DE19627507A1 - Process for multidimensional interpretation, representation and operationalization of harmonic structures and progressions - Google Patents

Abstract

The invention concerns a system for the multidimensional interpretation, operationalization and visualization according to primary factors of one-dimensional scale systems focusing on the representation of musical-acoustic structures and sequences to form a concrete concept of a music composition programme. Musical events are interpreted proportionally and these, in part, highly complex proportions are broken down into their smallest integral partial proportions, the primary numbers, and quantized accordingly: (a), fi being the i-th primary number and n being the dimensional complexity. The traditional musical forms have very great affinity for the primary factor vectors of the third degree. Each dimension is correlated with a primary factor and the exponents of the primary factor argument fi are correlated with the linear quantization of the resultant three axes. A further axis corresponds to the time dimension which is also relativized on the basis of the primary factor vector. By reducing the primary factor vector interpretation, chords and scales can be represented and instruments, such as keyboards, whose operation is based on the ordinal scale can be adapted for inputting highly proportional musical structures and the latter rendered functionally visualizable.

Description

The procedure generally enables interpretation and dar position of harmonious structures and progressions and is especially as a tool for functional musical interpretation conceptualized events.

I am aware of harmonious structures and progressions by algebraic characters and by means of the grading system as well other symbols in the field of music.

Therefore, important aspects of the harmony of the visual remained Cognition largely closed.

The object of the invention is to har the functional representation monical structures and progressions especially in the outward direction look at corresponding stimulus patterns that are perceptible to perception regarding the auditory perception of humans and possible Forms of systematic adaptation of traditional aids in music processing to this new technology and genes new processes and tools for the purpose of with this new technology.

The object is achieved according to the features of claims 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 and 13 solved.  

For the field of music there is the advantage that two important perceptual-irritable stimulus patterns of auditory truth acceptance, a) regarding the harmonics of complex vibrations events and b) in terms of frequency or pitch (as Continuum) in interpretation, (visual) representation and Operationalization in a clear and at the same time complete gene (isomorphic) functional relationship, even in relation to be brought to another continuum (the time dimension) can. Almost all relevant aspects of harmonics in the area the music can be represented and treated selectively - a revolution in the music industry and music hardware - and software development. The adaptation process of well-being pertaining to the prime factorial interpretation (and vice versa) on the one hand and the adaptation of the prime factorial Interpretation to the notation in the traditional grading system make the system highly compatible with the traditional ones Music processing techniques.

In addition, it is easily possible to use this procedure differently Adapt requirements from other technical areas sen. All technical fields in which the Analysis or synthesis (interpretation, operationalization, dar position) of harmonic structures is meaningful or through this procedure only makes sense in the first place. Appropriate Products can be as diverse as the areas of application with their special requirements.  

2. Description of the method based on an execution example

Show it:

Graphic 1

Three projective representations of different forms of interpretation harmonic structures using the example of a harmonic pro gression.

Graphic 2

The adaptation of a keyboard manual to the multidimensional one prime factorial interpretation.

To explain the procedure, the individual com components of the process, their modification options and the composition (possible combinations of the individual Systematized components based on graphic 1 and then A technique of interpretation and representation becomes harmless African progressions from the field of music based on graphics 2 presented.

2.1 One-dimensional definition, interpretation and representation (har monical) proportional events.

2.1.1 Harmonic proportions can be one-dimensional by each because expressing a factor.

2.1.2 The proportions expressed by a factor can be interpret logarithmically to the base ln (b) (spectral).

2.1.3 The proportions interpreted in this way can be explicitly represented by markings ( 1, 16 ) on a logarithmically interpreted axis ( 1, 6 ) (here on the basis of ln (2)).

2.2 Multidimensional definition, interpretation and representation of (harmonic) proportional events.

2.2.1 Harmonic proportions p are expressed by breaking them down into their prime factors. From this follows the mathematical interpretation and operationalization model of the prime factor vector: p = f₀ ^e1 _* f₂ ^e2 _* f _n ^en : where nε N / 0; where f _i corresponds to the i-th natural prime number.

2.2.2 The prime factor vector expressions are interpreted multidimensionally by correlating the prime factor vector argument with interpretation dimensions and correlating the integer exponents e _{i of} the prime factor vector arguments with the (linear) quantization of the corresponding interpretation dimension.

