DE19716937A1 - Spectral harmonic interpretation process - Google Patents

Abstract

A multi dimensional representation is provided by three fields 1,2,3. Continuously proportional scales 4,6,7,8 are presented for the information representations. For music the characteristics are represented in various forms.

Description

The invention is generally applicable in the field of measurement, control and process technology (cybernetics), wherever the continua (e.g. Frequency spectrum) with regard to their harmonic aspects (e.g. Reso financial laws) should be technically accessible.

A special visualization technique enables the handling of the Event structures and progressions interpreted harmoniously in a (spatial) virtual reality described in this patent application especially with regard to music production (composition) is highlighted.

I have no comparable methods of interpretation, visualization tion, analysis and synthesis of spectral event structures and processes known fractions.

Important aspects of harmony therefore remained visual cognition largely closed. Harmonious relationships axiomi Systematic technical use remained of high quality (especially when dealing with spectra and continua) by far Field withheld.

The object of the invention is the functional multidimensional primfak torial interpretation, operationalization, visualization and processes visualization (analysis / synthesis) of spectral event structures, the Establishment of appropriate graphical interfaces, "virtual reali "and other techniques especially for music composition software would.

The only advantage for music is that two important stimulus-appropriate stimulus patterns, a) regarding the Harmonics of more or less complex vibration events and b) in terms of frequency or pitch (as a continuum) in the Interpretation and visualization in a complete (isomor phem) brought functional connection, even in relation to egg can be brought to another continuum (the time dimension) nen. Almost all relevant aspects of harmonics in the area of Music can thus be controlled and explained - a revolution in the Music industry and music hardware and software development.  

The process of adaptation from a well-tempered mood to the prim factorial interpretation and on the one hand and the adaptation of the prim factorial multidimensional interpretation to the traditional Music notation leave the process as a real productive-kon structural progress in this sector appear.

Solution to the problem: Theoretical part

Show it:

Figure 1: Three projective representations of different interpretations harmonious structures.

Figure 2: Path regression technique for harmonic progressions.

Figure 3: Path regression technique for harmonic structures.

Figure 4: Area formation with a combined path regression technique for harmonious structures and progressions.

Integration of the temporary aspects
Spectral events p (proportions) with their intensities i _p and their temporal aspects t _p are both scaled one-dimensionally (1.16) and interpreted in a multivariate (1.12) manner.

The spectrum becomes one-dimensional logarithmic to the base 1n b (before preferably b = 2) (1.6) interpreted scaled, primary factor multidimensi onal by n-degree prime factor vectors in the form:

where f _i ; iε | N, which corresponds to the natural prime number correlated with the i-th (spatial) interpretation dimension (1.8) (1.9) (1.10) and n to the dimensional complexity (the graphics show third-degree prime factor vector interpretations) of the interpretation model .

Each prime factor correlates with a (spatial) interpretation onsdimension, the integer exponents of each prime factor argument The elements correlate with the (linear) quantizations of everyone Dimension.

(The 0th dimension (f₀ = 1) has primarily a theoretical character, possible values of the exponent eo have no practical significance for the technical processes presented here.)
The primary, multi-dimensional interpretation of the spectrum is interpreted coincidentally (explicitly or implicitly) with regard to the interpretation dimension of its one-dimensional (spectral) interpretation (1,14). The temporary aspects can be interpreted (e.g.) orthogonally (1,19) to the representation dimension of the one-dimensional representation.

The proportions interpreted in this way can be spatial interpreted and presented in perspective or projectively the.

Shortened harmonics

For the interpretation, the prime factor vector is shortened by means of one or more bases of its prime factor arguments f _j :

• 1. By ignoring the relevant arguments.
• 2. By functional shortening (2):
• 3. By algorithmic shortening (3), z. B .:

This sets the dimensions of the corresponding prime factors suppresses and determines the correlation between multidimensional and one-dimensional interpretation (2.10) (2.6) (2.4) (3.10) (3.6) (2.4).

Average values - harmonic centers

The mean value of a harmonic structure can be determined spectrally via the geometric mean of the relationship values (e.g. frequency values) or via the geometric complement of the arithmetic mean of the logarithmic (based on 1n 2) complements m _{1n b of} the relationship values.

