Mensural notation

The freeing of ligatures from considerations of context occurred during the early 13th century. Time values for ligatures, single notes, and rests were codified around 1260 by the influential theorist Franco of Cologne.

These note symbols provided the basis for notation from the late 13th to the late 15th century. This system, called mensural notation, was based on several fundamental principles that determined the value of a note relative to that of its neighbours. In the terminology of mensural notation a given note might be either perfect—i.e., divided into three notes of the next lesser time value; or imperfect—i.e., divided into two notes of the next lesser value. Thus, as in part (a) of the example below, a long might be perfect, containing three breves; or imperfect, containing two breves.

The time-value relationships between breve and long similarly applied to the other pairs of notes:

• maxima and long (longa)
• breve and semibreve (semibrevis)
• semibreve and minim (minima)

As the system of mensural notation evolved, another device, coloration, came into use. If a composer wished to render a potentially perfect note imperfect, he could write it in red, or as a hollow note:

...these two devices had, however, various other, less common meanings. About 1400, hollow note shapes were adopted where full black notes had hitherto been used, and full black served as coloration. The notes then current and their corresponding rests were as follows:

Transition to modern staff notation

In 16th-century manuscripts and, later, in printed music, the diamond-headed notes became rounded. Ligatures were used less often in the later 15th century. The principles of perfection and imperfection gave way to the modern relationship of 2 to 1 between adjacent note values, with the dot adding an extra half value to give a 3 to 1 relationship.

20th-century notation

Two developments in musical style in the 20th century have placed great strain upon staff notation: integral serialism—in which the music is controlled by a mathematical system—and indeterminacy, or chance music. In the former, every note in a texture may have its individual dynamic marking and type of attack (for example, in Olivier Messiaen's Mode de valeurs et d'intensités and in parts of Structures I and II of Pierre Boulez). There may also be extensive use of unconventional playing techniques. Since staff notation indicates volume, attack, and technical effects in a comparatively clumsy manner, the written page becomes cluttered and unclear. In indeterminate and aleatory music (the latter type allowing the performer a limited degree of freedom), the notation must offer choices to the performer or be deliberately imprecise. Staff notation is for these purposes often too specific. In addition, electronic music, composed with such devices as graphs, mathematical symbols, and diagrams, is not easily translated into a readable “score” for publication.

New systems

Notation, in face of this, has moved in two directions: toward adaptation of staff notation and toward the devising of new notational systems. Music using microtonal intervals (less than a semitone) has tended to adapt by modifying the standard accidental signs—meaning one-third sharp, two-thirds sharp, and so on (e.g., in Krzysztof Penderecki's Anaklasis).

So-called space time notation is a further adaptation that reasserts the graphic nature of staff notation. It abandons symbolic indication of note values and replaces it by the spacing of note heads horizontally on the staff, accompanied by an instruction such as “1 inch = 1 second.” The principle may then be amplified by using different note heads:

...to signify short, medium, and long sustaining of a note, thus obviating the use of rests, and by beaming together notes to be performed in one breath or bow (e.g., Witold Lutoslawski's Trois poèmes d'Henri Michaux and John Cage's Music of Changes).

Indeterminacy

Indeterminate music requires constant experimentation with notation. A composer may offer directions for one element of the music—as rhythm or pitch contour—and leave the performer to improvise the remaining elements. Or he may simply describe the general character of a passage by resorting to a specially designed symbol, a verbal description, or even an impressionistic drawing (as in Earle Brown's Hodograph). At the extreme, John Cage supplies “materials” (lines, dots, shapes) and leaves the performer to attach musical significance to them. For electronic music, published scores have so far adopted either strict graph form (Karlheinz Stockhausen, Electronic Study II; see illustration) or pictorial form, using patterned drawings to represent different timbres (as in Henri Pousseur's Electre [see ] and Stockhausen's Kontakte).

