- #91
Raphie
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And, Jeremy, Marin Mersenne, for whom Mersenne Primes are named, in relation to the pendulum...
via Wikipedia
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http://en.wikipedia.org/wiki/Marin_Mersenne
Mersenne's description in the 1636 Harmonie universelle of the first absolute determination of the frequency of an audible tone (at 84 Hz) implies that he had already demonstrated that the absolute-frequency ratio of two vibrating strings, radiating a musical tone and its octave, is 1 : 2. The perceived harmony (consonance) of two such notes would be explained if the ratio of the air oscillation frequencies is also 1 : 2, which in turn is consistent with the source-air-motion-frequency-equivalence hypothesis.
He also performed extensive experiments to determine the acceleration of falling objects by comparing them with the swing of pendulums, reported in his Cogitata Physico-Mathematica in 1644. He was the first to measure the length of the seconds pendulum, that is a pendulum whose swing takes one second, and the first to observe that a pendulum's swings are not isochronous as Galileo thought, but that large swings take longer than small swings.[2]
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To set a pendulum half period equal to exactly 1 second, then L/g = 1/pi^2.
To set a pendulum full period to exactly 1 second, then L/g = 1/4pi^2
pi^2 --> Idealized Acceleration due to Gravity * s^2/m (Denote g')
4pi^2 --> GM * s^2/m^3
...for GM the Gravitational Parameter and 4pi^2/GM the unit multiplier of Kepler's 3rd Law.
And bear in mind the following:
In an idealized mathematical environment, there is no mass, and, therefore, no friction to slow things down, although partial sums of infinite series do, in fact, often seem to simulate such.
84 Hz, by the way, maps to 1 Hz per note of the Circle of Fifths, which has 84 notes and 7 octaves. 84 is a Tetrahedral number, and at least up to 1.41*10^1504 (computer checked by CRGreathouse), the last one such that 1.5 times a Tetrahedral Number = 2^y - 2.
1.5 * 84 = 126 ==2^7 - 2 == Vertices of E_7 == totient(p_31)
31 (as is 7...) is a Doubly Mersenne Number (both a Mersenne Prime and Mersenne Prime Exponent) == Dimensions of E_8/8 == 496/16 for 496 the 3d Perfect Number.
Totient (496) = 240, the number of vertices of E_8, made famous by A. Garrett Lisi in his paper "An Exceptionally Simple Theory of Everything."
RELATED THREAD
A Tetrahedral Counterpart to Ramanujan-Nagell Triangular Numbers?
https://www.physicsforums.com/showthread.php?t=443958
A "fun" little related equivalency that may have nothing to do with physics, but at least a thing or two to do with building models:
299800649
= (G_(totient(56) + totient(70)) - (56 + 70))/10^2
= (G_48 - 126)/10^2
= (G_24-0*L_24+0 - 128)/10^2
= (G_24-6*L_24+6 + 128)/10^2
= 299792458 + 8191; [8191 = 2^13 - 1 is a Mersenne Prime = sqrt ((8190^2 + 2*8190) + 1) = sqrt (67092480 + 1)]
for G_n the Golden Scale
for L_n the Lucas Series
48 is the number of roots of F_4, while 56 & 70 are the (partitioned) roots of E_7. 24, of course, is the number of roots of D_4 [and p^2 - 1 == 0(mod 24) for all p>3] and 6 the number of roots of A_2.
- RF
via Wikipedia
-------------------------------------------------------------
http://en.wikipedia.org/wiki/Marin_Mersenne
Mersenne's description in the 1636 Harmonie universelle of the first absolute determination of the frequency of an audible tone (at 84 Hz) implies that he had already demonstrated that the absolute-frequency ratio of two vibrating strings, radiating a musical tone and its octave, is 1 : 2. The perceived harmony (consonance) of two such notes would be explained if the ratio of the air oscillation frequencies is also 1 : 2, which in turn is consistent with the source-air-motion-frequency-equivalence hypothesis.
He also performed extensive experiments to determine the acceleration of falling objects by comparing them with the swing of pendulums, reported in his Cogitata Physico-Mathematica in 1644. He was the first to measure the length of the seconds pendulum, that is a pendulum whose swing takes one second, and the first to observe that a pendulum's swings are not isochronous as Galileo thought, but that large swings take longer than small swings.[2]
-------------------------------------------------------------
To set a pendulum half period equal to exactly 1 second, then L/g = 1/pi^2.
To set a pendulum full period to exactly 1 second, then L/g = 1/4pi^2
pi^2 --> Idealized Acceleration due to Gravity * s^2/m (Denote g')
4pi^2 --> GM * s^2/m^3
...for GM the Gravitational Parameter and 4pi^2/GM the unit multiplier of Kepler's 3rd Law.
And bear in mind the following:
In an idealized mathematical environment, there is no mass, and, therefore, no friction to slow things down, although partial sums of infinite series do, in fact, often seem to simulate such.
84 Hz, by the way, maps to 1 Hz per note of the Circle of Fifths, which has 84 notes and 7 octaves. 84 is a Tetrahedral number, and at least up to 1.41*10^1504 (computer checked by CRGreathouse), the last one such that 1.5 times a Tetrahedral Number = 2^y - 2.
1.5 * 84 = 126 ==2^7 - 2 == Vertices of E_7 == totient(p_31)
31 (as is 7...) is a Doubly Mersenne Number (both a Mersenne Prime and Mersenne Prime Exponent) == Dimensions of E_8/8 == 496/16 for 496 the 3d Perfect Number.
Totient (496) = 240, the number of vertices of E_8, made famous by A. Garrett Lisi in his paper "An Exceptionally Simple Theory of Everything."
RELATED THREAD
A Tetrahedral Counterpart to Ramanujan-Nagell Triangular Numbers?
https://www.physicsforums.com/showthread.php?t=443958
A "fun" little related equivalency that may have nothing to do with physics, but at least a thing or two to do with building models:
299800649
= (G_(totient(56) + totient(70)) - (56 + 70))/10^2
= (G_48 - 126)/10^2
= (G_24-0*L_24+0 - 128)/10^2
= (G_24-6*L_24+6 + 128)/10^2
= 299792458 + 8191; [8191 = 2^13 - 1 is a Mersenne Prime = sqrt ((8190^2 + 2*8190) + 1) = sqrt (67092480 + 1)]
for G_n the Golden Scale
for L_n the Lucas Series
48 is the number of roots of F_4, while 56 & 70 are the (partitioned) roots of E_7. 24, of course, is the number of roots of D_4 [and p^2 - 1 == 0(mod 24) for all p>3] and 6 the number of roots of A_2.
- RF
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