Meaning of idealism.

The word "idealism" has multiple meanings. The philosophical meaning of idealism are those properties we discover in objects dependent on the way that those objects appear to us, as perceived subjects. These properties only belong to the perceived appearance of the objects, and not something they possess "in themselves". The term "idea-ism" is closer to this intended meaning than the common notion of idealism. The question of what properties a thing might have "independently of the mind" is thus unknowable and a moot point, within the idealist tradition.

History.

Immanuel Kant's work purported to bridge the two dominant philosophical schools in the 18th century: 1) rationalism, which held that knowledge could be attained by reason alone "a priori" (prior to experience), and 2) empiricism, which held that knowledge could be arrived at only through the senses "a posteriori" (after experience), as expressed by philosopher David Hume, whom Kant sought to rebut. Kant's solution was to propose that, while we depend on objects of experience to know anything about the world, we can investigate "a priori" the "form" that our thoughts can take, determining the boundaries of possible experience. Kant called his mode of philosophising "critical philosophy", in that it was supposedly less concerned with setting out positive doctrine than with critiquing the limits to the theories we "can" set out. The conclusion he presented, as above, he called "transcendental idealism". This distinguished it from classical idealism and subjective idealism such as George Berkeley's, which held that external objects have actual being or real existence only when they are perceived by an observer. Kant said that there are things-in-themselves ("noumena", that is), things that exist other than being merely sensations and ideas in our minds. Kant held in the "Critique of Pure Reason" (1781) that the world of appearances (phenomena) is empirically real and transcendentally ideal. The mind plays a central role in influencing the way that the world is experienced: we perceive phenomena through time, space and the categories of the understanding. It is this notion that was taken to heart by Kant's philosophical successors.  Arthur Schopenhauer considered himself to be a transcendental idealist. In his major work "The World as Will and Representation" (1818/1819) he discusses his indebtedness to Kant, and the work includes Schopenhauer's extensive analysis of the "Critique".  The best-known German idealist thinkers, besides Kant, were Johann Gottlieb Fichte, Friedrich Wilhelm Joseph Schelling, Georg Wilhelm Friedrich Hegel. The Young Hegelians, a number of philosophers who developed Hegel's work in various directions, were in some cases idealists. On the other hand, Karl Marx, who was numbered among them, had professed himself to be a materialist, in opposition to idealism. Another member of the Young Hegelians, Ludwig Feuerbach, advocated for materialism, and his thought was influential in the development of historical materialism, where he is often recognized as a bridge between Hegel and Marx.

Theorists.

Kant.

Immanuel Kant's transcendental idealism consisted of taking a point of view outside and above oneself (transcendentally) and understanding that the mind directly knows only phenomena or ideas. Whatever exists other than mental phenomena, or ideas that appear to the mind, is a thing-in-itself and cannot be directly and immediately known.  Kant criticized pure reason. He wanted to restrict reasoning, judging, and speaking only to objects of possible experience. The main German Idealists, who had been theology students, reacted against Kant's stringent limits. "It was Kant’s criticism of all attempts to prove the existence of God which led to the romantic reaction of Fichte, Schelling, and Hegel." "Kant sets out to smash not only the proofs of God but the very foundations of Christian metaphysics, then turns around and 'postulates' God and the immortality of the soul, preparing the way for Fichte and idealism."

Jacobi.

In 1787, Friedrich Heinrich Jacobi addressed, in his book "On Faith, or Idealism and Realism", Kant's concept of "thing-in-itself". Jacobi agreed that the objective thing-in-itself cannot be directly known. However, he stated, it must be taken on belief. A subject must believe that there is a real object in the external world that is related to the representation or mental idea that is directly known. This belief is a result of revelation or immediately known, but logically unproved, truth. The real existence of a thing-in-itself is revealed or disclosed to the observing subject. In this way, the subject directly knows the ideal, subjective representations that appear in the mind, and strongly believes in the real, objective thing-in-itself that exists outside the mind. By presenting the external world as an object of belief, Jacobi legitimized belief. "...[B]y reducing the external world to a matter of faith, he wanted merely to open a little door for faith in general..."

Reinhold.

Karl Leonhard Reinhold published two volumes of "Letters Concerning the Kantian Philosophy" in 1790 and 1792. They provided a clear explication of Kant's thoughts, which were previously inaccessible due to Kant's use of complex or technical language.  Reinhold also tried to prove Kant's assertion that humans and other animals can know only images that appear in their minds, never "things-in-themselves" (things that are not mere appearances in a mind). In order to establish his proof, Reinhold stated an axiom that could not possibly be doubted. From this axiom, all knowledge of consciousness could be deduced. His axiom was: "Representation is distinguished in consciousness by the subject from the subject and object, and is referred to both."  He thereby started, not from definitions, but, from a principle that referred to mental images or representations in a conscious mind. In this way, he analyzed knowledge into (1) the knowing subject, or observer, (2) the known object, and (3) the image or representation in the subject's mind. In order to understand transcendental idealism, it is necessary to reflect deeply enough to distinguish experience as consisting of these three components: subject, subject's representation of object, and object.

Schulze.

Kant noted that a mental idea or representation must be a representation of something, and deduced that it is of something external to the mind. He gave the name of "Ding an sich", or thing-in-itself to that which is represented. However, Gottlob Ernst Schulze wrote, anonymously, that the law of cause and effect only applies to the phenomena within the mind, not between those phenomena and any things-in-themselves outside the mind. That is, a thing-in-itself cannot be the cause of an idea or image of a thing in the mind. In this way, he discredited Kant's philosophy by using Kant's own reasoning to disprove the existence of a thing-in-itself.

Fichte.

After Schulze had seriously criticized the notion of a thing-in-itself, Johann Gottlieb Fichte produced a philosophy similar to Kant's, but without a thing-in-itself. Fichte asserted that our representations, ideas, or mental images are merely the productions of our ego, or knowing subject. For him, there is no external thing-in-itself that produces the ideas. On the contrary, the knowing subject, or ego, is the cause of the external thing, object, or non-ego.  Fichte's style was a challenging exaggeration of Kant's already difficult writing. Also, Fichte claimed that his truths were apparent to intellectual, non-perceptual, intuition. That is, the truth can be immediately seen by the use of reason.  Schopenhauer, a student of Fichte's, wrote of him:

Schelling.

Friedrich Wilhelm Joseph Schelling attempted to rescue theism from Kant's refutation of the proofs for God's existence. "Now the philosophy of Schelling from the first admitted the possibility of a knowledge of God, although it likewise started from the philosophy of Kant, which denies such knowledge."  With regard to the experience of objects, Friedrich Wilhelm Joseph Schelling (1775–1854) claimed that the Fichte's "I" needs the Not-I, because there is no subject without object, and vice versa. So the ideas or mental images in the mind are identical to the extended objects which are external to the mind. According to Schelling's "absolute identity" or "indifferentism", there is no difference between the subjective and the objective, that is, the ideal and the real.  In 1851, Arthur Schopenhauer criticized Schelling's absolute identity of the subjective and the objective, or of the ideal and the real. "...[E]verything that rare minds like Locke and Kant had separated after an incredible amount of reflection and judgment, was to be again poured into the pap of that absolute identity. For the teaching of those two thinkers [Locke and Kant] may be very appropriately described as the doctrine of the "absolute diversity of the ideal and the real, or of the subjective and the objective"."

Schleiermacher.

Friedrich Schleiermacher was a theologian who asserted that the ideal and the real are united in God. He understood the ideal as the subjective mental activities of thought, intellect, and reason. The real was, for him, the objective area of nature and physical being. Schleiermacher declared that the unity of the ideal and the real is manifested in God. The two divisions do not have a productive or causal effect on each other. Rather, they are both equally existent in the absolute transcendental entity which is God.

Maimon.

Salomon Maimon influenced German idealism by criticizing Kant's dichotomies, claiming that Kant did not explain how opposites such as sensibility and understanding could relate to each other. Maimon claimed that the dualism between these faculties was analogous to the old Cartesian dualism between the mind and body, and that all the problems of the older dualism should hold "mutatis mutandis" for the new one. Such was the heterogeneity between understanding and sensibility, Maimon further argued, that there could be no criterion to determine how the concepts of the understanding apply to the intuitions of sensibility. By thus pointing out these problematic dualisms, Maimon and the neo-Humean critics left a foothold open for skepticism within the framework of Kant’s own philosophy. For now the question arose how two such heterogeneous realms as the intellectual and the sensible could be known to correspond with one another. The problem was no longer how we know that our representations correspond with things in themselves but how we know that "a priori" concepts apply to "a posteriori" intuitions.  Schelling and Hegel, however, tried to solve this problem by claiming that opposites are absolutely identical. Maimon's concept of an infinite mind as the basis of all opposites was similar to the German idealistic attempt to rescue theism by positing an Absolute Mind or Spirit.  Maimon's metaphysical concept of "infinite mind" was similar to Fichte's "Ich" and Hegel's "Geist." Maimon ignored the results of Kant's criticism and returned to pre-Kantian transcendent speculation. What characterizes Fichte’s, Schelling’s, and Hegel’s speculative idealism in contrast to Kant's critical idealism is the recurrence of metaphysical ideas from the rationalist tradition. What Kant forbade as a violation of the limits of human knowledge, Fichte, Schelling, and Hegel saw as a necessity of the critical philosophy itself. Now Maimon was the crucial figure behind this transformation. By reviving metaphysical ideas from within the problematic of the critical philosophy, he gave them a new legitimacy and opened up the possibility for a critical resurrection of metaphysics.  Maimon is said to have Influenced Hegel's writing on Spinoza. "[T]here seems to be a striking similarity between Maimon’s discussion of Spinoza in the "Lebensgeschichte" (Maimon's autobiography) and Hegel’s discussion of Spinoza in the "Lectures in the History of Philosophy"."

