title: Lexicographic order

text: If formula_6 is a total order on formula_7 then so is the lexicographic order on the words of formula_8 However, in general this is not a well-order, even if the alphabet formula_9 is well-ordered. For instance, if , the language has no least element in the lexicographical order: .


title: Lexicographic order

text: Since many applications require well orders, a variant of the lexicographical orders is often used. This well-order, sometimes called or , consists in considering first the lengths of the words (if , then formula_10), and, if the lengths are equal, using the lexicographical order. If the order on is a well-order, the same is true for the shortlex order.


title: Lexicographic order

text: The lexicographical order defines an order on a Cartesian product of ordered sets, which is a total order when all these sets are themselves totally ordered. An element of a Cartesian product formula_11 is a sequence whose formula_12 element belongs to formula_13 for every formula_14 As evaluating the lexicographical order of sequences compares only elements which have the same rank in the sequences, the lexicographical order extends to Cartesian products of ordered sets.


title: Lexicographic order

text: The result is a partial order. If formula_9 and formula_20 are each totally ordered, then the result is a total order as well. The lexicographical order of two totally ordered sets is thus a linear extension of their product order.


title: Lexicographic order

text: One can define similarly the lexicographic order on the Cartesian product of an infinite family of ordered sets, if the family is indexed by the nonnegative integers, or more generally by a well-ordered set. This generalized lexicographical order is a total order if each factor set is totally ordered.


title: Lexicographic order

text: Unlike the finite case, an infinite product of well-orders is not necessarily well-ordered by the lexicographical order. For instance, the set of countably infinite binary sequences (by definition, the set of functions from non-negative integers to formula_21 also known as the Cantor space formula_22) is not well-ordered; the subset of sequences that have precisely one formula_23 (that is, ) does not have a least element under the lexicographical order induced by formula_24 because is an infinite descending chain. Similarly, the infinite lexicographic product is not Noetherian either because is an infinite ascending chain.


title: Lexicographic order

text: The functions from a well-ordered set formula_25 to a totally ordered set formula_26 may be identified with sequences indexed by formula_25 of elements of formula_28 They can thus be ordered by the lexicographical order, and for two such functions formula_29 and formula_30 the lexicographical order is thus determined by their values for the smallest formula_31 such that formula_32


title: Lexicographic order

text: If formula_26 is also well-ordered and formula_25 is finite, then the resulting order is a well-order. As shown above, if formula_25 is infinite this is not the case.


title: Lexicographic order

text: In combinatorics, one has often to enumerate, and therefore to order the finite subsets of a given set formula_36 For this, one usually chooses an order on formula_36 Then, sorting a subset of formula_38 is equivalent to convert it into an increasing sequence. The lexicographic order on the resulting sequences induces thus an order on the subsets, which is also called the .


title: Lexicographic order

text: In this context, one generally prefer to sort first the subsets by cardinality, such as in the shortlex order. Therefore, in the following, we will consider only orders on subsets of fixed cardinal.


title: Lexicographic order

text: For example, using the natural order of the integers, the lexicographical ordering on the subsets of three elements of formula_39 is


title: Lexicographic order

text: For ordering finite subsets of a given cardinality of the natural numbers, the order (see below) is often more convenient, because all initial segments are finite, and thus the colexicographical order defines an order isomorphism between the natural numbers and the set of sets of formula_40 natural numbers. This is not the case for the lexicographical order, as, with the lexicographical order, we have, for example, formula_41 for every formula_42


title: Lexicographic order

text: Let formula_43 be the free Abelian group of rank formula_44 whose elements are sequences of formula_40 integers, and operation is the addition. A group order on formula_43 is a total order, which is compatible with addition, that is


title: Lexicographic order

text: The lexicographical ordering may also be used to characterize all group orders on formula_48 In fact, formula_40 linear forms with real coefficients, define a map from formula_43 into formula_52 which is injective if the forms are linearly independent (it may be also injective if the forms are dependent, see below). The lexicographic order on the image of this map induces a group order on formula_48 Robbiano's theorem is that every group order may be obtained in this way.


title: Lexicographic order

text: More precisely, given a group order on formula_54 there exist an integer formula_55 and formula_56 linear forms with real coefficients, such that the induced map formula_57 from formula_43 into formula_59 has the following properties;


title: Lexicographic order

text: The colexicographic or colex order is a variant of the lexicographical order that is obtained by reading finite sequences from the right to the left instead of reading them from the left to the right. More precisely, whereas the lexicographical order between two sequences is defined by


title: Lexicographic order

text: In general, the difference between the colexicographical order and the lexicographical order is not very significant. However, when considering increasing sequences, typically for coding subsets, the two orders differ significantly.


title: Lexicographic order

text: For example, for ordering the increasing sequences (or the sets) of two natural integers, the lexicographical order begins by


title: Lexicographic order

text: The main property of the colexicographical order for increasing sequences of a given length is that every initial segment is finite. In other words, the colexicographical order for increasing sequences of a given length induces an order isomorphism with the natural numbers, and allows enumerating these sequences. This is frequently used in combinatorics, for example in the proof of the Kruskal–Katona theorem.


title: Lexicographic order

text: When considering polynomials, the order of the terms does not matter in general, as the addition is commutative. However, some algorithms, such as polynomial long division, require the terms to be in a specific order. Many of the main algorithms for multivariate polynomials are related with Gröbner bases, concept that requires the choice of a monomial order, that is a total order, which is compatible with the monoid structure of the monomials. Here "compatible" means that formula_64 if the monoid operation is denoted multiplicatively. This compatibility implies that the product of a polynomial by a monomial does not change the order of the terms. For Gröbner bases, a further condition must be satisfied, namely that every non constant monomial is greater than the monomial . However this condition is not needed for other related algorithms, such as the algorithms for the computation of the tangent cone.


title: Lexicographic order

text: As Gröbner bases are defined for polynomials in a fixed number of variables, it is common to identify monomials (for example formula_65) with their exponent vectors (here ). If is the number of variables, every monomial order is thus the restriction to formula_66 of a monomial order of formula_43 (see above formula_54 for a classification).


title: Lexicographic order

text: One of these admissible orders is the lexicographical order. It is, historically, the first to have been used for defining Gröbner bases, and is sometimes called for distinguishing it from other orders that are also related to a lexicographical order.


title: Lexicographic order

text: Another one consists in comparing first the total degrees, and then resolving the conflicts by using the lexicographical order. This order is not widely used, as either the lexicographical order or the degree reverse lexicographical order have generally better properties.


title: Lexicographic order

text: The consists also in comparing first the total degrees, and, in case of equality of the total degrees, using the reverse of the colexicographical order. That is, given two exponent vectors, one has


title: Lexicographic order

text: For this ordering, the monomials of degree one have the same order as the corresponding indeterminates (this would not be the case if the reverse lexicographical order would be used). For comparing monomials in two variables of the same total degree, this order is the same as the lexicographic order. This is not the case with more variables. For example, for exponent vectors of monomials of degree two in three variables, one has for the degree reverse lexicographic order:


title: Lexicographic order

text: A useful property of the degree reverse lexicographical order is that a homogeneous polynomial is a multiple of the least indeterminate if and only if its leading monomial (its greater monomial) is a multiple of this least indeterminate.


title: Sloop John B

text: "Sloop John B" (originally published as "The John B. Sails") is a Bahamian folk song from Nassau. A transcription by Richard Le Gallienne was published in 1916, and a version was included in Carl Sandburg's "The American Songbag" in 1927. Since the early 1950s there have been many recordings of the song with variant titles including "I Want to Go Home" and "Wreck of the John B".


title: Sloop John B

text: In 1966, the Beach Boys recorded a folk rock adaptation that was produced and arranged by their "de facto" leader, Brian Wilson. Released as the second single from their album "Pet Sounds", the record peaked at number three in the U.S., number two in the UK, and topped the charts in several other countries. It was innovative for containing an elaborate a cappella vocal section not found in other pop music of the era, and it remains one of the group's biggest hits.


title: Sloop John B

text: In 2011, the Beach Boys' version of "Sloop John B" was ranked on "Rolling Stone"s list of "The 500 Greatest Songs of All Time".


title: Sloop John B

text: "The John B. Sails" was transcribed by Richard Le Gallienne, with five verses and the chorus published in his article “Coral Islands and Mangrove-Trees” in the December 1916 issue of "Harper’s Monthly Magazine" (pp. 81–90). The first two verses and chorus were also published in Chapter IV of Gallienne's 1917 novel "Pieces of Eight". The lyrics describe a disastrous voyage on a sloop, with the vessel plagued by drunkenness, arrests and a pig eating the narrator's food. In the chorus, the narrator repeatedly expresses a desire to return home.


title: Sloop John B

text: Carl Sandburg included the first three verses and chorus of "The John B. Sails" in his 1927 collection of folksongs, "The American Songbag". He states that he collected it from John T. McCutcheon (a political cartoonist from Chicago) and his wife, Evelyn Shaw McCutcheon, who at the time owned Blue Lagoon Island, a Cay off of Nassau. The McCutcheons told him:


title: Sloop John B

text: The Kingston Trio's 1958 recording of "The John B. Sails" was recorded under the title "The Wreck of the John B." It was the direct influence on the Beach Boys' version. The Beach Boys' Al Jardine was a keen folk music fan, and he suggested to Brian Wilson that the Beach Boys should record the song. As Jardine explains:


title: Sloop John B

text: Jardine updated the chord progression by having the subdominant (D♭ major) move to its relative minor (B♭ minor) before returning to the tonic (A♭ major), thus altering a portion of the song's progression from IV — I to IV — ii — I. This device is heard immediately after the lyric "into a fight" and "leave me alone".


title: Sloop John B

text: Wilson elected to change some lyrics: "this is the worst trip since I've been born" to "this is the worst trip I've ever been on", "I feel so break up" to "I feel so broke up", and "broke up the people's trunk" to "broke in the captain's trunk". The first lyric change has been suggested by some to be a subtle nod to the 1960s psychedelia subculture.


