January 2018 - mathfun@mailman.xmission.com

Re: [math-fun] Touch Tone frequencies
by Dan Asimov 19 Jan '18

19 Jan '18

I'm curious to hear low-integer ratios that don't commonly occur in Western music. (Anyone for 7/K, 2 <= K <= 5 ???) —Dan

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[math-fun] Touch Tone frequencies
by Keith F. Lynch 18 Jan '18

18 Jan '18

Inspired by Henry Baker's musing about modem tones: Each Touch Tone button on a telephone sends two simultaneous audio frequencies through the phone line, one associated with the row it's on, and one associated with the column it's on. There are four rows and four columns (though only the first three columns exist on most telephones) for a total of eight frequencies. The frequencies were chosen such that the sums or differences of any two of the eight frequencies are as far as possible from any of the eight frequencies. Similarly with whole-number multiples ("harmonics") of the eight frequencies or of their sums or differences. This is to prevent the phone company equipment from getting confused, since non-linearities in the circuits could result in such sum and difference frequencies appearing. This is probably why Touch Tones are so unmusical. Music relies on frequencies with small whole-number ratios, almost the exact opposite of the Touch Tone criteria. (Or did they only concern themselves with sums or differences of row and column frequencies? Two different row frequencies, or two different column frequencies, should never appear together.) Also, the frequencies should be between 400 and 3400 Hz, so they'll fit in a phone line's bandwidth. See https://en.wikipedia.org/wiki/Dual-tone_multi-frequency_signaling for more information, including the eight frequencies. Can you do better on selecting eight frequencies with those constraints? What about other numbers of frequencies? Is there a good algorithm for selecting them?

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[math-fun] Math to court
by Brent Meeker 18 Jan '18

18 Jan '18

/ //Mathematics has helped a US court to overrule an example of partisan gerrymandering in North Carolina.Credit: Jonathan Drake/REUTERS// // //Mathematicians are no longer devices for turning coffee into theorems, as the Hungarian mathematics researcher (and caffeine addict) Alfréd Rényi is said to have claimed. They seem pretty useful for preserving democracy, too. In striking down the way that officials in North Carolina unfairly partitioned the state into electoral districts, a US federal court last week conspicuously cited the work of mathematicians including Jonathan Mattingly, an expert in mathematical modelling.// // //In a 200-page decision released on 9 January, the three-judge court in Richmond, Virginia, said that the districting had unfairly favoured the Republican Party. Maths played a key part in helping the court to reach that decision, by demonstrating the unlawful use of partisan gerrymandering — fiddling with district boundaries to include or exclude certain voters and steer the results of an election. Those apportioning districts might draw borders that pack large numbers of voters for an opposition party into a small number of districts, for example, limiting the number of seats that the opposition can win. The process has been likened to allowing lawmakers to choose their voters, rather than the other way around/. https://www.nature.com/articles/d41586-018-00661-x?utm_source=briefing&… Of course there are other ways to improve the process besides drawing the district lines so the elected officers more nearly match the electorate preferences, e.g. instant run-off, multiple seat districts,... Brent

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[math-fun] Polygons from x^3 - 7 x +7 =0 and x^3 - 9 x +9 =0
by Ed Pegg Jr 18 Jan '18

18 Jan '18

Pick a root of x^3 - 7 x +7 =0. I'll use ~1.3569 Take the dot product of {1,x,x^2} with the following, then take the sign-carry square root (as in sqrt(4) = 2, sqrt(-4)=-2). You'll get a heptagon. v = {{{0, 0, 0}, {4, 0, 0}}, {{7, -2, -1}, {-3, 2, 1}}, {{7, -1, -1}, {3, -1, -1}}, {{-7, 3, 2}, {-11, 3, 2}}, {{7, -3, -2}, {-11, 3, 2}}, {{-7, 1, 1}, {3, -1, -1}}, {{-7, 2, 1}, {-3, 2, 1}}}/4; Pick a root of x^3 - 9 x +9 =0. I'll use ~1.18479 Take the dot product of {1,x,x^2} with the following, then take the sign-carry square root (as in sqrt(4) = 2, sqrt(-4)=-2). You'll get a nonagon. v = {{{0,0,1},{-12,0,1}},{{9,0,0},{-3,0,0}},{{18,-3,-2},{-6,3,2}}, {{0,3,1},{12,-3,-1}},{{0,0,0},{12,0,0}},{{0,-3,-1},{12,-3,-1}}, {{-18,3,2},{-6,3,2}},{{-9,0,0},{-3,0,0}},{{0,0,-1},{-12,0,1}}}; In short x^3 = 7 (x -1) makes a 7-gon. x^3 = 9 (x -1) makes a 9-gon. Some similar things at https://math.stackexchange.com/questions/2609505/the-heegner-polynomials --Ed Pegg Jr