The mean of the multivariate interpretation of a structure or a progression is (advantageously) determined at the exponential level multidimensionally by the arithmetic mean of the exponents e _{i of} the prime factor vector arguments of the m spectral events involved in the structure or progressions.

and the following applies:

(1nb represents the index, the basis of the Logarithmization)

Level and class scaling

Spectra are usually quantized in classes or levels in the form s = b ^{(1 / w)} (1,4). 1n b represents the basis of the quantization, w the number of levels or classes that the spectrum is evenly divided with respect to the basis of its logarithmization. Level and class scales are interpreted differently.

Interpretation of the step scale

Primary intervals of ordinal quality are formed, which the Prim express numbers approximated by steps of the step scale (1.4) (1.6). They determine the adaptation of multidimensional and one-dimensional regional interpretation of events.

Interpretation of a scale level (provisional)

According to the logic of the primary intervals, there are theoretically un finally many interpretations for a level or a level interval the step scale.

Depending on the degree of dimensionality of the prime factorial space and the resolution of the step scale, there is a structure possible Interpretations of a scale level in the prime factorial space (level in interpretation structure).

There are different vek around an interpretation element (around) torial alignments different neighboring interpretations elements that (based on previous experience) in the degree of prime factor differentiate from each other (according to the stage interpretation axes).

In this way, an algorithmic search procedure can be set up that for a level selects the prime factorial interpretation that most harmoniously to a multidimensional prime factor pressure behaves e.g. B. to a multidimensionally expressed means worth a harmonious structure or a (complex) harmoni progression, a harmonious center.

Recursive approximation (provisional)

Here, for. B. recursively that neighboring inter pretationselement or the interpretation output element (for the selected recursive loop), which is the harmo niche relates to a chosen harmonic center until  twice the same element result of such a recursion loop is.

Functional approximation (provisional)

Alternatively, the dissonance potential can be interpreted from the original on the respective scale level to the current harmonic center can be expressed and the target interpretation of the scale level by breaking down (dissolution) this dissonance potential into the completely rational (multidimensionally expressed) harmonic pots tiale of the stage interpretation axis quantizations directly determined at least be approximated. This is with strong dissonance potentiometers alen makes sense because the functional approximation of the recursive Far superior in terms of computing effort and speed is.

Interpretation of the class scale (provisional)

Each measurement process does not represent an absolute isomorphism between data and feature forth.

Therefore, spectral events can occur with every further operationalization Never never from a point, but always from an interval interpolatively determined a proportion expressed in terms of prime factors will.

It follows that there is a range of information for each measured value. Accordingly, the multidimensional interpretation also has one Measured value, a range of information that varies depending on the spectral Selectivity as more or less dense multidimensional inter reference structure (theoretically infinite expansion) shows.

By identifying a relative harmonic center and the harmonious vectorial-polygonal inhibition of the periphery and exhibiti At the center it is (temporarily theoretically) possible to create a "harmo African shape "from a complex spectral structure approximate to extract iteratively.

Multidimensional paths

A path partial ( 20 ) is the smallest path unit from which paths in the multidimensional structure are composed. A path partial corresponds to a quant of a (prime, spatial) dimension of interpretation.

Each individual progression h, with the individual proportions e _{u, i} and e _{z, 1} (u: origin; Z: destination) involved in it, has a path length 1 _h in path parts:

For each individual progression there are k structural elements that can be affected by the path through the degrees of path freedom (progressions complexity structure ( 11 )):

There are c possible paths or possible per per individual progression mutations of the path partial arrangement involved in the individual progression mentions.

Complex path regression

For each individual progression, one of the c possible path permutations is selected so that the complex progression is interpreted by a path structure that is as concentrated as possible (spatially and structurally) ( 21 ). In the graphic, the individual progression (vocals) are ordered according to frequency (Vocal 1) ( 21 ).

For each individual progression the progression potentials are multidimensional by differences of the exponents e _i d _i = e _{z, 1} -e _{u, i} , d _i ε | R; of the prime factor arguments.

In each of the 1 _h partial path steps, if the path is (still) necessary for the formation, the progress is always made in accordance with the dimension at which the structural element that most harmoniously relates to the average of the entire progression is reached, whose prime factor vector is the prime factor vector expression of the mean (here the e _i values are not just whole numbers).

