 29
                       192–197 : 30
                       198–204 : 31
                       205–210 : 32
                       211–217 : 33

A line like "153–158 : 24" states that a body weight in the range 153–158 lbs means a BMI of 24. For weights outside that range: multiply the magnitude in lbs by 0.1543.  --Lambiam 22:18, 1 June 2021 (UTC)

@Lambiam: Thanks for the chart. Very helpful! Thank you. 32.209.55.38 (talk) 18:22, 2 June 2021 (UTC)

Thanks, all. This was very helpful. 32.209.55.38 (talk) 20:02, 5 June 2021 (UTC)

Resolved

Mean seasonal insolation contrast between the hemispheres

Greetings, is there a study that stipulates what the mean seasonal insolation contrast (in watt or watt/square metres) is between Earth's two hemispheres is? The graphs that show it by latitude are useless; I need an average across the hemisphere. Jo-Jo Eumerus (talk) 20:24, 29 May 2021 (UTC)

It's pretty complicated, since there's more landmass in the northern tropics and more ice (with high albedo) in the southern hemisphere. Try this. nagualdesign 21:20, 29 May 2021 (UTC)

That does not give the seasonal differences, only the annual ones. Jo-Jo Eumerus (talk) 08:23, 30 May 2021 (UTC)

If you have a table of some quantity of interest by latitude, you should be able to approximate that quantity as a function $f(\phi )$ of latitude $\phi$ by interpolation. Then you can estimate the mean value between two latitudes $\phi _{0}$ and $\phi _{1}$ by computing the value of the quotient of two integrals:

$\int _{\phi _{0}}^{\phi _{1}}f(\phi )\cos \phi \,d\phi \left/\int _{\phi _{0}}^{\phi _{1}}\cos \phi \,d\phi \right..$

 --Lambiam 00:24, 30 May 2021 (UTC)

Audible beat phenomenon for frequencies that aren't close together

I wasn't sure which board to ask this on, so please let me know if I should move this question to one of the others. When I generate multiple pure tones from an online tone generator (https://www.szynalski.com/tone-generator/) or from code, I notice a beat phenomenon that seems like it shouldn't be there. When I play 400hz and 400hz, or 400hz and 600hz, there is no beat as expected. When I play 400hz and 401hz, there is a strong beat as expected at 1hz. But when I play 400hz and 601hz, there is a beat audible at 2hz, though it is not as strong. I don't understand why that should happen, as the only interference frequency should be at 201hz (or 100.5hz depending on how you're counting it). Black Carrot (talk) 22:33, 29 May 2021 (UTC)

Any pair of frequencies can interfere. If you calculate the sum of two sines you get a waveform. If the resulting waveform has a period within the range of human hearing then you will hear that frequency. nagualdesign 22:55, 29 May 2021 (UTC)

Right. But as I said, the interference between 400hz and 601hz is not within the human hearing range (that is, the range in which we hear variations in volume, less than about 10hz). And in particular, is not in the range around 1-2hz. Black Carrot (talk) 22:56, 29 May 2021 (UTC)

To be more specific, the combined waveform cos(400x)+cos(601x) = 2*cos(100.5x)*cos(500.5x) Black Carrot (talk) 23:08, 29 May 2021 (UTC)

According to WolframAlpha (you'll have to insert the frequencies yourself) the frequency of beats would be 201hz. IIRC human hearing is from about 40hz to 40khz. I'm not sure where you got the figure of 2hz. nagualdesign 23:22, 29 May 2021 (UTC)

...Actually, it's 20hz to 20khz. nagualdesign 23:25, 29 May 2021 (UTC)

I got the figure of 2hz from my ears, I played the tones and watched a clock. There is a clearly audible cyclic change in volume (or "beat") at that frequency. I've already done the calculation for the 201hz interference, as you can see in my original question. Black Carrot (talk) 00:58, 30 May 2021 (UTC)

Perhaps your "pure tones" are not as pure as one might think, but are undergoing a non-linear distortion. The third harmonic of 400hz is at 1200hz, and the second harmonic of 601hz is at 1202hz, You may be hearing the interference between these two harmonics.  --Lambiam 23:51, 29 May 2021 (UTC)