title: Sloop John B

text: The instrumental section of the song was recorded on July 12, 1965, at United Western Recorders, Hollywood, California, the session being engineered by Chuck Britz and produced by Brian Wilson. The master take of the instrumental backing took fourteen takes to achieve.


title: Sloop John B

text: The vocal tracks were recorded over two sessions. The first was recorded on December 22, 1965, at Western Recorders, produced by Wilson. The second, on December 29, added a new lead vocal and Billy Strange's 12-string electric guitar part. Jardine explained that Wilson "lined us up one at a time to try out for the lead vocal. I had naturally assumed I would sing the lead, since I had brought in the arrangement. It was like interviewing for a job. Pretty funny. He didn't like any of us. My vocal had a much more mellow approach because I was bringing it from the folk idiom. For the radio, we needed a more rock approach. Wilson and Mike [Love] ended up singing it." On the final recording, Brian Wilson sang the first and third verses and Mike Love sang the second.


title: Sloop John B

text: Kent Hartman, in his book "", described Billy Strange's contribution to the song. Brian Wilson called Strange into the studio one Sunday, played him the rough recording, and told him he needed an electric twelve-string guitar solo in the middle of the track. When Strange replied that he did not own a twelve string, Wilson responded by calling Glenn Wallichs, the head of Capitol Records and owner of Wallichs Music City. A Fender Electric XII and Twin Reverb amplifier were quickly delivered (despite the shop they were ordered from being closed on Sundays), and Strange recorded the guitar part in one take. Wilson then gave Strange $2,000 to cover the cost of the equipment.


title: Sloop John B

text: A music video set to "Sloop John B" was filmed for the UK's "Top of the Pops", directed by newly employed band publicist Derek Taylor. It was filmed at Brian's Laurel Way home with Dennis Wilson acting as cameraman.


title: Sloop John B

text: The single, backed with the B-side "You're So Good to Me", was released on March 21, 1966. It entered the "Billboard" Hot 100 chart on April 2, and peaked at on May 7, remaining on the chart, in total, for 11 weeks. It charted highly throughout the world, remaining as one of the Beach Boys' most popular and memorable hits. It was in Germany, Austria, and Norway—all for five weeks each—as well as Sweden, Switzerland, the Netherlands, South Africa, and New Zealand. It placed in the UK, Ireland (where it was the group's highest charting single), Canada, and in "Record World". It was the fastest Beach Boys seller to date, moving more than half a million copies in less than two weeks after release. It had a three-week stay at number 1 in the Netherlands, making it the "Hit of the Year".


title: Sloop John B

text: "Cash Box" described the single as a "topflight adaptation" that treats "the folk oldie in a rhythmic, effectively-building warm-hearted rousing style."


title: Sloop John B

text: In 1968, the recording's instrumental was released on "Stack-O-Tracks". Along with sessions highlights, the box set "The Pet Sounds Sessions" includes two alternate takes, one with Carl Wilson singing lead on the first verse, and one with Brian singing all parts.


title: Sloop John B

text: In 2011, the song was sung by Fisherman's Friends at Cambridge Folk Festival. and released on "Suck'em and Sea". It was featured in the compilation album "Cambridge Folk Festival 2011"


title: Sloop John B

text: In 2016, to commemorate the fiftieth anniversary of "Pet Sounds", Brian Wilson and his touring band (including Al Jardine) performed "Sloop John B" live at Capitol Studios.


title: Sloop John B

text: In 2021, another UK based group, Isle 'Ave A Shanty sang the song at the 2021 Harwich Sea Shanty Festival and included the song on their 2022 debut album "Swinging the Lamp".


title: Sloop John B

text: It has been popular amongst English football fans since the mid-2000s when Liverpool adapted the song to sing about their 2005 Champions League final triumph in Istanbul. It was subsequently adopted by the supporters of English non-league team F.C. United of Manchester as a club anthem in 2007.


title: Sloop John B

text: Since then more high-profile teams have followed suit, usually with different lyrics for their own teams, including Watford, with Newcastle, Blackpool, Middlesbrough and Hull also adopting the song as their own. It was sung by Phil Brown, the manager of Hull City FC, shortly after Hull had avoided relegation from the Premier League in 2009.


title: Sloop John B

text: The melody of "Sloop John B" has been used as the basis for the "Famine Song", a sectarian anti-Irish Catholic song which refers to Irish migration to Great Britain in the context of the Great Irish Famine and contains the line "the famine's over, why don't you go home?". The song has been sung by fans of Rangers F.C. in reference to rival club Celtic F.C., which was established by Irish Catholic migrants in Glasgow and retains a large Irish supporter base. The song was first sung publicly by Rangers fans at a match at Celtic Park in April 2008. Rangers have repeatedly asked their fans not to sing the song. In 2009 Scotland's Justiciary Appeal Court ruled that the song was racist, with judge Lord Carloway stating that its lyrics "are racist in calling upon people native to Scotland to leave the country because of their racial origins".


title: Rockefeller Foundation

text: The Rockefeller Foundation is an American private foundation and philanthropic medical research and arts funding organization based at 420 Fifth Avenue, New York City. The second-oldest major philanthropic institution in America, after the Carnegie Corporation, the foundation was ranked as the 39th largest U.S. foundation by total giving as of 2015. By the end of 2016, assets were tallied at $4.1 billion (unchanged from 2015), with annual grants of $173 million. According to the OECD, the foundation provided US$103.8 million for development in 2019. The foundation has given more than $14 billion in current dollars.


title: Rockefeller Foundation

text: The foundation was started by Standard Oil magnate John D. Rockefeller ("Senior") and son "Junior", and their primary business advisor, Frederick Taylor Gates, on May 14, 1913, when its charter was granted by New York.


title: Rockefeller Foundation

text: The foundation has had an international reach since the 1930s and major influence on global non-governmental organizations. The World Health Organization is modeled on the International Health Division of the foundation, which sent doctors abroad to study and treat human subjects. The National Science Foundation and National Institute of Health are also modeled on the work funded by Rockefeller. It has also been a supporter of and influence on the United Nations.


title: Rockefeller Foundation

text: In 2020 the foundation pledged that it would divest from fossil fuel, notable since the endowment was largely funded by Standard Oil.


title: Rockefeller Foundation

text: The foundation also has a controversial past, including support of eugenics in the 1930s, as well as several scandals arising from their international field work. In 2021 the foundation's president committed to reckoning with their history, and to centering equity and inclusion.


title: Rockefeller Foundation

text: John D. Rockefeller Sr. first conceived the idea of the foundation in 1901. In 1906, Rockefeller's business and philanthropic advisor, Frederick Taylor Gates, encouraged him toward "permanent corporate philanthropies for the good of Mankind" so that his heirs should not "dissipate their inheritances or become intoxicated with power." In 1909 Rockefeller signed over 73,000 shares of his Standard Oil company worth $50 million, to his son, Gates and Harold Fowler McCormick as the third inaugural trustee, in the first installment of a projected $100 million endowment.


title: Rockefeller Foundation

text: The nascent foundation applied for a federal charter in the US Senate in 1910, with at one stage Junior even secretly meeting with President William Howard Taft, through the aegis of Senator Nelson Aldrich, to hammer out concessions. However, because of the ongoing (1911) antitrust suit against Standard Oil at the time, along with deep suspicion in some quarters of undue Rockefeller influence on the spending of the endowment, the result was that Senior and Gates withdrew the bill from Congress in order to seek a state charter from New York.


title: Rockefeller Foundation

text: On May 14, 1913, New York Governor William Sulzer approved a charter for the foundation with Junior becoming the first president. With its large-scale endowment, a large part of Senior's fortune was insulated from inheritance taxes. The first secretary of the foundation was Jerome Davis Greene, the former secretary of Harvard University, who wrote a "memorandum on principles and policies" for an early meeting of the trustees that established a rough framework for the foundation's work. It was initially located within the family office at Standard Oil's headquarters at 26 Broadway, later (in 1933) shifting to the GE Building (then RCA), along with the newly named family office, "Room 5600", at Rockefeller Center; later it moved to the Time-Life Building in the center, before shifting to its current Fifth Avenue address.


title: Rockefeller Foundation

text: In 1914, the trustees set up a new Department of Industrial Relations, inviting William Lyon Mackenzie King to head it. He became a close and key advisor to Junior through the Ludlow Massacre, turning around his attitude to unions; however the foundation's involvement in IR was criticized for advancing the family's business interests. The foundation henceforth confined itself to funding responsible organizations involved in this and other controversial fields, which were beyond the control of the foundation itself.


title: Rockefeller Foundation

text: Junior became the foundation chairman in 1917. Through the "Laura Spelman Rockefeller Memorial" (LSRM), established by Senior in 1918 and named after his wife, the Rockefeller fortune was for the first time directed to supporting research by social scientists. During its first few years of work, the LSRM awarded funds primarily to social workers, with its funding decisions guided primarily by Junior. In 1922, Beardsley Ruml was hired to direct the LSRM, and he most decisively shifted the focus of Rockefeller philanthropy into the social sciences, stimulating the founding of university research centers, and creating the Social Science Research Council. In January 1929, LSRM funds were folded into the Rockefeller Foundation, in a major reorganization.


title: Rockefeller Foundation

text: The Rockefeller family helped lead the foundation in its early years, but later limited itself to one or two representatives, to maintain the foundation's independence and avoid charges of undue family influence. These representatives have included the former president John D. Rockefeller III, and then his son John D. Rockefeller, IV, who gave up the trusteeship in 1981. In 1989, David Rockefeller's daughter, Peggy Dulany, was appointed to the board for a five-year term. In October 2006, David Rockefeller Jr. joined the board of trustees, re-establishing the direct family link and becoming the sixth family member to serve on the board.


title: Rockefeller Foundation

text: C. Douglas Dillon, the United States Secretary of the Treasury under both Presidents John F. Kennedy and Lyndon B. Johnson, served as chairman of the foundation.


title: Rockefeller Foundation

text: Stock in the family's oil companies had been a major part of the foundation's assets, beginning with Standard Oil and later with its corporate descendants, including Exxon Mobil. In December 2020, the foundation pledged to dump their fossil fuel holdings. With a $5 billion endowment, the Rockefeller Foundation was "the largest US foundation to embrace the rapidly growing divestment movement." CNN writer Matt Egan noted, "This divestment is especially symbolic because the Rockefeller Foundation was founded by oil money."