Hegel.

Georg Wilhelm Friedrich Hegel was a German philosopher born in Stuttgart, Württemberg, in present-day southwest Germany. Hegel responded to Kant's philosophy by suggesting that the unsolvable contradictions given by Kant in his Antinomies of Pure Reason applied not only to the four areas Kant gave (world as infinite vs. finite, material as composite vs. atomic, etc.) but in all objects and conceptions, notions and ideas. To know this he suggested makes a "vital part in a philosophical theory." Given that abstract thought is thus limited, he went on to consider how historical formations give rise to different philosophies and ways of thinking. For Hegel, thought fails when it is only given as an abstraction and is not united with considerations of historical reality. In his major work "The Phenomenology of Spirit" he went on to trace the formation of self-consciousness through history and the importance of other people in the awakening of self-consciousness (see master-slave dialectic). Thus Hegel introduces two important ideas to metaphysics and philosophy: the integral importance of history and of the Other person. His work is theological in that it replaces the traditional concept of God with that of an Absolute Spirit. Spinoza, who changed the anthropomorphic concept of God into that of an abstract, vague, underlying Substance, was praised by Hegel whose concept of Absolute fulfilled a similar function. Hegel claimed that "You are either a Spinozist or not a philosopher at all". Reality results from God's thinking, according to Hegel. Objects that appear to a spectator originate in God's mind.

Responses.

Neo-Kantianism.

Neo-Kantianism refers broadly to a revived type of philosophy along the lines of that laid down by Immanuel Kant in the 18th century, or more specifically by Schopenhauer's criticism of the Kantian philosophy in his work "The World as Will and Representation" (1818), as well as by other post-Kantian philosophers such as Jakob Friedrich Fries and Johann Friedrich Herbart. It has some more specific reference in later German philosophy.

Hegelianism.

Hegel was hugely influential throughout the nineteenth century; by its end, according to Bertrand Russell, "the leading academic philosophers, both in America and Britain, were largely Hegelian". His influence has continued in contemporary philosophy but mainly in Continental philosophy.

Schopenhauer.

Arthur Schopenhauer contended that Spinoza had a great influence on post-Kantian German idealists. Schopenhauer wrote: "In consequence of Kant's criticism of all speculative theology, almost all the philosophizers in Germany cast themselves back on to Spinoza, so that the whole series of unsuccessful attempts known by the name of post-Kantian philosophy is simply Spinozism tastelessly got up, veiled in all kinds of unintelligible language, and otherwise twisted and distorted."  According to Schopenhauer, Kant's original philosophy, with its refutation of all speculative theology, had been transformed by the German Idealists. Through the use of his technical terms, such as "transcendental," "transcendent," "reason," "intelligibility," and "thing-in-itself" they attempted to speak of what exists beyond experience and, in this way, to revive the notions of God, free will, and immortality of soul. Kant had effectively relegated these ineffable notions to faith and belief.

British idealism.

In England, during the nineteenth century, philosopher Thomas Hill Green embraced German Idealism in order to salvage Christian monotheism as a basis for morality. His philosophy attempted to account for an eternal consciousness or mind that was similar to Berkeley's concept of God and Hegel's Absolute. John Rodman, in the introduction to his book on Thomas Hill Green's political theory, wrote: "Green is best seen as an exponent of German idealism as an answer to the dilemma posed by the discrediting of Christianity...."

United States.

"German idealism was initially introduced to the broader community of American "literati" through a Vermont intellectual, James Marsh. Studying theology with Moses Stuart at Andover Seminary in the early 1820s, Marsh sought a Christian theology that would 'keep alive the heart in the head.' " Some American theologians and churchmen found value in German Idealism's theological concept of the infinite Absolute Ideal or "Geist" [Spirit]. It provided a religious alternative to the traditional Christian concept of the Deity. "...[P]ost–Kantian idealism can certainly be viewed as a religious school of thought...." The Absolute Ideal "Weltgeist" [World Spirit] was invoked by American ministers as they "turned to German idealism in the hope of finding comfort against English positivism and empiricism." German idealism was a substitute for religion after the Civil War when "Americans were drawn to German idealism because of a 'loss of faith in traditional cosmic explanations.' " "By the early 1870s, the infiltration of German idealism was so pronounced that Walt Whitman declared in his personal notes that 'Only Hegel is fit for America — is large enough and free enough.' "

Ortega y Gasset.

According to José Ortega y Gasset, with Post-Kantian German Idealism, "...never before has a lack of truthfulness played such a large and important role in philosophy." "They did whatever they felt like doing with concepts. As if by magic they changed anything into any other thing." According to Ortega y Gasset, "...the basic force behind their work was not strictly and exclusively the desire for truth...." Ortega y Gasset quoted Schopenhauer's "Parerga and Paralipomena", Volume II, in which Schopenhauer wrote that Fichte, Schelling, and Hegel forgot "the fact that one can feel an authentic and bitter seriousness" for philosophy. Schopenhauer, in Ortega y Gasset's quote, hoped that philosophers like those three men could learn "true and fruitful seriousness, such that the problem of existence would capture the thinker and bestir his innermost being."

George Santayana.

George Santayana had strongly-held opinions regarding this attempt to overcome the effects of Kant's transcendental idealism.

G. E. Moore.

In the first sentence of his "The Refutation of Idealism", G. E. Moore wrote: "Modern Idealism, if it asserts any general conclusion about the universe at all, asserts that it is spiritual," by which he means "that the whole universe possesses all the qualities the possession of which is held to make us so superior to things which seem to be inanimate." He does not directly confront this conclusion, and instead focuses on what he considers the distinctively Idealist premise that "esse is percipere" or that to be is to be perceived. He analyzes this idea and considers it to conflate ideas or be contradictory.

Slavoj Žižek.

Slavoj Žižek sees German idealism as the pinnacle of modern philosophy, and as a tradition that contemporary philosophy must recapture: "[T]here is a unique philosophical moment in which philosophy appears 'as such' and which serves as a key—as the only key—to reading the entire preceding and following tradition as philosophy... This moment is the moment of German Idealism..."

Hannah Arendt.

Hannah Arendt stated that Immanuel Kant distinguished between "Vernunft" ("reason") and "Verstand" ("intellect"): these two categories are equivalents of "the urgent need of" reason, and the "mere quest and desire for knowledge". Differentiating between reason and intellect, or the need to reason and the quest for knowledge, as Kant has done, according to Arendt "coincides with a distinction between two altogether different mental activities, thinking and knowing, and two altogether different concerns, meaning, in the first category, and cognition, in the second". These ideas were also developed by Kantian philosopher, Wilhelm Windelband, in his discussion of the approaches to knowledge named "nomothetic" and "idiographic".  Kant's insight to start differentiating between approaches to knowledge that attempt to understand meaning (derived from reason), on the one hand, and to derive laws (on which knowledge is based), on the other, started to make room for "speculative thought" (which in this case, is not seen as a negative aspect, but rather an indication that knowledge and the effort to derive laws to explain objective phenomena has been separated from "thinking"). This new-found room for "speculative thought" (reason, or thinking) touched-off the rise of German idealism. However, the new-found "speculative thought", reason or thinking of German idealism "again became a field for a new brand of specialists committed to the notion that philosophy's'subject proper' is 'the actual knowledge of what truly is'. Liberated by Kant from the old school of dogmatism and its sterile exercises, they erected not only new systems but a new'science' - the original title of the greatest of their works, Hegel's "Phenomenology of the mind", was "Science of the experience of consciousness" - eagerly blurring Kant's distinction between reason's concern with the unknowable and the intellect's concern with cognition. Pursuing the Cartesian ideal of certainty as though Kant had never existed, they believed in all earnest that the results of their speculations possessed the same kind of validity as the results of cognitive processes".

German idealism was a philosophical movement that emerged in Germany in the late 18th and early 19th centuries. It developed out of the work of Immanuel Kant in the 1780s and 1790s, and was closely linked both with Romanticism and the revolutionary politics of the Enlightenment. The best-known thinkers in the movement, besides Kant, were Johann Gottlieb Fichte, Friedrich Wilhelm Joseph Schelling, Georg Wilhelm Friedrich Hegel, and the proponents of Jena Romanticism (Friedrich Hölderlin, Novalis, and Karl Wilhelm Friedrich Schlegel). August Ludwig Hülsen, Friedrich Heinrich Jacobi, Gottlob Ernst Schulze, Karl Leonhard Reinhold, Salomon Maimon, Friedrich Schleiermacher, and Arthur Schopenhauer also made major contributions.

Als Deutscher Idealismus wird die Epoche der deutschen Philosophie von Kant bis zu Hegel und zum Spätwerk Schellings bezeichnet. Als zeitliche Rahmendaten gelten meist das Erscheinen von Kants "Kritik der reinen Vernunft" (1781) und der Tod Georg Wilhelm Friedrich Hegels (1831). Der Deutsche Idealismus gilt vielfach als Blütezeit der deutschen Philosophie und wird hinsichtlich seiner philosophiegeschichtlichen Bedeutung mit der klassischen griechischen Philosophie verglichen. Alternativ zur Bezeichnung „Deutscher Idealismus“ wird daher diese Epoche oft auch als „Klassische deutsche Philosophie“ bezeichnet.


Definitions.

Recursive definition.

To actually define a binary tree in general, we must allow for the possibility that only one of the children may be empty. An artifact, which in some textbooks is called an "extended binary tree" is needed for that purpose. An extended binary tree is thus recursively defined as:  Another way of imagining this construction (and understanding the terminology) is to consider instead of the empty set a different type of node—for instance square nodes if the regular ones are circles.

Using graph theory concepts.

