Uranus takes 84.016846years to orbit around sun, neptune takes 164.79132years. We can scale each orbital frequency up many octaves by doubling again and again. In this brainwave binaural beats meditation piece, the frequency of uranus = 103.672Hz and neptune = 52.856Hz

The difference tone or conjunction frequency when the planets align corresponds to = 50.816Hz and it is calculated that 25 conjunctions fit into a complete cycle again since 52.856-50.816 = 2.04hz 

50.816/2.04 = 24.9098 aprox = 25

So when constucting the sine waves for uranus, we used an amplitude variation = sine squared in left channel and cosine squared in right at frequency = 2.04hz.

For neptune, we did the same for uranus but inverted the stereo channel oscillation.

Since there are two peaks in a period of sin squared or cosine squared for the stereo oscillation, corresponding to 25 conjunctions in a great cycle of the uranus neptune relationship, the real frequency percieved = double = 4.08hz

Theta Brain Waves (4 to 8 cycles per second) Deep meditation. Deep inward thought. This is associated with life-like imagination. High state of mental concentration. A magical mind. Internal pictures / visualisation. Intuition, inner guidance. Access to unconscious material. Dreaming.

This frequency percieved of 4.08hz is possiblly also present with the 2.04hz percieved in a different way

Delta Brain Waves (0.5 to 4 cycles per second) Deep dreamless sleep. Deep relaxation. State of oneness, whole body feeling. Pure being & will.

Binaural Beats Planetary uranus neptune

Here is a simple piece for choir based on this balanced string principle with the connected trinity of intervals played together so if a string of length = 1 is divided into a string length = 1/r1 & 1/r2 then 1/r1 + 1/r2 = 1 so r1 = r2/(r2-1) &r2 = r1/(r1-1)
The trinity of frequencies, fundamental = 1 & r1 & r2 are sounded together in this piece in order, from one end of a hypothetical string to the other, then repeated. Kept it simple as a demonstration.
These abstract intervals seem harmonious and balanced in a way 
Heres the scale i used double checking i hadn't made an error, so very sure this is what it sounds like
Letting sum1 =(r1+1)
Sum2 = (r2+2)
diff1 = (r1-1)
diff2 = (r2-1)

r1 = 8/7 = 231.174c  r2 = 3600c


When diff2/sum1 = 3
r1 = 2/sqroot(3) = 249.022c r2= 3479.962c
 
r1 = 7/6= 266.8708 r2=3368.826

r1 = 6/5 = 315.641 r2= 3101.955

When 3*sum1 = sum2
r1 = (1+sqroot(7))/3 = 337.504cents r2= 2996.598

When diff2/sum1 = 2
r1 = sqroot(3/2) = 350.977cents r2=2935.345

r1 = 5/4 = 386.3137 r2=2786.314

When 3*(r1) = 1/diff1
r1 = (1+sqrt(7/3))/2 = 405.271cents r2=2712.496

When 2*diff1 = (r1)/sum1
r1 = (1+sqroot(17)/4 = 428.422cents r2=2627.43


r1 = 4/3 = 498.048 r2=2400

When 2*(r1) = 1/(diff1)
r1 = (1+sqrt(3))/2= 539.981cents r2=2279.962 yes

When (1/diff1) - (r1)/sum1 = 2
the same as saying r2/sum1 = 3/2
r1 = (1+sqroot(10))/3 = 566.893cents r2=2208.502

When 1/sum1 = diff1
r1 = sqroot(2) = 600.000cents r2= 2125.864 yes

When 3*r1 = sum2
r1 = (5+sqroot(13))/6 = 624.366cents r2= 2068.415


r1 = 3/2 = 701.955cents r2=1901.955


When r1 = 1/diff1 {Golden ratio phi}
r1 = (1+sqroot(5))/2 = 833.090cents r2= 1666.180


r1 = 5/3 = 884.3586 r2=1586.314


When diff2 / diff1 = 2
r1 = 1+1/sqroot(2) = 925.864cents r2= 1525.864


When sum1 = 2*diff2
r1 = sqroot(3) = 950.978cents r2= 1490.959


When diff1/diff2 = 2/sum2

r1 = (3+sqr(17))/4 = 999.008cents r2= 1427.430

When 3*diff1 = 2*diff2
r1 = 1+sqroot(2/3) = 1033.390cents r2=1384.367

When 3*diff2 = 4*diff1
r1 = 1+sqroot(3)/2 = 1079.962cents r2=1328.984

Trinity Abstract (balanced divisions )

I uploaded a wrong version After listening a few times parts beating oscillation rhythm doesn't seem to align with the percussion , I can't see where the mistake , I checked the data so It must be out of phase or something in parts. But then i found out there is an option in U-He Zebra VST to turn Voice drift off, .This is what was causing the error ! So now it works.