2.2.3 The proportions interpreted in this way can also be used with spatially interpreted using this method of interpretation, space Lich perspective or according to the example below be presented projectively.

In the example a harmonic structure, which can be described by prime factor vector expressions of the third degree, consisting of the partial proportions 2 , 3 and 5, is represented in such a way that all harmonic proportions ( 6 , 14 ) are taken into account, the same greatest common denominator, the proportion 1 , and at the same time the same common smallest counter, the proportion 30 , as the partial proportions of the harmonic structure, the proportions 1 , 2 , 3 , 5 , 6 , 10 , 15 , 30 have ( 6 , 14 ) by accordingly multidimensional interpretation are connected by lines ( 12 ) and the partial proportions of the harmonic structure are highlighted by markings ( 16 , 12 ).

This procedure can also be used for interpretation, Ope rationalization and representation of harmonic structures applied, which are prime factor vector expressions higher and have lower grades described.

2.3 Correspondence between one-dimensional and multidimensional Interpretation or representation of harmonic structures.

2.3.1 The adaptation of one-dimensional ( 6 ) and multidimensional ( 12 ) interpretation or representation is achieved by a coincidence between multidimensional and one-dimensional interpretation or representation is achieved in accordance with the representation dimension of the one-dimensional representation of harmonic proportions or by the fact that the relevant context is explicitly ( 6 , 16 , 12 ) clarified or implicitly assumed.

2.4 Special forms of adapting one-dimensional and multidi dimensional interpretation or representation.

2.4.1 The one-dimensional (based on ln (b) logarithmic) scaling is divided into stages and ordinally scaled. According to the basis ln (b) (bε N ) of the logarithmization, the scale is scaled in corresponding logarithmics into x (xε N ) classes per proportion b with a respective class width of b ^{(1 / x)} , so that the levels b ^{(1 / x) · c} (cε N ) give the middle of the class (according to the logarithmic metric) of the c-th class - logarithm-class scale ( 4 ).

2.4.1.1 The scaling of the pitches preferred in Western music, the well-tempered mood, can be described as a step scale, which is proportionally divided into steps of 2 ^(1/12) and corresponds to a proportion of 2, p = 2, 12 steps. Using this scale, approximate harmonic structures can be generated, which can be described in a more perceptual manner using third-degree prime factor vector expressions.

2.4.2 Primary intervals of ordinal quality are formed, which the Proportions based on the respective prime factor vectorial inter pretation or representation involved prime factors accordingly the metric of a logarithm as described in point 2.4.1 Most closely approximate the Sche class scale.

2.4.2.1 In drawing 1, the ordinal scale ( 4 ) described in point 2.4.1.1 is shown schematically, from which primary intervals of original quality are derived. In the uninterrupted interpretation ( 1 ) of prime factor vectorial events, these are the ordinary levels 0, 12, 19 and 28 (4) corresponding to the proportions 1, 2, 3 and 5 (6) corresponding to the prime numbers, etc.

2.4.3 The multidimensional interpretation or representation of an out intersection of the prime vector space can be done by means of logical (algorithms sher) assignment happen. Each quant corresponds to the in the representation (linear) quantized, each with a prim factor correlated, representation dimension a primary interval the one-dimensional representation of the one described in point 2.4.1 ordinal scale (logarithmic class scale) or this Context is emphasized implicitly or explicitly.

2.4.3.1 In accordance with the primary interval of ordinal quality described in point 2.4.2.1, the procedural step described in point 2.4.3 is used to interpret structural or pro-gressive events of the well-tempered mood ( 20 , 21 , 22 , 23 , 24 ). More details under point 3.1.

2.5 Multidimensional interpretation and representation more harmonious Progressions.

2.5.1 The problem of multidimensional representation more harmonious Progression is according to claims 5, 6, 7, 8 and 9 solved.  

2.6 Shortened multidimensional interpretation and representation of the Harmony.

In order not to overload the (visual) stimulus processing capacity and to explicitly emphasize certain aspects of harmony, techniques for shortened interpretation, operationalization and presentation are presented ( 2 , 3 ).

2.6.1 When interpreting harmonic structures that can be described by n-degree prime factor vector expressions, it is possible to disregard one or more dimensions that correlate with prime factors f _i , where 0 <in.

