Prosodic Structure - Analog Signal Block Coding and Vowel's VK Attribute

 tinyimin Nanjing China ; yimintin@163.com

 

Do you like the music,and understand the relations between seven Nodes “CDEFGAB”. Ancient Greece Pythagoras said: Four degree five degree sounds are nature sounds under universe, therefore we study to this magical Prosodic Structure .

The core of the ProsodicStructure is the relation of "Four_way Ratio"(2/3,3/4,4/3,3/2).

[Main Text]:

1. The Prosodic Structure

The Prosodic Structure is a data relation structural, called " PS "，and it is a block coding way for Analog Signal too.

1.1 The Prosodic Ratio and Prosodic Network (See Fig.1)

In the positive real number domain, assume: X1 is a any point ,then ,always have four point X2, X3, X4, X5, that to X1 ,the Ratio of their numerical values is [2/3, 3/4, 4/3, 3/2], that called "Prosodic Ratio" or  "Four_way Ratio", these four points are called "Four-way point" of X1. Between them is “Prosodic Relations” or called “Prosodic Correlation”, it is two-way (See Fig.1). Similarly, these four points respectively have four points that with it are Prosodic Relations too. Similarly ....It is easy to prove that, by the “Prosodic Ratio”, these Prosodic Relations points are convergence at the same Network, that called "Prosodic Network” or "pW ". 

In the field of positive real numbers, there are many pW's co-exist logically, and they are all similar.

1.2 Prosodic Area

On the number axis, there are many points that have prosodic relationship between them, and they are formed a Prosodic Network (pW). That pW likes "fishing net" when folding, and after "spread out" ,its topological relation is like Fig.2 (local).  Fig.1 is a part of Fig.2.

[Description]: The short arc in the figure is expressing “Four_way Ratio” relationship (logical relationship).

 

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Assume: X1 is any point of the pW, Set X1 as the Base Frequency, call it "C1" (the same below), then, there is "Prosodic Area Coefficient Table" or called “pCT” that:

pCT= [1,k, k*k, 4/3, 3/2,3/2*k, 3/2*k*k] *2 ^ n   ( n is the integer; k=9/8;  MATLAB form; same below)

Assume again: Xi is any point of the pW too, and the ratio that is Xi to X1 is in this pCT table, this subset of Xi is called " Prosodic Area " or “pA” for short, or called "main pA" of X1 (C1). They are  represented by solid black dot.

For example: X2, X3, X4, X5 are "Four-way point" of X1, Their ratio that to X1 are in pCT, they're all black real points, they are  nodes of the main Prosodic Area of X1; But the "Four-way point" of X2, there are Only two pW points that on the right, their ratio to X1 is in the pCT, and is solid black dot in the main Prosodic Area; However,the ratio of the  other two pW points to X1 is not in the pCT, They are hollow point, and not in the main pA of X1.

According to the "Four-way Ratio", tracking point by point, it is not difficult to see: In the Fig.2  ①The Ratio that is black real point than X1, are in the pCT all. ② If the Ratio of the Xi and X1 is in this pCT, then, that Xi must be a black real point (local). This black real points set is the "main pA" of X1 (C1). In a pW, any dot can be as the base frequency C1, and have a pA accompanied in logically. the pA is subset of pW。

1.3 Prosodic Area and Music

At first, please refer to “Pythagorean tuning”.

In Fig.1 above , [X2,X3,X4,X5] = X1*[2/3,3/4,4/3,3/2]  (Four_way Ratio).

If take X1 as the Node C1 (musical "1 do"), according to the “Five_degree law” in Music:

The upper fourth(4 degree) of C1 is the notes F1 (musical "4 fa"),and F1=C1*4/3, it can be seen that the node X4 of  "Four_way Ratio" is the notes F1 of music;

The upper fifth(5 degree) of C1 is the notes G1 (musical "5 So"),and G1=C1*3/2, it can be seen that the node X5 of  "Four_way Ratio" is the notes G1 of music;

The lower fifth of C1 is the notes F0, F0=C1*2/3, it can be seen that the node X2 of  "Four_way Ratio" is the notes F0 of music;

The lower fourth of C1 is the notes G0, G0=C1*3/4, it can be seen that the node X3 of  "Four_way Ratio" is the notes G0 of music.

It can be seen that the essence of “Pythagorean tuning” is "Four_way Ratio".