title: Rockefeller Foundation

text: Public health, health aid, and medical research are the most prominent areas of work of the foundation. On December 5, 1913, the Board made its first grant of $100,000 to the American Red Cross to purchase property for its headquarters in Washington, D.C.


title: Rockefeller Foundation

text: The foundation established the Johns Hopkins School of Public Health and Harvard School of Public Health, two of the first such institutions in the United States, and established the School of Hygiene at the University of Toronto in 1927, and the London School of Hygiene and Tropical Medicine in the United Kingdom;. they spent more than $25 million in developing other public health schools in the US and in 21 foreign countries. In 1913, it also began a 20-year support program of the "Bureau of Social Hygiene", whose mission was research and education on birth control, maternal health and sex education. In 1914, the foundation set up the China Medical Board, which established the first public health university in China, the Peking Union Medical College, in 1921; this was subsequently nationalized when the Communists took over the country in 1949. In the same year it began a program of international fellowships to train scholars at many of the world's universities at the post-doctoral level. The Foundation also maintained a close relationship with Rockefeller University (also known as the Rockefeller Institute for Medical Research) with many faculty holding overlapping positions between the institutions.


title: Rockefeller Foundation

text: The Sanitary Commission for the Eradication of Hookworm Disease was a Rockefeller-funded campaign from 1909 to 1914 to study and treat hookworm disease in 11 Southern states. Hookworm was known as the "germ of laziness." In 1913, the foundation expanded its work with the Sanitary Commission abroad and set up the International Health Division (also known as International Health Board), which began the foundation's first international public health activities. The International Health Division conducted campaigns in public health and sanitation against malaria, yellow fever, and hookworm in areas throughout Europe, Latin America and the Caribbean including Italy, France, Venezuela, Mexico, and Puerto Rico, totaling fifty-two countries on six continents and twenty-nine islands. The first director was Wickliffe Rose, followed by F.F. Russell in 1923, Wilbur Sawyer in 1935, and George Strode in 1944. A number of notable physicians and field scientists worked on the international campaigns, including Lewis Hackett, Hideyo Noguchi, Juan Guiteras, George C. Payne, Livingston Farrand, Cornelius P. Rhoads, and William Bosworth Castle. The World Health Organization, seen as a successor to the IHD, was formed in 1948, and the IHD was subsumed by the larger Rockefeller Foundation in 1951, discontinuing its overseas work.


title: Rockefeller Foundation

text: While the Rockefeller doctors working in tropical locales such as Mexico emphasized scientific neutrality, they had political and economic aims to promote the value of public health to improve American relations with the host country. Although they claimed the banner of public health and humanitarian medicine, they often engaged with politics and business interests. Rhoads was involved in a racism whitewashing scandal in the 1930s during which he joked about injecting cancer cells into Puerto Rican patients, inspiring Puerto Rican nationalist and anti-colonialist leader Pedro Albizu Campos. Noguchi was also involved in an unethical human experimentation scandal. Susan Lederer, Elizabeth Fee, and Jay Katz are among the modern scholars who have researched this period. Researchers with the foundation including Noguchi developed the vaccine to prevent yellow fever. Rhoads later became a significant cancer researcher and director of Memorial Sloan-Kettering, though his eponymous award for oncological excellence was renamed after the scandal reemerged.


title: Rockefeller Foundation

text: During the late-1920s, the Rockefeller Foundation created the Medical Sciences Division, which emerged from the former Division of Medical Education. The division was led by Dr. Richard M. Pearce until his death in 1930, to which Alan Gregg succeeded him until 1945. During this period, the Division of Medical Sciences made contributions to research across several fields of psychiatry. In 1935 the foundation granted $100000 to the Institute for Psychoanalysis in Chicago. This grant was renewed in 1938, with payments extending into the early-1940s. This division funded women's contraception and the human reproductive system in general, but also was involved in funding controversial eugenics research. Other funding went into endocrinology departments in American universities, human heredity, mammalian biology, human physiology and anatomy, psychology, and the studies of human sexual behavior by Dr. Alfred Kinsey.


title: Rockefeller Foundation

text: In the interwar years, the foundation funded public health, nursing, and social work in Eastern and Central Europe.


title: Rockefeller Foundation

text: In 1950, the foundation expanded their international program of virus research, establishing field laboratories in Poona, India, Trinidad, Belém, Brazil, Johannesburg, South Africa, Cairo, Egypt, Ibadan, Nigeria, and Cali, Colombia, among others. The foundation funded research into the identification of human viruses, techniques for virus identification, and arthropod-borne viruses.


title: Rockefeller Foundation

text: Bristol-Myers Squibb, Johns Hopkins University and the Rockefeller Foundation are currently the subject of a $1 billion lawsuit from Guatemala for "roles in a 1940s U.S. government experiment that infected hundreds of Guatemalans with syphilis". A previous suit against the United States government was dismissed in 2011 for the Guatemala syphilis experiments when a judge determined that the U.S. government could not be held liable for actions committed outside of the U.S.


title: Rockefeller Foundation

text: An experiment was conducted by Vanderbilt University in the 1940s where they gave 800 pregnant women radioactive iron, 751 of which were pills, without their consent. In a 1969 article published in the "American Journal of Epidemiology", it was estimated that three children had died from the experiment.


title: Rockefeller Foundation

text: John D. Rockefeller Jr. was an outspoken supporter of eugenics. Even as late as 1951, John D. Rockefeller III and John Foster Dulles, who was chairman of the foundation at the time, established the Population Council to advance family planning, birth control, and population control, and goals of the eugenics movement.


title: Rockefeller Foundation

text: The Rockefeller Foundation, along with the Carnegie Institution, was the primary financier for the Eugenics Record Office, until 1939. The foundation also provided grants to Margaret Sanger and Alexis Carrel, who supported birth control, compulsory sterilization and eugenics. Sanger went to Japan in 1922 and influenced the birth control movement there.


title: Rockefeller Foundation

text: By 1926, Rockefeller had donated over $400,000, which would be almost $4 million adjusted for inflation in 2003, to hundreds of German researchers, including Ernst Rüdin and Otmar Freiherr von Verschuer, through funding the Kaiser Wilhelm Institute of Anthropology, Human Heredity, and Eugenics, (also known as the Max Planck Institute for Medical Research) which conducted eugenics experiments in Nazi Germany and influenced the development of Nazi racial scientific ideology. Rockefeller spent almost $3 million between 1925 and 1935, and also funded other German eugenicists, Herman Poll, Alfred Grotjahn, Eugen Fischer, and Hans Nachsteim, continuing even after Hitler's ascent to power in 1933; Rüdin's work influenced compulsory sterilisation in Nazi Germany. Josef Mengele worked as an assistant in Verschuer's lab, though Rockefeller executives did not know of Mengele and stopped funding that specific research before World War II started in 1939.


title: Rockefeller Foundation

text: The Rockefeller Foundation continued funding German eugenics research even after it was clear that it was being used to rationalize discrimination against Jewish people and other groups, after the Nuremberg laws in 1935. In 1936, Rockefeller fulfilled pledges of $655,000 to Kaiser Wilhelm Institute, even though several distinguished Jewish scientists had been dropped from the institute at the time. The Rockefeller Foundation did not alert the world about the racist implications of Nazi ideology, but furthered and funded eugenic research through the 1930s. Even into the 1950s, Rockefeller continued to provide some funding for research borne out of German eugenics.


title: Rockefeller Foundation

text: The foundation also funded the relocation of scholars threatened by the Nazis to America in the 1930s, known as the Refugee Scholar Program and the Emergency Committee in Aid of Displaced Foreign Scholars. Some of the notable figures relocated or saved, among a total of 303 scholars, were Thomas Mann, Claude Lévi-Strauss and Leó Szilárd. The foundation helped The New School provide a haven for scholars threatened by the Nazis.


title: Rockefeller Foundation

text: After World War II the foundation sent a team to West Germany to investigate how it could become involved in reconstructing the country. They focused on restoring democracy, especially regarding education and scientific research, with the long-term goal of reintegrating Germany into the Western world.


title: Rockefeller Foundation

text: The foundation also supported the early initiatives of Henry Kissinger, such as his directorship of Harvard's "International Seminars" (funded as well by the Central Intelligence Agency) and the early foreign policy magazine "Confluence", both established by him while he was still a graduate student.


title: Rockefeller Foundation

text: In 2021, Dr. Rajiv J. Shah, president of the Rockefeller Foundation, released a statement condemning eugenics and supporting the anti-eugenics movement. He stated that


title: Rockefeller Foundation

text: "[...]we commend the Anti-Eugenics Project for their essential work to understand[...] the harmful legacies of eugenicist ideologies. [...] examine the role that philanthropies played in developing and perpetuating eugenics policies and practices. The Rockefeller Foundation is currently reckoning with our own history in relation to eugenics.  This requires uncovering the facts and confronting uncomfortable truths, [...] The Rockefeller Foundation is putting equity and inclusion at the center of all our work: [...] confronting the hateful legacies of the past [...] we understand that the work we engage in today does not absolve us of yesterday’s mistakes. [...]"


title: Rockefeller Foundation

text: Although the United States never joined the League of Nations, the Rockefeller Foundation was involved, and by the 1930s the foundations had changed the League from a "Parliament of Nations" to a modern think tank that used specialized expertise to provide in-depth impartial analysis of international issues. After the war, the foundation was involved in the establishment of the United Nations.


title: Rockefeller Foundation

text: In the arts the Rockefeller Foundation has supported the Stratford Shakespeare Festival in Ontario, Canada, and the American Shakespeare Festival in Stratford, Connecticut, Arena Stage in Washington, D.C., Karamu House in Cleveland, and Lincoln Center in New York. The foundation underwrote of Spike Lee's documentary on New Orleans, "When the Levees Broke". The film has been used as the basis for a curriculum on poverty, developed by the Teachers College at Columbia University for their students.


