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Galactic Center

The Galactic Center is the barycenter of the Milky Way and a corresponding point on the rotational axis of the galaxy. Its central massive object is a supermassive black hole of about 4 million solar masses, which is called Sagittarius A*,  a compact radio source which is almost exactly at the galactic rotational center. The Galactic Center is approximately 8 kpc away from Earth in the direction of the constellations Sagittarius, Ophiuchus, and Scorpius, where the Milky Way appears brightest, visually close to the Butterfly Cluster (M6) or the star Shaula, south to the Pipe Nebula.

There are around 10 million stars within one parsec of the Galactic Center, dominated by red giants, with a significant population of massive supergiants and Wolf–Rayet stars from star formation in the region around 1 million years ago. The core stars are a small part within the much wider galactic bulge.

Discovery

Because of interstellar dust along the line of sight, the Galactic Center cannot be studied at visible, ultraviolet, or soft (low-energy) X-ray wavelengths. The available information about the Galactic Center comes from observations at gamma ray, hard (high-energy) X-ray, infrared, submillimetre, and radio wavelengths.

Immanuel Kant stated in Universal Natural History and Theory of the Heavens (1755) that a large star was at the center of the Milky Way Galaxy, and that Sirius might be the star. Harlow Shapley stated in 1918 that the halo of globular clusters surrounding the Milky Way seemed to be centered on the star swarms in the constellation of Sagittarius, but the dark molecular clouds in the area blocked the view for optical astronomy.

In the early 1940s Walter Baade at Mount Wilson Observatory took advantage of wartime blackout conditions in nearby Los Angeles, to conduct a search for the center with the 100 in Hooker Telescope. He found that near the star Alnasl (Gamma Sagittarii), there is a one-degree-wide void in the interstellar dust lanes, which provides a relatively clear view of the swarms of stars around the nucleus of the Milky Way Galaxy. This gap has been known as Baade's Window ever since.

At Dover Heights in Sydney, Australia, a team of radio astronomers from the Division of Radiophysics at the CSIRO, led by Joseph Lade Pawsey, used "sea interferometry" to discover some of the first interstellar and intergalactic radio sources, including Taurus A, Virgo A and Centaurus A. By 1954 they had built an 80 ft fixed dish antenna and used it to make a detailed study of an extended, extremely powerful belt of radio emission that was detected in Sagittarius. They named an intense point-source near the center of this belt Sagittarius A, and realised that it was located at the very center of the Galaxy, despite being some 32 degrees south-west of the conjectured galactic center of the time.

In 1958 the International Astronomical Union (IAU) decided to adopt the position of Sagittarius A as the true zero coordinate point for the system of galactic latitude and longitude. In the equatorial coordinate system the location is: RA 17:45:40.04, Dec -29°00′28.1″ (J2000 epoch).

In July 2022, astronomers reported the discovery of massive amounts of prebiotic molecules, including some associated with RNA, in the Galactic Center of the Milky Way Galaxy.

Distance to the Galactic Center
The exact distance between the Solar System and the Galactic Center is not certain, although estimates since 2000 have remained within the range 24 - 28.4 kly. The latest estimates from geometric-based methods and standard candles yield the following distances to the Galactic Center:
 * $(stat) ± 0.2(syst)7.4$ or $7.4 kpc$ ($≈24 kly$)
 * 7.94 or $7.62 kpc$ ($≈24.8 kly$)
 * $7.7 kpc$ or $≈25.1 kly$ ($8 kpc$)
 * $≈26 kly$ ($(stat) ± 0.20(syst)7.98$)
 * $8 kpc$ kpc ($≈26 kly$)
 * $8.33 kpc$ ($≈27 kly$)
 * $8 kpc$ kpc ($≈25.96 kly$)
 * $8.7 kpc$ ($≈28.4 kly$)
 * $8.122 kpc$ kpc ($≈26.49 kly$)
 * $(stat) ± 0.022(syst)8.178$ kpc ($≈26.67 kly$)

An accurate determination of the distance to the Galactic Center as established from variable stars (e.g. RR Lyrae variables) or standard candles (e.g. red-clump stars) is hindered by numerous effects, which include: an ambiguous reddening law; a bias for smaller values of the distance to the Galactic Center because of a preferential sampling of stars toward the near side of the Galactic bulge owing to interstellar extinction; and an uncertainty in characterizing how a mean distance to a group of variable stars found in the direction of the Galactic bulge relates to the distance to the Galactic Center.

The nature of the Milky Way's bar, which extends across the Galactic Center, is also actively debated, with estimates for its half-length and orientation spanning between 1–5 kpc (short or a long bar) and 10–50°. Certain authors advocate that the Milky Way features two distinct bars, one nestled within the other. The bar is delineated by red-clump stars (see also red giant); however, RR Lyrae variables do not trace a prominent Galactic bar. The bar may be surrounded by a ring called the 5-kpc ring that contains a large fraction of the molecular hydrogen present in the Milky Way, and most of the Milky Way's star formation activity. Viewed from the Andromeda Galaxy, it would be the brightest feature of the Milky Way.

