Mathematical logical techniques resemble repetition. In the context of these techniques additional and logically deductible considerations are made concerning the grade and the frequency of repetition instead of serially arranging the existing elements without modifying them. While repetition is closely connected to the material, the mathematical logical method focuses more on the structure of the work.

The three most famous methods of mathematical logic in Minimal Art and Minimal Music are addition, subtraction and substitution. They are applied either jointly to smaller units or to the work as a whole.
Musical addition means adding one tone to the composition. For example, if a third is added to an existing fifth, a decision has to be taken between a major and a minor key. Thus, the importance as well as the effect changes significantly. Vice versa this applies also to subtraction which is frequently used as a kind of mirrored reflection after an addition, like in Steve Reich’s Drumming (1971).
Musical substitution is not limited to replacing notes but is also applied for pauses. Similarly, in Minimal Art the technique of substitution refers to single elements and empty spaces. The same holds true for adding and subtracting as well. All of these methods can change the objects or its characteristics in fine arts, and, in music, the tone or the length of a tone. The techniques used, however, should bring along only slight changes in order to maintain the Minimalist form even when applying complex techniques. After all, the prerequisite of Minimalist work is that at least either the concept or the resulting implementation complies with the criteria of Minimalism.

Mathematical logical approaches
Since composing his work Nine Bells (1979) the music critic and later composer Tom Johnson, who was the first apart from Michael Nyman to coin the term Minimal Music, remains committed to his own style of Minimalist Music which is guided by mathematical logical phenomena. The best example of his rationally predicable compositions is the religious, serious oratorio Bonhoeffer Oratorium (1988-92).
In his work The Chord Catalogue (1986) Johnson creates a close relation between chords and tone scales and he writes down all the 8178 possible chords of one octave in chromatic tone scales. He follows a previously exactly defined process which determines the whole composition without further interventions from the artist. The use of certain processes is a typical feature of mathematical logical methods. They foster the reduction of the artistic ego, an element already widely applied in Minimal Art rather long before.
Hanne Darbovens (born in Munich in 1941) deals with mathematics not so much on a conceptual but on a meta-level and reduces her artistic material to numbers. Her painting 4868 (1969) consists solely of the four numbers mentioned in the title. They are repeated according to their numerical value in rows with a length of 104 characters. Since 1980 she applies the mathematical structure of her paintings to her musical pieces where she assigns a specific tone pitch to each visual element.
The writer and art collector Donald Karsham defines Minimalist artworks as mostly being arranged mathematically in space, in their own seriality of intervals.[1] Sol LeWitt is one of the most renowned representatives of this technique. With his serially structured paintings, cubic objects, grids and, above all, with his strongly conceptual approach he has gained international recognition.
In his pencil walldrawings (1969) sized 190x190 cm Sol LeWitt works with vertical, horizontal and diagonal lines, all placed very closely next to each other. The whole series is composed of 15 quadratic drawings and is divided into groups of three that are again arranged within themselves. When adding the first element to the second the result one gets is the last drawing of each row. Surprisingly though, LeWitt breaks his own rules by creating a hardly identifiable divergence in the combinations of his drawings.
Usually artists distance themselves from the artistic act when applying mathematical logical methods, and they imperturbably finish a certain process. Sol LeWitt, by contrast, intervenes deliberately in the process of addition of his walldrawings. In doing so he responds to the criticism that holds the artists’ withdrawal from their works responsible for the lack of emotions that Minimalist works evoke.
The cube takes a central role in LeWitt’s whole oeuvre. His series Variations of Incomplete Open Cubes (1974) consists of a schematic drawing showing all the different permutations of a cube, describing a cubical form with three to eleven straight lines. LeWitt translates the drafts of this series also into the third dimension. The resulting objects made from enamelled aluminium automatically bring the recipients to complete the cubic form in their minds. Later on, LeWitt applies the idea of the puristic geometry of a cube also to photography. In his work Cube (1988) he exposes the same cube in more than 500 different light situations.
LeWitt’s works often seem like the visual solution of a simple mathematical question. The concept is for him the essential component of his artistic work, which is reflected by the fact that he does not make any modifications during the production process as he wants to emphasise the disproportion of the object and the idea.
Donald Judd is considered to be the most famous representative of Minimal Art and the master of conveying the tension between visuality and material. He agrees with Sol LeWitt who says that irrational thoughts should be followed strictly and logically.[2] Judd’s specific objects are invented rationally as confirmed by the use of mathematical series to determine the exact distance between the single elements of an object. Robert Morris, too, uses this technique but the effects of his objects are often irrational. The reason for that is that it seems impossible to grasp the different perspectives as a clear entity as e.g. with Mirrored Cube (1963).