A binary tree is a rooted tree that is also an ordered tree (a.k.a. plane tree) in which every node has at most two children. A rooted tree naturally imparts a notion of levels (distance from the root), thus for every node a notion of children may be defined as the nodes connected to it a level below. Ordering of these children (e.g., by drawing them on a plane) makes possible to distinguish left child from right child. But this still doesn't distinguish between a node with left but not a right child from a one with right but no left child.  The necessary distinction can be made by first partitioning the edges, i.e., defining the binary tree as triplet (V, E, E), where (V, E ∪ E) is a rooted tree (equivalently arborescence) and E ∩ E is empty, and also requiring that for all "j" ∈ { 1, 2 } every node has at most one E child. A more informal way of making the distinction is to say, quoting the Encyclopedia of Mathematics, that "every node has a left child, a right child, neither, or both" and to specify that these "are all different" binary trees.

Types of binary trees.

Tree terminology is not well-standardized and so varies in the literature.

Combinatorics.

In combinatorics one considers the problem of counting the number of full binary trees of a given size. Here the trees have no values attached to their nodes (this would just multiply the number of possible trees by an easily determined factor), and trees are distinguished only by their structure; however the left and right child of any node are distinguished (if they are different trees, then interchanging them will produce a tree distinct from the original one). The size of the tree is taken to be the number "n" of internal nodes (those with two children); the other nodes are leaf nodes and there are of them. The number of such binary trees of size "n" is equal to the number of ways of fully parenthesizing a string of symbols (representing leaves) separated by "n" binary operators (representing internal nodes), so as to determine the argument subexpressions of each operator. For instance for one has to parenthesize a string like, which is possible in five ways: The correspondence to binary trees should be obvious, and the addition of redundant parentheses (around an already parenthesized expression or around the full expression) is disallowed (or at least not counted as producing a new possibility).  There is a unique binary tree of size 0 (consisting of a single leaf), and any other binary tree is characterized by the pair of its left and right children; if these have sizes "i" and "j" respectively, the full tree has size. Therefore, the number formula_15 of binary trees of size "n" has the following recursive description formula_16, and formula_17 for any positive integer "n". It follows that formula_15 is the Catalan number of index "n".  The above parenthesized strings should not be confused with the set of words of length 2"n" in the Dyck language, which consist only of parentheses in such a way that they are properly balanced. The number of such strings satisfies the same recursive description (each Dyck word of length 2"n" is determined by the Dyck subword enclosed by the initial '(' and its matching ')' together with the Dyck subword remaining after that closing parenthesis, whose lengths 2"i" and 2"j" satisfy ); this number is therefore also the Catalan number formula_15. So there are also five Dyck words of length 6:  These Dyck words do not correspond to binary trees in the same way. Instead, they are related by the following recursively defined bijection: the Dyck word equal to the empty string corresponds to the binary tree of size 0 with only one leaf. Any other Dyck word can be written as (formula_21)formula_22, where formula_21,formula_22 are themselves (possibly empty) Dyck words and where the two written parentheses are matched. The bijection is then defined by letting the words formula_21 and formula_22 correspond to the binary trees that are the left and right children of the root.  A bijective correspondence can also be defined as follows: enclose the Dyck word in an extra pair of parentheses, so that the result can be interpreted as a Lisp list expression (with the empty list () as only occurring atom); then the dotted-pair expression for that proper list is a fully parenthesized expression (with NIL as symbol and '.' as operator) describing the corresponding binary tree (which is in fact the internal representation of the proper list).  The ability to represent binary trees as strings of symbols and parentheses implies that binary trees can represent the elements of a free magma on a singleton set.

Methods for storing binary trees.

Binary trees can be constructed from programming language primitives in several ways.

Nodes and references.

In a language with records and references, binary trees are typically constructed by having a tree node structure which contains some data and references to its left child and its right child. Sometimes it also contains a reference to its unique parent. If a node has fewer than two children, some of the child pointers may be set to a special null value, or to a special sentinel node.  This method of storing binary trees wastes a fair bit of memory, as the pointers will be null (or point to the sentinel) more than half the time; a more conservative representation alternative is threaded binary tree.  In languages with tagged unions such as ML, a tree node is often a tagged union of two types of nodes, one of which is a 3-tuple of data, left child, and right child, and the other of which is a "leaf" node, which contains no data and functions much like the null value in a language with pointers. For example, the following line of code in OCaml (an ML dialect) defines a binary tree that stores a character in each node.  type chr_tree = Empty | Node of char * chr_tree * chr_tree

Arrays.

Binary trees can also be stored in breadth-first order as an implicit data structure in arrays, and if the tree is a complete binary tree, this method wastes no space. In this compact arrangement, if a node has an index "i", its children are found at indices formula_27 (for the left child) and formula_28 (for the right), while its parent (if any) is found at index "formula_29" (assuming the root has index zero). This method benefits from more compact storage and better locality of reference, particularly during a preorder traversal. However, it is expensive to grow and wastes space proportional to 2 - "n" for a tree of depth "h" with "n" nodes.  This method of storage is often used for binary heaps.

Encodings.

Succinct encodings.

A succinct data structure is one which occupies close to minimum possible space, as established by information theoretical lower bounds. The number of different binary trees on formula_1 nodes is formula_31, the formula_1th Catalan number (assuming we view trees with identical "structure" as identical). For large formula_1, this is about formula_34; thus we need at least about formula_35 bits to encode it. A succinct binary tree therefore would occupy formula_36 bits.  One simple representation which meets this bound is to visit the nodes of the tree in preorder, outputting "1" for an internal node and "0" for a leaf. If the tree contains data, we can simply simultaneously store it in a consecutive array in preorder. This function accomplishes this:  The string "structure" has only formula_37 bits in the end, where formula_1 is the number of (internal) nodes; we don't even have to store its length. To show that no information is lost, we can convert the output back to the original tree like this:  More sophisticated succinct representations allow not only compact storage of trees but even useful operations on those trees directly while they're still in their succinct form.

Encoding general trees as binary trees.

There is a one-to-one mapping between general ordered trees and binary trees, which in particular is used by Lisp to represent general ordered trees as binary trees. To convert a general ordered tree to binary tree, we only need to represent the general tree in left-child right-sibling way. The result of this representation will automatically be a binary tree, if viewed from a different perspective. Each node "N" in the ordered tree corresponds to a node "N' " in the binary tree; the "left" child of "N' " is the node corresponding to the first child of "N", and the "right" child of "N' " is the node corresponding to "N"'s next sibling --- that is, the next node in order among the children of the parent of "N". This binary tree representation of a general order tree is sometimes also referred to as a left-child right-sibling binary tree (also known as LCRS tree, doubly chained tree, filial-heir chain).  One way of thinking about this is that each node's children are in a linked list, chained together with their "right" fields, and the node only has a pointer to the beginning or head of this list, through its "left" field.  For example, in the tree on the left, A has the 6 children {B,C,D,E,F,G}. It can be converted into the binary tree on the right.  The binary tree can be thought of as the original tree tilted sideways, with the black left edges representing "first child" and the blue right edges representing "next sibling". The leaves of the tree on the left would be written in Lisp as:  which would be implemented in memory as the binary tree on the right, without any letters on those nodes that have a left child.

Common operations.

There are a variety of different operations that can be performed on binary trees. Some are mutator operations, while others simply return useful information about the tree.

Insertion.

Nodes can be inserted into binary trees in between two other nodes or added after a leaf node. In binary trees, a node that is inserted is specified as to which child it is.

Leaf nodes.

To add a new node after leaf node A, A assigns the new node as one of its children and the new node assigns node A as its parent.

Internal nodes.

Insertion on internal nodes is slightly more complex than on leaf nodes. Say that the internal node is node A and that node B is the child of A. (If the insertion is to insert a right child, then B is the right child of A, and similarly with a left child insertion.) A assigns its child to the new node and the new node assigns its parent to A. Then the new node assigns its child to B and B assigns its parent as the new node.

Deletion.

Deletion is the process whereby a node is removed from the tree. Only certain nodes in a binary tree can be removed unambiguously.

Node with zero or one children.

Suppose that the node to delete is node A. If A has no children, deletion is accomplished by setting the child of A's parent to null. If A has one child, set the parent of A's child to A's parent and set the child of A's parent to A's child.

Node with two children.

In a binary tree, a node with two children cannot be deleted unambiguously. However, in certain binary trees (including binary search trees) these nodes "can" be deleted, though with a rearrangement of the tree structure.

Traversal.

Pre-order, in-order, and post-order traversal visit each node in a tree by recursively visiting each node in the left and right subtrees of the root.

Depth-first order.

In depth-first order, we always attempt to visit the node farthest from the root node that we can, but with the caveat that it must be a child of a node we have already visited. Unlike a depth-first search on graphs, there is no need to remember all the nodes we have visited, because a tree cannot contain cycles. Pre-order is a special case of this. See depth-first search for more information.

Breadth-first order.

Contrasting with depth-first order is breadth-first order, which always attempts to visit the node closest to the root that it has not already visited. See breadth-first search for more information. Also called a "level-order traversal".  In a complete binary tree, a node's breadth-index ("i" − (2 − 1)) can be used as traversal instructions from the root. Reading bitwise from left to right, starting at bit "d" − 1, where "d" is the node's distance from the root ("d" = ⌊log2("i"+1)⌋) and the node in question is not the root itself ("d" > 0). When the breadth-index is masked at bit "d" − 1, the bit values 0 and 1 mean to step either left or right, respectively. The process continues by successively checking the next bit to the right until there are no more. The rightmost bit indicates the final traversal from the desired node's parent to the node itself. There is a time-space trade-off between iterating a complete binary tree this way versus each node having pointer/s to its sibling/s.