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Re: [math-fun] Touch Tone frequencies
by Dan Asimov 18 Jan '18

18 Jan '18

IF by "it" is meant a major third instead of the devil's interval, then: Yes, it is 1/3 of an octave since it is 4 half-tones. (If "it" means the devil's interval, then it's 1/2.) —Dan -----Original Message----- >From: Mike Stay <metaweta(a)gmail.com> >Sent: Jan 17, 2018 5:50 PM >To: Dan Asimov <dasimov(a)earthlink.net>, math-fun <math-fun(a)mailman.xmission.com> >Subject: Re: [math-fun] Touch Tone frequencies > >It's not 1/3 of an octave, it's the third note up (using 1-indexing). > >On Wed, Jan 17, 2018 at 6:11 PM, Dan Asimov <dasimov(a)earthlink.net> wrote: >> In what sense is it 1/3 of an octave??? >> >> —Dan >> >> >> -----Original Message----- >>>From: Cris Moore <moore(a)santafe.edu> >>>Sent: Jan 17, 2018 2:56 PM >>>To: Dan Asimov <dasimov(a)earthlink.net>, math-fun <math-fun(a)mailman.xmission.com> >>>Subject: Re: [math-fun] Touch Tone frequencies >>> >>> >>>Right, but it’s 1/3 of an octave, so in the equal-tempered scaled it would be 2^(1/3). >>> >>>- Cris >>> >>>> On Jan 17, 2018, at 12:23 PM, Dan Asimov <dasimov(a)earthlink.net> wrote: >>>> >>>> The "devil's interval" of sqrt(2) was certainly considered dissonant. >>>> >>>> But I never heard that about the major third, whose ratio is 1.25, >>>> which is more like 10% off sqrt(2) than 1%. >>>> >>>> —Dan >>> >>>_

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Re: [math-fun] Touch Tone frequencies
by Dan Asimov 18 Jan '18

18 Jan '18

In what sense is it 1/3 of an octave??? —Dan -----Original Message----- >From: Cris Moore <moore(a)santafe.edu> >Sent: Jan 17, 2018 2:56 PM >To: Dan Asimov <dasimov(a)earthlink.net>, math-fun <math-fun(a)mailman.xmission.com> >Subject: Re: [math-fun] Touch Tone frequencies > > >Right, but it’s 1/3 of an octave, so in the equal-tempered scaled it would be 2^(1/3). > >- Cris > >> On Jan 17, 2018, at 12:23 PM, Dan Asimov <dasimov(a)earthlink.net> wrote: >> >> The "devil's interval" of sqrt(2) was certainly considered dissonant. >> >> But I never heard that about the major third, whose ratio is 1.25, >> which is more like 10% off sqrt(2) than 1%. >> >> —Dan > >_______________________________________________ >math-fun mailing list >math-fun(a)mailman.xmission.com >https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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Re: [math-fun] Touch Tone frequencies
by Dan Asimov 17 Jan '18

17 Jan '18

The "devil's interval" of sqrt(2) was certainly considered dissonant. But I never heard that about the major third, whose ratio is 1.25, which is more like 10% off sqrt(2) than 1%. —Dan ----- That is a famous tritone (in the equal-tempered scale, the square root of 2) but I don’t think it’s a joke. Another classic use is in the Tristan chord. I was surprised to learn from a friend that in the middle ages, the major third was considered very dissonant. We consider it quite consonant, and associate it with the ratio 5:4. In the 12-tone scale this is about 1% off from the cube root of 2. Cris > On Jan 17, 2018, at 9:36 AM, Henry Baker <hbaker1(a)pipeline.com> wrote: > > Leonard Bernstein famously (in musical circles, at least) > included a musical joke in "West Side Story" with the first > two notes of the song "Maria", which form a *tritone* > (the two "Ma ri" notes of the "Ma ri a"). -----