The step can be algorithms for each individual progression express mixed.

The spelling of mathematics must be temporarily expanded at this point.
Execution loop: e.g.

sets variables e _i for i continuously i = 1 from 1 to n to the value 0.
Colon:
closes a loop.
min _i : is a local condition hierarchically subordinated by the index variable i of an execution loop (or sum symbol etc.): results in the smallest value that is determined within the respective loop run through the arguments of the function min _i .
sng (x): returns x / | x | if x <0 and 0 if x = 0.

Embodiment 1: "music composition system"

The harmony and rhythm of traditional music are immense to the third degree prime factor vector.

Through the primary factorial multidimensional interpretation of the spectra it is now possible to make music related to perceptual and cognitive irritation patterns relevant to pitch and harmony of the Visualize and simultaneously operationalize music in a strictly functional manner sieren. The music can be in instead of in the well-tempered mood pure mood, the composer and arranger enabled to visually cybernetically almost all relevant To control aspects of the music in a way that is perceptible to him clarify.

Solution to embodiment 1

Harmonic structures (e.g. chords) are interpreted as multiple spectral events (notes, pitch)) with their intensities i _p (volume, velocity) (logarithmically scaled to the base 1n 2) and simultaneously adapted in a multi-dimensional manner. The prime factor arguments are correlated with the multidimensional (spatial) representation axes ( 8 ) ( 9 ) ( 10 ) and the exponents e _{i of} the prime factor arguments are correlated with the linear quantizations of the respective axes in such a way that the prime factorially multidimensional representation of the spectral events with respect to the representation dimension the logarithmic representation (based on 1n 2) of the spectrum coincides.

The scaling of the pitches preferred in Western music, the well-tempered mood, can be described as a step scale (s: step), which is divided into proportional steps p _s of p _s = 2 ^{s (1/12)} (1.4) (1.5) ( 22 ).

From this scale, primary intervals of ordinary quality are derived for each prime factor, the ordinal levels 0, 12, 19, 28 (etc.), which most closely approximate the proportions of the prime numbers 1 , 2 , 3 , 5 (etc.) (1, 4) (1.5).

The representations of level scaling and that of the multidimensional in are interpreted according to the logic of the primary intervals ordinal quality matched.

Alternatively, other step scales (s = b ^{(1 / w)} ; wε | R; preferably wε | N⁺ bε | R⁺ preferably bε | N⁺) can also be selected. Through them special effects such. B. Generate clusters with many tones at very close intervals or the spectrum can be controlled almost continuously (provisionally).

Harmonious structures and progressions can be simple, multible or complex. They are relativized by the temporal aspect t _p , which in turn is represented with a further axis of interpretation or representation (here orthogonal to the spectral representation dimension).

Format of harmonic progressions of music

Harmonic proportional events (here individual notes or voices) are one-dimensionally frequency-related as proportions or their logarithmic complements (based on 1n2) primary intervals (or levels of ordinal quality) by prime factors vectors with their intensity i _p and their expressed temporal aspect t _p and can in this or a similar format z. B. are mapped in the memory of a computer. The events are organized hierarchically (e.g. double-linked list). This enables fast, time-determined access to the data, for input, output, visualization etc.

Shortened harmonics - chords and scales (keys)

The three-dimensional version of the algorithmic shortening interpretation with j = 1 has proven to be particularly useful for music. f _j = 2; and b = 2 proved ( 3 ) ( 23 ):

respectively.

As a result, the dimension of the first prime factor is suppressed here and also determines the correlation between multidimensional and one-dimensional representation. As a result, traditional functional level functionality (with tonic, dominate, parallel, etc.) and at the same time concrete chord symbols (such as C major, G major, E minor) can be functionally visualized ( 24 ) ( 23 ).

The formation of scales ( 25 ) is also based on the same shortening. The scales represent the functional mood of the step scale (here keyboard manual), here controlled or visualized by a fifth of a third. The assignment of an element of the fifth-third structure to one of the 12 original keys implies the octave addition for all keys of the same original key relationship. By marking the elements of the fifth-third field, notes and chords can be displayed functionally within the scale (fifth-third field - primary keys - assignment) very simply and clearly ( 23 ) ( 24 ).

