Wikipedia:Reference desk/Archives/Mathematics/2007 September 27

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September 27

Encryption using musical notation

I hope this is the right place to be asking a question like this, which combines disciplines. Can anybody cite any references to known instances of (i.e. textual) information being encrypted in musical notation? What I am looking for is aesthetically pleasing music which also contains an encrypted message.

Thank you, in anticipation of your help.

--Lil Miss Picky 02:59, 27 September 2007 (UTC)

Check out Edward Elgar and his Enigma Variations — Kieff | Talk 03:17, 27 September 2007 (UTC)

It's fairly well known that Barrington Pheloung's theme for Inspector Morse included the letters MORSE prominently in Morse code. It's less well known that Pheloung also sometimes used to put other short passages of Morse code into his scores for particular episodes of the show, for instance spelling out another name. Xn4 06:31, 27 September 2007 (UTC)

In a slightly different form 'encryption', both Bach and Cage have included their name in musical pieces, by simply spelling it out with musical notes (H used to be a note in german musical notation). risk 13:30, 27 September 2007 (UTC)

It still very much is. In German, B means what we call B-flat; H means what we call B (natural). See also B-A-C-H. -- JackofOz 04:47, 1 October 2007 (UTC)

The opening of Beethoven's Fifth Symphony was used by the BBC as morse code for V (meaning victory) during World War II. Alban Berg was fond of using the names or initials of his friends, and also the four-note BACH melody mentioned by Risk above, as motifs in his music. A variety of 20th century composers have encoded various things into their music, though not necessarily with the intention that they be decoded later. Trying to encrypt something as a melody is slightly problematic, as pitches in music are percieved relative to eachother and not with absolute values (unless the listener posesses Absolute pitch, which is a rare ability). For instance, I could encode my telephone number as a melody using pitch class notation, but to decipher it without ambiguity you would need to know one of the digits (or pitches) beforehand with certainty. - Rainwarrior 16:28, 27 September 2007 (UTC)

I don't know where to find it, but I've heard of musicians that hide pictures in their recordings. If you image the sound using some sort of standard tool, it flashes up a clear picture of something partway through. I saw a picture of the guitarist's face embedded this way. Black Carrot 03:50, 29 September 2007 (UTC)

There was Aphex Twin's face hidden in his "Equation" song by filtering white noise with that image[1]. (Thank link also shows a picture of cats stored in a Venetian Snares song.) - Rainwarrior 18:41, 30 September 2007 (UTC)

And then there's Solresol. —Tamfang 06:12, 30 September 2007 (UTC)

Most algorithmic compositions could be used for this purpose by using the plaintext for a random number generator. You have to be careful to make sure no information is lost. — Daniel 21:52, 30 September 2007 (UTC)

For for, read in place of. —Tamfang 03:12, 1 October 2007 (UTC)

• Rimsky-Korsakov, Borodin, Lyadov, Glazunov and Kopylov collectively wrote a string quartet in honour of the music publisher Mitrofan Belyayev's 50th birthday. They used the theme B-A-F (in Russian pronounced be-la-ef).
• DSCH was Shostakovich's personal musical autograph. He used it in many of his works.
• Schumann used such techniques, most particularly in Carnaval where almost all the sections are based on either AsCH (= A flat-C-B), ASCH (= A-E flat-C-B), or SCHA (= E flat-C-B-A). The town of Asch was where his girlfriend lived, and the letters SCHA also appear in Schumann's own name. Schumann had as keen an interest in puzzles and codes as Elgar did, and he died not long before Elgar was born. I once read a serious contribution to a musical journal hypothesising that Elgar could have been the reincarnation of Schumann (!)
• It has been claimed (although I disbelieve it) that the phrase "DUM-da-da-dum" with which Rachmaninoff's works sometimes begin (eg. the Prelude in G minor, Op. 23, No. 5) or end (eg. the 2nd and 3rd Piano Concertos), was chosen as a rhyme for RACH-ma-ni-noff. However, the stress in "Rachmaninoff" is on the second syllable, not the first. I can see how they can be made to resemble each other, but I suspect it's just a (half-)coincidence. -- JackofOz 04:47, 1 October 2007 (UTC)