In computer science, a binary tree is a tree data structure in which each node has at most two children, which are referred to as the'and the '. A recursive definition using just set theory notions is that a (non-empty) binary tree is a tuple ("L", "S", "R"), where "L" and "R" are binary trees or the empty set and "S" is a singleton set. Some authors allow the binary tree to be the empty set as well.

Binärbäume sind in der Informatik die am häufigsten verwendete Unterart der Bäume. Im Gegensatz zu anderen Arten von Bäumen können die Knoten eines Binärbaumes nur höchstens zwei direkte Nachkommen haben.


Role.

As the Constitution of Austria (B-VG) draws a strict distinction between federal and state legislation, its Article 42 provides the "Bundesrat" only with the right to veto federal laws passed by the National Council. Moreover, in most cases a Federal Council's veto is just suspensive, meaning the National Council can override it, passing the law again by ordinary resolution of at least half of its members. Therefore, the decisions of the "Bundesrat" can only delay legislation.  In the following cases, though, the Federal Council's approval is mandatory:  Since its inauguration on 10 November 1920, the deputies of "Bundesrat" have never achieved the status of a counterbalance in relation to the National Council. Over the decades the role of the Federal Council as a mere adjunct of the Austrian parliament has led to several discussions upon regulatory reforms, towards an actual representation of the states' governments modeled on the German Bundesrat or the complete abolition of the second chamber. So far, the concept has been maintained as a manifestation of Austria's federal system.  The Federal Council and the National Council, if in joint session, form a third parliamentary body: the Federal Assembly that convenes for the oath of office of the President of Austria.

Composition.

The 61 members of the Federal Council ("Mitglieder des Bundesrats", colloquially called "Bundesräte") are elected according to proportional representation by each of the Austrian states' legislatures ("Landtage") for 5- to 6-year terms. The composition of the "Bundesrat" therefore changes after every state election and the distribution of seats in the Austrian Landtage. The second largest faction of the particular "Landtag" has the right to designate at least one deputy. The number of representatives delegated by each "Bundesland" ranges between three and twelve, depending on its population as ascertained by a regular census; it is fixed per presidential decree.  The deputies may ally along party lines and form parliamentary groups, which have to meet a quorum of five seats, if not admitted by particular resolution. Currently the MPs of the Austrian People's Party (ÖVP), the Social Democratic Party of Austria (SPÖ), the Freedom Party of Austria (FPÖ) and the GREENS form political groups in the "Bundesrat": The President of the Federal Council is nominated by the largest party of each state in half-yearly intervals.

Presidium.

The Federal Council is made up of a president (elected for a term of half a year) and 2 vice-presidents.

The Federal Council (, ) is the upper house of the Austrian Parliament, representing the nine States of Austria on federal level. As part of a bicameral legislature alongside of the National Council, it can be compared with an upper house or a senate. In fact, however, it is far less powerful than the National Council: although it has to approve every new law decided for by this "lower" chamber, the latter can—in most cases—overrule the Federal Council's refusal to approve.

Der Bundesrat bildet in Österreich neben dem Nationalrat die zweite Kammer des österreichischen Parlaments. Er ist der Vertretungskörper der Bundesländer auf Bundesebene. Der Vorsitzende des Bundesrates wird als Bundesratspräsident bezeichnet. Die Mandatare führen den Titel "Mitglied des Bundesrates", gebräuchlich sind aber auch die Bezeichnungen Bundesrat beziehungsweise Bundesrätin.


Biography.

Early years.

Otto Grotewohl was born on 11 March 1894 in Braunschweig, the son of a master tailor. Grotewohl started his political career following World War I as a leader of the Independent Social Democratic Party of Germany (USPD) and in 1920 was elected to the "Landtag" of the Free State of Brunswick in the Weimar Republic. Grotewohl served as a minister in numerous cabinets of the Brunswick state government, including Minister of Justice and Education from March to May 1922, and Minister of Justice from February 1923.  In 1922, Grotewohl and the majority of the USPD members joined the Social Democratic Party of Germany (SPD), and on 31 October 1925 he became a member of parliament in the national Reichstag to replace the SPD representative Elise Bartles after her death. Grotewohl was elected to the Reichstag in his own right in the September 1930 election, and re-elected in the July 1932, November 1932, and March 1933 elections.

Nazi era.

Grotewohl was eventually dismissed as a representative in the Reichstag after the "Machtergreifung", the establishment of Nazi Germany, and like other SPD members was subject to discrimination. On 23 March 1933, Grotewohl had voted against Chancellor Adolf Hitler's Enabling Act, a constitutional amendment allowing Hitler to enact laws without the Reichstag's approval, which passed. Grotewohl was imprisoned several times and forced to leave Braunschweig, first moving to Hamburg then from 1938 to Berlin, where he worked as a grocer and industrial representative. Grotewohl joined a resistance group centered around Erich Gniffke, an SPD politician he knew from Braunschweig, but the group ended up ensuring the contact and economic survival of its members rather than resisting Nazi rule. In August 1938, Grotewohl was arrested and charged with treason before the People's Court. On March 4, 1939, Grotewohl was released from pre-trial detention and the court's procedure against him was discontinued after seven months. Grotewohl was arrested after Georg Elser's attempted assassination of Adolf Hitler and other high-ranking Nazis on 8 November 1939, spending about eight weeks in custody before being released. Grotewohl worked as a clerk in Berlin after his release and increasingly devoted his time to painting. Grotewohl had been scheduled for arrest again on 20 July 1944, but the Gestapo was unable to locate him because he was now living off-the-grid. According to Heinz Voßke's 1979 biography of Grotewohl, this lifestyle allowed him to avoid being conscripted into the "Volkssturm" during the closing months of World War II.

Formation of the Socialist Unity Party.

After German defeat in World War II in May 1945, the country was occupied by the Allied forces and divided into four zones governed by the United States, the Soviet Union, the United Kingdom, and France, respectively. Grotewohl and several other former SPD politicians founded a branch of the re-established Social Democratic Party of Germany in the Soviet Occupation Zone, and he became the branch leader as Chairman of the Central Committee. Immediately after the war, the Soviets believed the Communist Party of Germany (KPD), rebuilt by the "Ulbricht Group" and led by Wilhelm Pieck, would naturally develop into the strongest political force in their zone with some guidance. However, Soviet leader Joseph Stalin and KPD deputy leader Walter Ulbricht began to push for party unification after the poor performance of communist parties in elections in Hungary and Austria in November 1945. The SPD in the Soviet zone faced increasing pressure from the Soviet Military Administration (SVAG) to merge with the KPD, despite historic animosity between the two parties. Unification was pushed by some members of Grotewohl's SPD in the Soviet zone and Berlin, under the belief that division between the main left-wing parties had led to Nazi rise to power. Grotewohl opposed the merger, but under duress from Ulbricht and SVAG soon yielded and became a proponent of a quick unification. Grotewohl's decision was fiercely opposed by Kurt Schumacher, a prominent member of the eastern SPD and anti-communist, who subsequently became leader of the western SPD after the merger.  In April 1946, the KPD and the eastern branch of the SPD merged as the Socialist Unity Party (SED), with Pieck and Grotewohl serving as co-Chairmen. Grotewohl's hand appeared on the SED's "handshake" logo derived from the SPD-KPD congress establishing the party where he symbolically shook hands with Pieck. Grotewohl's position allowed him to avoid the systematic sidelining and exclusion of former SPD members that began soon after the merger. Former KPD members almost exclusively held the influential positions except for Grotewohl, whose supporters were now condemned as "Agents of Schumacher", and consolidated the SED as a communist party. In 1948, Grotewohl became Chairman of the constitutional committee of the German People's Council, the predecessor of the "Volkskammer".

Prime Minister.

On 12 October 1949, Grotewohl became the first prime minister ("Ministerpräsident") of the German Democratic Republic (commonly known as East Germany or the GDR), five days after its establishment from the Soviet Occupation Zone with the SED as the ruling party. Grotewohl was appointed Chairman of the Council of Ministers ("Ministerrat"), the "de jure" government of the GDR, while Pieck served as State President. Although Grotewohl and Pieck were officially on equal footing, Grotewohl wielded far more real political power in state affairs than Pieck. In the East German political hierarchy, the prime minister was the highest state official, with the president nominally ranking second. Thus, for the GDR's first year of existence, Grotewohl was the most powerful politician in the country.  Grotewohl's power was significantly reduced in July 1950, when the SED restructured along more orthodox Soviet lines. Ulbricht became First Secretary of the SED's Central Committee, the "de facto" government of the GDR, and thus the "de facto" leader of East Germany. Grotewohl remained as Chairman of the Council of Ministers and the official head of government without challenge from the SED, but effectively became a figurehead with little power in state affairs. Grotewohl was a member of the delegation that signed the Treaty of Zgorzelec on the recognition of the Oder-Neisse border as a border between the GDR and the People's Republic of Poland. In 1957, Grotewohl advocated for the Rapacki Plan for a nuclear weapon-free zone in Central Europe.  Unlike Ulbricht and most of his other SED colleagues, Grotewohl was known to openly favour a less repressive way of governing. Grotewohl condemned abuses in the legal system in a major speech at the SED party conference on 28 March 1956. Grotewohl also denounced illegal arrests, called for more respect for civil rights, and even asked the parliament to develop lively debate. Grotewohl also made a veiled criticism of Justice Minister Hilde Benjamin's notoriously heavy-handed handling of political trials. Despite his open criticism of the SED's increasingly heavy-handed rule, Grotewohl retained his posts without reprisal because the Soviet leadership trusted in him.

Last years and death.