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Re: [math-fun] "Baby" dimensional analysis
by Keith F. Lynch 17 Jan '18

17 Jan '18

Eugene Salamin <gene_salamin(a)yahoo.com> > Math-fun seems to have garbled up the line feeds in this message, so > I'll try again. I'm pretty sure the problem is at your end, i.e. you're sending each paragraph as one long line. Also, whenever you have two spaces in a row, one of them gets turned into what I assume is a UTF-8 non-breaking space, which math-fun turns into a question mark. I've noticed that the "smarter" software gets, the more mangled messages get. That's why I use very simple software. > Here's a nice one. What is the energy flux = power per unit area F > (W m^-2 = kg s^-3) carried by a gravitational wave of strain h? I solved that one too. Except I too was off by a small factor. After a live talk at the Philosophical Society of Washington on the topic of LIGO, I mentioned to one of the speakers that LIGO wasn't very sensitive at all, given that if the gravitational wave energy had been visible light instead, it would have been bright enough, not just to see with unaided eyes, but to read by, even though the source was more than a billion light years away. He asked me where I found the equation for converting strain into flux. Apparently they don't publicize that equation much, as it makes LIGO look bad. (On double checking, I found that a video of the whole talk is online, at https://www.youtube.com/watch?v=y6_NpIRhoWg That post-meeting discussion isn't on it, but I do ask a question at 1:52:40, if anyone wants to know what I look and sound like.) There is an element of arbitariness in dimensional analysis. For instance notice that the dimensions of electric charge are completely different in SI and in cgs. See https://en.wikipedia.org/wiki/Statcoulomb Is vertical distance the same thing as horizontal distance? In most contexts, yes, but not to interior decorators, since it's against the rules of interior decorating to place a piece of furniture on its side, back, or front. People used to measure the vacuum speed of light, and debate whether it was constant, or whether it might vary slightly with time, direction, or frequency. But in the 1980s, the length of the meter was redefined to depend on the duration of the second, which caused c to be defined as exactly 299792458 meters per second. It feels like sleight of hand to me. Or has SI changed the definition, not just of the *unit* of distance, but also of what's *meant* by distance? They plan to make further changes, which will turn some numbers that were measured into defined constants, and vice versa. See https://en.wikipedia.org/wiki/Proposed_redefinition_of_SI_base_units They're coming perilously close to *defining* General Relativity as correct. I'm most fascinated by dimensionless constants, such as the fine structure constant, alpha. Those can never have defined values; they have to be measured. If alpha turns out not to be constant after all, it's an interesting question whether we can make an arbitrary choice of whether the speed of light (c), Planck's constant (h), the charge of the electron (e), or the permeability of space (mu-0), had changed, or whether there's some non-arbitrary way to determine which had changed. With current SI, c and mu-0 can't change. With the new SI, c, h, and e can't change, but mu-0 can. But that may just mean that either the old or new SI, or both, are inconsistent with a world in which alpha changes.

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Re: [math-fun] Bell 202 1200 baud modem emulation
by Keith F. Lynch 17 Jan '18

