20 February 2019; Feast of Sts Francisco Marto & Jacinta Marto; 25 February 2019, Feast of Sts Tharasius, Walburga, Ethelbert

Book 1 Music - Fundamental Transcendental Frequencies [in Hertz i.e. cycles per second].

**The first article in 2-3 part series. The first part is Transcendental Music, the second part is Angelic Music (for the lack of a better word).**

Transcendental Function described in Book 1-b “Transcendental Constants - Introduction” [ link: Book 1-b "Transcendental Constants - Introduction ] produces music scale similar to modern 88-key or 108-key piano, with only small differences.

It is a major proof that Transcendental Constants rule Music, and other sectors of science, such as Quantum and Cosmos. I would not be surprised if Transcendental Constants can be applied to the DNA structure (this will become clearer at the end of this article).

First, a bit of math is necessary to calculate all the fundamental frequencies. The formula is similar to **the official formula for modern piano:**

The frequency of the nth piano key equals to:

f (n)= {2^{(n-49)/12}} * 440 [Hz] [ Eqn. 1 ]

where n is the key number, 440 Hz is the 49th key A4.

You can read more about it in this Wikipedia article [ link: Piano Key Frequencies ].

Transcendental Music Formula is a little bit more precise. Here it is.

First of all, one has to notice that a Transcendental Function with integer values (i.e. values of function @...-3,-2,-1,0-,1,2,3,... give of course Transcendental Constants).

You can see the graph of the Transcendental Function at the link at the beginning of this article, or you can get the values of Transcendental Function (at Integers) going up from pi [ link to the article: Book 4b - Updated Table of Transcendental Constants going Up ] or going down from pi [ link to the article: Book 4a - Table of Transcendental Constants going Down ]. As you can see there is are two sequences of numbers: given constant and its inverse.

The second step is to notice that these values of Transcendental Constants may form a music scale.

How to calculate fundamental frequencies:

We start with some fundamental frequency e.g. A_{4}= 44.000Hz, so we can use the official formula (Eqn. 1) which uses as a point of reference 49th key i.e. A_{4}= 440.000 Hz. (Starting in the middle of the scale won’t show some possible DNA relation - I will write about it in the separate article).

Now we have to create a simple proportion showing the relation between the Constant and associated with it frequency. I have chosen as a point of reference Transcendental Constant C, and I decided to use reciprocals, but it doesn’t really matter since the same results are obtained. Only the direction will be opposite.

**Here is the proportion for the fundamental frequencies going up from 440.000 Hz = A _{4}:**

(inverse of C_{0}= 1.0131955032) / (f_{0}=440.000Hz) = (inverse of C_{-2} = 1.35333101602) / f_{-2} [ Eqn. 2 ]

and the value of frequency f_{-2} at Inverse of the Constant i.e. C_{-2} equals:

f_{-2} = 587.710511 Hz and piano key frequency gives D_{5} = 587.3295 Hz with difference d = 0.38 Hz and relative error (or rather difference since there is no error) is equal to drel = 0.00065

Next frequency is at the inverse of the Constant C_{-4}( first part of the proportion is the same as in Eqn. 2):

(inverse of C_{0}= 1.0131955032) / (f_{0}=440.000Hz) = (inverse of C_{-4}= 1.807651961634) / f_{-4}

from this the frequency f_{-4} = 785.008284 Hz, and piano key frequency is G_{5}= 783.9909 Hz; with difference d = 1.02 Hz, and drel (relative difference) = 0.0013

Going down with even frequencies gives in a similar way following values:

f_{-6} = 1048.54004 Hz and piano gives C_{6} = 1046.502 Hz, with difference d = 2.04 Hz, and relative difference drel = 0.002

f_{-8} = 1400.54092 Hz, and piano gives F_{6} = 1396.913 Hz, with difference d = 3.63 Hz and relative difference drel = 0.0026

f_{-10} = 1870.71050 Hz, and piano gives A_{#6} = 1864.655 Hz, with difference d = 6.06 Hz, and relative difference drel = 0.0033

f_{-12} = 2498.71869 Hz, and piano gives D_{#7} = 2489.016 Hz, with difference d = 9.7 Hz and relative difference drel = 0.0039

f_{-14} = 3337.55282 Hz, and piano gives G_{#7} = 3322.438 Hz, with difference d = 15.11 Hz and relative difference drel = 0.0045

f_{-16} = 4457.98835 Hz, and piano gives C_{#8} = 4434.922 Hz, with difference d = 23.1 Hz and relative difference drel = 0.0052

f_{-18} = 5954.56048 Hz, and piano gives F_{#8} = 5919.911 Hz, with difference d = 34.7 Hz, and relative difference = 0.0059

f_{-20} = 7953.54042 Hz, and piano gives = B_{8} = 7902.133 Hz, with absolute difference d = 51.4 Hz, and relative difference drel = 0.0065

That’s end of scale for piano.