Supermassive black hole

The complex astronomical radio source Sagittarius A appears to be located almost exactly at the Galactic Center and contains an intense compact radio source, Sagittarius A*, which coincides with a supermassive black hole at the center of the Milky Way. Accretion of gas onto the black hole, probably involving an accretion disk around it, would release energy to power the radio source, itself much larger than the black hole.

A study in 2008 which linked radio telescopes in Hawaii, Arizona and California (Very-long-baseline interferometry) measured the diameter of Sagittarius A* to be 44 million kilometers (0.3 AU). For comparison, the radius of Earth's orbit around the Sun is about 150 million kilometers (1.0 AU), whereas the distance of Mercury from the Sun at closest approach (perihelion) is 46 million kilometers (0.3 AU). Thus, the diameter of the radio source is slightly less than the distance from Mercury to the Sun.

Scientists at the Max Planck Institute for Extraterrestrial Physics in Germany using Chilean telescopes have confirmed the existence of a supermassive black hole at the Galactic Center, on the order of 4.3 million solar masses. Later studies have estimated a mass of 3.7 million or 4.1 million solar masses.

On 5 January 2015, NASA reported observing an X-ray flare 400 times brighter than usual, a record-breaker, from Sagittarius A*. The unusual event may have been caused by the breaking apart of an asteroid falling into the black hole or by the entanglement of magnetic field lines within gas flowing into Sagittarius A*, according to astronomers.

Gamma- and X-ray emitting Fermi bubbles
In November 2010, it was announced that two large elliptical lobe structures of energetic plasma, termed bubbles, which emit gamma- and X-rays, were detected astride the Milky Way galaxy's core. Termed Fermi or eRosita bubbles, they extend up to about 25,000 light years above and below the Galactic Center. The galaxy's diffuse gamma-ray fog hampered prior observations, but the discovery team led by D. Finkbeiner, building on research by G. Dobler, worked around this problem. The 2014 Bruno Rossi Prize went to Tracy Slatyer, Douglas Finkbeiner, and Meng Su "for their discovery, in gamma rays, of the large unanticipated Galactic structure called the Fermi bubbles".

The origin of the bubbles is being researched. The bubbles are connected and seemingly coupled, via energy transport, to the galactic core by columnar structures of energetic plasma termed chimneys. In 2020, for the first time, the lobes were seen in visible light and optical measurements were made. By 2022, detailed computer simulations further confirmed that the bubbles were caused by the Sagittarius A* black hole.

Stellar population

The central cubic parsec around Sagittarius A* contains around 10 million stars. Although most of them are old red giant stars, the Galactic Center is also rich in massive stars. More than 100 OB and Wolf–Rayet stars have been identified there so far. They seem to have all been formed in a single star formation event a few million years ago. The existence of these relatively young stars was a surprise to experts, who expected the tidal forces from the central black hole to prevent their formation.

This paradox of youth is even stronger for stars that are on very tight orbits around Sagittarius A*, such as S2 and S0-102. The scenarios invoked to explain this formation involve either star formation in a massive star cluster offset from the Galactic Center that would have migrated to its current location once formed, or star formation within a massive, compact gas accretion disk around the central black-hole. Current evidence favors the latter theory, as formation through a large accretion disk is more likely to lead to the observed discrete edge of the young stellar cluster at roughly 0.5 parsec. Most of these 100 young, massive stars seem to be concentrated within one or two disks, rather than randomly distributed within the central parsec. This observation however does not allow definite conclusions to be drawn at this point.

Star formation does not seem to be occurring currently at the Galactic Center, although the Circumnuclear Disk of molecular gas that orbits the Galactic Center at two parsecs seems a fairly favorable site for star formation. Work presented in 2002 by Antony Stark and Chris Martin mapping the gas density in a 400-light-year region around the Galactic Center has revealed an accumulating ring with a mass several million times that of the Sun and near the critical density for star formation.

They predict that in approximately 200 million years, there will be an episode of starburst in the Galactic Center, with many stars forming rapidly and undergoing supernovae at a hundred times the current rate. This starburst may also be accompanied by the formation of galactic relativistic jets, as matter falls into the central black hole. It is thought that the Milky Way undergoes a starburst of this sort every 500 million years.

In addition to the paradox of youth, there is a "conundrum of old age" associated with the distribution of the old stars at the Galactic Center. Theoretical models had predicted that the old stars—which far outnumber young stars—should have a steeply-rising density near the black hole, a so-called Bahcall–Wolf cusp. Instead, it was discovered in 2009 that the density of the old stars peaks at a distance of roughly 0.5 parsec from Sgr A*, then falls inward: instead of a dense cluster, there is a "hole", or core, around the black hole.

Several suggestions have been put forward to explain this puzzling observation, but none is completely satisfactory. For instance, although the black hole would eat stars near it, creating a region of low density, this region would be much smaller than a parsec. Because the observed stars are a fraction of the total number, it is theoretically possible that the overall stellar distribution is different from what is observed, although no plausible models of this sort have been proposed yet.