title: Rockefeller Foundation

text: The foundation also owns and operates the Bellagio Center in Bellagio, Italy. The center has several buildings, spread across a property, on the peninsula between lakes Como and Lecco in Northern Italy. The center is sometimes referred to as the "Villa Serbelloni", the property bequeathed to the foundation in 1959 under the presidency of Dean Rusk (who was later to become U.S. President Kennedy's secretary of state). The Bellagio Center operates both a conference center and a residency program. Numerous Nobel laureates, Pulitzer winners, National Book Award recipients, Prince Mahidol Award winners and MacArthur fellows, as well as several acting and former heads of State and Government, have been in residence at Bellagio.


title: Rockefeller Foundation

text: The Cultural Innovation Fund is a pilot grant program that is overseen by Lincoln Center. The grants are to be used towards art and cultural opportunities in the underserved areas of Brooklyn and the South Bronx with three overarching goals.


title: Rockefeller Foundation

text: The Rockefeller Foundation supported the art scene in Haiti in 1948 and a literacy project with UNESCO.


title: Rockefeller Foundation

text: Rusk was involved with funding the humanities and the social sciences during the Cold War period, including study of the Soviet Union.


title: Rockefeller Foundation

text: Agriculture was introduced to the Natural Sciences division of the foundation in the major reorganization of 1928. In 1941, the foundation gave a small grant to Mexico for maize research, in collaboration with the then new president, Manuel Ávila Camacho. This was done after the intervention of vice-president Henry Wallace and the involvement of Nelson Rockefeller; the primary intention being to stabilise the Mexican Government and derail any possible communist infiltration, in order to protect the Rockefeller family's investments.


title: Rockefeller Foundation

text: By 1943, this program, under the foundation's "Mexican Agriculture Project", had proved such a success with the science of corn propagation and general principles of agronomy that it was exported to other Latin American countries; in 1956, the program was then taken to India; again with the geopolitical imperative of providing an antidote to communism. It wasn't until 1959 that senior foundation officials succeeded in getting the Ford Foundation (and later USAID, and later still, the World Bank) to sign on to the major philanthropic project, known now to the world as the Green Revolution. It was originally conceived in 1943 as CIMMYT, the International Maize and Wheat Improvement Center in Mexico. It also provided significant funding for the International Rice Research Institute in the Philippines. Part of the original program, the funding of the IRRI was later taken over by the Ford Foundation. The International Rice Research Institute and the International Maize and Wheat Improvement Center are part of a consortium of agricultural research organizations known as CGIAR.


title: Rockefeller Foundation

text: Costing around $600 million, over 50 years, the revolution brought new farming technology, increased productivity, expanded crop yields and mass fertilization to many countries throughout the world. Later it funded over $100 million of plant biotechnology research and trained over four hundred scientists from Asia, Africa and Latin America. It also invested in the production of transgenic crops, including rice and maize. In 1999, the then president Gordon Conway addressed the Monsanto Company board of directors, warning of the possible social and environmental dangers of this biotechnology, and requesting them to disavow the use of so-called terminator genes; the company later complied.


title: Rockefeller Foundation

text: In the 1990s, the foundation shifted its agriculture work and emphasis to Africa; in 2006, it joined with the Bill & Melinda Gates Foundation in a $150 million effort to fight hunger in the continent through improved agricultural productivity. In an interview marking the 100 year anniversary of the Rockefeller Foundation, Judith Rodin explained to This Is Africa that Rockefeller has been involved in Africa since their beginning in three main areas – health, agriculture and education, though agriculture has been and continues to be their largest investment in Africa.


title: Rockefeller Foundation

text: A total of 100 cities across six continents were part of the 100 Resilient Cities program funded by the Rockefeller Foundation. In January 2016, the United States Department of Housing and Urban Development announced winners of its National Disaster Resilience Competition (NDRC), awarding three 100RC member cities – New York, NY; Norfolk, VA; and New Orleans, LA – with more than $437 million in disaster resilience funding. The grant was the largest ever received by the city of Norfolk.


title: Rockefeller Foundation

text: In April 2019, it was announced that the foundation would no longer be funding the 100 Resilient Cities program as a whole. Some elements of the initiative's work, most prominently the funding of several cities' Chief Resilience Officer roles, continues to be managed and funded by the Rockefeller Foundation, while other aspects of the program continue in the form of two independent organizations, Resilient Cities Catalyst (RCC) and the Global Resilient Cities Network (GRCN), founded by former 100RC leadership and staff.


title: Philippa Gregory

text: Philippa Gregory (born 9 January 1954) is an English historical novelist who has been publishing since 1987. The best known of her works is "The Other Boleyn Girl" (2001), which in 2002 won the Romantic Novel of the Year Award from the Romantic Novelists' Association and has been adapted into two films.


title: Philippa Gregory

text: Philippa Gregory was born on 9 January 1954 in Nairobi, at that time serving as capital city of the Colony and Protectorate of Kenya (modern-day Republic of Kenya), the second daughter of Elaine (Wedd) and Arthur Percy Gregory, a radio operator and navigator for East African Airways. When she was two years old, her family moved to Bristol, England.


title: Philippa Gregory

text: She was a "rebel" at Colston's Girls' School where she obtained a B grade in English and two E grades in History and Geography at A-level. She then went to journalism college in Cardiff and spent a year as an apprentice with the "Portsmouth News" before she managed to gain a place on an English literature degree course at the University of Sussex, where she switched to a history course. In 1982, she received a B.A. degree in history from Sussex University.


title: Philippa Gregory

text: She worked for BBC radio for two years before attending University of Edinburgh, where she obtained a Ph.D. degree in 18th-century literature in 1985 for her thesis entitled "The popular fiction of eighteenth-century commercial circulating libraries". Gregory has taught at the University of Durham, University of Teesside, and the Open University, and was made a Fellow of Kingston University in 1994.


title: Philippa Gregory

text: She has written novels set in several different historical periods, though primarily the Tudor period and the 16th century. Reading a number of novels set in the 17th century led her to write the best-selling Lacey trilogy "Wideacre", which is a story about the love of land and incest, "The Favoured Child" and "Meridon". This was followed by "The Wise Woman". "A Respectable Trade", a novel of the slave trade in England, set in 18th-century Bristol, was adapted by Gregory for a four-part drama series for BBC television. Gregory's script was nominated for a BAFTA, won an award from the Committee for Racial Equality, and the film was shown worldwide.


title: Philippa Gregory

text: Two novels about a gardening family are set during the English Civil War: "Earthly Joys" and "Virgin Earth", while she has in addition written contemporary fiction – "Perfectly Correct", "Mrs Hartley and the Growth Centre", "The Little House" and "Zelda's Cut". She has also written for children.


title: Philippa Gregory

text: Some of her novels have won awards and have been adapted into television dramas. The most successful of her novels has been "The Other Boleyn Girl," published in 2001 and adapted for BBC television in 2003 with Natascha McElhone, Jodhi May and Jared Harris. In the year of its publication, "The Other Boleyn Girl" also won the Romantic Novel of the Year and it has subsequently spawned sequels – "The Queen's Fool," "The Virgin's Lover," "The Constant Princess," "The Boleyn Inheritance," and "The Other Queen". Miramax bought the film rights to "The Other Boleyn Girl" and released a film of the same name in February 2008.


title: Philippa Gregory

text: Gregory has also published a series of books about the Plantagenets, the ruling houses that preceded the Tudors, and the Wars of the Roses. Her first book "The White Queen", published in 2009, centres on the life of Elizabeth Woodville the wife of Edward IV. "The Red Queen", published in 2010, is about Margaret Beaufort the mother of Henry VII and grandmother to Henry VIII. "The Lady of the Rivers" (2011), is the life of Jacquetta of Luxembourg, mother of Elizabeth Woodville. "The Kingmaker's Daughter", published in 2012, is about Anne Neville, the wife of Richard III, and "The White Princess" (2013) centres on the life of Elizabeth of York, wife of Henry VII and the mother of Henry VIII. The latest work is the 2017 novel "The Last Tudor". The 2013 BBC One television series "The White Queen" is a 10-part adaptation of Gregory's novels "The White Queen", "The Red Queen" and "The Kingmaker's Daughter" (2012).


title: Philippa Gregory

text: In 2013, Helen Brown of "The Telegraph" wrote that "Gregory has made an impressive career out of breathing passionate, independent life into the historical noblewomen whose personalities had previously lain flat on family trees, remembered only as diplomatic currency and brood mares." She added, "Gregory’s historical fiction has always been entertainingly speculative (those tempted to sneer should note that she’s never claimed otherwise) and comes with lashings of romantic licence."


title: Philippa Gregory

text: In 2011 she contributed a short story "Why Holly Berries are as Red as Roses" to an anthology supporting the Woodland Trust. The anthology, "Why Willows Weep" has so far helped The Woodland Trust plant approximately 50,000 trees.


title: Philippa Gregory

text: Gregory was appointed Commander of the Order of the British Empire (CBE) in the 2021 Birthday Honours for services to literature and to charity in the UK and the Gambia.


title: Philippa Gregory

text: Gregory has said that her "commitment to historical accuracy" is a hallmark of her writing. This is disputed by historians. Historian David Starkey, appearing alongside Gregory in a documentary about Anne Boleyn, described her work as "good Mills and Boon", adding that: "We really should stop taking historical novelists seriously as historians. The idea that they have authority is ludicrous." Susan Bordo criticised Gregory's claims to historical accuracy as "self-deceptive and self-promoting chutzpah", and notes that it is not so much the many inaccuracies in her work as "Gregory's insistence on her meticulous adherence to history that most aggravates the scholars."


title: Philippa Gregory

text: In her novel "The Other Boleyn Girl", her portrayal of Henry VIII's second wife Anne Boleyn drew criticism. The novel depicts Anne as cold and ruthless, as well as strongly implying that the accusations that she committed adultery and incest with her brother were true, despite it being widely accepted that she was innocent of the charges. Novelist Robin Maxwell refused on principle to write a blurb for this book, describing its characterisation of Anne as "vicious, unsupportable".