Grotewohl, who was 55 years old upon coming to power, faced rapidly declining health during his premiership. Grotewohl was repeatedly taken to a government hospital during the 1950s, typically minor examinations in which he was released on the same day, but also multi-day stays. However, Grotewohl was not only examined by specialist physicians in the GDR, who identified arteriosclerosis and incipient calcification of the coronary system in his heart in 1953, but also took advantage of the medical care of top Soviet politicians in Moscow. On 12 November 1953, Grotewohl visited the Kremlin Polyclinic at the Kremlin in Moscow. Afterwards, Grotewohl completed a three-and-a-half-week cure on the Black Sea. He reportedly took advantage of these unofficial stays in Moscow to conduct political talks with the Kremlin, but there are no official records. From 1955, Grotewohl's doctors were worried about the condition of his cardiovascular system. In 1959, they finally diagnosed incipient heart failure and pushed for a reduction in workload. Due to the persistent high blood pressure and the chronic arrhythmia, the physicians feared a heart attack.  In 1960, Grotewohl was diagnosed with leukemia, and the course of the year saw his health deteriorate rapidly to the point he was barely able to participate in daily political business. On 4 April 1960, Grotewohl traveled to a four-week relaxing holiday on the Black Sea; eight months later, he arrived again for several weeks in the Soviet sanatorium in Barwicha. After his return from the Soviet Union, he reluctantly moved with his wife from Pankow to Wandlitz, giving in to a previous decision from Ulbricht. At the end of October 1960, Grotewohl had commissioned his top deputy, Willi Stoph, as acting prime minister, although he officially remained in office. The permanent cardiovascular disorders prevented Grotewohl's return to politics and he was no longer able to participate actively in the meetings of the leadership committees of the party and the government. As his eyesight also faded, he could no longer read any script, which is why there are hardly any public speeches from him at the beginning of 1961. Despite the clear medical situation, his resignation was out of the question. On the contrary, in September 1960 Grotewohl became Deputy Chairman of the State Council, the collective body that was created on the base of the Presidency which was abolished after Wilhelm Pieck's death in 1960.  Grotewohl died on September 21, 1964, at 12:35 noon, in the Niederschönhausen area of Pankow, East Berlin, from the complications of a brain haemorrhage. A few hours later, the GDR flag on the Brandenburg Gate was lowered half-mast and the Deutscher Fernsehfunk broadcasting was interrupted. The GDR Council of Ministers ordered 3 day mourning period and Grotewohl was lying in state in the SED Headquarters. On the 15th of October, his ashes were placed in an urn at the Zentralfriedhof Friedrichsfelde.

Legacy.

After his death, the "Wilhelmstrasse" in East Berlin was renamed "Otto-Grotewohl-Straße" in his honor; the street retained this name until 1991, following German reunification. On 15 April 1986, the present-day Mohrenstraße U-Bahn station in eastern Berlin, then known as the "Thälmannplatz" station, was also renamed "Otto-Grotewohl-Straße". The Third German School in Chapayesky Lane, Moscow, was named the Otto Grotewohl School.

Personal life.

Grotewohl married to Marie Martha Louise (née Ohst) from 1919 until 1949. Grotewohl and Ohst had two children, a son Hans and daughter Dorle. Hans Grotewohl (1924–1999), was an architect who was sent by his father to lead a German Work Team for rebuilding Hamhung, North Korea, in 1954 after the Korean War. Grotewohl married his secretary, Johanna Schumann (née Danielzik), the same year as his divorce from Ohst. Grotewohl was an avid painter and amateur filmmaker.

Otto Grotewohl (; 11 March 1894 – 21 September 1964) was a German politician who served as the first prime minister of the German Democratic Republic (GDR or East Germany) from 1949 to 1964.

Otto Emil Franz Grotewohl (* 11. März 1894 in Braunschweig; † 21. September 1964 in Ost-Berlin) war ein deutscher Politiker (SPD, ab 1946 SED). Er war von 1949 bis 1964 Ministerpräsident der Deutschen Demokratischen Republik.


History.

The letter 'G' was introduced in the Old Latin period as a variant of 'C' to distinguish voiced from voiceless. The recorded originator of 'G' is freedman Spurius Carvilius Ruga, the first Roman to open a fee-paying school, who taught around 230 BCE. At this time, 'K' had fallen out of favor, and 'C', which had formerly represented both and before open vowels, had come to express in all environments.  Ruga's positioning of 'G' shows that alphabetic order related to the letters' values as Greek numerals was a concern even in the 3rd century BC. According to some records, the original seventh letter, 'Z', had been purged from the Latin alphabet somewhat earlier in the 3rd century BC by the Roman censor Appius Claudius, who found it distasteful and foreign. Sampson (1985) suggests that: "Evidently the order of the alphabet was felt to be such a concrete thing that a new letter could be added in the middle only if a'space' was created by the dropping of an old letter." The 3rd-century-BC addition of the letter G to the Roman alphabet is credited to Spurius Carvilius Ruga.  George Hempl proposed in 1899 that there never was such a "space" in the alphabet and that in fact 'G' was a direct descendant of zeta. Zeta took shapes like ⊏ in some of the Old Italic scripts; the development of the monumental form 'G' from this shape would be exactly parallel to the development of 'C' from gamma. He suggests that the pronunciation > was due to contamination from the also similar-looking 'K'.  Eventually, both velar consonants and developed palatalized allophones before front vowels; consequently in today's Romance languages, and have different sound values depending on context (known as hard and soft C and hard and soft G). Because of French influence, English language orthography shares this feature.

Typographic variants.

The modern lowercase 'g' has two typographic variants: the single-storey (sometimes "opentail") 'g' and the double-storey (sometimes "looptail") 'g'. The single-storey form derives from the majuscule (uppercase) form by raising the serif that distinguishes it from 'c' to the top of the loop, thus closing the loop, and extending the vertical stroke downward and to the left. The double-storey form (g) had developed similarly, except that some ornate forms then extended the tail back to the right, and to the left again, forming a closed bowl or loop. The initial extension to the left was absorbed into the upper closed bowl. The double-storey version became popular when printing switched to "Roman type" because the tail was effectively shorter, making it possible to put more lines on a page. In the double-storey version, a small top stroke in the upper-right, often terminating in an orb shape, is called an "ear".  Generally, the two forms are complementary, but occasionally the difference has been exploited to provide contrast. In the International Phonetic Alphabet, opentail has always represented a voiced velar plosive, while was distinguished from and represented a voiced velar fricative from 1895 to 1900. In 1948, the Council of the International Phonetic Association recognized and as typographic equivalents, and this decision was reaffirmed in 1993. While the 1949 "Principles of the International Phonetic Association" recommended the use of for a velar plosive and for an advanced one for languages where it is preferable to distinguish the two, such as Russian, this practice never caught on. The 1999 "Handbook of the International Phonetic Association", the successor to the "Principles", abandoned the recommendation and acknowledged both shapes as acceptable variants.  Wong et al. (2018) found that native English speakers have little conscious awareness of the looptail 'g' (). They write: "Despite being questioned repeatedly, and despite being informed directly that G has two lowercase print forms, nearly half of the participants failed to reveal any knowledge of the looptail 'g', and only 1 of the 38 participants was able to write looptail 'g' correctly."

Use in writing systems.

English.

In English, the letter appears either alone or in some digraphs. Alone, it represents  In words of Romance origin, is mainly soft before (including the digraphs and ),, or, and hard otherwise. Soft is also used in many words that came into English through medieval or modern Romance languages from languages without soft (like Ancient Latin and Greek) (e.g. "fragile" or "logic"). There are many English words of non-Romance origin where is hard though followed by or (e.g. "get", "gift"), and a few in which is soft though followed by such as "gaol" or "margarine".  The double consonant has the value (hard ) as in "nugget", with very few exceptions: in "suggest" and in "exaggerate" and "veggies".  The digraph has the value (soft ), as in "badger". Non-digraph can also occur, in compounds like "floodgate" and "headgear".  The digraph may represent Non-digraph also occurs, with possible values  The digraph (in many cases a replacement for the obsolete letter yogh, which took various values including,, and ) may represent Non-digraph also occurs, in compounds like "foghorn", "pigheaded"  The digraph may represent Non-digraph also occurs, as in "signature", "agnostic"  The trigraph has the value as in "gingham" or "dinghy". Non-trigraph also occurs, in compounds like "stronghold" and "dunghill".  G is the tenth least frequently used letter in the English language (after Y, P, B, V, K, J, X, Q, and Z), with a frequency of about 2.02% in words.

Other languages.

Most Romance languages and some Nordic languages also have two main pronunciations for, hard and soft. While the soft value of varies in different Romance languages ( in French and Portuguese, in Catalan, in Italian and Romanian, and in most dialects of Spanish), in all except Romanian and Italian, soft has the same pronunciation as the.  In Italian and Romanian, is used to represent before front vowels where would otherwise represent a soft value. In Italian and French, is used to represent the palatal nasal, a sound somewhat similar to the in English "canyon". In Italian, the trigraph, when appearing before a vowel or as the article and pronoun "gli", represents the palatal lateral approximant.  Other languages typically use to represent regardless of position.  Amongst European languages, Czech, Dutch, Finnish, and Slovak are an exception as they do not have in their native words. In Dutch, represents a voiced velar fricative instead, a sound that does not occur in modern English, but there is a dialectal variation: many Netherlandic dialects use a voiceless fricative ( or ) instead, and in southern dialects it may be palatal. Nevertheless, word-finally it is always voiceless in all dialects, including the standard Dutch of Belgium and the Netherlands. On the other hand, some dialects (like Amelands) may have a phonemic.  Faroese uses to represent, in addition to, and also uses it to indicate a glide.  In Māori, is used in the digraph which represents the velar nasal and is pronounced like the in "singer".  In older Czech and Slovak orthographies, was used to represent, while was written as ( with caron).