A piece based around the concept of having beating oscillation constant and flattened & sharpened (4/3) , (3/2) & (2/1) in that order 
The displacement ratio between the notes in the mistuned interval with the LFO beat frequency starts with 1/4 then 1/6, 1/8,1/12,1/16,1/24,1/32, 1/48

It is known that if interval I=(b/a)+ζ where a & b are integers & ζ is a small change from just, then:

Secondary order beat frequency = a*ζ 
If we want to have a ratio such that 
R/(a*ζ) = (b/a)+ζ and R is reciprocal ratio such as eg, R = 1/4 or R = 1/6 or R = 1/8 etc

then a*ζ^2 + b*ζ - R = 0 
So solving quadratic equation gives: 
ζ+ = (+-sqroot[(b^2)+(4*a*R)] -b)/(2*a)

but for interval I- = (b/a)-ζ- the form should be: 
ζ- = [b+-sqroot((b^2)-(4aR))]/2a 

The beating oscillation is chosen in this piece is related to the Scumann resonance of Saturn. If one imagines a standing wave of light around the equitorial circumerence c=f*2*pi*R 
So f = c/(2*pi*R)
f = 0.79169Hz
Saturn has a perculiar hexagonal structure around the poles, this could be due to the 1/3 & 1/6 resonance of 2 of it's inner moons.
So for this piece 6*0.79169 = 4.750Hz was chosen as a constant beat frequency.

 the tempo of the subtle drumbeat is aligned too at 60seconds*4.75/octave = 142.50Bpm

I'm Still learning the theory though. 
Thanks for listening.

Vibrational Relativity (Saturn6)

Here is a demo of a sequence of these symmetric triads about (3/2)
in the order they are played and later some sequences start backwards for the up and down progression.
The drone note is played with each pair of notes
Inital three pairs are
....8/7
....13/7
....9/7
....12/7
....10/7
....11/7
then    
 
  1:          9/8             203.910  major whole tone
  2:         15/8            1088.269  classic major seventh
  3:          5/4             386.314  major third
  4:          7/4             968.826  harmonic seventh
  5:         11/8             551.318  undecimal semi-augmented fourth
  6:         13/8             840.528  tridecimal neutral sixth
  7:         10/9             182.404  minor whole tone
  8:         17/9            1101.045  septendecimal major seventh
  9:         11/9             347.408  undecimal neutral third
 10:         16/9             996.090  Pythagorean minor seventh
 11:          4/3             498.045  perfect fourth
 12:          5/3             884.359  major sixth, BP sixth
 13:         13/9             636.618  tridecimal diminished fifth
 14:         14/9             764.916  septimal minor sixth
 15:         11/10            165.004  4/5-tone, Ptolemy's second
 16:         19/10           1111.199  undevicesimal major seventh
 17:          6/5             315.641  minor third
 18:          9/5            1017.596  just minor seventh, BP seventh
 19:         13/10            454.214  tridecimal semi-diminished fourth
 20:         17/10            918.642  septendecimal diminished seventh
 21:          7/5             582.512  septimal or Huygens' tritone, BP fourth
 22:          8/5             813.686  minor sixth
 23:         12/11            150.637  3/4-tone, undecimal neutral second
 24:         21/11           1119.463  undecimal major seventh
 25:         13/11            289.210  tridecimal minor third
 26:         20/11           1034.996  large minor seventh
 27:         14/11            417.508  undecimal diminished fourth or major third
 28:         19/11            946.195
 29:         15/11            536.951  undecimal augmented fourth
 30:         18/11            852.592  undecimal neutral sixth
 31:         16/11            648.682  undecimal semi-diminished fifth
 32:         17/11            753.637
 33:         13/12            138.573  tridecimal 2/3-tone
 34:         23/12           1126.319  vicesimotertial major seventh
 35:          7/6             266.871  septimal minor third
 36:         11/6            1049.363  21/4-tone, undecimal neutral seventh
 37:          5/4             386.314  major third
 38:          7/4             968.826  harmonic seventh
 39:          4/3             498.045  perfect fourth
 40:          5/3             884.359  major sixth, BP sixth
 41:         17/12            603.000  2nd septendecimal tritone
 42:         19/12            795.558  undevicesimal minor sixth
 43:         14/13            128.298  2/3-tone
 44:         25/13           1132.100
 45:         15/13            247.741  tridecimal 5/4-tone
 46:         24/13           1061.427  tridecimal neutral seventh
 47:         16/13            359.472  tridecimal neutral third
 48:         23/13            987.747
 49:         17/13            464.428
 50:         22/13            910.790  tridecimal major sixth
 51:         18/13            563.382  tridecimal augmented fourth
 52:         21/13            830.253
 53:         19/13            656.985
 54:         20/13            745.786  tridecimal semi-augmented fifth
 55:         15/14            119.443  major diatonic semitone
 56:         27/14           1137.039  septimal major seventh
 57:          8/7             231.174  septimal whole tone
 58:         13/7            1071.702  16/3-tone
 59:         17/14            336.130  supraminor third
 60:         25/14           1003.802  middle minor seventh
 61:          9/7             435.084  septimal major third, BP third
 62:         12/7             933.129  septimal major sixth
 63:         19/14            528.687
 64:         23/14            859.448
 65:         10/7             617.488  Euler's tritone
 66:         11/7             782.492  undecimal augmented fifth
 67:          2/1            1200.000  octave