Criticising the lack of emotions
When arranged the single elements lose some of their individuality and become interchangeable. Thus, the individualising form of the work is shifted from the concept to the implementation. The majority of Minimalist Artists, however, abstains from demonstrating their own emotions. Instead, they create mathematical logical entities in order to ban their artistic ego as much as possible from their works.
Sigmund Freud (born in Freiberg in 1856; died in London in 1939) also deals with the relationship between the artwork and its creator, stating in one of his late writings that the force of creation does not always obey the artist’s will. He says that the work gets the way it can get and confronts the author in an unruly and even unfamiliar manner.[3] Freud implies a previously given distance between the work and the artist. This distance deliberately reaches its limit in Minimalism.
Emotions are a much discussed problem in artistic and musical works. Already in the early 20th century intellectualism blames the so called new music to originate in the head and not in the heart or the ear.[4] While this criticism refers mainly to serial composition techniques and to atonal music, objections in Minimalist art predominantly refer to mathematical logic, repetitive elements and restricted forms.
In music already the philosopher Georg Wilhelm Friedrich Hegel (born in Stuttgart in 1770; died in Berlin in 1831) criticised the increasing lack of emotions of the instrumental virtuosos of his days. He says that, as a consequence, the composer can concentrate fully on the musical structure of his work and the witty elements of its structure. He complains that musical production could become something lacking thoughts and emotions and therefore not requiring a deeper consciousness of knowledge or mind.[5]
The lack of emotions, however, can also be traced back to the expectations of consumers who apparently are not as concerned with the conceptual background as with the returned benefit in terms of delight as Adorno says.[6] Hegel examines in more detail the recipients’ perception which differs substantially from the object i.e. the proper work. According to Hegel the reason for this is that feelings belong to the region of mind that is unascertained and dull. He says that a feeling is hidden in the most abstract form of individual subjectivity and therefore the differences in feelings are totally abstract and do not refer to differences in the work itself.[7]
In general the relationship between expression and construction, between pathos and logos[8] seems to be one of the central problems of music in the 20th century. At the end of the 20th century an increasing number of composers become opposed to rigorously concentrating on the form and structure of music. Instead of that they prefer an emotional representation of their ideas that goes beyond material reality. Nevertheless, even at the climax of Minimalism there are some artists who feel related to the roots of Minimalist ideas but produce emotional works as well. The most intensive discussions about emotions as part of the artistic work take place, above all, during the second generation of Minimal Art.
In 1968 e.g. Bruce Nauman shows fiberglass objects that take the form of loaves in his promising debut exhibition in New York. They reveal Minimalist elements but are combined with disturbing overtones of organic life.[9] The same holds true for his body casts, neon tubing as well as for the use of rarely applied materials such as styrofoam, felt and grease in his work Collection of Various Materials Separated by Layers of Grease with Holes the Size of My Waist and Wrists (1966). His apparently organic forms and materials instigate a debate in Minimal Art concerning the emotional value, a debate unheard-of at that time.
Another artist that ranks among the second generation of Minimalists is Eva Hesse. Her works glow with a strong feeling of intimacy or well a feeling of self-confident sexuality like in Ingeminate (1965). She attaches great importance to the production process and therefore makes visible all the decisions and actions during this process. Thereby she concedes herself a lot of space in the context of her artistic work. So Hesse broadens the artistic result by the emotional component and like Nauman she clearly delimits the scope of her work compared to the work of the first generation of Minimalist Artists.
Although these artists meet the claims for more emotions in their artworks at the end of the 1960s they cannot prevent the public from abandoning Minimal Art. One of the reasons for this is certainly the sinister and partly menacing effect of Minimalist works.
Dan Flavin’s oeuvre ranks among the few works that succeed to create a balance between the inexorable nature of geometry and oddly sentimental feelings. His work The Diagonal of May 25, 1963 (1963) consists solely of a fluorescent light. Nonetheless, it already joins all the essential aspects of Flavin’s future works such as the look of Non-Art of the unprocessed commercial light, the art-historical nostalgy of diagonals (the definite metaphor of Constructivism) and the urban, diary-like glamour of the title.[10]