Integral_of_secant_cubed

Why not use substitution using tan^2 x + ¹ = sec^2 x and u = sec x, du = sec x tan x dx  ?--Mostargue 11:58, 27 September 2007 (UTC)

adding another derivation won't hurt - I always like to see more than one..83.100.254.236 12:08, 27 September 2007 (UTC)

y = x^x

a few questions on the function $y=x^{x}$

1. for positive values of x the function appears as a line, but for various values of x there are possible negative values, i.e. for x=1/2 is root(1/2) and therefore should there not be sporadic points on the reflection of the function across the y-axis
2. for negative values of x integers obviously form real answers, but also which in other values, because if for example you take -1/2 you get ${\frac {-1}{2}}^{\frac {-1}{2}}={\frac {1}{\sqrt {\frac {-1}{2}}}}$ which obviously gives complex roots, whereas also ${\frac {-1}{2}}^{\frac {-1}{2}}={\frac {-2}{4}}^{\frac {-2}{4}}={\frac {1}{\sqrt[{4}]{({\frac {-2}{4}})^{2}}}}={\frac {1}{\sqrt[{4}]{\frac {4}{16}}}}$ which obviously gives real roots, which implies that the formulae gives real positive and negative roots for all rational values of x
3. can you geometrically interpret ${\sqrt {2}}^{\sqrt {2}}$

thank you. ΦΙΛ Κ 18:55, 27 September 2007 (UTC)

1.y = x^x is not a function unless the domain is restricted x>0, then it is a continuous smooth curve, which behaves like the exponential function for small x, but its growth rate is greater than the exponential function for large x. So reflection across the y axis isn't needed.

2. see 1

3. Yes. it's approximately 1.6325269194381528447734953810247. But put more symbolically, it is the solution to $x^{2}=2^{\sqrt {2}}$ or

• $\ln {x}={\frac {{\sqrt {2}}\ln {2}}{2}}$

--Mostargue 19:15, 27 September 2007 (UTC)

Youd didn't really provide satisfactory answers to any of the questions. Whay cant f(x)=x^x be a function? ΦΙΛ Κ 11:45, 28 September 2007 (UTC)

Please see Function(mathematics) to learn why it cannot have multiple outputs for each input.--Mostargue 21:14, 28 September 2007 (UTC)

You can have "one to many" functions. Complex logs for example. ΦΙΛ Κ 19:17, 9 October 2007 (UTC)

1. I know that x^1/2, x^1/3, etc. can be defined to be multivalued functions (with 2 and 3 values respectively). But I don't think you consider that when you look at single values. And anyway if you do that, then x^{(any irrational number)} would have infinite values along a circle in the complex plane, and it wouldn't show up very usefully in a graph.
2. You are adding additional roots that weren't there before. Consider x¹ which has 1 value. If you re-write it as above as x^42/42 for example; and then you say it is ${\sqrt[{42}]{x^{42}}}$; and then if you consider the multi-valued-ness of the 42th root, now it suddenly has 42 values.

--Spoon! 19:17, 27 September 2007 (UTC)

But why is -2/4 an any less valid interpretation of the value 0.5 than -1/2. And what is the meaning of the corresponding real value if it only arose due to inclusion of extra roots. Notice that y = |x|^x provides a continuous curve that is the same as x^x with the inclusion of "extra roots".ΦΙΛ Κ 11:45, 28 September 2007 (UTC)

The function x^x is usually defined for positive values of x as being e^{x ln(x)}. This definition makes x^x single valued for positive x. It avoids the complication of choosing between multiple roots for non-integer rational values of x, and it avoids even more complications at irrational values of x. Note that the graph of y = x^x is not monotonic - it has a minimum value between 0 and 1. It is a simple but interesting exercise in calculus to find the value of this minimum. Gandalf61 08:53, 28 September 2007 (UTC)