Demo Of Symmetric Triads About (3/2)

Using fibonacci 13 Limit intervals, 2,3,5,8,13 although 16/15 is used too.

Quiet Symphony 53ET: Part1- Shelter

Jokim Bang Larsen wished to write a piece making beating frequencies have a tempo of 1cps, 2cps, 3cps so here is the demonstration i have done. All mistuned fourths from 1, (6/5) & (5/4) to make beat rates quavers or triplets in a (1/96) ratio from first lower note in piece

Beats Synchronicity (Ratio 1/96)

in 53 equal temperament this is Part 2 - Fire and Warmth
Had the vision in mind of expressing trying to start a fire by friction of wood and then when the fire starts the music is tried to be expressive of the fire.
Wrote down sketches of possible chords all stacked (5/4) (6/5) (3/2) and there octave inversions in a variety of combinations.
However there is a 13/8 at the End to make a fibonacci chord 1,2,3,5,8,13

Quiet Symphony 53ET- Part2 - Fire And Warmth

This is a demonstration of The beats or psychoacoustical myriad of frequencies that crop up when The orbital frequency  of Venus 224.7 days and Earth 365.256 days (doubled 32 times for Earth = 136.097Hz & Venus = 221.23 Hz ) is fed into an expression function for angular frequency.
ψ(ω) = sin(2*pi*ψ(y))
= sin(2*pi*μ *(cos(2*pi*f*t)+(1/x)*cos(2*pi*f*x*t)-(1/x)*sin(2*pi*f*t*(x-1))))
μ  is a constant to be varied but was made = 2 for this soundscape
and a slow doubled tanh plateau filter was used to rise through the frequencies. It begins with the difference frequency which you can hear about 0.625527 then 1 for earth then 1.625527 for venus.
The low frequency wobbling corresponds to the period between venus transits every 235 years.


Excerpt from:http://www.nanovibrationalrelations.com/theory-of-planetary-cycles--sound.html
"What is psychoacoustically perceived could be velocity which is analogous to angular frequency, Energy or something else.
Imagine a pair of orbiting planets exerting a sinusoidal pressure on the Galactic plane field
In dealing with velocity which has reciprocal of time units matching with angular frequency, ω,
 We can construct a displacement wave function    ψ(y)
and from ω = v/r  velocity over radius, velocity is the differential of displacement with time
Letting r = 1 for convenience.

​For when frequency is changing and not constant, a displacement expression

y = sin(2*pi*∫ f.dt)       ∫f.dt is the integral of frequency with respect to time.

So the Angular frequency expression function is: ψ(ω) = sin(2*pi*∫ω.dt)

ψ(ω) = sin(2*pi*ψ(y))

It's just a case of finding the possible displacement wave function.
In the meantime we can theorize from our planetary model various types of displacement wave function
formats that yields interesting results.

ψ(y) = e^(i*2*pi*f*t)+(1/k)*e^(i*2*pi*f*k*t)

e is the exponential constant and i is the imaginary unit, the square root of negative 1
e^(i*2*pi*f*t) = cos(2*pi*f*t)+ i*sin(2*pi*f*t)

angular frequency expression function:
ψ(ω) = sin(2*pi*ψ(y)) = sin(2*pi*μ *( cos(2*pi*f*t)+(1/k)*cos(2*pi*f*k*t) )  )
μ  is a constant to be varied.