The same can be deduced:

The lower fourth of G1 is the notes D1, D1=G1*3/4=C1*3/2*3/4 =C1*9/8;

The upper fifth of D1 is the notes A1, A1=D1*3/2 =C1*9/8*3/2 =C1*3/2*9/8;

The lower fourth of A1 is the notes E1, E1=A1*3/4 = C1*3/2*9/8*3/4 =C1*9/8*9/8;

The upper fifth of E1 is the notes B1, B1=E1*3/2   = C1*9/8*9/8*3/2; So:

 [C,D,E,F,G,A,B] / C = [1,k, k*k,4/3, 3/2,3/2*k, 3/2*k*k]   (k=9/8)

So, all seven notes are in the pCT. Seven notes of music are  seven  nodes of a same Prosodic Area. And the same is true of the other octave levels, they are one-to-one correspondences.

In a same pA, the nodes(Black real point) is feeled as Musical Notes (CDEFGAB) by the mankind. According to the beat, take these Nodes to compose a array to play that at their frequency, is the music that people like. It follows that: The music is a Auditory phenomena that is the Node of the "Prosodic Area (PA)".

 

1.4  In a pW, on both sides of the main pA are “Shadow Areas” .The Shadow Areas likes Shadow of the main pA that one by one. For example, the first of the left side is "Shadow Areas-1 " (in blue arc), the first of the right side is "Shadow Areas+1" (in green arc).... Any pA has all properties of main pA, they are similar.The Ratio of Between the adjacent pA is 3^7/2^11.

In Fig.2 there are two red arcs, which are boundary line that made by F or B. Between two boundary line is main pA of X1 (C1).

For example: the node F1, the two red relationship arcs to its right (3/4,3/2) are the left boundary of the main Prosodic Area, and the two blue relationship arcs to its left (2/3,4/3) are the right boundary of the lower Prosodic Area-1.

For example again, the node B1, the two red relationship arcs to its left (2/3,4/3) are the right boundary of the main Prosodic Area, and the two green relationship arcs to its right (3/4,3/2) are the left boundary of the upper Prosodic Area+1.

This is the physical meaning of the two semitones (of music) F and B in the Prosodic Structure, which naturally and logically exist and define the Prosodic Areas. All this provides a window and reference for us to further study human consciousness.

So far, the feature of Prosodic Structure that as “a Block Coding way for Analog Signal”, is tentatively present. In fact, the prosodic structure is also a data structure, and is also a block encoding method for audio signals (analog signals).

 

[Analysis and Summary]: It is well known that human consciousness is inspired by music. And each has his own understanding. But, the essence of music is the Prosodic Ratio between these points. It indicates that Human Consciousness has ability to recognize Prosodic Structures (PS).

In addition, many people have known that there can be no fundamental frequency C1 in a certain piece of music, which does not affect the listening effect of the music. This shows that the auditory recognition of music does not directly depend on the ratio of each sound to C1, but, it relies on the “Frame” of Prosodic Structure that called "PS Frame" or "PSFrame". PSFrame is a Prosodic Network (pW) or a Prosodic Area (pA), that exists in the human brain and is used by consciousness. It is the PS architecture, in which it maps the prosodic relationship between each sound and C1.

Due to the similarity of PS: Any frequency point can be used as the fundamental frequency; All Prosodic Networks are similar; All Prosodic Areas are similar.

So that, PS's "pattern matching" (move scaling rotate,etc.) is particularly easy and efficient. Due to the PSFrame architecture pattern exists, any music or sound that floats by, the hearing is quickly able to locate, recognize and discriminate.

This view will be further demonstrated in the following article: The language function of Human Consciousness, whose formants of vowels are closely related to "VK (Volute Knot)" that is the element of Prosodic Structure.

[Consultation]:

There is a Prosodic Structure in human consciousness: "PSFrame", and it is "innate". The form of this PSFrame is not clearly, but there is one reference, There is only one operation in PS that is "contrast", in which both sides of the contrast (numerator/denominator) have only two factors of 2 and 3 ( See above: "Prosody Coefficient table pCT").

At this point, please to go back to see the upper part of this article again, and then move on to the following.

 

2. VK Sampling and Vowel's VK Attribute (this is the second application of PS)

2.1 Block coding of speech signals: Volute Knot (VK); vkPD; VK System

In Fig.2, in "one Octave"(Music): main pA [C, D, E, F, G, A, B], and together with [d, e] that in previous pA, and together with [f, g, a] that in next pA, they are composed a set "VK":

VK=[d, e, C, D, E, F, G, A, B, f, g, a], this Set is called "Volute Knot " or "VK" for short .These 12 points are called “vkP”,there are from the adjacent pA, so:

  [d, e] =[D, E]*2^11/3^7;  

  [f, g, a] =[F, G, A]*3^7/2^11.