Gallery
In May 2021, NASA published new images of the Galactic Center, based on surveys from Chandra X-ray Observatory and other telescopes. Images are about 2.2 degrees (1,000 light years) across and 4.2 degrees (2,000 light years) long.

title: Galactic Center


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Department for Culture, Media and Sport

The Department for Culture, Media and Sport (DCMS) is a ministerial department of the Government of the United Kingdom. It holds the responsibility for culture and sport in England, and some aspects of the media throughout the UK, such as broadcasting. Its main offices are at 100 Parliament Street, occupying part of the building known as Government Offices Great George Street.

It also has responsibility for the tourism, leisure and creative industries (some jointly with the Department for Business and Trade). The department was also responsible for the delivery of the 2012 Olympic Games and Paralympic Games.

From 2017 to 2023, the department had responsibility for the building of a digital economy and the internet and was known as the Department for Digital, Culture, Media and Sport. The responsibilities for digital policy were transferred to the newly created Department for Science, Innovation and Technology in the February 2023 cabinet reshuffle.

History and responsibilities
DCMS originates from the Department of National Heritage (DNH), which itself was created on 11 April 1992 out of various other departments, soon after the Conservative election victory. The former ministers for the Arts and for Sport had previously been located in other departments.

DNH was renamed as the Department for Culture, Media and Sport (DCMS) on 14 July 1997, under the premiership of Tony Blair. It was renamed to Department for Digital, Culture, Media and Sport on 3 July 2017, staying DCMS under the premiership of Theresa May to reflect the department's increased activity in the digital sector. The department was renamed back to the Department for Culture, Media and Sport (DCMS) in February 2023, with responsibility for digital moving to the new Department for Science, Innovation and Technology.

2012 Olympics
DCMS was the co-ordinating department for the successful bid by London to host the 2012 Olympics and appointed and oversees the agencies delivering the Games' infrastructure and programme, principally the Olympic Delivery Authority (ODA) and LOCOG.

The June 2007 Cabinet reshuffle led to Tessa Jowell MP taking on the role of Paymaster General and then Minister for the Cabinet Office while remaining Minister for the Olympics. Ministerial responsibility for the Olympics was shared with Ms Jowell in the Cabinet Office, but the staff of the Government Olympic Executive (GOE) remained based in DCMS.

2010–2022
Following the 2010 general election, ministerial responsibility for the Olympics returned to the Secretary of State. Although Jeremy Hunt's full title was Secretary of State for Culture, Olympics, Media and Sport, the department's name remained unchanged. On 4 September 2012, Hunt was appointed Health Secretary in a cabinet reshuffle and replaced by Maria Miller. Maria Miller later resigned due to controversy over her expenses. Her replacement was announced later that day as Sajid Javid.

After the 2015 general election, John Whittingdale was appointed as Secretary of State, tasked with initiating the BBC Charter review process. DCMS received full responsibility for the digital economy policy, formerly jointly held with BIS, and sponsorship of the Information Commissioner's Office from the Ministry of Justice.

Whittingdale was replaced by Karen Bradley after the referendum on the UK's membership of the EU in July 2016. The Office for Civil Society moved from the Cabinet Office to DCMS as part of the same reshuffle.

In January 2018, Matthew Hancock, previous Minister of State for Digital, was appointed Secretary of State as part of a Cabinet reshuffle. In the 9 July 2018 reshuffle, Jeremy Wright became the Secretary of State. Nicky Morgan became Secretary of State in July 2019; she stood down as an MP at the 2019 United Kingdom general election but was ennobled as Baroness Morgan of Cotes and retained her position from within the House of Lords. As part of the 13 February 2020 reshuffle, Oliver Dowden MP was appointed Secretary of State for Digital, Culture, Media and Sport. Nadine Dorries succeeded on 15 September 2021. In July 2022, Dorries personally granted Grade II-listed status to a plaque of Cecil Rhodes which she believed is of "special historic interest". This decision attracted controversy. On 5 September 2022, in anticipation of the appointment of Liz Truss as Prime Minister, Dorries tendered her resignation as culture secretary.

Policy areas
It is responsible for government policy in the following areas:
 * The arts
 * Broadcasting, including the BBC and Channel 4
 * Civil society
 * Charities
 * Creative industries
 * Advertising
 * Design
 * Fashion
 * Film
 * Music industry
 * Publishing
 * Historic environment
 * Architecture and design
 * Arts market
 * Cultural property and heritage
 * Entertainment licensing
 * Gambling and racing
 * Press freedom and regulation
 * Libraries
 * Museums and galleries
 * The National Lottery
 * Tourism
 * Sport
 * Olympic legacy

Other responsibilities
Other responsibilities of DCMS include listing of historic buildings, scheduling of ancient monuments, export licensing of cultural goods, and management of the Government Art Collection (GAC).

The Secretary of State has responsibility for the maintenance of the land and buildings making up the historic Royal Estate under the Crown Lands Act 1851. These inherited functions, which were once centralised in the Office of Works, are now delivered as follows:

 * The Royal Parks are maintained by an executive agency within DCMS, the Royal Parks Agency;
 * The unoccupied royal palaces in England are managed by a contract with Historic Royal Palaces;
 * Maintenance of the occupied royal palaces in England was funded by an annual grant-in-aid to the Royal Household until 31 March 2012. The Secretary of State for Culture retains legal responsibility for these palaces, but from 1 April 2012 this funding was amalgamated with the Civil List into a single Sovereign Grant administered by HM Treasury.  DCMS continues to make a separate small grant to the Royal Household for the maintenance of Marlborough House

The department also has responsibility for state ceremonial occasions and royal funerals. However, responsibility for the Civil List element of head-of-state expenditure and income from the separate Crown Estate remains with the Chancellor of the Exchequer.