G or g is the seventh letter of the ISO basic Latin alphabet. Its name in English is "gee" (pronounced ), plural "gees".

G bzw. g (gesprochen: []) ist der siebte Buchstabe des klassischen und modernen lateinischen Alphabets. Er wurde um 230 v. Chr. von dem römischen Schreibschulbetreiber Spurius Carvilius Ruga als Variante des C eingeführt. Der Buchstabe G hat in deutschen Texten eine durchschnittliche Häufigkeit von 3,01 %. Er ist damit der 13.-häufigste Buchstabe in deutschen Texten.


Etymology.

The name of the castle is probably derived from, a watchtower, in spite of a tradition which holds that the castle's founder, on first laying eyes on the site, exclaimed, ""Warte, Berg—du sollst mir eine Burg tragen!" ("Wait, mountain—you shall bear my castle!"). It is a German play on words for mountain ("Berg") and fortress ("Burg"), coupled with a further play on "warte" = wait and "Warte" = watchtower.

Location.

Wartburg is located on a precipice to the southwest of, and overlooking the town of Eisenach, in the state of Thuringia, Germany. The hill is an extension of Thuringian Forest, overlooking "Mariental" to the south-east and the valley of the Hörsel to the north, through which passed the historical Via Regia. The Rennsteig passes not far to the south of the castle.

History.

The castle's foundation was laid about 1067 by the Thuringian count of Schauenburg, Louis the Springer ( "Ludwig der Springer" ), a relative of the Counts of Rieneck in Franconia. Together with its larger sister castle Neuenburg in the present-day town of Freyburg, the Wartburg secured the extreme borders of his traditional territories. Louis the Springer is said to have had clay from his lands transported to the top of the hill, which was not quite within his lands, so he might swear that the castle was built on his soil.  The castle was first mentioned in a written document in 1080 by Bruno, Bishop of Merseburg, in his "De Bello Saxonico" ("The Saxon War") as "Wartberg".  During the Investiture Controversy, Louis's henchmen attacked a military contingent of King Henry IV of Germany. The count remained a fierce opponent of the Salian rulers, and upon the extinction of the line, his son Louis I was elevated to the rank of a Landgrave in Thuringia by the new German king Lothair of Supplinburg in 1131.  From 1172 to 1211, the Wartburg was one of the most important princes' courts in the German Reich. Hermann I supported poets like Walther von der Vogelweide and Wolfram von Eschenbach who wrote part of his "Parzival" here in 1203.  The castle thus became the setting for the legendary "Sängerkrieg", or Minstrels' Contest in which such "Minnesänger" as Walther von der Vogelweide, Wolfram von Eschenbach, Albrecht von Halberstadt (the translator of Ovid) and many others supposedly took part in 1206/1207. The legend of this event was later used by Richard Wagner in his opera Tannhäuser.  At the age of four, St. Elisabeth of Hungary was sent by her mother to the Wartburg to be raised to become consort of Landgrave Ludwig IV of Thuringia. From 1211 to 1228, she lived in the castle and was renowned for her charitable work. In 1221, Elisabeth married Ludwig. In 1227, Ludwig died on the Crusade and she followed her confessor Father Konrad to Marburg. Elisabeth died there in 1231 at the age of 24 and was canonized as a saint of the Roman Catholic Church just five years after her death.  In 1247, Heinrich Raspe, the last landgrave of Thuringia of his line and an anti-king of Germany, died at the Wartburg. He was succeeded by Henry III, Margrave of Meissen.  In 1320, substantial reconstruction work was done after the castle had been damaged in a fire caused by lightning in 1317 or 1318. A chapel was added to the Palas.  The Wartburg remained the seat of the Thuringian landgraves until 1440.  From May 1521 to March 1522, Martin Luther stayed at the castle under the name of "Junker Jörg" (the Knight George), after he had been taken there for his safety at the request of Frederick the Wise following his excommunication by Pope Leo X and his refusal to recant at the Diet of Worms. It was during this period that Luther translated the New Testament from ancient Greek into German in just ten weeks. Luther's was not the first German translation of the Bible but it quickly became the most well known and most widely circulated.  From 1540 until his death in 1548,, an Anabaptist farmer from Herda, was held captive in the dungeon of the south tower, because he refused to abjure anabaptism. After his death, he was buried in the Wartburg near the chapel of St. Elisabeth. In 1925, a handwritten signature of Fritz Erbe was found on the prison wall.  Over the next centuries, the castle fell increasingly into disuse and disrepair, especially after the end of the Thirty Years' War when it had served as a refuge for the ruling family.  In 1777, Johann Wolfgang von Goethe stayed at the Wartburg for five weeks, making various drawings of the buildings.  On 18 October 1817, the first Wartburg festival took place. About 500 students, members of the newly founded German Burschenschaften ("fraternities"), came together at the castle to celebrate the German victory over Napoleon four years before and the 300th anniversary of the Reformation, condemn conservatism and call for German unity under the motto "Honour - Freedom - Fatherland". Speakers at the event included Heinrich Hermann Riemann, a veteran of the Lützow Free Corps, the philosophy student Ludwig Rödiger, and Hans Ferdinand Massmann.  With the permission of the absent chaplain Friedrich Ludwig Jahn, the Code Napoléon and other books were burned 'in effigy': instead of the costly volumes, scraps of parchment with the titles of conservative books (including August von Kotzebue's "History of the German Empires") were placed on the bonfire. Karl Ludwig Sand, who would assassinate Kotzebue two years later, was among the participants.  This event and a similar gathering at Wartburg during the Revolutions of 1848 are considered seminal moments in the movement for German unification.  During the rule of the House of Saxe-Weimar-Eisenach, Grand Duke Karl Alexander ordered the reconstruction of Wartburg in 1838. The lead architect was, for whom it became a life's work. In fact, it was finished only a year after his death in 1889.  Drawing on a suggestion by Goethe that the Wartburg would serve well as a museum, Maria Pavlovna and her son Karl Alexander also founded the art collection ("Kunstkammer") that became the nucleus of today's museum.  The reign of the House of Saxe-Weimar-Eisenach ended in the German Revolution in 1918. In 1922, the "Wartburg Stiftung" (Wartburg Foundation) was established to ensure the castle's maintenance.  After the end of World War II, Soviet occupation forces took the renowned collection of weapons and armour. Its whereabouts still remain unknown. The "" (the armoury) of the Wartburg used to contain a notable collection of about 800 pieces, from the splendid armour of King Henry II of France, to the items of Frederick the Wise, Pope Julius II and Bernhard von Weimar. All these objects were taken by the Soviet Occupation Army in 1946 and have disappeared in the Soviet Union. Two helmets, two swords, a prince's and a boy's armour, however, were found in a temporary store at the time and a few pieces were given back by the USSR in the 1960s. The new Russian Government has been petitioned to help locate the missing treasures.  Under communist rule during the time of the GDR extensive reconstruction took place in 1952-54. In particular, much of the palas was restored to its original Romanesque style. A new stairway was erected next to the palas.  In 1967, the castle was the site of celebrations of the GDR's national jubilee, the 900th anniversary of the Wartburg's foundation, the 450th anniversary of the beginning of Luther's Reformation and the 150th anniversary of the Wartburg Festival.  On June 13, 1980, Devo performed at the castle during their Freedom of Choice tour.  In 1983, it was the central point of the celebrations on account of the 500th birthday of Martin Luther.

Architecture.

Palas.

The largest structure of the Wartburg is the "Palas", originally built in late Romanesque style between 1157 and 1170. It is considered the best-preserved non-ecclesial Romanesque building north of the Alps.  The Palas features rooms like the "Rittersaal" and the "Speisesaal" which have been reconstructed as closely as possible to the original Romanesque style and which contain original structures (pillars or roof elements). However, many of the rooms mostly reflect the tastes of the 19th and 20th centuries and the image of the Middle Ages prevalent at the time: the "Elisabeth-Kemenate" was fitted with mosaics showing the life of St. Elisabeth (created in 1902-06) on behalf of Kaiser Wilhelm II, the "Sängersaal" (with frescoes of the "Sängerkrieg" by Moritz von Schwind) and the "Festssaal" on the top floor. The latter also features frescoes by Schwind (on the triumph of Christianity) and served as the inspiration for the "Sängerhalle" at Neuschwanstein Castle. The Wartburg's "Sängersaal" is the setting for Act II of Tannhäuser. None of the wallpaintings, including those in the "Landgrafenzimmer" or the "Elisabethengalerie", are actually medieval in origin, but were created in the 19th century.

Other structures.

The drawbridge and barbican offer the only access to the castle and have been largely unchanged since medieval times.  "Vorburg" is the area immediately inside the first gate. It dates to the 14th/15th century and is made up of several half-timbered buildings: the "Elisabethengang" (covered walkway), the "Vogtei" (Bailiff's lodge), the "Margarethengang" (covered walkway) and the "Ritterhaus" (Knights' House).  The "Lutherstube" in the "Vogtei", where Martin Luther stayed when he was in the castle, also features paintings by Lucas Cranach.  The "Bergfried" (donjon) was completed in 1859 and sits on the foundations of a medieval keep. It is topped by a landmark three-meter-tall.  The "Neue Kemenate" (New Bower, 1853–1860) today exhibits the art treasures of the Wartburgsammlung, including paintings by Lucas Cranach the Elder and sculptures from the workshop of Tilman Riemenschneider.  The Romanesque "Südturm" or South Tower was built in 1318. Together with the Palas it is the oldest part of the castle. A dungeon is located below.  In 1999, UNESCO added Wartburg Castle to the World Heritage List as an "Outstanding Monument of the Feudal Period in Central Europe", citing its "Cultural Values of Universal Significance".  For a while, the status of Wartburg as a World Heritage site was endangered by plans to build very tall wind turbines on Milmesberg near Marksuhl. However, in November 2013, the investor agreed not to build the turbines and a regional planning update has banned such structures within sight of Wartburg in the future.

