Imagine a series of positive real numbers defined so that each next number is a direct math function of the one preceding it such that
every Xn = f(Xn-1)
Require further that after the function is applied about 5 to 25 times the number ends up exactly doubled, that is every Xn = 1/2 Xn+j where 5 <= j <= 25
In other words, I want all series members to always have the same relationship to their adjacent members AND I want the series to require about j steps to achieve a doubling.
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Puzzle #1:
Can you find a function that would accomplish this for any given j in the range?
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Puzzle #2:
Consider the j members that fall between any pair of members in a 2:1 ratio.
Can you select an ingenious value of j such that some of the in-between members are also in nice small integer ratios to the starting member, e.g. in a 3:2, 4:3, or 5:4 ratio?
i.e. where the series looks something like
X1, . . Xa~1.25X1, . . Xb~1.33 X1, . . Xc~1.5 X1, . . Xj=2 X1
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Puzzle # 3:
How is it that almost everyone has listened to this hundreds of times before but few have ever heard it put this way?
5 comments:
This puzzle was from mrLee.
I should clarify that in puzzle 2, the in-between members' ratios needn't be exactly in those ratios, just near to them, but the jth member must always be exactly 2:1.
Here's a clue. If we had interest in j<5, then one solution for j=3 is that f(x) = kx where k = the cube root of 2.
This is approximately 1.26. If one starts with the first member being x, we can generate the next 3 members:
x, 1.26 x, 1.59 x, 2 x
i.e. where each member is 1.26 times the preceding member. Any member is exactly double the one 3 members before.
Even with the hint you gave me the other day, my brain just didn't want to figure out this one.
I probably worded this one a little too obscurely.
Puzzle # 1 solution is:
f(x) = kx
where k = the jth root of 2
= 2^(1/j)
Puzzle #2 solution is:
Choose j = 12
then k = 1.05946...
so the series will follow the pattern:
x,
x*k = 1.06 x,
x*k^2 = 1.12 x,
x*k^3 = 1.19 x,
x*k^4 = 1.26 x ~ 5/4 x,
x*k^5 = 1.33 x ~ 4/3 x,
x*k^6 = 1.41 x,
x*k^7 = 1.49 x ~ 3/2 x,
x*k^8 = 1.59 x,
x*k^9 = 1.68 x,
x*k^10 = 1.78 x,
x*k^11 = 1.89 x,
x*k^12 = 2x,
x*k^13 = 2.12 x,
x*k^14 = 2.24 x,
x*k^15 = 2.38 x,
x*k^16 = 2.52 x,
x*k^17 = 2.67 x,
x*k^18 = 2.83 x,
x*k^19 = 3.99 x,
x*k^20 = 3.17 x,
x*k^21 = 3.36 x,
x*k^22 = 3.56 x,
x*k^23 = 3.78 x,
x*k^24 = 4x, . . .
This solution is independent of x and is more a statement about the relationships of series members to ones before or after.
For arbitrary x = 440, the series would be:
440, 466, 494, 523, 554, 587,
622, 659, 698, 740, 784, 831,
880, 932, 988, 1047, 1109, 1175,
1245, 1319, 1397, 1480, 1568, 1661, 1760, . . .
where no matter where you start, if you count 4 members later you get a value approx 5/4 the original one, 5 members later gives 4/3, 7 members later gives 3/2, and 12 members later gives 2/1
Puzzle #3 solution:
This is the precise relationship of frequencies of the musical notes in western music's scale system. Any 12 steps higher is an octave, where the frequency is doubled (although they call my steps, half-steps).
Due to the way sound waves constructively interfere, when the frequency of two waves have ratio in small integers (5:4, 3:2 4:3, 2:1 etc), they reinforce in a beat pattern that we can hear, giving us the basis for most chords. . . what they call consonance. It sounds good, like the notes go together.
But if you tuned your instrument so that you had these perfect ratios, it would be difficult to play in another key. However if you tuned them to the mathematical relationship above, they are almost perfect, and the octaves are perfect, no matter what key you start on. This makes this scale extremely flexible.
The amazing thing is that the musicians who perfected this, knew nothing of the math, but could hear the relationships naturally.
btw, middle C on a piano is 440 Hz
The notes that are most consonant with this frequency correspond to E, F, G, and C an octave higher. These are important keys in some main chords in that key.
middle C = 440 Hz
E = 554 ~ 550 = 5/4 * 440
F = 587 ~ 4/3 * 440
G = 659 ~ 660 = 3/2 * 440
next C = 880 = exactly 2*440
This scale design is the best choice of having many steps within an octave to give a rich range of dissonant notes for creating melodies, while always having consonant notes nearby to form chords, and at the same time allowing a continuous ability to shift to a slightly higher or lower key.
I think that is unbelieveable.
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