DCMS works jointly with the Department for Business, Innovation and Skills (BIS) on design issues, including sponsorship of the Design Council, and on relations with the computer games and publishing industries.

DCMS works with local community system providers in broadcasting service via TV and Internet channels. In September 2022, it managed to hold a home office visit activity for media and data security knowledge sharing across different government departments.

DCMS organises the annual Remembrance Day Ceremony at the Cenotaph and has responsibility for providing humanitarian assistance in the event of a disaster. In the government's response to the 7 July 2005 London bombings the department coordinated humanitarian support to the relatives of victims and arranged the memorial events.

DCMS has also supported cyber initiatives such as Cyber Discovery and the UK Cyber Security Forum to support innovation in the cyber industry.

Ministers
The DCMS ministers are as follows, with cabinet ministers in bold:

On 7 February 2023 it was announced that the Department's Permanent Secretary role would be performed on an interim basis by Ruth Hannant and Polly Payne (job share).

Bodies sponsored by DCMS
The DCMS has policy responsibility for three statutory corporations and two public broadcasting authorities. These bodies and their operation are largely independent of government policy influence.

Non-ministerial departments
DCMS works with two non-ministerial departments:
 * The National Archives
 * Charity Commission for England and Wales

Statutory corporations
The statutory corporations are:
 * Channel Four Television Corporation
 * Historic Royal Palaces
 * Office of Communications (Ofcom)

The department was responsible for the Horserace Totalisator Board (The Tote) until the sale of the Tote's business to Betfred in July 2011.

Public corporations
The public corporations are:
 * British Broadcasting Corporation
 * Sianel Pedwar Cymru – and the S4C Authority which regulates and manages S4C

Non-departmental public bodies
The DCMS sponsors the following executive non-departmental public bodies including a number of museums and galleries:
 * Arts Council England
 * British Film Institute
 * British Library
 * British Museum
 * Equality and Human Rights Commission
 * Gambling Commission
 * Museum of the Home (formerly the Geffrye Museum)
 * Historic England (separated from English Heritage in 2015, formally the Historic Buildings & Monuments Commission for England)
 * Horniman Museum
 * Horserace Betting Levy Board
 * Imperial War Museum
 * National Gallery
 * National Heritage Memorial Fund (the Trustees of the NHMF also administer the Heritage Lottery Fund)
 * National Maritime Museum
 * National Museums Liverpool
 * National Portrait Gallery
 * Natural History Museum
 * Royal Armouries
 * Science Museum Group
 * Sir John Soane's Museum
 * Sport England (formally the English Sports Council)
 * Sports Grounds Safety Authority
 * Tate
 * UK Anti-Doping
 * UK Sport (formally the UK Sports Council)
 * Victoria and Albert Museum
 * VisitBritain (formally the British Tourist Authority)
 * VisitEngland
 * Wallace Collection

The DCMS sponsors the following advisory non-departmental public bodies:
 * Reviewing Committee on the Export of Works of Art and Objects of Cultural Interest
 * Theatres Trust
 * Treasure Valuation Committee

DCMS also has responsibility for two other bodies classified by the Office for National Statistics as being within the central government sector:

 * The London Organising Committee for the Olympic Games (LOCOG) is a company limited by guarantee, established by a joint venture agreement between the Secretary of State for Culture, Media and Sport, the Mayor of London and the British Olympic Association.
 * Churches Conservation Trust

DCMS is also the major financial sponsor of the following bodies, which are not classed as part of the UK central government
 * Chatham Historic Dockyard Trust
 * Greenwich Foundation for the Old Royal Naval College for the Old Royal Naval College
 * Tyne and Wear Museums

Sponsorship of the National Endowment for Science, Technology and the Arts (NESTA) transferred to the Department for Innovation, Universities and Skills in June 2007. The Museum of London transferred to the Greater London Authority from 1 April 2008.

DCMS formerly sponsored eight Regional Cultural Consortiums with NDPB status. In July 2008, DCMS announced that the consortiums would be phased out over a twelve-month period and replaced by a new alliance of the regional teams of Arts Council England, Sport England, English Heritage and the MLA.

Devolution
Culture, sport and tourism are devolved matters, with responsibility resting with corresponding departments in the Scottish Government in Scotland, the Welsh Government in Wales and the Northern Ireland Executive in Northern Ireland.

Media-related policy is generally reserved to Westminster i.e. not devolved. These areas include:

Scotland
Reserved matters: Scotland's comparability factor (the proportion of spending in this area devolved to the Scottish Government) was 68% for 2021/22.
 * Film classification
 * Broadcasting
 * Public lending right
 * Entertainment licensing
 * National Lottery
 * Digital economy
 * Telecommunications and broadband
 * Internet
 * press freedom and regulation

Northern Ireland
Reserved matters:
 * Broadcasting
 * The National Lottery
 * Digital economy
 * Telecommunications and broadband
 * Internet

The department's main counterparts in Northern Ireland are as follows:
 * Department of Culture, Arts and Leisure (architecture, arts, culture, galleries, libraries, museums, sport)
 * Department of the Environment (historic built environment)
 * Department of Enterprise, Trade and Investment (tourism)
 * Department for Social Development (gambling, liquor licensing)

Northern Ireland's comparability factor (the proportion of spending in this area devolved to the Northern Ireland Executive) was 69.9% for 2021/22.