17 Jan '18

Interesting, but not very physical. The frequencies present in a waveform are unchanged if you replace the waveform with its derivative. As such, it makes no difference whether you have a whole number of cycles before changing frequencies. The derivative of a sine wave that changes frequency when it crosses zero will change frequency at a different amplitude. A sudden change of frequency will splatter all over the spectrum, even if the change happens at an instantaneous amplitude of zero. For a two-frequency modem on a standard analog telephone line, that doesn't matter, since the telephone circuit blocks extreme frequencies, and since the receiving modem is listening only for those two frequencies. Frequency shift keying on radio is generally done by simply adding a radio frequency to the two tones. So instead of 1200 and 2200 Hz, it might be at 3501200 and 3502200 Hz. Obviously that will change whether the (unchanged) duration of one bit is a whole number of waves for one or both frequencies. Fortunately, that doesn't matter. One way to avoid splattering all over the radio spectrum, which would of course be a bad thing, is to change the frequency gradually rather than abruptly. Slide smoothly from 1200 to 2200 or vice versa. Another way is to gradually decrease the amplitude of the 1200 tone while gradually increasing the amplitude of the 2200 tone or vice versa. Yet another way is to just run the signal through a narrow-pass analog filter. None of these depend on whether there are a whole number of waves, or on when the waveform crosses zero, or on what the radio frequency is. Similarly with the even older and simpler method of on-off keying (e.g. Morse code). The amplitude should gradually increase and decrease, since suddenly starting or stopping a radio signal will produce "key click" noise all over the spectrum. Of course by "gradually" I mean typically on a millisecond time scale. Though the time scale can be anything. The shorter it is, the more bandwidth the signal uses, and the more bits can be sent per second. The link between channel capacity (bits per second) and bandwidth (difference between the highest and lowest frequency used) is so strong that most people say "bandwidth" when they mean "channel capacity." The link between channel capacity and bandwidth is due to the Heisenberg uncertainty principle. The more precisely you know the frequency of a signal, the less precisely you can know its time of arrival, and vice versa. What frequencies are present in a signal depends as strongly on the receiver as on the transmitter. Alternate once per second between 1000000-1 Hz and 1000000+1 Hz. Is there any energy at exactly 1000000 Hz? If you have a narrow band receiver, yes, nearly all the signal power is there. If you have a wide band receiver, no, none is. You can see this effect for yourself at sdr.hu, a website that lets you see and hear, in real time, radio signals in the HF band and below as picked up at various places all over the world. You can select the receiver location, frequency, bandwidth, and mode. It's one of the truly wonderful websites, right up there with Wikipedia, Google, IMDB, and OEIS. (Similarly with the signal's direction versus where on your antenna it landed. If you know very precisely where it landed, you know its position on the axis at right angles to its direction of propagation to a high precision, which means you know its momentum on that axis, i.e. the direction it came from, to a very low precision. This is why directional antennas have to be large.) A few nitpicks: The first digital modem in widespread use was the Bell 103 (and compatibles), dating to 1962, and still in widespread use through the 1980s. It worked only up to 300 bits per second. It wasn't as fast as the later Bell 202, but unlike the 202 it was full duplex, meaning that signals could be sent in both directions at once. In one direction, it switched between 1070 and 1270 Hz, in the other direction, it switched between 2025 and 2225 Hz. It was always more common than the 202, though perhaps not as common as the later Bell 212, which was a full duplex 1200 bits per second modem. I don't think ham radio operators ever used Bell 202, at least in the US. For one thing, ASCII wasn't allowed on the air until 1980, by which time Bell 202 was long obsolete. Hams have been using the older five bit ITA2 code (often wrongly called Baudot) since 1946. Hams typically use 45 bits per second and a 170 Hz frequency difference between mark (1 bit) and space (0 bit) when sending ITA2 ("RTTY"). I think deaf people also still use ITA2 over the telephone.

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[math-fun] "Baby" dimensional analysis
by James Propp 16 Jan '18

16 Jan '18

In high school I was taught a baby version of dimensional analysis in which the quantity you're trying to compute is either directly or inversely proportional to each of the quantities you're given, and the only pitfall is that you might divide when you should multiply or vice versa. In such a case, you keep yourself on the right path by including units along with the numbers, and checking that your big product-quotient expression has the right units. (For instance, you don't multiply distance by velocity if you want an answer that has units of time. m meters times v meters-per-second has units of meters-squared-per-second, while m meters DIVIDED BY v meters-per-second has units of seconds.) The Wikipedia page https://en.wikipedia.org/wiki/Dimensional_analysis does a pretty good job of explaining this. Anyway, I recall from high school chemistry that there were some pretty complex problems involving reaction rates of different chemical species where the keeping-track-of-units trick was really helpful because the complexity of the problem overloaded my short-term memory. Can anyone think of a problem (maybe chemical, maybe not) where keeping track of units, and trying to get them to cancel out, is really helpful? Of course, this is not to deprecate the sense-making approach that determines whether a number should be put "up top" or "down below" by asking "If this number were larger, would my final answer get bigger or smaller?" (My February "Mathematical Enchantments" essay will be about dimensional analysis, and I want to discuss these sorts of "baby" applications before I get into deeper ones where the exponents aren't all +1 and -1.) Thanks, Jim Propp

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