These 12 vkP are composed one Octave, so, every vkP has a short frequency area that called “vkP’s Domain", or “vkPD” . The Domain of the vkPD is that starting from this vkP and ending to next vkP (open),

e.g. vkPD(C)=[1:256/243). .

VK*2^n is “VK System” that is the Block coding of speech signals, and that is a series of the Prosodic Ratio. According to arrays of VK System, and use their every vkPD to sample the speech signal, that called “VK Sampling”.

[Note]: The "twelve_equal law" of music and the 12 points of PS VK are only numerical approximations.  (a vkPD is approximate to "semitone" of music). But have nothing to do with each other in nature.

2. 2 VK Sampling and Vowel's VK Attribute

This text takes six Vowels of Chinese as an example to analyze. The Vowel of other languages such as English,etc...[a],[e],[o],[i],[u], can be contrasted and analyzed too.

[Explain]: Following two Data Table are extracted from two Vowel's examples. Separately is: "Frequency Row" (divided into three sections to display); "Amplitude Row" ( red or pink is main Formant); "vkPD Row " (continuously same letter are same vkPD sampled. blue or green).

[Symbol Explain]: Such as “1248~1296 (B)" represent the Formant that in 1248~1296Hz is sampled by vkPD(B); "12th" represent 12th time period.

2.2.1 "Me1" is a Example of the Vowel's [e] , this is 12th time period of it. (goal peak >12.0. the max 1312 Hz(70.2) is in 9th time period):  Table1:

Analysis: In “ Me1” the main Formant is in close with vkPD.

In which, there are three main Formants: “896~928(F)”; “1072~1104(g)”; “1248~1296(B)”, that sampled by three vkPD that is F,g,B, that can writed “vkPD(F,g,B)”. In these vkPD the energy compared to their left and right sides, these vkPD are stronger a lot. In which: Three main Formant  (their main part of energy) (Red) are respectively sent in the three vkPD(F,g,B)(blue); Three vkPD (F,g,B) respectively sampled the three Formants (their main part).  

In other, Formant "352~368~384" is jointly sampled by two adjacent vkPD (d,D) , that formed "left hillside", "right hillside"; the Formant “528~544~576” is jointly sampled by two adjacent vkPD (g, A) , that formed "left hillside", "right hillside" too.

These sampling facts,have provided two aspects information for us: 1)The pronunciation of this Formant is generated from this "vkPD" (frequency area), this vkPD is called "goal vkPD" of this Formant (its pronounce goal); 2)These 5 main Formants (its main part) are 7 vkPD of the VK System. (These has provided important clue for follow-up study) .

Detailed materials please look at the Attachment: " T-VK Sampling-[a][e]" ,at first please look at “Guide " .

2.2.2  “za7" is a Example of the Vowel's [a] , below are 13th, 14th time period of it ,Table2:

Analysis: In Table2 main Formants( red or pink) are in close with vkPD(blue or green).

At first, look at the 14th period. In which there are 4 main Formants: 832~880(F), 1056~1120(A), 1248~1312(C), 1408~1472(D), with 4 vkPD (F,A,C,D), they are all that one Formant is Sampled by one vkPD, and the main energy of Formants (red) are separately falling in vkPD area (blue). These facts is expressing distinctly that this example of Vowel has "VK Attribute". These 4 sound of Formant are generated from 4 vkPD area, and 4 frequency band of Formant are 4 vkPD of “VK System”.

Sometimes, the Formant is a Big Wave (The frequency band is wide, the energy is strong. In this paper only discusses the Big Wave that greater than 350Hz), they are sampled by several vkPD. For example, in the 13th time period in above, 1008~1040~1120(g,A) and 1200~1248~1312(B,C) are Big Waves. They are pushed and squeezed in last time period, and changed from the last vkPD. They are shrink to 1056~1120(A), 1248~1312(C) in 14th. In many examples we can see the developing process of the Big Wave: Some of them, or born in a vkPD, or shrink into a vkPD that at midway or tail of sound, e.g. (" za7", " Ma2" ," re8" ). In numerous examples, although the peak of the Big Wave is changing constantly, but some peak of Big Wave is always that, or sampled by one vkPD, or sampled by two adjacent vkPD that separately formed "left hillside" and "right hillside",e.g. ("Mu2" 320~416; "zo8" 432~448; ”oo12” 368~384). Especially the latter that sampled by two vkPD, which seems to only emphasize both sides vkPD identity, but intensity of both sides is often asymmetric. All these are emphasizing that the Big Wave of the Formant has the VK Attribute. Under VK system, the change of the pronunciation seems organized and organized. As a block encoding of speech signals, "VK sampling" is a natural tool for speech signal processing. Please refer to the attachment for more details: "T-VK Sampling-[a][e]".