The inner ear contains parts (the non-auditory labyrinth or vestibular organ) that are sensitive to acceleration in space, rotation, and orientation in the gravitational field. Rotation is signaled by way of the semicircular canals, three bony tubes in each ear that lie embedded in the skull roughly at right angles to each other. These canals are filled with fluid called endolymph; in the ampulla of each canal are receptor cells with fine hairs that project up into the fluid to be displaced as the endolymph lags behind when rotation begins. When rotation is maintained at a steady velocity, the fluid catches up, and stimulation of the hair cells no longer occurs until rotation suddenly stops, again circulating the endolymph. Whenever the hair cells are thus stimulated, one normally experiences a sensation of rotation in space. During rotation one exhibits reflex nystagmus (back-and-forth movement) of the eyes. Slow displacement of the eye occurs against the direction of rotation and serves to maintain the gaze at a fixed point in space; this is followed by a quick return to the initial eye position in the direction of the rotation. Stimulation of the hair cells in the absence of actual rotation tends to produce an apparent “swimming” of the visual field, often associated with dizziness and nausea.

Bone is found only in vertebrates, and, among modern vertebrates, it is found only in bony fish and higher classes. Although ancestors of the cyclostomes and elasmobranchs had armoured headcases, which served largely a protective function and appear to have been true bone, modern cyclostomes have only an endoskeleton, or inner skeleton, of noncalcified cartilage and elasmobranchs a skeleton of calcified cartilage. Although a rigid endoskeleton performs obvious body supportive functions for land-living vertebrates, it is doubtful that bone offered any such mechanical advantage to the teleost (bony fish) in which it first appeared, for in a supporting aquatic environment great structural rigidity is not essential for maintaining body configuration. The sharks and rays are superb examples of mechanical engineering efficiency, and their perseverance from the Devonian Period attests to the suitability of their nonbony endoskeleton.

In modern vertebrates, true bone is found only in animals capable of controlling the osmotic and ionic composition of their internal fluid environment. Marine invertebrates exhibit interstitial fluid compositions essentially the same as that of the surrounding seawater. Early signs of regulability are seen in cyclostomes and elasmobranchs, but only at or above the level of true bone fishes does the composition of the internal body fluids become constant. The mechanisms involved in this regulation are numerous and complex and include both the kidney and the gills. Fresh and marine waters provide abundant calcium but only traces of phosphate; because relatively high levels of phosphate are characteristic of the body fluids of higher vertebrates, it seems likely that a large, readily available internal phosphate reservoir would confer significant independence of external environment on bony vertebrates. With the emergence of terrestrial forms, the availability of calcium regulation became equally significant. Along with the kidney and the various component glands of the endocrine system, bone has contributed to development of internal fluid homeostasis—the maintenance of a constant chemical composition. This was a necessary step for the emergence of terrestrial vertebrates. Furthermore, out of the buoyancy of water, structural rigidity of bone afforded mechanical advantages that are the most obvious features of the modern vertebrate skeleton.

Chemical composition and physical properties

Depending upon species, age, and type of bone, bone cells represent up to 15 percent of the volume of bone; in mature bone in most higher animals, they usually represent only up to 5 percent. The nonliving intercellular material of bone consists of an organic component called collagen (a fibrous protein arranged in long strands or bundles similar in structure and organization to the collagen of ligaments, tendons, and skin), with small amounts of proteinpolysaccharides, glycoaminoglycans (formerly0 known as mucopolysaccharides) chemically bound to protein and dispersed within and around the collagen fibre bundles, and an inorganic mineral component in the form of rod-shaped crystals. These crystals are arranged parallel with the long axes of collagen bundles and many actually lie in voids within the bundles themselves.

Collagen is the structural protein of bones, tendons, ligaments, and skin. For many years collagen was considered to be insoluble in water. Part of the collagen of calf skin, however, can be extracted with citrate buffer at pH 3.7. A precursor of collagen called procollagen is converted in the body into collagen. Procollagen has a molecular weight of 120,000. Cleavage of one or a few peptide bonds of procollagen yields collagen, which has three subunits, each with a molecular weight of 95,000; therefore, the molecular weight of collagen is 285,000 (3 × 95,000). The three subunits are wound as spirals around an elongated straight axis. The length of each subunit is 2,900 angstroms, and its diameter is approximately 15 angstroms. The three chains are staggered, so that the trimer has no definite terminal limits.