Harmonious auxiliary structures

Harmonic structures ( 16 ) can be represented in such a way that all harmonic partial proportions are taken into account in the multidimensional representation, which (together) have the same largest common denominator and at the same time the smallest common numerator as the partial proportions of the harmonic structure ( 14 ). They can be interpreted in a multidimensional, spatially interpreted perspective or projectively ( 12 ) by connecting the individual points representing the harmonic partial proportions with lines, identifying them and highlighting the partial proportions of the harmonic structure ( 16 ).

Harmonic progressions can from origin ( 12 ) to target structures ( 13 ) by means of encompassing progression complexity structures ( 11 ) and / or by means of structures enclosed by source and target structures ( 18 ) (e.g. according to the criterion of the minimum Use of auxiliary lines for the display) of progressions in the display.

Regression and visualization

To visualize the proportions and corresponding structures and progressions, a path regression technique is used that the complex harmonic structures (Intervocale) and Progressio NEN (intravocale) with as few paths as possible under the condition represents that all individual structures and progression involved in it can be represented with as few paths as possible.

The duration of the harmonic events (of the notes) is represented by vertical bars (paths) ( 26 ). The length of the thick paths here corresponds to the length of the notes.

The time interval from the end of the note to the beginning of the following note of the same voice (musical term) is represented by a thin auxiliary line ( 27 ).

The path progression is only shown when a new harmonic follow-up event occurs, even if that Original event of the individual progression (the grade) before has expired.

Progression regression

For this purpose, the differences of the exponents e _{z, i} -e _{u, i are} formed (for each individual progression). This creates a progression potential for each argument of prime factors. The paths are then formed by, for each of the 1 _h partial path steps,

Always choose the path partial that is still required for completing the path, which at the same time achieves the element of the multidimensional structure that most harmoniously relates to the other progression elements of the original and target structure - that is, in harmony with the current harmonic center behaves ( 28 ) (all thicker lines in graphic 2 (except the vertical lines).

The algorithm is at the end of the theoretical part "into a formula cast "(→ from line 251).

Structural regression

All harmonic events involved in a structure determine the harmonic center and, similar to the progression regression of each event element (multidimensional), is connected to each other by paths in such a way that each intrastructural path has as few path parts as possible and at the same time the entire structure has as few path parts , that is, as concentrated as possible ( 29 ) (all thick lines of the visualization in graphic 3 (except the vertical lines)).

Furthermore, vertical lines indicate how many quanta of a dimension a path (continuously) progresses ( 30 ).

Areas and transparencies

It shows those surfaces (transparent) that are completely enclosed by the path and have the same (here vertical) spatial orientation as the time ( 31 ).

In this way, the harmony of the music is multidimensionally continuous visualizable and all important aspects of harmony and structure complexity, progression complexity, voice guidance, etc.) visually understandable.

Colours

The different paths (voices) or path groups (e.g. multiple voices) instruments) can be colored differently. Also the Areas can have different spatial orientations be colored differently. Here the users can get the Possibility to be opened, the "Klangwel ten "individually to color.

Voice guidance (provisional)

All possible tuning styles are in this composition system definable primarily. Furthermore, in the manufacture of a polyphonic set for each chord progression the structure real and progressive complexities of the chords involved be calculated and z. B. the one Voice guidance variant (automatically) can be selected, the total including the least complexity factor.

Adaptation of the prime factorial interpretation to the tra traditional notation of musical events

The traditional notation can be developed algorithmically. The six root names b, f, c, g, d, a, e, (American spelling) are in the circle of fifths (corresponding to the prime factor f₂ = 3) arranges and considers it a repeating sequence.

Each octave step (p₁ = 2) does not result in a change in the parent name men. Each original fifth step marks a step in the stem name sequence to the right. Each major third step has four steps right. As soon as the parent name sequence exceeded to the right , it will be taken up again on the left and thus determined Trunk names added a cross "#". This is how the note name is created. If the parent name sequence is exceeded to the left, ent is entered neither removed a cross or added a "b" etc.