Today.

Wartburg is a popular tourist destination, the most-visited site in Thuringia after Weimar. It is accessible to visitors and guided tours offer access to the interior of the buildings. In addition, there is a museum in the castle. Children can ride donkeys up the hill. The "Festsaal" is used regularly for staging the opera "Tannhäuser", as well as concerts and other events. There is also a hotel, located right next to the castle, originally built during the castle's reconstruction in the 19th century.  Bill Clinton, 42nd president of the United States of America, visited the Wartburg castle, as well as the Bach House, on May 14, 1998 during his state visit to Germany.

Legacy.

For centuries, the Wartburg has been a place of pilgrimage for many people from within and outside Germany, for its significance in German history and in the development of Christianity. Several places (especially US towns founded by Lutherans) and a local brand of automobile have been named after the Wartburg. Wartburg College in Iowa, United States, is named in commemoration of Martin Luther's receiving refuge at the castle and because of the college's forest location and its Thuringian heritage. Wartburg Theological Seminary, also located in Iowa was named in commemoration of Wartburg Castle.  The Wartburgkreis is named after the castle, although Wartburg is located outside the district. Eisenach, originally part of the district, became "kreisfrei" ("district-free") in 1998.

The Wartburg is a castle originally built in the Middle Ages. It is situated on a precipice of to the southwest of and overlooking the town of Eisenach, in the state of Thuringia, Germany. In 1999, UNESCO added Wartburg Castle to the World Heritage List. It was the home of St. Elisabeth of Hungary, the place where Martin Luther translated the New Testament of the Bible into German, the site of the Wartburg festival of 1817 and the supposed setting for the possibly legendary "Sängerkrieg". It was an important inspiration for Ludwig II when he decided to build Neuschwanstein Castle. Wartburg is the most-visited tourist attraction in Thuringia after Weimar. Although the castle today still contains substantial original structures from the 12th through 15th centuries, much of the interior dates back only to the 19th century.

Die Wartburg ist eine Burg in Thüringen, über der Stadt Eisenach am nordwestlichen Ende des Thüringer Waldes gelegen. Sie wurde um 1067 von Ludwig dem Springer gegründet und gehört seit 1999 zum UNESCO-Weltkulturerbe.


History.

KDevelop 0.1 was released in 1998, with 1.0 following in late 1999. 1.x and 2.x were developed over a period of four years from the original codebase.  It is believed that Sandy Meier originated KDevelop. Ralf Nolden is also known to be an early developer of the project. In 1998 Sandy Meier started KDevelop and worked 8 weeks alone on this project. Since then, the KDevelop IDE is publicly available under the GPL and supports many programming languages.  Bernd Gehrmann started a complete rewrite and announced KDevelop 3.x in March 2001. Its first release was together with K Desktop Environment 3.2 in February 2004, and development of KDevelop 3.x continued until 2008.  KDevelop 4.x, another complete rewrite with a more object-oriented programming model, was developed from August 2005 and released as KDevelop 4.0.0 in May 2010. The last feature update of this branch was version 4.7.0 in September 2014, with bugfix releases continuing until KDevelop 4.7.4 in December 2016  KDevelop 5 development began in August 2014 as a continuation of the 4.x codebase, ported to Qt5 and KDE Frameworks 5. The custom C++ parser used in earlier versions, which had poor support for C++11 syntax, was replaced by a new Clang-based backend. The integrated CMakeFile interpreter was also removed in favour of JSON metadata produced by the upstream CMake tool.  Semantic language support was added for QML and JavaScript, using the parser from Qt Creator, alongside a new QMake project-manager backend.  The first stable 5.x release was KDevelop 5.0.0 in August 2016. In October 2016, official Microsoft Windows builds were released for the first time.

Features.

KDevelop uses an embedded text editor component through the KParts framework. The default editor is KDE Advanced Text Editor, which can optionally be replaced with a Qt Designer-based editor. This list focuses on the features of KDevelop itself. For features specific to the editor component, see the article on Kate.  KDevelop 4 is a completely plugin-based architecture. When a developer makes a change, they only must compile the plugin. There is a possibility to keep several profiles each of which determines which plugins to be loaded. KDevelop does not come with a text editor, but instead uses a plugin for this purpose as well. KDevelop is programming language independent and build system-independent, supporting KDE, GNOME, and many other technologies such as Qt, GTK+, and wxWidgets.  KDevelop has supported a variety of programming languages, including C, C++, Python, PHP, Java, Fortran, Ruby, Ada, Pascal, SQL, and Bash scripting. Supported build systems include GNU (automake), cmake, qmake, and make for custom projects (KDevelop does not destroy user Makefiles if they are used) and scripting projects which don't need one.  "Code completion" is available for C and C++. Symbols are kept in a Berkeley DB file for quick lookups without re-parsing. KDevelop also offers a developer framework which helps to write new parsers for other programming languages.  An integrated debugger allows graphically doing all debugging with breakpoints and backtraces. It even works with dynamically loaded plugins unlike command line GDB.  "Quick Open" allows quick navigation between files.  Currently, around 50 to 100 plugins exist for this IDE. Major ones include persistent project-wide code bookmarks, "Code abbreviations" which allow expanding text quickly, a "Source formatter" which reformats code to a "style guide" before saving, "Regular expressions search", and project-wide "search/replace" which helps in refactoring code.

KDevelop is a free and open-source integrated development environment (IDE) for Unix-like computer operating systems and Windows. It provides editing, navigation and debugging features for several programming languages, and integration with build automation and version-control systems, using a plugin-based architecture.

KDevelop ist eine freie grafische Entwicklungsumgebung für Unix-basierte Betriebssysteme und Microsoft Windows. Der Fokus liegt auf der Unterstützung von C++ als Programmiersprache, einige weitere Sprachen werden aber durch Zusatzmodule („Plugins“) ebenfalls unterstützt.


History.

In German, the word may as well refer to Protestant congregational singing as to other forms of vocal (church) music, including Gregorian chant. The English word which derived from this German term, that is "chorale", however almost exclusively refers to the musical forms that originated in the German Reformation.

17th century.

The bulk of Lutheran hymn texts and chorale melodies was created before the end of the 17th century.  Johann Pachelbel's "Erster Theil etlicher Choräle", a set of organ chorales, was published in the last decade of the 17th century. Johann Sebastian Bach's earliest extant compositions, works for organ which he possibly wrote before his fifteenth birthday, include the chorales BWV 700, 724, 1091, 1094, 1097, 1112, 1113 and 1119.

18th century.

In the early 18th century Erdmann Neumeister introduced the cantata format, originally consisting exclusively of recitatives and arias, in Lutheran liturgical music. Within a few years, the format was combined with other pre-existing liturgical formats such as the chorale concerto, resulting in church cantatas that consisted of free poetry, for instance used in recitatives and arias, dicta and/or hymn-based movements: the "Sonntags- und Fest-Andachten" cantata libretto cycle, published in Meiningen in 1704, contained such extended cantata texts. The chorale cantata, called "per omnes versus" (through all verses) when its libretto was an entire unmodified Lutheran hymn, was also a format modernised from earlier types. Dieterich Buxtehude composed six "per omnes versus" chorale settings. BWV 4, an early Bach-cantata composed in 1707, is in this same format. Later, for his 1720s second cantata cycle, Bach developed a chorale cantata format where the inner movements paraphrased (rather than quoted) text of the inner verses of the hymn on which the cantata was based.  Each of the Meiningen cantata librettos contained a single chorale-based movement, on which it ended. Composers of the first half of the 18th century, such as Bach, Stölzel and Georg Philipp Telemann, often closed a cantata with a four-part chorale setting, whether or not the libretto of the cantata already contained verses of a Lutheran hymn. Bach set several of the Meiningen librettos in 1726, and Stölzel expanded the librettos of Benjamin Schmolck's "Saitenspiel" cycle with a closing chorale for each half cantata, when he set that cycle in the early 1720s. Two of such closing chorales by Telemann inadvertently ended up in the "Bach-Werke-Verzeichnis" (BWV): the fifth movements of the cantatas BWV 218 and 219, in the catalogue of Telemann's vocal works adopted as Nos. 1:634/5 and 1:1328/5 respectively. These closing chorales almost always conformed to these formal characteristics: Around 400 of such settings by Bach are known, with the colla parte instrumentation surviving for more than half of them. They do not only appear as closing movements of church cantatas: they can appear in other places in cantatas, even, exceptionally, opening a cantata (BWV 80b). Bach's "Jesu, meine Freude" motet contains several such chorales. Larger-scale compositions, such as Passions and oratorios, often contain multiple four-part chorale settings which in part define the composition's structure: for instance in Bach's "St John" and "St Matthew" Passions they often close units (scenes) before a next part of the narrative follows, and in the "Wer ist der, so von Edom kömmt" Passion pasticcio the narrative is carried by interspersed four-part chorale settings of nearly all stanzas of the "Christus, der uns selig macht" hymn.  Vocal church music of this period also contained other types of chorale settings, the general format of which is indicated as chorale fantasia: one voice, not necessarily the voice with the highest pitch, carries the chorale tune, with the other voices rather contrapuntal than homorhythmic, often with other melodies than the chorale tune, and instrumental interludes between the singing. For instance, the four cantatas with which Bach opened his second cantata cycle each start with a choral movement in chorale fantasia format, where the chorale tune is respectively sung by the soprano (BWV 20, 11 June 1724), alto (BWV 2, 18 June 1724), tenor (BWV 7, 24 June 1724) and bass (BWV 135, 25 June 1724) voices. Chorale fantasia settings are not necessarily choral movements: for instance, the fifth movement of the cantata BWV 10 is a duet for alto and tenor voices in that format. Quarter of a century after Bach had composed that duet, he published it in an arrangement for organ, as fourth of the "Schübler Chorales", showing that the chorale fantasia format adapts itself very well to purely instrumental genres such as the chorale prelude for organ. Around 200 of Bach's chorale preludes are extant, many of them in the chorale fantasia format (others are fugues, or homorhythmic settings).  In the first half of the 18th century, chorales also appear in (music performance in family circle), e.g. BWV 299 in "Notebook for Anna Magdalena Bach", and/or are used for didactical purposes, e.g. BWV 691 in the.  Most of Bach's four-part chorales, around 370 of them, were published for the first time between 1765 and 1787: these were the only works by the composer published between "The Art of Fugue" (1751) and the 50th anniversary of the composer's death in 1800. In the late 18th century symphonies could include a chorale movement: for instance the third movement of Joseph Martin Kraus's 1792 "Symphonie funèbre" is a chorale on (the Swedish version of) "Nun lasst uns den Leib begraben".