Yes the root is at e^-1, I have a feeling you think I am looking for more simplistic answers than I actually am. Even though this function is usually defined with a restricted domain, in the case of the question posed, it is not. ΦΙΛ Κ 11:45, 28 September 2007 (UTC)

Okay, if you don't want to use the usual definition of x^x, can you explain what definition you want to use instead ? How would you rigourously and consistently define x^x in a way that works for both negative and positive values of x ? You appear to be giving real negative roots a special status that ranks them above complex roots - how do you justify that ? How do you assign a value (or even a finite set of values) to x^x when x is negative and irrational - for example, what is $-{\sqrt {2}}^{(-{\sqrt {2}})}$ ? What value does your method assign to 0⁰ ? The usual definition sidesteps all of these problems by restricting the range of x^x to positive values of x, and I don't know of any other way of handling them. Gandalf61 13:29, 28 September 2007 (UTC)

You want to extend x^x to negative reals. It is pretty clear that the absolute value should satisfy $|(-x)^{-x}|={\frac {1}{|x^{x}|}}$, so all you need to do now is figure out how to calculate the phase of the complex number. Basically, you need to define a function "$g(x)=(-1)^{x}$" that agrees with the usual definition on negative integers. It will have to have values in the set of complex numbers of absolute value 1. You'll have to think whether you want it to have any other properties or whether simply using $g(x)=\cos(\pi x)+isin(\pi x)$ or a similar simple periodic construction will do what you want it to do (I'm not quite sure what you want to do with your function). Be careful with exponentiation laws, though, see Exponentiation#Failure_of_power_and_logarithm_identities. Kusma (talk) 14:01, 28 September 2007 (UTC)

If you really want to examine the expression x^x for negatives (and fractions) I suggest to you to relplace x with the rational fraction n/m eg n/m^n/m.. This gives m values for each x, and when x is negative (taking the primary root) n/m^1/m as i|n/m|^1/m gives the primary value of iⁿ|n/m|^n/m However when x is irrational eg sqrt(2), pi etc then this method fails - as other have said when x<0 the function is non-continuos87.102.83.163 17:38, 28 September 2007 (UTC)

So questions 1,2 yes and question 3 no I can't easily do that (but will try)- though I might interpret sqrt(2)^sqrt(2) as a circle of radius r=e^{sqrt(2)ln(sqrt(2))} with the points a+ib where a²+b²=r², and as sqrt (2) is irrational as a fraction (ie m above can not be a finite integer) I would assume that there are infinite values of a+ib that could be used eg the circle is continuous... I'm guessing that this analysis which works for rational fractions is not applicable in this was to irrationals -so I'd take the latter part of this paragraph with a pinch of salt87.102.83.163 17:45, 28 September 2007 (UTC)

weight distribution

I am wanting to build a car trailer that will carry my motorcycles and my jeep. I need to place the axles so the weight will be correctly distributed on the deck of the trailer so it will tow well. How do I figure the center of gravity (balance point) front to rear of a vehicle? Or in this case I need to figure the balance point of the motorcycles, then the jeep, then I'm guessing that I could use the same formula to figure the balance point of the trailer, depending on the placement of the vehicles and the actual axle weight of each. —Preceding unsigned comment added by 207.69.139.147 (talk) 21:09, 27 September 2007 (UTC)

Did you read center of mass? --Spoon! 23:55, 27 September 2007 (UTC)

It's a lot of work to do it mathematically, since every component must be weighed and must have it's center of gravity determined, then you must do the math to figure out it's contribution to the overall center of gravity. You could also do it experimentally, by loading the trailer in various configurations and seeing what the balance point is (say by putting a fulcrum under it that just lifts the trailer an inch off the ground, at various points, until it easily teeter-totters back and forth). Also, you could just assume the midpoint of the trailer is the COG, as that's probably close enough (unless you load it very unevenly or the tow vehicle is supporting some of the weight). Finally, if you put four wheels at the outside edges of the trailer, it really doesn't matter where the COG is. StuRat 19:17, 28 September 2007 (UTC)