for the curious - planetary orbits and musical intervals
The Golden Section arises as the ratio between the side and the diagonal of a regular pentagon. For a pentagon size of 1, the size of the diagonal is ~1.618. The Golden Section naturally forms a self-similar geometric series (each stage forms a Golden Section ratio with the preceding one. The classical well-tempered system is based on the Golden section. The relationship of the frequencies of a minor triad exhibits this principle: in reverse order, octave to 5th is 1, 5th to flat 3rd is 1/φ, flat 3rd to root is 1/φ^2. The Lydian sequence of root, 3rd, sharp 4th, and 5th forms the same series, except this time going up.
The isoperimetric principle simply states that the circle, as a geometric shape, encloses the greatest possible area compared to any other shape with the same perimeter. Thus, circular action is the most efficient form of action. Using synthetic geometry (folding the circle onto itself) we produce a straight line; folding it again vertically produces a point. Thus, all geometric shapes that are taken as axiomatic in Eucledian geometry can actually be derived and proven – and generated with nothing but the circle as a starting point.
Taking the synthetic geometry bit further we generate three-dimensional shapes – by rotation (normally called an integral). Further constructions generate regular polygons for three sides, constituting the faces of the five regular solids. The polygons demonstrate the self-dividing principle of circular action. First fold forms the octave (divide into 2), and the ratios of intervals proceed by this division: 1:2, 2:3, 3:4, 4:5 – respectively octave, fifth, fourth, and major third.
The isoperimetric principle simply states that the circle, as a geometric shape, encloses the greatest possible area compared to any other shape with the same perimeter. Thus, circular action is the most efficient form of action. Using synthetic geometry (folding the circle onto itself) we produce a straight line; folding it again vertically produces a point. Thus, all geometric shapes that are taken as axiomatic in Eucledian geometry can actually be derived and proven – and generated with nothing but the circle as a starting point.
Taking the synthetic geometry bit further we generate three-dimensional shapes – by rotation (normally called an integral). Further constructions generate regular polygons for three sides, constituting the faces of the five regular solids. The polygons demonstrate the self-dividing principle of circular action. First fold forms the octave (divide into 2), and the ratios of intervals proceed by this division: 1:2, 2:3, 3:4, 4:5 – respectively octave, fifth, fourth, and major third.
Comments
Mercury: 384’/164’ = 12:5 = minor 10th (octave + min3rd)
Venus: 97’37”/94’50” = 25:24 = E to Eb (in a well-tempered system, the ratios change)
Earth: 61’18”/57’03” = 16:15 = B to C
Mars: 38’01”/26’14” = 3:2 = fifth
Ceres: 15’06”/11’00” = 1:0.7111 = diminished 5th (using Kepler’s numbers, not modern data)
Jupiter: 5’30”/4’30” = 6:5 = minor 3rd
Saturn: 2’15”/1’46” = 5:4 = major 3rd
Uranus: 46”/39” = 6:5 again
Neptune: 22”/21” = 25:24 again
Pluto: 24”/8.7” = octave + dim5th
These values hold true at well-tempered tuning of middle C=256Hz, not the usual A=440Hz. The period of one cycle of C=256 can be constructed by using the period of Earth’s rotation: division by 24 yields one hour, division by 60 will yield a minute, etc. Dividing a second into 256 yields that interval. So the Earth rotates at a G twenty-four octaves below the middle C. Between the harmonic mean of an octave from middle C (256 to 512) the arithmetic mean yields the G, the geometric mean yields the F# (first register shift for the soprano voice) and harmonic mean yields the F natural. Respectively, these represent the semi-major axis, semi-minor axis, and the semi-latus rectum of the rotational action (self-similar spiral) of the orbit projected onto a surface. The two tetrachords in the C major scale are divided by the F# tone.
From the table above, we may say that the asteroid belt is the equivalent of the register shift for the solar system. Below it, planets are small and have few moons. Outer planets are large, gaseous by contrast, and have many moons and even rings.
This principle is easy to verify. Begin at the outer layer of the sun, and construct a logarithmic spiral making exactly one rotation in passing from that layer to the orbit of the innermost planet, Mercury. Continue the spiral exactly one full cycle passing from Mercury to the region defined by the overlapping orbits of Neptune and Pluto; the geometric mean is exactly at the outer boundary of the asteroid belt. Now, if we compare the planetary spiral with simple spiral action of the tuning system, and let the interval from Mercury to the overlap represent a C-C octave, the planetary orbits will correspond exactly in angular displacement to the principal steps of the scale, with the asteroid belt in the angular position corresponding to the interval between F and F#.
Mitogenic radiation @ wavelength 200nm = F (341) + 42 octaves
DNA @ wavelength 265nm = C (256) + 42 octaves
Vision @ lower bound 360nm = G (188) + 42 octaves
Guess what happens here? Register shift of the F-F# interval falls exactly between ultraviolet and visible light! What a surprise, the space/time continuum does curve! If you look at it from the other end, same thing happens between visible and infrared - again between F and F#.
Go outside and see the curvature of the girls. 8)
40\" Double D on a 24\" waist = 1.6
That's golden !!!
Poor Pluto has been demoted so time to retune. http://www.flightnest.com/2006/08/24/pluto-no-longer-a-planet/
There is Biblical support for singing creation. See Job 38:7 and the Allstar Rock Band in Luke 19:40 :shock:
Thus the interval order matters, a perfect fifth, for instance (C,G), ordered <C,G>, 2:3, gives an \"effect of indicated continuation\", while <G,C>, 3:2, gives an \"effect of finality.\"
This helps me understand now exactly how to \"look into the future.\" !!! :shock:
I will need to flesh this out further...but this all makes more sense now.
....OH...and I will need a few things...so I'm shopping on eBay. Here are some things I bought. on eBay
:shock: :shock: :shock:
(\"OK Terriffic!\")
Nurse. It's time for the medication.
Hey, you guys asked for it! Some of you did, anyway... :roll:
And I appreciate it.
I find it quite interesting.
I wish I had this level of theory training, but I never was \"pushed\" musically by my parents, who were unwilling to spend anymore than they had to for lessons
and am trying to analyse that depth of information. Thanks for all of that.
Initially I feel that there are so many notes on the scale, all just 6% apart.
This means that almost anything we measure can be considered to be very close to one of those \"markings on the scale\" which are just 6% apart. Your maximum error can only be 3% which is not too bad for most.
For example, If is check the height of my three dogs. I find their ratios and round it up to the nearest 6%. and I find that my Dalmatian is an octave higher than my Spitz.
And my Dobermann \"Pluto\" is an inverted Octave + Dim5th from my Dalmatian. But thats because he is only 3 months old. when he grows up, he will be an inverted Octave + third.
Also, I understand that planetary speeds and distances are changing all the while. So while music remains music forever, the planetary ratios you mentioned could be only for this century ?
I love science.
After extensive research I have found a vid of a young Frank Zappa and Steve Allen, demonstrating the \"Golden Section\" on a concerto for 2 bicycles...
God Bless!!
Partch
A little Alien to western ears, but strangely familiar with the 17 tone scale playing rag time in places...
Would I retune my guitars to get this sound???
Not in a million years..
God Bless!!
Partch