title: Philippa Gregory

text: She is a frequent contributor to magazines and newspapers, with short stories, features and reviews. She is also a frequent broadcaster and a regular contestant on "Round Britain Quiz" for BBC Radio 4 and the Tudor expert for Channel 4's "Time Team". She won the 29 December 2008 edition of "Celebrity Mastermind" on BBC1, taking Elizabeth Woodville as her specialist subject.


title: Philippa Gregory

text: Gregory runs a small charity building wells in school gardens in The Gambia. Gardens for The Gambia was established in 1993 when Gregory was in The Gambia, researching for her book "A Respectable Trade".


title: Philippa Gregory

text: Since then the charity has dug almost 200 low technology, low budget and therefore easily maintained wells, which are on-stream and providing water to irrigate school and community gardens to provide meals for the poorest children and harvest a cash crop to buy school equipment, seeds and tools.


title: Philippa Gregory

text: In addition to wells, the charity has piloted a successful bee-keeping scheme, funded feeding programmes and educational workshops in batik and pottery and is working with larger donors to install mechanical boreholes in some remote areas of the country where the water table is not accessible by digging alone.


title: Philippa Gregory

text: Philippa Gregory is a patron of The UK Chagos Support Association, which supports the Chagos islanders in their legal disputes with the British government. The people of Chagos were relocated by the British government when the archipelago in the Indian Ocean was cleared in the 1960s and 1970s to make way for an important U.S. airbase. Gregory often speaks about the Chagossians' situation and lobbies the government to take action.


title: Philippa Gregory

text: Gregory wrote her first novel "Wideacre" while completing her doctorate and lived during that time in a cottage on the Pennine Way with her first husband Peter Chislett, editor of the "Hartlepool Mail", and their baby daughter, Victoria. They divorced before the book was published.


title: Philippa Gregory

text: Following the success of "Wideacre" and the publication of "The Favoured Child", she moved south to near Midhurst, West Sussex, where the "Wideacre" trilogy was set. Here Gregory married Paul Carter, her second husband, with whom she has a son, named Adam. She divorced for a second time.


title: Philippa Gregory

text: After the break-up of her second marriage, she met and married Anthony Mason, whom she had first met during her time in Hartlepool.


title: Philippa Gregory

text: Gregory now lives on a farm in the North York Moors National Park, with her husband, children and stepchildren (six in all). Her interests include riding, walking, skiing, and gardening.


title: Philippa Gregory

text: Previously separated as the "Tudor Court" and "Cousins' War" series, as of August 2016 Gregory lists these novels as one series, "The Plantagenet and Tudor Novels".


title: Philippa Gregory

text: Gregory has suggested a "reading order" for the series, based on the real-world chronology of historical figures and events.


title: Single-sideband modulation

text: In radio communications, single-sideband modulation (SSB) or single-sideband suppressed-carrier modulation (SSB-SC) is a type of modulation used to transmit information, such as an audio signal, by radio waves. A refinement of amplitude modulation, it uses transmitter power and bandwidth more efficiently. Amplitude modulation produces an output signal the bandwidth of which is twice the maximum frequency of the original baseband signal. Single-sideband modulation avoids this bandwidth increase, and the power wasted on a carrier, at the cost of increased device complexity and more difficult tuning at the receiver.


title: Single-sideband modulation

text: Radio transmitters work by mixing a radio frequency (RF) signal of a specific frequency, the carrier wave, with the audio signal to be broadcast. In AM transmitters this mixing usually takes place in the final RF amplifier (high level modulation). It is less common and much less efficient to do the mixing at low power and then amplify it in a linear amplifier. Either method produces a set of frequencies with a strong signal at the carrier frequency and with weaker signals at frequencies extending above and below the carrier frequency by the maximum frequency of the input signal. Thus the resulting signal has a spectrum whose bandwidth is twice the maximum frequency of the original input audio signal.


title: Single-sideband modulation

text: SSB takes advantage of the fact that the entire original signal is encoded in each of these "sidebands". It is not necessary to transmit both sidebands plus the carrier, as a suitable receiver can extract the entire original signal from either the upper or lower sideband. There are several methods for eliminating the carrier and one sideband from the transmitted signal. Producing this single sideband signal can be done at high level in the final amplifier stage as with AM


title: Single-sideband modulation

text: but it is usually produced at a low power level and linearly amplified. The lower efficiency of linear amplification partially offsets the power advantage gained by eliminating the carrier and one sideband. Nevertheless, SSB transmissions use the available amplifier energy considerably more efficiently, providing longer-range transmission for the same power output. In addition, the occupied spectrum is less than half that of a full carrier AM signal.


title: Single-sideband modulation

text: SSB reception requires frequency stability and selectivity well beyond that of inexpensive AM receivers which is why broadcasters have seldom used it. In point to point communications where expensive receivers are in common use already they can successfully be adjusted to receive whichever sideband is being transmitted.


title: Single-sideband modulation

text: The first U.S. patent application for SSB modulation was filed on December 1, 1915 by John Renshaw Carson. The U.S. Navy experimented with SSB over its radio circuits before World War I. SSB first entered commercial service on January 7, 1927, on the longwave transatlantic public radiotelephone circuit between New York and London. The high power SSB transmitters were located at Rocky Point, New York, and Rugby, England. The receivers were in very quiet locations in Houlton, Maine, and Cupar Scotland.


title: Single-sideband modulation

text: SSB was also used over long-distance telephone lines, as part of a technique known as frequency-division multiplexing (FDM). FDM was pioneered by telephone companies in the 1930s. With this technology, many simultaneous voice channels could be transmitted on a single physical circuit, for example in L-carrier. With SSB, channels could be spaced (usually) only 4,000 Hz apart, while offering a speech bandwidth of nominally 300 Hz to 3,400 Hz.


title: Single-sideband modulation

text: Amateur radio operators began serious experimentation with SSB after World War II. The Strategic Air Command established SSB as the radio standard for its aircraft in 1957. It has become a de facto standard for long-distance voice radio transmissions since then.


title: Single-sideband modulation

text: Single-sideband has the mathematical form of quadrature amplitude modulation (QAM) in the special case where one of the baseband waveforms is derived from the other, instead of being independent messages:


title: Single-sideband modulation

text: where formula_1 is the message (real-valued), formula_2 is its Hilbert transform, and formula_3 is the radio carrier frequency.


title: Single-sideband modulation

text: To understand this formula, we may express formula_4 as the real part of a complex-valued function, with no loss of information:


title: Single-sideband modulation

text: where formula_6 represents the imaginary unit.  formula_7 is the analytic representation of formula_8  which means that it comprises only the positive-frequency components of formula_4:


title: Single-sideband modulation

text: where formula_11 and formula_12 are the respective Fourier transforms of formula_7 and formula_14  Therefore, the frequency-translated function formula_15 contains only one side of formula_16  Since it also has only positive-frequency components, its inverse Fourier transform is the analytic representation of formula_17


title: Single-sideband modulation

text: and again the real part of this expression causes no loss of information.  With Euler's formula to expand  formula_19  we obtain :


title: Single-sideband modulation

text: Coherent demodulation of formula_21 to recover formula_4 is the same as AM: multiply by formula_23  and lowpass to remove the "double-frequency" components around frequency formula_24. If the demodulating carrier is not in the correct phase (cosine phase here), then the demodulated signal will be some linear combination of formula_4 and formula_26, which is usually acceptable in voice communications (if the demodulation carrier frequency is not quite right, the phase will be drifting cyclically, which again is usually acceptable in voice communications if the frequency error is small enough, and amateur radio operators are sometimes tolerant of even larger frequency errors that cause unnatural-sounding pitch shifting effects).


title: Single-sideband modulation

text: formula_4 can also be recovered as the real part of the complex-conjugate, formula_28 which represents the negative frequency portion of formula_16 When formula_3 is large enough that formula_31 has no negative frequencies, the product formula_32 is another analytic signal, whose real part is the actual "lower-sideband" transmission:


title: Single-sideband modulation

text: One method of producing an SSB signal is to remove one of the sidebands via filtering, leaving only either the upper sideband (USB), the sideband with the higher frequency, or less commonly the lower sideband (LSB), the sideband with the lower frequency. Most often, the carrier is reduced or removed entirely (suppressed), being referred to in full as single sideband suppressed carrier (SSBSC). Assuming both sidebands are symmetric, which is the case for a normal AM signal, no information is lost in the process. Since the final RF amplification is now concentrated in a single sideband, the effective power output is greater than in normal AM (the carrier and redundant sideband account for well over half of the power output of an AM transmitter). Though SSB uses substantially less bandwidth and power, it cannot be demodulated by a simple envelope detector like standard AM.


title: Single-sideband modulation

text: An alternate method of generation known as a Hartley modulator, named after R. V. L. Hartley, uses phasing to suppress the unwanted sideband. To generate an SSB signal with this method, two versions of the original signal are generated, mutually 90° out of phase for any single frequency within the operating bandwidth. Each one of these signals then modulates carrier waves (of one frequency) that are also 90° out of phase with each other. By either adding or subtracting the resulting signals, a lower or upper sideband signal results. A benefit of this approach is to allow an analytical expression for SSB signals, which can be used to understand effects such as synchronous detection of SSB.


title: Single-sideband modulation

text: Shifting the baseband signal 90° out of phase cannot be done simply by delaying it, as it contains a large range of frequencies. In analog circuits, a wideband 90-degree phase-difference network is used. The method was popular in the days of vacuum tube radios, but later gained a bad reputation due to poorly adjusted commercial implementations. Modulation using this method is again gaining popularity in the homebrew and DSP fields. This method, utilizing the Hilbert transform to phase shift the baseband audio, can be done at low cost with digital circuitry.


title: Single-sideband modulation

text: Another variation, the Weaver modulator, uses only lowpass filters and quadrature mixers, and is a favored method in digital implementations.


title: Single-sideband modulation

text: In Weaver's method, the band of interest is first translated to be centered at zero, conceptually by modulating a complex exponential formula_35 with frequency in the middle of the voiceband, but implemented by a quadrature pair of sine and cosine modulators at that frequency (e.g. 2 kHz). This complex signal or pair of real signals is then lowpass filtered to remove the undesired sideband that is not centered at zero. Then, the single-sideband complex signal centered at zero is upconverted to a real signal, by another pair of quadrature mixers, to the desired center frequency.