This is frequency modulation but under Fast fourier transform frequency analysis for μ =0.2
, a plethora of constant psychoacoustic frequencies exist. But for this angular frequency expression
function, the Secondary order beat frequencies are missing when (b-a) is an even number from predicted
formula from before:
If we call an Interval, k = b/a + ζ
secondary order frequency,      Fβ  = fo. (a.k - b)    if  k > b/a
If we call an Interval, k = b/a - ζ
secondary order frequency,      Fβ  = fo. (b - a.k)    if  k < b/a
An alternative displacement wave function could be including the conjunction frequency when two planets
align which is the difference frequency.

ψ(y) = e^(i*2*pi*f*t)+(1/k)*e^(i*2*pi*f*k*t) + (1/k)*i*e^(i*2*pi*f*(k-1)*t)

The orbital conjunction term (1/k)*i*e^(i*2*pi*f*(k-1)*t) has i as a factor since the waveform result is more in
normal symmetry later. And also has (1/k) as a factor for assuming the difference
between radius = 1 for e^(i*2*pi*f*t) term and radius = (1/k) for e^(i*2*pi*f*k*t) term
in circular orbits yields the total distance separating the orbiting particles =1 - (1 - 1/k) = 1/k

angular frequency expression function:
ψ(ω) = sin(2*pi*ψ(y))
= sin(2*pi*μ *(cos(2*pi*f*t)+(1/x)*cos(2*pi*f*x*t)-(1/x)*sin(2*pi*f*t*(x-1))))
μ  is a constant to be varied.

In this case under frequency analysis of this alternative angular frequency expression function,
the Secondary order beat frequencies are all visible but with some doubling and tripling anomalies for high value of  μ
 μ = 0.3 the results look good."

Earth And Venus (beats Wave Function)

An expression function (one of many ) that amplifies intangible beat frequencies to become tangible and detected by Frequency analysis FFT is y = sin(2*pi*0.3*(cos(2*pi*f*t)+(1/x)*cos(2*pi*f*x*t)-(1/x)*sin(2*pi*f*(x-1)*t)))
the 0.3 can be altered or exchanged for a sinusoidal form buzz modulation of A*(1-cos(2*pi*t/Tw)) where Tw is the time period of the  buzz modulation and A is a parameter to be altered sounds ok < 0.25

31et rising chromatic ( Enhanced Beats Expression Demo)

18.752754 ET found by the Mean summed method of very near perfect harmony point of Major 3rd and harmony point of minor 3rd
Wendy Carlos Tunings, there are alot of them out there it seems.
Source:- Lumma.org leading to further info online.

Merci..

Short Improvisation Of 18.752754ET (Classical Guitar)

A bit of a improvization sketch in 20.50488 ET, a Wendy Carlos based tuning reformed in the 60's or 70's yet we know of course these tunings were played in Ancient Civilizations.
The tuning is designed for perfect 5/3's, 8/5's and 3/2's all splendid fibonacci number ratios.

Moon After Tomorrow - ( Sketch In 20.50488 ET Classical Guitar)

The Balmer lines for Hydrogen comprise the scale in this piece.
of the order , frequency = (1/n^2 - 1/m^2)
where n & m are integers such that n=2, m >=3

Incuded are a few key JI intervals that differ from the Balmer lowest balmer line  by a 4/5, a 9/10, a 6/5. a 27/25

Hydrogen Sun

On first introduction to this tuning, an  exploration of cycles of pythagorian (3/2)'s & (4/3)'s  is evident here.

Beginning With Threes (29Et)

19 ET electric guitar on acoustic setting.
This is is a short improvization of a sequence of chords written down in tab form that seemed to sound fitting.
The phase of the moon is on the same day time of the year every 19 years as used in some Calendars around the world. I thought the instrument felt oriental while playing it too , hence the title of the piece.

Moon Of The Orient (19ET electric guitar)

At 13.62474 Equal divisions per octave the interval 7/4 is perfect and is in between   an accurate (3/2) and (5/3) on either side.
This piece is a mix of improvization and melodic harmony motifs for a classical guitar and strings using Omnisphere VST.
The piece used primarily (5/3) & (3/2) stacks with a tritone and also on occasion use of a beating octave.

NanoHarmony

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