**Now, start from A _{4} = 440.000 Hz and this time going down the scale. You will notice that the absolute difference d < 0.5 Hz and relative difference is as before drel _{max} < 0.007**

First relation:

(inverse of C_{0}= 1.0131955032) / (f_{0}=440.000Hz) = (inverse of C_{2} = 0.758546959764) / f_{2} [ Eqn. 3 ]

The above proportion gives the value of frequency f_{2} at the inverse of the Constant C_{2}.

So, the frequency f_{2} = 329.41388 Hz, and piano gives E_{4} = 329.6276 Hz, with the difference d = -0.214 Hz, and the relative difference drel = -0.00065

f_{4} = 246.621601 Hz, and piano gives B_{3} = 246.9417 Hz, with difference d = -0.32 Hz, and relative difference drel = -0.0013

f_{6} = 184.637679 Hz, and piano gives F_{#3} = 184.9972 Hz, with absolute difference d = -0.36 Hz, and relative difference drel = -0.0019

f_{8} = 138.232305 Hz, and piano gives C_{#3} = 138.5913 Hz, with difference d = -0.36 Hz, and relative difference drel = -0.0026

f_{10} = 103.4900091 Hz, and piano gives G_{#2} = 103.8262 Hz, with absolute difference d = -0.34 Hz, and relative difference drel = -0.0032

f_{12} = 77.4797101 Hz, and piano gives D_{#2} = 77.78175 Hz, with difference d = -0.30 Hz, and relative difference drel = -0.0039

f_{14} = 58.0065726 Hz, and piano gives A_{#1} = 58.27047 Hz, with absolute difference d = -0.26 Hz, and relative difference drel = -0.0045

f_{16} = 43.4276595 Hz, and piano gives F_{1} = 43.65353 Hz, with difference d = -0.23 Hz, and relative difference drel = -0.0052

f_{18} = 32.512895 Hz, and the piano gives C_{1} = 32.70320 Hz, with absolute difference d = -0.19 Hz, and the relative difference drel = -0.0058

f_{20} = 24.3413612 Hz, and the piano gives G_{0} = 24.49971 Hz, with absolute difference d = -0.16 Hz, and relative difference drel = -0.0065

f_{22} = 18.223596 Hz, and the piano gives D_{0} = 18.35405 Hz, with absolute difference d = -0.13 Hz, and the relative difference drel = -0.0071

**From piano key C _{6} = 1046.502 Hz down to D_{0} = 18.35405 Hz the absolute difference in Hz is less than 2 Hz, which means it cannot be heard.**

**As a percentage, the relative difference is less than 0.007 i.e. 0.7%, very small indeed.**

As you could notice, the notes jump every 5th key:

D_{0} to G_{0}jumps 5 keys,

G_{0} to C_{1} jumps 5 keys,

C_{1}to F_{1} jumps 5 keys, etc.

Deriving the general formula for music scale from Transcendental Constants:

The formula for piano is:

f (n)= {2^{(n-49)/12}} * 440 [Hz] [ Eqn. 1 ]

We keep the keys (n) the same way as on the piano, so the only different part will base 2 of power.

We have exact frequency, key number so the value of the base can be easily calculated as follows:

Let us choose -6th key D_{0}

( D_{0} = 18.223596 Hz ) = {X^{(-6-49)/12}} * 440.000 Hz

Dividing LHS by ( A_{4} = 440.000 Hz ) we get::

X^{(-6-49)/12}= 18.223596 Hz / 440.000 Hz

Taking logarithm of both sides gives:

LN X = 0.694703452

X = EXP^{0.694703452}= 2.00311496627229543134954...

This is our base instead of base = 2 as it is in original piano frequencies formula.

**Transcendental Constants Music Formula:**

F(n)= {(2.00311496627229543134954...)^{(n-49)/12}} * 440 [Hz] [ Eqn. 4 ]

The above formula can be used to get any note.

The ratio between the five keys is:

e.g. f_{16} / f_{11} = 1.33570570705474757456485... = approx. 4/3

Now, the question is: “Which scale is better i.e. more real?”

**I believe that the Transcendental Music Scale is much better, this is the language of God** if you believe, or if you don’t this will be the language of the Universe. There is no doubt in my mind about it.

**Second thing is that the point of reference A _{4}= 440.000Hz should be changed a little bit**. That would eliminate high difference at the high end of the scale.

This leads to the conclusion that the real musical scale (Transcendental) is slightly different and of course much better. Ours is as usual only the approximation.

The next article will be about internal mechanics of the Transcendental Constant Music Scale calculation, showing possible DNA build links.

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