title: Single-sideband modulation

text: Conventional amplitude-modulated signals can be considered wasteful of power and bandwidth because they contain a carrier signal and two identical sidebands. Therefore, SSB transmitters are generally designed to minimize the amplitude of the carrier signal. When the carrier is removed from the transmitted signal, it is called "suppressed-carrier SSB".


title: Single-sideband modulation

text: However, in order for a receiver to reproduce the transmitted audio without distortion, it must be tuned to exactly the same frequency as the transmitter. Since this is difficult to achieve in practice, SSB transmissions can sound unnatural, and if the error in frequency is great enough, it can cause poor intelligibility. In order to correct this, a small amount of the original carrier signal can be transmitted so that receivers with the necessary circuitry to synchronize with the transmitted carrier can correctly demodulate the audio. This mode of transmission is called reduced-carrier single-sideband.


title: Single-sideband modulation

text: In other cases, it may be desirable to maintain some degree of compatibility with simple AM receivers, while still reducing the signal's bandwidth. This can be accomplished by transmitting single-sideband with a normal or slightly reduced carrier. This mode is called "compatible (or full-carrier) SSB" or "amplitude modulation equivalent (AME)". In typical AME systems, harmonic distortion can reach 25%, and intermodulation distortion can be much higher than normal, but minimizing distortion in receivers with envelope detectors is generally considered less important than allowing them to produce intelligible audio.


title: Single-sideband modulation

text: A second, and perhaps more correct, definition of "compatible single sideband" (CSSB) refers to a form of amplitude and phase modulation in which the carrier is transmitted along with a series of sidebands that are predominantly above or below the carrier term. Since phase modulation is present in the generation of the signal, energy is removed from the carrier term and redistributed into the sideband structure similar to that which occurs in analog frequency modulation. The signals feeding the phase modulator and the envelope modulator are further phase-shifted by 90° with respect to each other. This places the information terms in quadrature with each other; the Hilbert transform of information to be transmitted is utilized to cause constructive addition of one sideband and cancellation of the opposite primary sideband. Since phase modulation is employed, higher-order terms are also generated. Several methods have been employed to reduce the impact (amplitude) of most of these higher-order terms. In one system, the phase-modulated term is actually the log of the value of the carrier level plus the phase-shifted audio/information term. This produces an ideal CSSB signal, where at low modulation levels only a first-order term on one side of the carrier is predominant. As the modulation level is increased, the carrier level is reduced while a second-order term increases substantially in amplitude. At the point of 100% envelope modulation, 6 dB of power is removed from the carrier term, and the second-order term is identical in amplitude to carrier term. The first-order sideband has increased in level until it is now at the same level as the formerly unmodulated carrier. At the point of 100% modulation, the spectrum appears identical to a normal double-sideband AM transmission, with the center term (now the primary audio term) at a 0 dB reference level, and both terms on either side of the primary sideband at −6 dB. The difference is that what appears to be the carrier has shifted by the audio-frequency term towards the "sideband in use". At levels below 100% modulation, the sideband structure appears quite asymmetric. When voice is conveyed by a CSSB source of this type, low-frequency components are dominant, while higher-frequency terms are lower by as much as 20 dB at 3 kHz. The result is that the signal occupies approximately 1/2 the normal bandwidth of a full-carrier, DSB signal. There is one catch: the audio term utilized to phase-modulate the carrier is generated based on a log function that is biased by the carrier level. At negative 100% modulation, the term is driven to zero (0), and the modulator becomes undefined. Strict modulation control must be employed to maintain stability of the system and avoid splatter. This system is of Russian origin and was described in the late 1950s. It is uncertain whether it was ever deployed.


title: Single-sideband modulation

text: A second series of approaches was designed and patented by Leonard R. Kahn. The various Kahn systems removed the hard limit imposed by the use of the strict log function in the generation of the signal. Earlier Kahn systems utilized various methods to reduce the second-order term through the insertion of a predistortion component. One example of this method was also used to generate one of the Kahn independent-sideband (ISB) AM stereo signals. It was known as the STR-77 exciter method, having been introduced in 1977. Later, the system was further improved by use of an arcsine-based modulator that included a 1-0.52E term in the denominator of the arcsin generator equation. E represents the envelope term; roughly half the modulation term applied to the envelope modulator is utilized to reduce the second-order term of the arcsin "phase"-modulated path; thus reducing the second-order term in the undesired sideband. A multi-loop modulator/demodulator feedback approach was used to generate an accurate arcsin signal. This approach was introduced in 1984 and became known as the STR-84 method. It was sold by Kahn Research Laboratories; later, Kahn Communications, Inc. of NY. An additional audio processing device further improved the sideband structure by selectively applying pre-emphasis to the modulating signals. Since the envelope of all the signals described remains an exact copy of the information applied to the modulator, it can be demodulated without distortion by an envelope detector such as a simple diode. In a practical receiver, some distortion may be present, usually at a low level (in AM broadcast, always below 5%), due to sharp filtering and nonlinear group delay in the IF filters of the receiver, which act to truncate the compatibility sideband – those terms that are not the result of a linear process of simply envelope modulating the signal as would be the case in full-carrier DSB-AM – and rotation of phase of these compatibility terms such that they no longer cancel the quadrature distortion term caused by a first-order SSB term along with the carrier. The small amount of distortion caused by this effect is generally quite low and acceptable.


title: Single-sideband modulation

text: The Kahn CSSB method was also briefly used by Airphone as the modulation method employed for early consumer telephone calls that could be placed from an aircraft to ground. This was quickly supplanted by digital modulation methods to achieve even greater spectral efficiency.


title: Single-sideband modulation

text: While CSSB is seldom used today in the AM/MW broadcast bands worldwide, some amateur radio operators still experiment with it.


title: Single-sideband modulation

text: The front end of an SSB receiver is similar to that of an AM or FM receiver, consisting of a superheterodyne RF front end that produces a frequency-shifted version of the radio frequency (RF) signal within a standard intermediate frequency (IF) band.


title: Single-sideband modulation

text: To recover the original signal from the IF SSB signal, the single sideband must be frequency-shifted down to its original range of baseband frequencies, by using a product detector which mixes it with the output of a beat frequency oscillator (BFO). In other words, it is just another stage of heterodyning. For this to work, the BFO frequency must be exactly adjusted.


title: Single-sideband modulation

text: If the BFO frequency is off, the output signal will be frequency-shifted (up or down), making speech sound strange and "Donald Duck"-like, or unintelligible.


title: Single-sideband modulation

text: For audio communications, there is a common agreement about the BFO oscillator shift of 1.7 kHz. A voice signal is sensitive to about 50 Hz shift, with up to 100 Hz still bearable. Some receivers use a carrier recovery system, which attempts to automatically lock on to the exact IF frequency. The carrier recovery doesn't solve the frequency shift. It gives better S/N ratio on the detector output.


title: Single-sideband modulation

text: As an example, consider an IF SSB signal centered at frequency formula_36 = 45000 Hz. The baseband frequency it needs to be shifted to is formula_37 = 2000 Hz. The BFO output waveform is formula_38. When the signal is multiplied by (aka "heterodyned with") the BFO waveform, it shifts the signal to  formula_39, "and" to formula_40, which is known as the "beat frequency" or "image frequency". The objective is to choose an formula_41 that results in  formula_42 = 2000 Hz. (The unwanted components at formula_43 can be removed by a lowpass filter; for which an output transducer or the human ear may serve).


title: Single-sideband modulation

text: There are two choices for formula_44: 43000 Hz and 47000 Hz, called "low-side" and "high-side" injection. With high-side injection, the spectral components that were distributed around 45000 Hz will be distributed around 2000 Hz in the reverse order, also known as an inverted spectrum. That is in fact desirable when the IF spectrum is also inverted, because the BFO inversion restores the proper relationships. One reason for that is when the IF spectrum is the output of an inverting stage in the receiver. Another reason is when the SSB signal is actually a lower sideband, instead of an upper sideband. But if both reasons are true, then the IF spectrum is not inverted, and the non-inverting BFO (43000 Hz) should be used.


title: Single-sideband modulation

text: If formula_45 is off by a small amount, then the beat frequency is not exactly formula_37, which can lead to the speech distortion mentioned earlier.


title: Single-sideband modulation

text: SSB techniques can also be adapted to frequency-shift and frequency-invert baseband waveforms (voice inversion). This voice scrambling method was made by running the audio of one side band modulated audio sample through its opposite (e.g. running an LSB modulated audio sample through a radio running USB modulation).


title: Single-sideband modulation

text: These effects were used, in conjunction with other filtering techniques, during World War II as a simple method for speech encryption. Radiotelephone conversations between the US and Britain were intercepted and "decrypted" by the Germans; they included some early conversations between Franklin D. Roosevelt and Churchill. In fact, the signals could be understood directly by trained operators. Largely to allow secure communications between Roosevelt and Churchill, the SIGSALY system of digital encryption was devised.


title: Single-sideband modulation

text: Today, such simple inversion-based speech encryption techniques are easily decrypted using simple techniques and are no longer regarded as secure.


title: Single-sideband modulation

text: Limitation of single-sideband modulation being used for voice signals and not available for video/TV signals leads to the usage of vestigial sideband. A vestigial sideband (in radio communication) is a sideband that has been only partly cut off or suppressed. Television broadcasts (in analog video formats) use this method if the video is transmitted in AM, due to the large bandwidth used. It may also be used in digital transmission, such as the ATSC standardized 8VSB.


title: Single-sideband modulation

text: The broadcast or transport channel for TV in countries that use NTSC or ATSC has a bandwidth of 6 MHz. To conserve bandwidth, SSB would be desirable, but the video signal has significant low-frequency content (average brightness) and has rectangular synchronising pulses. The engineering compromise is vestigial-sideband transmission. In vestigial sideband, the full upper sideband of bandwidth W2 = 4.0 MHz is transmitted, but only W1 = 0.75 MHz of the lower sideband is transmitted, along with a carrier. The carrier frequency is 1.25 MHz above the lower edge of the 6MHz wide channel. This effectively makes the system AM at low modulation frequencies and SSB at high modulation frequencies. The absence of the lower sideband components at high frequencies must be compensated for, and this is done in the IF amplifier.