Wales
Reserved matters: Wales' comparability factor (the proportion of spending in this area devolved to the Welsh Government) was 67.7% for 2021/22.
 * Film classification
 * Broadcasting, BBC
 * Public lending right
 * Digital economy
 * Entertainment licensing
 * Betting, Gambling and Lotteries
 * Telecommunications and broadband
 * Internet
 * press freedom and regulation

Video clips

 * DCMS YouTube channel

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Department for Culture, Media and Sport


text:
Talk:Harmonic mean

Electrical Resistance?
Is this true?: "in an electrical circuit you have two resistors connected in parallel, one with 40 ohms and the other with 60 ohms, then the average resistance is 48 ohms" 210.224.218.13 15:51, Nov 20, 2003 (UTC)

 * The total resistance is given by the harmonic mean divided by the number of resistors, not just the harmonic mean. The resistance should then be around 24 ohms. 128.84.45.4 20:55, Jul 06, 2004 (UTC)

 * Exactly. The average resistance is 48 ohms, while the equivalent resistance is 24 ohms. (I don't think "average resistance" is a term any physicist/electrician/electrical engineer would use, since it would have different meanings for resistors in parallel as for resistors in series, but that's how this article is using the term, and I think that's fine.) Ruakh 02:32, 8 Mar 2005 (UTC)

 * Shouldn't that read arithmetic mean resistance rather than average?. Also resistance should strictly be impedance Smidoid 12:18, 23 Feb 2008 (UTC)

 * See my section on "taking the average of ratios", The ratios are voltages/currents which is the definition of resistance.  In parallel, the voltages are all the same, but the currents are different.  So the problem falls in the case of denominators unknown which implies use of the harmonic mean.  We obtain the "ratio of averages" the voltage average is fixed at V.  But what we need is the ratio of V to total current so we must multiply the average current denominator by N (the number of resistors) to get total current (same as dividing the "average of ratios" by N).   Thus we get the value of the equivalent resistance V/I(total).  Note that we can do this without assigning any numerical values to voltage and currents.David S. Lawyer 10:23, 23 November 2015 (UTC)  — Preceding unsigned comment added by Dlawyer (talk • contribs)

Negative Numbers?
What if the two numbers are negative? The harmonic mean of -1 and -2 is 2/(-1 + -(1/2)), which equals 2/(-1.5) or 2 * -(2/3) or -4/3, which is larger than the arithmetic mean, which is -1.5 (as compared to the geometric mean being -1.3......). However, the geometric mean would actually be the square root of 2... messed up for negative numbers, I suppose. ugen64 03:26, Dec 16, 2003 (UTC)

 * The article explicitly restricts the notion of harmonic mean to the positive reals. If it didn't, then we'd have the bigger issue of finding the harmonic mean of -1 and 1 (since 2/(1/-1 + 1/1) = 2/(-1 + 1) = 2/0, which is not defined). Ruakh 02:32, 8 Mar 2005 (UTC)

stats comment
added the definition when frequency is not unary. Nova67 16:16, 12 June 2005 (UTC)

 * I don't understand. Are you saying that you added such a definition? If so, there's really no need to mention it here; the article history will show that. Are you saying that such a definition needs to be added? Please explain what you mean. Ruakh 14:27, 15 Jun 2005 (UTC)

Removed example.
I removed the following text that a non-registered user inserted into the article, as its placement and internal organization were confusing and ill-suited to the article:

 * ===An example===

 * * An experiment yields the following data: 34,27,45,55,22,34 To get the harmonic mean

 * # How many items? There are 6. Therefore n=6

 * # What is the sum on the bottom of the fraction? It is 0.181719152307

 * # Get the reciprocal of that sum. It is 5.50299727522

 * # To get the the harmonic mean multiply that by n to get 33.0179836513

If anyone wants to rescue it, feel free. It needs to be rewritten, given lead text, and placed appropriately (perhaps in a sidebar; the "ExampleSidebar" template could be used). Ruakh 19:48, 8 September 2005 (UTC)

Average speed
This statement warrants a little qualification:

if for half the distance of a trip you travel at 40 miles per hour and for the other half of the distance you travel at 60 miles per hour, then your average speed for the trip is given by the harmonic mean of 40 and 60, which is 48; that is, the total amount of time for the trip is the same as if you traveled the entire trip at 48 miles per hour.

it may be true, of course, but not independent of the premises of the problem which are rather subtely expressed only. It is equally true that:

if for half the time of a trip you travel at 40 miles per hour and for the other half of the time you travel at 60 miles per hour, then your average speed for the trip is given by the arithmetic mean of 40 and 60, which is 50; that is, the total distance you covered is the same as if you traveled the entire trip at 50 miles per hour.