Human ears can easily detect an amplitude modulation at 2 Hz, in fact I worked on a problem that was 0.2 Hz. Anyone who has heard a WW2 bomber film, or tuned a guitar will be familiar with this. Greglocock (talk) 05:27, 30 May 2021 (UTC)

That would make sense. I'm surprised though, I thought that a computer generated sound would be more faithful than that, otherwise ordinary recordings should sound distorted as well. Do you know if that sort of effect is common, or what might cause it? Black Carrot (talk) 00:58, 30 May 2021 (UTC)

Computers can generate very pure tones. Thinking about my mistake, I remembered back to when I was 7 years old, programming my Amstrad, and learned that middle A was 440 Hz. So 220 Hz would be an octave lower, and 201 Hz would be what, an F or G (ish)? Certainly audible. nagualdesign 23:59, 29 May 2021 (UTC)

But is, in this case, the addition of the signals performed digitally from the computed sine waves, or is the addition that of the physical waves? In the latter case, the digital purity of the computer may be compromised by non-linear aspects of analog components. (In equal temperament a G near 200 Hz is about 196 Hz. A semitone higher, G sharp or enharmonically equivalent A flat, is about 208 Hz. So for 201 Hz, think quarter tones. The lowest key of an 88-key piano is an A₀ with a pitch of 27.5 Hz.)  --Lambiam 00:52, 30 May 2021 (UTC)

The website that Black Carrot linked to appears to only produce one tone, so I'm guessing that they opened two browser tabs to create two tones at once. Assuming that both tones came out of both speakers simultaneously (rather than one tone from the left channel and one from the right) then the waveforms are combined digitally at a very high sample rate before being fed to the digital-analogue converter. Good quality DACs have been available at very low cost for decades. Having said that, Bluetooth speakers can struggle with high frequencies, but these are not high frequencies. nagualdesign 01:13, 30 May 2021 (UTC)

I don't think it should be audible at 201hz, though. My understanding of how the ear works is, any frequency components greater than about 20hz are separated out from each other. So the interference pattern between two tones should only matter if it produces a beat within that range of <20hz. Black Carrot (talk) 01:03, 30 May 2021 (UTC)

Okay, perhaps we're talking at cross purposes. It's my understanding that the waveform isn't as important as the period. If that's 201 Hz then you'll hear a G (ish). The difference between a square wave, sawtooth or whatever you can imagine only affects the quality of the sound, but a G is a G regardless. In other words, you'll hear the 400 Hz, the 601 Hz and the resulting 201 Hz as 3 separate tones. nagualdesign 01:13, 30 May 2021 (UTC)

I just repeated the experiment using the link you provided (opening 2 tabs) and what I heard was the 201 Hz tone pulsing at around 2 Hz. nagualdesign 01:41, 30 May 2021 (UTC)

Playing a bit around A = 400 Hz and B = 600 Hz, I hear a pulsation with a frequency of │3A − 2B│.  --Lambiam 12:21, 30 May 2021 (UTC)

I agree with Lambiam. Your computer can generate digital signals for perfect pure tones with no harmonics, but your earphones or speakers might not be able to play them. – b_jonas 14:05, 30 May 2021 (UTC)

Thanks for the help, everyone. If I could ask a second question, do you know if there is an audio setup (particular speakers, etc) that would allow me to play pure tones without noticeable overtones, to prevent this beating effect? On google at least, it does not seem to be a common question. Black Carrot (talk) 03:03, 1 June 2021 (UTC)

Humans can hear beats with combination tones, so even if your waveform has no overtones, and even if your audio setup has no overtones, your inner ear will still create them. (Even if your inner ear doesn't do it, it seems that your brain will do it anyway!) There's some discussion here: [3]. See also Beat_(acoustics)#Binaural_beats, missing fundamental and psychoacoustics. It might be an interesting experiment to try introducing the 400 Hz and 601 Hz pure tones separately into each ear, and see whether or not you still perceive beats. --Amble (talk) 17:29, 1 June 2021 (UTC)

See also [4], [5], [6], [7]. --Amble (talk) 23:05, 1 June 2021 (UTC)

A short explanation with references: [8] --Amble (talk) 23:14, 1 June 2021 (UTC)