19th century.

Early in the 19th century Ludwig van Beethoven chose a chorale-like ending for his Sixth Symphony (1808). Chorale analogies are even stronger in the choral finale of his Ninth Symphony (1824). Felix Mendelssohn, champion of the 19th-century Bach Revival, included a chorale ("Ein feste Burg ist unser Gott") in the finale of his "Reformation Symphony" (1830). His first oratorio, "Paulus", which premièred in 1836, featured chorales such as "Allein Gott in der Höh sei Ehr" and "Wachet auf, ruft uns die Stimme". His "Lobgesang" Symphony-Cantata (1840) contained a movement based on the Lutheran chorale "Nun danket alle Gott". Lutheran hymns also appear in the composer's chorale cantatas, some of his organ compositions, and the sketches of his unfinished "Christus" oratorio.  In the first half of the 19th century, chorale-like symphony finales were also composed by Louis Spohr ("Begrabt den Leib in seiner Gruft" concludes his 1832 Fourth Symphony, named "Die Weihe der Töne"), Niels Gade (Second Symphony, 1843) and others. Otto Nicolai wrote concert overtures on "Vom Himmel hoch, da komm ich her" ("Christmas Overture", 1833) and on ""Ein feste Burg ist unser Gott"" ("Ecclesiastical Festival Overture", 1844). Giacomo Meyerbeer set "Ad nos, ad salutarem undam" to a chorale melody of his own invention in his 1849 opera "Le prophète". The chorale tune was the basis for Franz Liszt's organ composition Fantasy and Fugue on the chorale "Ad nos, ad salutarem undam" (1850).  Joachim Raff included Luther's "Ein feste Burg ist unser Gott" in his Overture Op. 127 (1854, revised 1865) and had his Fifth Symphony ("Lenore", Op. 177, 1872) end on a chorale. The Finale of Camille Saint-Saëns's contains a homorhythmic chorale. One of the themes in the Finale of his 1886 Third Symphony, that is the theme that was adopted in the 1978 "If I Had Words" song, is a chorale. Anton Bruckner's 1873 Third Symphony and his 1876 Fifth Symphony both end on a chorale played by brass instruments. Bruckner also used the chorale as a compositional device in Two Aequali. Further, he included chorales in masses and motets (e.g. "Dir, Herr, dir will ich mich ergeben", "In jener letzten der Nächte"), and in part 7 of his festive cantata "Preiset den Herrn". In his setting of Psalm 22 and in the Finale of his Fifth Symphony he used a chorale in contrast to and combination with a fugue. One of the themes in the Finale of Johannes Brahms's First Symphony (1876) is a chorale.  In 1881 Sergei Taneyev described chorale harmonisations, such as those ending Bach's cantatas, rather as a necessary evil: inartistic, but unavoidable, even in Russian church music. From the 1880s Ferruccio Busoni was adopting chorales in his instrumental compositions, often adapted from or inspired by models by Johann Sebastian Bach: for example BV 186 (), an introduction and fugue on "Herzliebster Jesu was hast verbrochen", No. 3 of Bach's "St Matthew Passion". In 1897 he transcribed Liszt's Fantasy and Fugue on the chorale "Ad nos, ad salutarem undam" for piano. César Franck emulated the chorale in compositions for piano ("Prélude, Choral et Fugue", 1884) and for organ ("", 1990). Johannes Zahn published an index and classification of all known Evangelical hymn tunes in six volumes from 1889 to 1893.  A chorale-like theme appears throughout the last movement of Gustav Mahler's Third Symphony (1896): <score vorbis="1"> \relative c' { \clef treble \numericTimeSignature \time 4/4 \key d \major \partial 4*1 a4\pp( | d2 cis4 b) | a( b cis d) | e2( fis4 e) | e2( d4) } </score>

20th to 21st century.

In his Fifth Symphony, the first version of which was composed 1901–1902, Gustav Mahler included a chorale near the end of Part I (2nd movement). The chorale melody reappears in a transformed version in the last movement of the symphony (Part III, 5th movement). Shortly after Mahler had completed the symphony, his wife Alma reproached him to have included a dreary church-like chorale in the work. Mahler replied that Bruckner had included chorales in his symphonies, to which she replied ""Der" darf, du nicht!" ("He" [Bruckner] can do that, you shouldn't). In her memoir, she continues that she then tried to convince her husband that his strength lay elsewhere than in the adoption of churchy chorales in his music.  Busoni continued to compose Bach-inspired chorales in the 20th century, for instance including chorale subsections in his "Fantasia contrappuntistica" (1910s). "Sports et divertissements", written by Erik Satie in 1914, opens with "Choral inappétissant" (unsavoury chorale), in which the composer put, according to his preface, everything he knew about tedium, and which he dedicated to all who disliked him. As with much of Satie's music, it was written down without metre.  Igor Stravinsky included chorales in some of his compositions: among others, a "Little Chorale" and a "Great Chorale" in his "L'Histoire du soldat" (1918) and a chorale concluding his "Symphonies of Wind Instruments" (1920, rev. 1947). "By the leeks of Babylon" is a chorale in "The Seasonings", an oratorio which appeared on "", a 1966 P. D. Q. Bach album. Chorales appear in Olivier Messiaen's music, for instance in "" (1986–1988) and "La ville d'en haut" (1989), two late.  Stand-alone orchestral chorales were adapted from works by Johann Sebastian Bach: for instance Leopold Stokowski orchestrated, among other similar pieces, the sacred song BWV 478 and the fourth movement of the cantata BWV 4 as chorales "Komm, süsser Tod" (recorded 1933) and "Jesus Christus, Gottes Sohn" (recorded 1937) respectively. Recordings of all of Bach's chorales—vocal as well as instrumental—appeared in the three complete works box sets that were issued around the 250th anniversary of the composer's death in 2000.

Types.

Chorale melodies are often in Bar form, that is, consisting of a repeated first phrase, called, and a concluding second phrase. The harmonisation of such a chorale melody may repeat the same harmonisation for both passes of the, or may present a variant harmonisation on the second pass of the first phrase of the melody.

Vocal.

Part song.

Hymnals:  Collections, e.g. Bach's four-part chorale editions  "Colla parte" accompaniment, e.g. closing chorales of Bach-cantatas

Elaborate choral settings.

Chorale fantasia, e.g. opening movement of "St Matthew Passion" (in English rather called Chorus than Chorale)

Monodic with instrumental accompaniment.

Voice and continuo, e.g. "Schemellis Gesangbuch" (1736) – rather called "Lied" in German

Instrumental.

In instrumental chorale settings, as well emulations of four-part homophony, as chorale fantasia type of approaches exist.  Originally "Choralbearbeitung", i.e. setting of a pre-existing chorale melody

Organ.

Chorale preludes, e.g. "Erster Theil etlicher Choräle" (Pachelbel), "Clavier-Übung III" (Bach)  Not based on pre-existing hymn tunes, e.g. César Franck's "Trois chorals"

Orchestra.

In symphonies, e.g. Mendelssohn, Bruckner, Saint-Saëns, Mahler

Other.

Chorales for solo piano are included in, for instance, Franck's "Prélude, Choral et Fugue" (1884), Satie's "Sports et divertissements" (1914, published ), and Busoni's "Fantasia contrappuntistica" (multiple versions, early 1910s). That last composition also exists in the composer's arrangement for two pianos (early 1920s).

Chorale is the name of several related musical forms originating in the music genre of the Lutheran chorale: The chorale originated when Martin Luther translated sacred songs into the vernacular language (German), contrary to the established practice of church music near the end of the first quarter of the 16th century. The first hymnals according to Luther's new method were published in 1524. Luther and his followers not only wrote metrical hymn lyrics, but also composed metrical musical settings for these texts. This music was partially based on established melodies of church hymns and known secular songs. In the 17th century the repertoire was enriched with more choral and organ settings of the chorale tunes. By the end of the century a four-part setting for SATB voices had become the standard for the choral settings, while the congregational singing of chorales was tending towards monody with an instrumental accompaniment. The prolific creation of new Lutheran chorale tunes ended around that time.

Das Wort Choral (lat. Adjektiv "choralis" zu gr. χορός "chorós" „Chor“) bezeichnet ursprünglich die in der Liturgie der Westkirche einstimmige Kirchenmusik, den auch Gregorianischer Choral genannten "cantus Romanus". Im 14. Jahrhundert war vom "cantus choralis sive ecclesiasticus" („chormäßiger oder kirchlicher Gesang“) die Rede, wobei "choralis" sich auf die ausführende Schola bezog. Die Choralmusik unterscheidet sich so von der Figuralmusik. In dieser Bedeutung wird das Wort bis heute im katholischen Bereich verwendet.