title: Single-sideband modulation

text: When single-sideband is used in amateur radio voice communications, it is common practice that for frequencies below 10 MHz, lower sideband (LSB) is used and for frequencies of 10 MHz and above, upper sideband (USB) is used. For example, on the 40 m band, voice communications often take place around 7.100 MHz using LSB mode. On the 20 m band at 14.200 MHz, USB mode would be used.


title: Single-sideband modulation

text: An exception to this rule applies to the five discrete amateur channels on the 60-meter band (near 5.3 MHz) where FCC rules specifically require USB.


title: Single-sideband modulation

text: Extended single sideband is any J3E (SSB-SC) mode that exceeds the audio bandwidth of standard or traditional 2.9 kHz SSB J3E modes (ITU 2K90J3E) to support higher-quality sound.


title: Single-sideband modulation

text: Amplitude-companded single sideband (ACSSB) is a narrowband modulation method using a single sideband with a pilot tone, allowing an expander in the receiver to restore the amplitude that was severely compressed by the transmitter. It offers improved effective range over standard SSB modulation while simultaneously retaining backwards compatibility with standard SSB radios. ACSSB also offers reduced bandwidth and improved range for a given power level compared with narrow band FM modulation.


title: Single-sideband modulation

text: The generation of standard SSB modulation results in large envelope overshoots well above the average envelope level for a sinusoidal tone (even when the audio signal is peak-limited). The standard SSB envelope peaks are due to truncation of the spectrum and nonlinear phase distortion from the approximation errors of the practical implementation of the required Hilbert transform. It was recently shown that suitable overshoot compensation (so-called controlled-envelope single-sideband modulation or CESSB) achieves about 3.8 dB of peak reduction for speech transmission. This results in an effective average power increase of about 140%.


title: Single-sideband modulation

text: Although the generation of the CESSB signal can be integrated into the SSB modulator, it is feasible to separate the generation of the CESSB signal (e.g. in form of an external speech preprocessor) from a standard SSB radio. This requires that the standard SSB radio's modulator be linear-phase and have a sufficient bandwidth to pass the CESSB signal. If a standard SSB modulator meets these requirements, then the envelope control by the CESSB process is preserved.


title: Single-sideband modulation

text: In 1982, the International Telecommunication Union (ITU) designated the types of amplitude modulation:


title: Focus (band)

text: Focus are a Dutch progressive rock band formed in Amsterdam in 1969 by keyboardist, vocalist, and flutist Thijs van Leer, drummer Hans Cleuver, bassist Martijn Dresden, and guitarist Jan Akkerman. The band has undergone numerous formations in its history; since December 2016, it has comprised Van Leer, drummer Pierre van der Linden, guitarist Menno Gootjes, and bassist Udo Pannekeet. They have sold one million RIAA-certified albums in the United States.


title: Focus (band)

text: After the addition of Akkerman to Van Leer's rock trio in late 1969, the band named themselves "Focus" and initially worked for a Dutch production of the rock musical "Hair". Their debut album "Focus Plays Focus" (1970) gained little attention but the follow-up, "Moving Waves" (1971), and its lead single "Hocus Pocus", earned the band international recognition. Their success continued with "Focus 3" (1972) and "Hamburger Concerto" (1974), the former containing their second hit single, "Sylvia". After recording two albums with various musicians, including guitarist Phillip Catherine, singer P. J. Proby, and drummers Colin Allen, David Kemper, and Steve Smith, Focus dissolved in 1978. They briefly reunited in 1990 and 1997.


title: Focus (band)

text: In 2002, Van Leer reformed Focus with a new line-up that saw Van der Linden rejoining the group in 2004. The albums "Focus 8" (2002), "Focus 9 / New Skin" (2006), and "Focus X" (2012) were well received, and Focus continue to perform worldwide. Their most recent album is "Focus 11" (2018). They received a renewed interest after Nike used "Hocus Pocus" in its 2010 World Cup commercial Write The Future. Focus remain one of the most successful and influential rock bands from the Netherlands.


title: Focus (band)

text: Focus formed in mid-1969 by keyboardist, vocalist, and flautist Thijs van Leer, who recruited bass guitarist Martijn Dresden and drummer Hans Cleuver after he met them at sessions for the "Jazz and Poetry" radio program in Hilversum, Netherlands. The three went on to start a new three-piece band initially known as Thijs van Leer and the Rebaptised, playing a set formed mostly of cover songs by Traffic and original material mostly written by van Leer. In November 1969, during rehearsals at the theatre where van Leer performed as part of Ramses Shaffy's theatre group, they were joined by guitarist Jan Akkerman of the rock band Brainbox after Shaffy invited him to play with the trio. Van Leer later recalled the first try out session: "Jan came in and we jammed for hours, and it was really kicking". They then settled on the name Focus for the new band; Akkerman later said, "Focus is a Latin word that is the same in many languages. It means concentration, which is the meaning of what Focus does". Their first live gig as Focus followed at the Bird's Club in Rembrandtplein, and they soon secured a weekly residency playing two shows a night on two days. Early sets were mainly formed of covers, including "I Shall Be Released" by Bob Dylan, "A Whiter Shade of Pale" by Procol Harum and "Nights in White Satin" by The Moody Blues, mixed with original material, including van Leer's first song written with the group in mind, the instrumental "Focus".


title: Focus (band)

text: In their search for more work and a steady income, Focus was chosen to play as part of the pit band for the Dutch production of the rock musical "Hair" produced by Welsh actor Victor Spinetti. They were invited to audition at the Victoria Ballroom, London after Dutch lyricist Lennaert Nijgh suggested them to Del Newman, the musical's director. Cast member Robin Lent claimed the production was suffering and Focus, taking part only for the money, "were also pretty rusty ... but this changed and everybody got into it". The show, launched in December 1969, involved six nightly performances a week and gave them space to rehearse in the afternoons for free and store their equipment. An album of the soundtrack featuring the band was recorded in February 1970 and released soon after by Polydor Records.


title: Focus (band)

text: After "Hair" ended its run in June 1970, Focus declined an offer to tour the musical across the Netherlands for a year and a half and become a full time band. They had picked up more local gigs and dates across the country by this time, and had performed their first international gigs in Belgium and Spain. Their weekly earnings from gigs, plus their earnings from "Hair", had reached 400 guilders though Akkerman recalled the group failed to appreciate its value and were quick to spend it. To manage their finances more carefully, Cleuver was chosen to oversee their earnings and expenses.


title: Focus (band)

text: In 1970, Focus reached contact with Hubert Terheggen, director of Radio-Tele-Music Belgium-Holland, a music publishing division of Radio Luxembourg, through connections with Dresden's father. Terheggen enjoyed their music and signed them to his production company, secured deals with music publishers worldwide, and booked studio time for them to record their first album, which took place in January 1970 at Sound Techniques in Chelsea, London during time off from "Hair". The result was "Focus Plays Focus", better known as its international release title "In and Out of Focus", with Terheggen credited as producer and Jerry Boys the engineer. A mix of pop-oriented songs and instrumentals were recorded for the album; van Leer felt the vocals suffered as a result of singing English lyrics with a foreign accent, which inspired the group to become stronger instrumentally.


title: Focus (band)

text: After recording "Focus Plays Focus", the band struggled to find a label willing to release it. Their fortunes changed once they had recorded "House of the King", an instrumental by Akkerman with different accounts of its origin. Focus biographer Peet Johnson gives the general consensus that it was inspired by their appearance at the Barbarela de Conjuntos music contest in Majorca on 11 June 1970, where in the final round, their rendition of "Concierto de Aranjuez" by Joaquín Rodrigo ran over the 15-minute time limit, causing their plugs to be pulled. Focus proceeded to retaliate with van Leer playing the flute to the audience as a distraction while the rest of the group trashed the dressing room, after which they were arrested and spent a night in jail. Van Leer claims Akkerman came up with the track the day after their arrest, but the guitarist said he wrote it "on a mountain, with a beautiful Spanish stewardess ... in about five minutes". Focus recorded it without management's permission in one evening at Heemstede, yet it reached staff at Imperial Records who thought it was strong enough as a hit single and signed the group.


title: Focus (band)

text: "Focus Plays Focus" was released in September 1970, eight months after it was recorded. "Why Dream" and "Happy Nightmare" were released as their first single in the Netherlands. The album saw an American release in October by Sire Records, who had secured the rights to the band's international distribution, with "House of the King" added to the track listing. Prior to signing Focus, Sire founder Seymour Stein had flown to the Netherlands to see the band perform: "Without a doubt they were the most original band I had ever heard". Though the album received little commercial attention it earned Focus their first Edison Award, and the January 1971 release of "House of the King" went to number 10 on the home chart.


title: Focus (band)

text: Late in 1970, Akkerman had become increasingly unhappy with Cleuver and Dresden as a rhythm section and missed his former Brainbox bandmate, drummer Pierre van der Linden. He thought the pair struggled to incorporate their own identity or musicianship into the music. Akkerman then presented an ultimatum to van Leer by threatening to quit unless he asked Cleuver and Dresden to leave. Van Leer refused to part ways with his co-founders and instead told Akkerman to leave, which increased tension between the two as they were often at odds with each other. Cleuver sensed panic from Imperial as it knew "House of the King" would be a hit and wanted van Leer and Akkerman to stay together. The situation reached Terheggen, who asked Yde de Jong to become Focus's manager if he could reunite the band. After six weeks of driving to both members' homes to negotiate, van Leer agreed to fire Cleuver and Dresden and joined Akkerman, van der Linden, and newcomer bassist Cyril Havermans, who had played on Akkerman's second solo album "Profile", on the condition that the group continue as Focus.


title: Focus (band)

text: With the new line-up secured, Stein convinced English producer Mike Vernon to witness the band perform in the Netherlands in late 1970. Vernon was not so impressed with their first album, but enjoyed them as a live act: "I couldn't stop talking or thinking about them! They were formidable ... the power of Jan and Thijs alone gave me heart palpitations". Vernon subsequently agreed to produce their next studio album.