While this isn't central to a discussion of harmonic means, as I discovered recently people are easily lulled, by reading such concise statements, into the mistaken belief that the harmonic mean is THE correct manner of averaging independent speed measures. And this, it is not. It is A correct measure given certain premises concerning the problem. Bwechner 05:00, 25 November 2005 (UTC)

The statement as it stands isn't even strictly correct. The harmonic mean of 40 and 60 is hardly an integer -- in fact, it is closer to 49 than it is to 48! 142.162.10.169 (talk) 01:33, 20 March 2008 (UTC)

The arithmetic mean of the speed example is stated to be 50 in the "Relationship with other means" section, but 40 in the "Examples" section. — Preceding unsigned comment added by 67.249.89.107 (talk) 15:15, 2 October 2019 (UTC)
 * Thank you for pointing that out. I've changed the 50 to 40.   Dbfirs  15:47, 2 October 2019 (UTC)

Deriving arithmetic and geometric means when n>2
OK, so in the article, there is a derived method of finding arithmetic mean and geometric mean when n=2. What about when n=3 or 4? I;m sorry I don't know this- I'm just a high school student at the Illinois Mathematics and Science Academy —The preceding unsigned comment was added by 143.195.150.93 (talk • contribs).

 * The article gives the relationship that for just two numbers, H = G2/A. This relationship doesn't hold for more than two numbers, and there's no corresponding relationship for more than two numbers: for any n > 2 and any A and G, there are infinitely many possibilities for H, unless A = G, in which case A = G = H. Ruakh 01:58, 8 March 2006 (UTC)
 * In the example area, it's shown that for pumps p1 p2 p3 ... working to empty the same pool, it takes $$\frac {{p_1}\cdot{p_2}\cdot{p_3}}{p_1+p_2+p_3}$$ hours to empty it, I believe this is the part you're mentioning? I think this only works for n==2. Assume you have n pumps which take 2 hours to empty it. A) for n==1, this gives $$\frac{2}{2}=1 hour$$. B) for n==2 this gives $$\frac{{2}\cdot{2}}{2+2}=1 hour$$ (correct) C) For n==1000 we get $$\frac{2^{1000}}{2000}=5.36*10^{297} hours$$. For this problem, it makes more sense to have it be 2/n for this problem. If the pool is 2m gallons, they drain it in 2 hours, giving a rate of m gallons per hour, then the rate at which n pumps can drain a pool of 2m gallons is 2m/mn=2/n. The general version seems to be exactly the harmonic mean over n (as intended), I.E.
 * $$\frac{H}{n} = \frac{1}{\frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_n}} = \frac{1}{\sum_{i=1}^n \frac{1}{a_i}}$$

Cassiline (talk) 19:59, 13 April 2008 (UTC)

”Draining Pool” example
I don’t believe the pool-draining example highlights a special case, and I suspect the example might warrant clarification or removal. As this is not my area of expertise, I submit the following for your review: The rates provided (4 and 6) are “hours per pool”. The harmonic mean, then, is the average number of hours it would take one pump to empty the pool (4.8 hours). With twice as many pumps working at that average rate, it would of course take only half the time, or 2.4 hours. —Preceding unsigned comment added by Jenburton (talk • contribs) 00:31, 25 August 2009 (UTC)

Trigonometric example

 * I added this part and I want to thank the person who did the editing. It really flows well.

 * This example only work with the double angle tangent identity. I tried $$tan A = tan (3x)$$ and the formula failed to work out. It will fail to work out for any $$tan(nx)$$ where $$n /= 2$$.

Opinionhead (talk) 20:09, 10 May 2011 (UTC)

Omission - Relationship with other means.
I just deleted a statement that cannot be correct, or is at least not properlyy explained/defined. DrMicro wrote that

"If all the real variables in the set are > 0 then H ≥ M2 / m where m is the minimum in the set."

If this were the case (and M is the maximum as defined in the main article above) then the H would be greater than the maximum, meaning that it no longer qualified as a mean. If someone knows the correct statement of this inequality as stated in the paper (it's too late for me to be reading a maths paper right now) feel free to re-add and correct.

I'll message DrMicro about it too.

Cheers,

JPBrod (talk) 23:28, 21 April 2012 (UTC)

 * Thank you. It must have been a typo. I have read the reference and I cannot find this inequality in this paper. Im afraid I do not have an explanation for its inclusion. Well spotted. DrMicro (talk) 11:14, 22 April 2012 (UTC)

"Theoretical values" section is confused
Currently the subsection "Theoretical values" of the section "Statistics" reads

 * Theoretical value

 * The variance of the harmonic mean is

 * $$ \operatorname{Var}\left( \frac { 1 } { x } \right) = \frac { m \left[ \operatorname{E}( 1 / x - 1 ) \right] } { n m^2 } $$

 * where m is the arithmetic mean of the reciprocals, x are the variates, n is the population size and E is the expectation operator. Asymptotically E(1 / x) is distributed normally.

 * The mean of the sample m is also distributed normally with variance s2.

 * $$ s^2 = \frac  { m [ \operatorname{E}( 1 / x - 1 ) ] }{ m^2 n } $$''

Unfortunately the citation for this is a paper at a 1972 conference, so there's no way to check the source to confirm that it's been copied correctly (or even that it was right in the first place since it appears not to have been refereed or published). Moreover, it's been changed in the present article from what it said earlier.