2.6.2 Another method of shortening the interpretation of harmonic structures (and progressions), which can be described by n-degree prime factor vector expressions, is to base every base f _{i of} each prime factor argument of the prime factor vector expressions by one or more of the prime factors f contained in the prime factor vector shorten _j if applicable; 0 <jn. Accordingly, the prime factor expressions

p = f₁ ^e1 _* f₂ ^e2 _* . . . _* f _n ^s

for interpretation, operationalization and presentation forms z. B.

p _{Interpretation} = (f₁ / f _j ) ^e1 _* (f₂ / f _j ) ^e2 _* . . . _* (f _n / f _j ) ^en

2.6.2.1. B. with the prime factor f₁ = 2, f _j = 2, correlating dimension ( 1 , 8 ) of the three-dimensional prim factor interpretation of harmonic events ( 1 , 12 ) in the operationalization, interpretation and representation suppress and the other dimensions are shortened primvectorial argumentative interpreted interpreted ( 2 , 12 ), ( 2 , 16 ), ( 2 , 6 ), ( 2 , 4 ) and determines the correlation between multidimensional and one-dimensional interpretation, operationalization and representation of all under points 2.1, 2.2 , 2.3, 2.4 and 2.5 mentioned process steps analogously ( 2 ).

2.6.3 Another possibility for the shortened interpretation of harmonic structures, which can be expressed by a n-degree prime factor vector, is to shorten each base f _{i of} each prime factor argument of the prime factor vector expression differently.

2.6.3.1 Each basis f _{i of} each prime factor argument of the prime factor vector is calculated by the algorithmically determined factor k; e.g. B.

k _i = f _j ^{int (ln (fi) / ln (b))}

abbreviated, where f _{j represents} a prime factor, which determines the mode of this algorithm indexed with i as a determinant and can produce the following prime factor vector modification:

p _{Interpretation} = (f₀ / k₀) ^e0 _* (f₁ / k₁) ^e1 _* . . . _* (f _n / k _n ) ^s

2.6.3.1.1 This shortening method results in the conversion of a third factor prime factor vector expression at f _j = f 1 = 2

p = 2 ^e1 _* ^e2 _* 5 ^e3

in the expression

p _{Interpretation} = 2/2 ^e1 _* 3/2 ^e2 _* (5/4) ^e3

Even with this form of shortened representation, the dimension correlating with the prime factor f₁ = 2 is suppressed and thereby also determines the correlation between multidimensional and one-dimensional interpretation, operationalization and representation of all the procedural steps mentioned under points 2.1, 2.2, 2.3, 2.5 and 2.6 ( 3 ).

3. Prime factor vector representation of musical-harmonic structures and progressions and adaptations the traditional notation

The possible uses of the method described above are very versatile. Therefore, here is only one possible form the implementation in an exemplary but detailed manner tert, namely for the field of application of music (analysis, -synthesis).

3.1 The graphic 2 shows four schematically represented keyboard manuals ( 20 ), between the manuals are the multidimensionally interpreted progression levels, each with origin ( 22 ) and target harmonics ( 23 ) and the progression complexity structure ( 24 ) surrounding them. The target harmonic ( 23 ) of a progression level forms the original harmonic ( 22 ) of the next progression level, this is explicitly expressed by connecting lines ( 21 ), the length of each connecting line ( 21 ) correlates with time, the connecting lines ( 21 ) are here The schematic representation of the manuals ( 20 ) is interrupted, which shows the adaptation of the manual and the unabridged prime factor vector structure as described under points 2.4.3.1 and 2.5.1.

3.2 The traditional notation can be determined algorithmically.

The six note master names F, C, G, D, A, E, H are in the Circle of fifths (corresponding to the prime factor f₂ = 3) arranged and are considered a repeating sequence.

Each octave step (p₁ = 2) does not change the stem names. Each original fifth step (p₂ = 3) marks a step in the Trunk name sequence to the right. Every major third step leads to time four steps to the right. Once the stem name sequence after is exceeded on the right, it is started again on the left and A cross "#" is added to the stem name determined in this way adds, so the note name is created. If exceeded after on the left the sequence is resumed on the right and the A "b" is added to trunk names. etc.