Depending on which strain name is used to determine the entire prime factor vector system (e.g. the chamber tone A≈440Hz), the strain factor A is assigned to the prime factor vector (third degree) (1 = p = 1 ° _* 2 ° _* 3 ° _* 5 ° ≈440 Hz) and the algorithm starts with this base note.

In the traditional grading system, an original octave (p = f₁ = 2) 7 pri mitive notation steps, a prime fifth (p = f₂ = 3) 11 notation steps and a major third (p = f₃ = 5) 16 notation steps. According to this  Notation steps become the relative notation level differences of a prime factor vector expression representing a note averaged and the noteheads are assigned to the notation levels and give them the possible increase, as described above, or a leading sign ("#" and "b"). The absolute Position depends on which grade is defined as the root grade is.

Primary ^factorial interpretation of rhythmic events p = f₀ ^e0 _* f₁ ^e1 _* . . . corresponds to the basic tempo (e.g. a recurring measure). Changes in tempo can also be understood proportionally. The clock is quantized using p2 of another prom factor vector (in the example, p2 / p1 = 72 = 2³ × 3²) and all the partial proportions contained in it (32). With music sequencers, it is customary to subsequently quantize "freely recorded passages" and to adapt them to "perfect cycle times. Here, the circular representation of the metric (p₂ / p₁) is appropriate. The quantizations can then, for example, as stars or n - Logically linked corner - diverse mixed quantifications are possible. "Tactile" and rhythmic shape are visualized in a more meaningful way.

The quantization of the time or the clock is realized here by horizontal lines. They are only valid for one element of the multi-dimensional prime factor space. When implementing this using a computer program, this element should be explicitly highlighted. B. represent a relative tonal center ( 33 ).

Claims (18)

1. Process for the interpretation, operationalization, visualization, processualization of complex harmonic structures and progressions, characterized in that (complex harmonic proportions in interpreted event structures and progressions are primarily interpreted, multidimensionally operationalized and visualized (1). The proportions involved are broken down into their smallest prime factor components and expressed in the form of the prime factor vector, where f _i , iε | N, the one with the i-th (spatial) interpretation dimension (1.8) (1.9) (1.10) correlated natural prime number and n corresponds to the dimensional complexity of the interpretation model, each (integer) exponent e _i correlating with the linear quantization of the i-th dimension. 2. The method according to claim 1, characterized in that traditional music is interpreted functionally in accordance with the prime factor model of the third degree, operationalized and visualized, by combined representation accordingly:

• - The scale and the mood of the original tones of a manual using a fifth-third structure ( 25 ).
• - the shortened structural progressions, which replace or complement the chord and step function symbols of traditional music ( 24 )
• - The unabridged functional representation of the harmonic (octave-fifth of a third), which is able to depict the harmonic event isomorphically and which replaces or supplements the traditionally written music notation ( 34 ).
• - the functional representation of the metric of time, which replaces or supplements the previous time signature and rhythmic note values ( 32 ).