(1) This looks suspicious to me because the variance of the harmonic mean is given as the same expression as the variance of the arithmetic mean of the reciprocals.

(2) It looks suspicious because surely the original source would not have left something with m not cancelled from the numerator and denominator.

(3) It says the variance of the harmonic mean is denoted as var(1/x) where "x are the variates". That doesn't make sense to me -- if 1/x is the harmonic mean, then x is the mean of the reciprocals of the variates, not "the variates". So this expression, even if correct, may refer to the variance of the reciprocals of the variates rather than the variance of the sample harmonic mean.

Does anyone have a source for the true mean and variance of the sample harmonic mean? Duoduoduo (talk) 15:08, 5 June 2013 (UTC)

Harmonics / history / music?
Anyone mind if I add a musical (harmonic ratio) example, since that's where the name of the mean comes from? Thanks! -- Michael Scott Cuthbert (talk) 21:26, 19 June 2014 (UTC)

Taking an average of ratios(rates). New article? Add to "Rate" article?
The harmonic mean is sometimes used to take "true' averages of ratios. In many cases, weighting is required.  In some cases, only the arithmetic mean is needed and gives the same results as the harmonic mean (since different weightings were used).   Shouldn't this (or another article) explain how to do this in general.  A formula showing weights is given in this article, but no rules for determining weights.  I've formulated this and it's really pretty simple.  What one often wants to determine when averaging ratios Ai/Bi (i terms from 1 to N) is (SumAi)/(SumBi) = "Ratio of averages" (both numerator and denominator are divided by N to get an "average" but the two N's cancel out).  Let Ri = Ai/Bi, the ratios to average.  If one knows only the Ri's and Bi's, then to get the "ratio of averages" one takes the arithmetic mean of the Ri's using the Bi's as weights (the denominator of the ratios). If one knows only the Ri's and Ai's then one uses Ai's as weights and takes the harmonic mean. The sum of the weights must add to 1 so each Ai weight is divided by sumAi. Likewise for Bi's. These rules show how to determine the weights. Proving these rules takes only several lines and is very simple algebra. For electrical components in series and parallel, there exist implied ratios. For example resistance is voltage/current (a ratio). Another case is where only the Ri's are known. Then one can use the above rules by assuming that all the Ai's (or all the Bi's) are the same number (say 1). This results in an unweighted harmonic mean. David S. Lawyer 09:31, 23 November 2015 (UTC)

 * We already have the use of harmonic mean in electronics given in the article. I'm not sure what you are suggesting.  There are lots of applications where a good understanding of ratio is required to calculate an appropriate average, but the method will vary according to the situation and type of weightings given.    D b f i r s   10:59, 23 November 2015 (UTC)""

 * I'm suggesting that we need an article giving a few rules for finding the mean of rates. Many "situations" have a lot in common and an utilize common principles to determine what type of mean to use (arithmetic or harmonic) and what weights to use.

 * Would this new article be just a list of methods for different situations? Are there many practical situations where the geometric mean is used?  You could create it in draft space so that we can see what it would look like.    D b f i r s   08:55, 24 November 2015 (UTC)

I've found a small article Rate (mathematics) and plan to add to it: taking the mean of rates. It might more than double the size of that article. It would be referenced in this article and the Arithmetic mean article. It would show when to use the harmonic mean and when to use the arithmetic mean and what weights to use for finding certain results. If and when I do this, I'll post in this Talk since someone here might be willing to review it, edit it, etc.David S. Lawyer 00:16, 25 November 2015 (UTC)
 * That sounds appropriate. Doubling the size will be no problem because it is too short as it stands.  Thank you for letting us know.    D b f i r s   21:52, 25 November 2015 (UTC)

Need proof of inequality
The article needs a new section entitled "Proof that the harmonic mean is less than the geometric mean", preferably with a source. Or, if there is enough to say, there could be a new article Inequality of harmonic and geometric means, analogous to the article Inequality of arithmetic and geometric means. Loraof (talk) 16:27, 13 December 2015 (UTC)

I put in a link to an article with a proof. Maybe that will help someone write a new section or article. Loraof (talk) 17:16, 13 December 2015 (UTC)

 * Really, this is another special case of Generalized mean inequality. Boris Tsirelson (talk) 19:48, 11 June 2016 (UTC)

Finance example is Not even wrong and subsection should be removed or heavily revised
The [| reference to "Fairness Opinions", on page 217] presents an example with two equal-sized positions. This is rarely the case in real life and the weighted average should be recommended instead.

Suppose you have a $100-billion firm with a P/E of 20 ($5 billion in earnings). Average this with a $1-billion firm with a P/E of 1000 ($1 million in earnings). The aggregate price/earnings of this combination should be dominated by the large firm.