Depending on which stem name is used to determine the entire prime vector system (e.g. the chamber tone A≈440 Hz), the third degree prime vector is assigned root note A without "#" or "b" (1 = p = 2⁰ _* 3⁰ _* 5⁰ ≈ 440 Hz), the algorithm starts with this parent name.

The staff consists primarily of lines with spaces, Notes are noted on or between the lines, the distance from a line to the nearest space a notation level. Accordingly, an original octave (p = f1 = 2) a 7 notation steps, a prime fifth (p = f₂ = 3) 11 notati ons steps and a major third (p = f₃ = 5) 16 notation steps. Ent The relative notati are speaking of these notation steps Level differences of one representing a grade Prime factor vector expression is determined and the noteheads are so assigned to the notation levels and the same as above wrote, the possibly determined increase or decrease characters preceded by the corresponding notation level. The off solute position depends on which grade as above Trunk note is defined.

3.3 The prime factorial interpretation of event structures and -Progressions in the quality of the well-tempered mood can be implemented using an algorithmic search method. Theo are ordinalska due to the quality differences and the proportional skailer (multidimensional) scaling an infinite number of interpretations possible of ordinal events. These interpretations but differ in terms of the degree of harmony and can be arranged hierarchically according to this criterion and a search algorithm should also be set up so that the simplest harmonious interpretations can be found first. In most cases, the simplest interpretation is that right is what, according to human perception is felt harmoniously and what is not.

The most important criteria that this algorithm must take into account are the prime factor vector interpretations of the origin structure, the target structure and the harmony progression comm complexity structure.  

4. Primary interpretation of rhythmic events

4.1 Events are interpreted rhythmically by going through a Continuum, according to their appearance, their length or with regard to other aspects.

4.2 The metric of time, which appears linear to humans, is interpreted proportionally. Any time interval defined in everyday understanding is interpreted as a proportion (e.g. if one minute corresponds to proportion 2 , then 2 minutes correspond to proportion 4 ). The everyday metric of time is thus converted for further operationalization:

t _proportional = b ^t ; (b is usually 2, according to the naturalis logarithm)

and to maintain the relation to the uniformity of the procedure, is defined:

p = t _proportional

From this follows the operationalization and interpretation model of the prime factor vector with respect to the Analysis of harmonic structures, that regarding the time proportions ones affine can be applied.

p = f₀ ^e0 _* f₁ ^e1 _* . . . _* f _n ^s .

The following matrix is provided to help you understand the patent application thought.

Reference symbol matrix for graphic 1

Claims (13)