Collagen differs from all other proteins in its high content of proline and hydroxyproline. Hydroxyproline does not occur in significant amounts in any other protein except elastin. Most of the proline in collagen is present in the sequence glycine–proline-X, in which X is frequently alanine or hydroxyproline. Collagen does not contain cystine or tryptophan and therefore cannot substitute for other proteins in the diet. The presence of proline causes kinks in the peptide chain and thus reduces the length of the amino acid unit from 3.7 angstroms in the extended chain of the β-structure to 2.86 angstroms in the collagen chain. In the intertwined triple helix, the glycines are inside, close to the axis; the prolines are outside.

(ATP is an) energy-carrying molecule found in the cells of all living things. ATP (adenosine triphosphate) captures chemical energy obtained from the breakdown of food molecules and releases it to fuel other cellular processes.

Cells require chemical energy for three general types of tasks: to drive metabolic reactions that would not occur automatically; to transport needed substances across membranes; and to do mechanical work, such as moving muscles. ATP is not a storage molecule for chemical energy; that is the job of carbohydrates, such as glycogen, and fats. When energy is needed by the cell, it is converted from storage molecules into ATP. ATP then serves as a shuttle, delivering energy to places within the cell where energy-consuming activities are taking place.

ATP is a nucleotide that consists of three main structures: the nitrogenous base, adenine; the sugar, ribose; and a chain of three phosphate groups bound to ribose.

First, an atom may complete its octet by sharing more than one pair of electrons with a bonded neighbour. Two shared pairs of electrons, represented by a double dash (=), form a double bond. Double bonds are found in numerous compounds, including carbon dioxide:

Three shared pairs of electrons are represented by a triple dash (≡) and form a triple bond. Triple bonds are found in, for example, carbon monoxide, nitrogen molecules, and acetylene, shown, respectively, as:

A double bond is stronger than a single bond, and a triple bond is stronger than a double bond. However, a double bond is not necessarily twice as strong as a single bond, nor is a triple bond necessarily three times as strong. Quadruple bonds, which contain four shared pairs of electrons, are rare but have been identified in some compounds in which two metal atoms are bonded directly together.

Resonance

There is sometimes an ambiguity in the location of double bonds. This ambiguity is illustrated by the Lewis structure for ozone (O3). The following are two possible structures:

In such cases, the actual Lewis structure is regarded as a blend of these contributions and is written:

The blending together of these structures is actually a quantum mechanical phenomenon called resonance, which will be considered in more detail below. At this stage, resonance can be regarded as a blending process that spreads double-bond character evenly over the atoms that participate in it. In ozone, for instance, each oxygen-oxygen bond is rendered equivalent by resonance, and each one has a mixture of single-bond and double-bond character (as indicated by its length and strength).

Catenation

Carbon is unique among the elements in the almost infinite capacity of its atoms to bond to each other in long chains, a process called catenation (Latin catena, chain). This characteristic reflects the strength of the bond between adjacent carbon atoms in the molecule, both in relationship to similar bonds involving other elements of the carbon family and in relationship to bonds between carbon atoms and atoms of many other elements. Only the carbon–hydrogen, carbon–fluorine, and carbon–oxygen single bonds (C-H, C-F, and C-O) are stronger than the carbon–carbon single bond (C-C), and each of these is weaker than the carbon–carbon multiple bonds (C=C or C≡C). On the other hand, the silicon–silicon single bond (Si-Si) is weaker than other single bonds involving an atom of other elements with the silicon atom. The same is undoubtedly true of the germanium–germanium and tin–tin single bonds (Ge-Ge, Sn-Sn) in relationship to single covalent bonds between atoms of these elements and atoms of other elements. Experimentally, there appears to be no practical upper limit to catenation involving carbon. This phenomenon in three dimensions produces the diamond and in two dimensions the layers in graphite. Catenation is also exhibited to a high degree by elemental silicon, germanium, and tin, but it is strictly limited in compounds of these elements; silicon may have up to 14 atoms in a chain; germanium, 9; and tin, 2 or 3 only, largely in hydrides (compounds containing hydrogen). Double and triple bonds in catenated arrangements are limited to carbon.