3. The method according to claim 1 and 2, characterized in that the events interpreted in terms of prime factors are interpreted in an abbreviated manner, operationalized and presented accordingly (2) (3) by each basis _{i i of} each prime factor argument by a (prime) factor or an algo rithmically determined parameter a _{i is} argumentatively shortened (e.g. a _i = f _j ^{int (1n (fi) / 1n (fj))} ; e.g. j = 1 ⇔ f _j = 2). 4. The method according to claim 1 and 2, characterized in that the multidimensional interpretation or representation (1, 2, 3) with respect to the representation dimension (at least least) a logarithmic, preferably to the base In (2) scaled axis ( 6 ) or coincides is taken into account in the presentation. 5. The method according to claim 2, characterized in that of an ordinal or step scale ( 4 ) that divides the logarithmic scale ( 6 ) into a certain number of classes per a factor defined interval (here I [p, 2p]) represents, for each prime factor ( 16 ) taken into account in the respective interpretation, a class or step interval corresponding to the metric of the ordinal scale ( 4 ) is derived, which most closely approximates the prime factor ( 16 ) and the relationship between multidimensional representation ( 12 ) and one-dimensional ordinal scaling ( 4 ) is taken into account implicitly or explicitly in the representation in accordance with the logic of the derived intervals. 6. The method according to claim 1, characterized in that harmonic structures ( 16 ) are so Darge that all harmonic partial proportions are taken into account in the multi-dimensional representation, the (together) the same greatest common denominator and at the same time the same smallest common numerator as the partial proportions have the harmonic structure ( 14 ), they are interpreted multidimensionally ( 8, 9, 10 ) interpreted derived perspective or projective ( 12 ) by z. For example, the individual points representing the harmonic partial proportions can be identified by lines ( 16 ). 7. The method according to claim 1, characterized in that multidimensionally interpreted complex harmonic progressions from source ( 12 to target structures ( 13 ) with the help of, enclosing progression complexity structures ( 11 ) or / and with the help of, source and target structures included structures ( 18 ) taken into account in the presentation. 8. The method according to claim 1 and 2, characterized in that the progression events are gradual and / or interpreted in relation to a continuum (e.g.) of time) and are shown, the target structure being the starting structure the next stage of progression.   9. The method according to claim 2, characterized in that the multidimensional interpretation and Representation z. B. by means of a corresponding practical surface a computer program with ordinally scaled input instruments, e.g. B. Operationally linked or correlated keyboard manual and a quasi-interval scaled (high-resolution ordinal scaling) Output instrument, e.g. B. with a synthesizer (tone generator), correct be connected or operatively connected. 10. The method according to claim 2, characterized in that the usual traditional (classic) Music notation regarding the Tobhöhe and the harmonious reference algorithmically unique from the third degree prime factor vector and vice versa, musical notation primarily multi-dimensional analyzed, interpreted, visualized multidimensionally and primarily acoustically in a pure mood (e.g. through electronic synthesizers) can be reproduced. 11. The method according to claim 1 and 2, characterized in that ordinally scaled progressions are interpreted by means of the interpretation by ordinally scaled primary intervals ( 4 ) algorhitically mixed primary factorial, operationalized and displayed or vice versa, that corresponding ordinally scaled progression are derived from primary factorial data. 12. The method according to claim 1, characterized in that spectral events ( 6, 14, 16 ) can be interpreted, operationalized and displayed in a prime, multi-dimensional manner. Complex spectral event characteristics (intensity characteristics) (interpolative) are loaded into a storage space, which is arranged in a factorially multidimensional and thus operationalized way, then a relative harmonic center is determined and, through peripheral inhibition and central exhibition according to the laws of resonance, a harmonic expression, the harmonic Shape of the spectrum regresses from the interference structure that inevitably results. 13. The method according to claim 1 and 2, characterized in that rhythmic events by means of prime factors vector expressions with respect to the tempo (the cycle time) and the quantization of time intervals are operationalized and represented ( 32 ) and quantizations are also circular by stars and n-corners by bool ' can be put together to form mixed quantizations. 14. The method according to claim 1, 2 and 3, characterized in that ordinal scale levels depending to a harmonic center (expressed multidimensionally) from a prime factorial interpretation of prime factor interpro be done, either through a recursively gradual Approach or functional by dissolving the dissonance potential tial in step interpretation axis quanta. 15. The method according to claim 1, 2, 3, characterized in that harmonic structures and progressions are interpreted and visualized by means of a path regression technique which represents the complex harmonic structures (intervocals) and progressions (intravocals) with as few paths as possible under the condition, that all individual structures ( 29 ) or progressions ( 28 ) involved in it are represented with as few paths as possible. 16. The method according to claim 1, 2, 3 and 15, characterized in that those surfaces are highlighted (transparent) ago, which are completely enclosed by paths and have the edges that have the same spatial orientation as the dimension of time ( 31st ). 17. The method according to claim 1, 2, 3, 15, 16, characterized in that the intravocal paths depending on "Stim me "or can be colored by instrument group and itself the user of a corresponding music software that he created Color music to your own taste and that the under Proof 15 addressed areas according to their differences Chen spatial orientation are colored differently. 18. The method according to claim 2, 3, characterized in that voting leadership laws are prime factorial can be defined and the "inversion" of a chord structure as well as the appropriate choice of a harmonic progression regarding their dissonances and their common dissonance para can be described metrically and z. B. corresponding conditions can be fed back with voice guidance algorithms.