1. Process for the interpretation, operationalization or presentation of complex harmonic structures and progressions, characterized in that (complex) harmonically proportioned interpreted event structures and progressions are factorially broken down and expressed, interpreted, operationalized or represented multidimensionally ( 1 ).
The proportions involved are broken down into their smallest prime factorial components and into the form of the prime factor vector expression p = f₁ ^e1 _* f₂ ^e2 _* . . . _* f _n ^en brought, where f _i corresponds to the i-th natural prime number / prime factor, each prime factor correlates with an interpretation dimension, the integer exponents of each prime factor argument correlate with the (linear) quantization of each dimension and n corresponds to the dimensional one Complexity. 2. The method according to claim 1, characterized in that the events interpreted in a prime factorial manner are interpreted in a shortened manner, operationalized and represented accordingly ( 2 , 3 ), in that each basis e _{i determines} each prime factor argument of the prime factor vector by a (prime) factor or an algorithmic one Factor k is shortened argumentatively (e.g. k = f _j ^{int (ln (fi) / ln (fj)} ; ^e.g .: j = 1 ↔ f _j = 2). 3. The method according to claim 1 or 2, characterized in that the multidimensional interpretation or representation ( 1 , 2 , 3 ) with respect to the representation dimension (at least) one logarithmic (to the base ln (b), preferably to the base ln (2) ) scaled axis ( 6 ) coincides or the corresponding correlation is implicitly or explicitly ( 16 , 14 ) taken into account in the display. 4. The method according to claim 1, 2 or 3, characterized in that of an ordinal scale ( 4 ), the logarithmic (to the base ln (b)) scale ( 6 ) (corresponding to the respective logarithmic metric) in a certain number of Classes are subdivided for each interval defined by a prime factor, for each prime factor ( 16 ) considered in the respective interpretation, a class interval is derived according to the metric of the ordinal scale ( 4 ) that represents the prime factor ( 16 ) most closely and the relationship between multidimensional Representation ( 12 ) and one-dimensional ordinal scaling ( 4 ) is taken into account implicitly or explicitly in the representation according to the logic of the derived intervals ( 20 , 21 , 22 , 23 , 24 ). 5. The method according to claim 1, 2, 3 or 4, characterized in that harmonic structures ( 16 ) are presented so that all harmonic partial proportions are taken into account in the multidimensional representation, the (together) the same largest common counter and at the same time the same have the smallest common counter such as the partial proportions of the harmonic structure ( 14 ), they are interpreted multidimensionally ( 8 , 9 , 10 ), derived in perspective or projectively ( 12 ) by z. B. the individual, the harmonic African partial proportions points are connected by lines, and that the partial proportions of the harmonic structure are marked in contrast ( 16 ). 6. The method according to claim 1, 2, 3, 4 or 5, characterized in that multidimensionally interpreted complex harmonic progressions from an origin ( 12 ) to target structures ( 13 ) with the help of, enclosing Progressi ons complexity structures ( 11 ) or / and with the aid of structures ( 18 ) which are included ( 12 ) and target structures ( 13 ) (e.g. according to the criterion of the minimal use of auxiliary lines for the display of progressions) in the display. 7. The method according to claim 1, 2, 3, 4, 5 or 6, characterized in that the progression events are interpreted and displayed in stages and / or in relation to a continuum (for example time), the target structure ( 23 ) represents the initial structure ( 22 ) of the next progression event and this is expressed implicitly ( 21 ) or explicitly ( 19 ). 8. The method according to claim 1, 2, 3, 4, 5, 6 or 7, characterized in that the multidimensional interpretation and representation ( 22, 23, 24 ) z. B. by means of a corresponding graphical surface of a computer program with ordi nalskalierter input instruments, for. B. keyboard manual ( 20 ) are operatively linked or correlated ( 21 ) and a (semi-) interval scaled, ie quasi-interval scaled output instrument, e.g. B. with a synthesizer (tone generator), correlated or operatively connected and this implicitly or explicitly ( 20 , 21 , 22 , 23 , 24 ) in the display or output (e.g. control signals regarding the pitch (pure Tuning) to a synthesizer) is taken into account. 9. The method according to claim 1, 2, 3, 4, 5, 6, 7 or 8, characterized in that the display elements of the multidimensional display (z. B. representation of the origin ( 12 ) and target structure ( 13 ) enclosing ( 11) (and enclosed (18)) harmony progression complexity of structure and other elements for multilevel representation (16,14, 15, 17), etc.) may be modified lung in depicting, by ignoring or only partly taken into account in the illustration, by color, line thickness , brightness levels, the transparency etc. by geometrical Fig. (eg. B. surfaces or rhomboid) or by the introduction of additional, derived from the above mentioned elements, lines and characters, the specific aspects of the harmonic events depending on the specific requirements accentuate by projective perspective or spatial (e.g. holographic) representation. 10. The method according to claim 1, 2, 3, 4, 5, 6, 7, 8, or 9, characterized in that the usual musical notation regarding the pitch and tonal reference algorithmically correct from Third degree prime factor vector and vice versa Musical notations interpreted primarily as multi-dimensional, be represented and analyzed and the frequency of each note can be calculated, provided the chamber tone or a similar Basis of voting is defined. 11. The method according to claim 1, 2, 3, 4, 5, 6, 7, 8, 9, or 10, characterized in that ordinally scaled progressions ( 20 ) by means of the interpretation by ordinally scaled primary intervals ( 4 ) interpreted, operationalized and algorithmically prime factorially ( 21 , 22 , 23 , 24 ), or vice versa, are derived from prime factorial events. 12. The method according to claim 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 or 11, characterized in that spectral events ( 6 , 14 , 16 ) are interpreted, operationalized and displayed primarily factor-vectorially multidimensionally ( 14 , 16 , 12 ), or vice versa, derived from prime factor vector events. 13. The method according to claim 1, 2, 3, 5, 6, 7, 9 and 11, characterized in that rhythmic events as harmoni events are described using prime factor vector expressions, can be operationalized and represented.