title: Focus (band)

text: Following rehearsals at Groeneveld Castle in Baarn, the band recorded "Focus II", better known as its international title "Moving Waves", in April and May 1971 at Sound Techniques and Morgan Studios, London. It showcased the band exploring progressive rock with jazz fusion and classical music elements with extended pieces and lengthy solos. Akkerman changed his sound, moving from a Gretsch White Falcon to a 1957 Gibson Les Paul Custom which enabled him to "'sing' on the guitar" to compensate for the group's weak vocals. Its opening track, "Hocus Pocus", became one of the band's best known tracks, featuring a recognisable rock guitar riff and van Leer's yodelling, whistling, and nonsensical vocals that developed through an improvised jam session. Akkerman wrote "Janis" as a tribute to Janis Joplin, and its closer, "Eruption", is a 23-minute multi-part track inspired by the operas "Orfeo ed Euridice" by Christoph Willibald Gluck and "L'Orfeo" by Claudio Monteverdi, combining rock, jazz and classical influences.


title: Focus (band)

text: Released in October 1971, "Moving Waves" became the band's international breakthrough, reaching number 2 in the UK, number 4 in the Netherlands, and number 8 in the US. The album awarded Focus with a Conamus Export Award, their second Edison Award, and gold certifications in the three aforementioned countries; the US award was issued in 1973 for selling 500,000 copies. Its success was helped by the release of "Hocus Pocus" as a single, released in the Netherlands in July 1971 which peaked at number 12 and exposed the band to a new audience. It peaked at number 20 in the UK in January 1973.


title: Focus (band)

text: Between the recording and release of "Moving Waves", Focus underwent further line-up changes. Shortly before a tour of France in July 1971, Van der Linden temporarily left the band after he found out Van Leer and Akkerman would get paid more than himself. He was replaced by Akkerman's younger brother, the 19-year-old Jacob "Cocky" Akkerman, before the rift was sorted and Van der Linden resumed on the drums at its conclusion. This was followed by Havermans' departure in September 1971, having handed in his notice in late June. He wished to sing on more tracks but was unable to do so within the confines of the group and pursued a solo career. The remaining members would reunite with the bassist in the US in early 1973 to play on his first solo album, "Cyril". Focus found their new bassist in Bert Ruiter, who had previously rehearsed with van der Linden prior to a gig in Texel in June 1971, and went on to see the band live around ten times thereafter. He was invited by the drummer to a tryout session, after which he became a full-time member.


title: Focus (band)

text: Focus resumed as a live act in September 1971; among their first gigs with Ruiter was in Rotterdam as part of the European Newport Jazz Festival on 29 October. Towards the end of the year, de Jong organised the band's first UK tour, a 15-gig trek between 15 February–10 March 1972, to increase exposure and record sales. Van Leer felt insecure about the tour at first and expressed the lack of courage from the group when they compared themselves to other popular UK rock acts. They arrived in the UK during a period of nationwide energy shortages, yet de Jong resolved the issue by having Focus tour with their own power generator. Despite receiving little money for their performances, they received radio airplay on BBC Radio 1, earned positive reviews from critics, and gained new fans. Akkerman recalled the attendance of the gigs "were packed because it was probably the only thing that was going on". On 28 May 1972, Focus were booked to play the Lincoln Festival for an estimated 40,000 people, but the band arrived too late for their spot and were unable to reschedule, leaving them furious at the missed opportunity. Two days later, Focus made their British television debut with a pre-recorded performance of excerpts of "Eruption" and "Hocus Pocus" on the BBC music show "The Old Grey Whistle Test". In the following month they were voted Brightest Hope by readers of "Melody Maker" and Best New Talent by readers of "New Musical Express".


title: Focus (band)

text: The band spent four days at Olympic Studios in Barnes, London in July 1972 recording their third album, "Focus 3". They had written a considerable amount of material by this time which led to the decision to make a double album, mostly by van Leer and Akkerman. "Focus 3" saw the group produce short and extended pieces, including the three-minute instrumental "Sylvia" and the group devised, 26-minute "Anonymous II" that devotes a solo spot for each member, and the addition of "House of the King". Upon its release in November 1972, the album went to number one in the Netherlands for one week, number 6 in the UK, and number 35 in the US. "Sylvia" was released as a single and reached number 4 in the UK in January 1973, the week after "Hocus Pocus" reached its peak on the same chart. This marked the first time in a decade that a band had two concurrent songs without lyrics in the UK top 40.


title: Focus (band)

text: To support the album, Focus began their largest tour since their formation with spots at the Reading Festival on 12 August 1972 followed by the "Melody Maker" Poll Awards show at The Oval, London on 30 September. This was followed by an exhaustive 31-date UK tour in 36 days that included their return appearance on "The Old Grey Whistle Test" on 12 December which caused a surge in interest for their records; host Bob Harris said the show was inundated with calls and letters about the band and Polydor's record plant printed nothing but Focus for ten days in order to meet demand. A performance of "Sylvia" from the Marquee Club, London aired on the BBC's "Top of the Pops" show. After a sell out UK tour in January 1973 that spanned 24 days, Focus completed their first of three North American tours in 1973 from February to April, supporting various acts including Gentle Giant, Frank Zappa and Yes. Upon their return, they toured the UK once more which included two sold out shows at the Rainbow Theatre, London in May 1973 which aired on BBC television.


title: Focus (band)

text: In early May 1973, Focus returned to Olympic Studios at Vernon's request and recorded a new cut of "Hocus Pocus" titled "Hocus Pocus II", released exclusively for the American market. Later in the month, the band retreated to Chipping Norton Recording Studios, Oxfordshire to record a fourth studio album. Akkerman had expressed a wish to put down "great ideas" he had written over the past year, but extensive touring had left the band physically and mentally exhausted which reduced their eagerness to write and record new group material. Vernon recalled Akkerman's reluctance to take part in the sessions which to him, "led to the final dissolving of the Akkerman/van Leer partnership". Around 40 minutes of songs was eventually recorded, yet the pair refused to co-operate and recorded their parts without the other present. Vernon later called this period as "probably the worst ten days I've ever spent in a studio". As a new studio album could not be released, Vernon took the recording from the Rainbow Theatre concerts and released it as a live album, "At the Rainbow". The album went to number 23 in the UK in October 1973. The band went on to receive a "Billboard" Trendsetter Award for their success after notching up two gold albums, combining sales of one million copies sold in the US, and one gold single. Recordings from the Chipping Norton sessions remained unreleased, rearranged years later, or included in the collection of previously unreleased Focus studio recordings "Ship of Memories", in 1976.


title: Focus (band)

text: Upon their return from their second North American tour from June to September 1973, the band returned to Groeneveld Castle for rehearsals. Van der Linden failed to turn up, and the group soon learned of his decision to quit the band. Van Leer claimed rock music was a step in the wrong direction for the drummer, who wished to pursue jazz. Vernon suggested English players Mitch Mitchell, Aynsley Dunbar, and Colin Allen as replacements. Allen was the only one available as his previous group, Stone the Crows had recently broken up. He flew to the Netherlands to meet the band and after a successful try out session, he joined the group. "It all happened pretty quickly ... I fitted in." His arrival came eight days before the band's upcoming North American tour was to start.


title: Focus (band)

text: From January to March 1974, Focus recorded "Hamburger Concerto" at Olympic Studios with Vernon. Recording was met with growing tension, with van Leer, Ruiter and Allen working together during the day and Akkerman recording his parts in the evening. The album presents further classical music influences, opening with "Delitæ Musicæ", a lute piece Akkerman adapted from "Delitæ Musicæ Cantiones" by Dutch composer Joachim van den Hove. It closes with a six-part, 20-minute conceptual title track inspired by Akkerman once eating a hamburger while watching cartoons at his hotel room in New York City and was put down in basic form during the 1973 Chipping Norton sessions. Released in May 1974, "Hamburger Concerto" peaked at number 5 in the Netherlands and number 20 in the UK. Van Leer claimed it was the band's "First proper work. It has strength and purpose", and Vernon later said the album and "Moving Waves" were the most rewarding on a second listen. Akkerman later felt the album suffered as his position in the group had weakened with van der Linden gone and claimed van Leer "broke up the tandem I had with Pierre rhythmically". A single version of "Harem Scarem" was released but did not chart in the UK.


title: Focus (band)

text: Focus toured "Hamburger Concerto" from March 1974, starting with dates across Europe, including two in London that were filmed for broadcast on the BBC2 and US television music series "In Concert". They began their first tour of Japan and Australia in June, followed by an appearance at the Reading Festival in August and the start of their fourth North American tour in September. They were featured on the nationally syndicated "Don Kirshner's Rock Concert" show aired live from Long Beach, California, giving the band widespread exposure. At one concert during the US leg, van Leer and Akkerman noticed a young Michael Jackson sitting in the front row. Upon their return, the band completed a 24-day tour of Europe and two compilation albums were released, "The Story of Focus" and "Masters of Rock".


title: Focus (band)

text: In 1975, the band gathered at Morgan Studios, Brussels to rehearse and record "Mother Focus". They reunited with Terheggen to oversee the project as executive producer, but the group found themselves without a collective direction; van Leer put its failure down to the band's recording contract which required them to produce one studio album each year, the pressures of touring had affected the songs, and a lack of overall interest from the band. In addition, Akkerman continued to record parts separately, showing no sign of an improvement in his relationship with van Leer. Ruiter became a more prominent songwriter and contributed more ideas than he had on previous Focus albums; he and Allen, after recording "I Need a Bathroom", began to experiment with a drum machine to aid their ideas, but the drummer recalled Akkerman got angry, shouted at him, and tossed the machine across the room. Allen was fired the next day, not knowing who made the decision. He was required to pay what he owed to the group during his tenure which amounted to roughly £10,000, using his earnings from royalties to pay much of it. Despite the problems, he "will always remain proud to have been a member". In May 1975, at the suggestion of their studio engineer, American drummer David Kemper was brought in to complete the rest of the album.