Incorrect: unweighted harmonic mean

 * $$ P/E = \frac  {1}{1/20 + 1/1000} \approx 19.6 $$

Correct: ratio of aggregates, equivalent to market-cap-weighted harmonic mean

 * $$ P/E = \frac  { \$100\ billion + \$1\ billion }{\$5\ billion + \$1\ million} \approx 20.2$$

11:50, 20 April 2016 198.180.162.5 (talk)

 * The finance section is correct, but under the assumption that it is an equally weighted portfolio; otherwise, you need to use the weighted harmonic mean. So you are right, I have made the change to clarify that. Also note that in your example of the calculation of the unweigthed harmonic mean, you forgot to divide the denominator by 2, which of course leads to a wrong P/E.7804j (talk) 09:28, 29 September 2016 (UTC)

The weights used in the article depend only on the investment fraction. There is nothing precisely wrong, but the reference to market capitalization may be misleading (it is used only to calculate the P/E, not the susbsequent weights). — Preceding unsigned comment added by 163.1.165.0 (talk) 12:35, 24 October 2022 (UTC)

Image label incorrect?
This image is from the article "Harmonic Mean:"

It appears to me, that the length of the purple line H is actually TWICE the harmonic mean for the following reason:

H = (G*G) / A = (a*b) / ( (a+b) / 2 ) = 2 * ( 1 / ( (1/a) + (1/b) ) )

Ergo:

H is TWICE as big as the harmonic mean.

Is this true? Please correct me, if I am wrong.

73.4.14.51 (talk) 14:19, 6 May 2016 (UTC)

 * Yes, you are wrong, just at the end of your calculation: $$H=\frac{2}{(1/a)+(1/b)}$$, that is, 1/H is the arithmetic mean (rather than the sum!) of 1/a and 1/b. Boris Tsirelson (talk) 15:53, 6 May 2016 (UTC)

You are right. Thank you for the explanation. — Preceding unsigned comment added by 73.4.14.51 (talk) 17:21, 10 May 2016 (UTC)

Without doing any calculations, this figure is incorrect by inspection: The rank order of lengths, from the drawing, is Q > A = H > M, not what's asserted (correctly) in the legend (Q>A>G>H). Note that H = A here because both are the radius of the same circle. — Preceding unsigned comment added by Memeri (talk • contribs) 18:22, 23 November 2022 (UTC)

"In trigonometry" section
The section "In trigonometry" says

 * In the case of the double-angle tangent identity, if the tangent of an angle A is given as $a⁄b$, then the tangent of 2A is the product of (1) the harmonic mean of the numerator and denominator of tan A and (2) the reciprocal of (the denominator minus the numerator of tan A).

 * In general, the double-angle formula can be written as

 * $$\tan(2A) = \frac{2ab}{a+b}\cdot\frac1{b-a}$$

and then gives a numerical example. I find this pointless—all it says is that the double angle formula does not use the harmonic mean, but can be expressed by starting with the harmonic mean and then adjusting it for the fact that the harmonic mean doesn't give the right answer. So I'm deleting it. Loraof (talk) 16:31, 12 June 2016 (UTC)

Dr. Anderson's comment on this article
Dr. Anderson has reviewed this Wikipedia page, and provided us with the following comments to improve its quality:

"The harmonic mean is the preferable method[dubious – discuss] for averaging multiples, such as the price–earnings ratio, in which price is in the numerator. If these ratios are averaged using an arithmetic mean (a common error), high data points are given greater weights than low data points. The harmonic mean, on the other hand, gives equal weight to each data point.[6] The simple arithmetic mean when applied to non-price normalized ratios such as the P/E is biased upwards and cannot be numerically justified, since it is based on equalized earnings; just as vehicles speeds cannot be averaged for a roundtrip journey.[7]

This discussion is misleading. Neither the arithmetic nor the harmonic mean of P/E ratios is useful for typical financial purposes. A typical concern is with a portfolio of assets with differing P/E ratios. The weighted arithmetic average P/E would typically be used to evaluate the P/E of the portfolio, the weight on each P/E ratio being the proportion of total assets allocated to that individual asset.

The harmonic mean is related to the other Pythagorean means, as seen in the third formula in the above equation. This is noticed if we interpret the denominator to be the arithmetic mean of the product of numbers n times but each time we omit the j-th term. That is, for the first term, we multiply all n numbers except the first; for the second, we multiply all n numbers except the second; and so on. The numerator, excluding the n, which goes with the arithmetic mean, is the geometric mean to the power n. Thus the nth harmonic mean is related to the nth geometric and arithmetic means.

This paragraph should go, with some editing, to immediately follow the equation defining H."

We hope Wikipedians on this talk page can take advantage of these comments and improve the quality of the article accordingly.

Dr. Anderson has published scholarly research which seems to be relevant to this Wikipedia article:

 * Reference : Anderson, James E & Neary, J Peter, 2014. "Revenue Tariff Reform," CEPR Discussion Papers 9838, C.E.P.R. Discussion Papers.

ExpertIdeasBot (talk) 17:56, 27 June 2016 (UTC)

 * The weighted harmonic mean seems to be the correct way to average P/E to me, but I agree that the section was misleading because it was referring to the unweighted harmonic mean. I have clarified the section and added an example. The weighted arithmetic average is definitely not the right way to calculate the P/E of the portfolio though, so I think you are wrong on that point. 7804j (talk) 14:23, 29 September 2016 (UTC)

In Section 3.1 "Two numbers", could it be that A, G, H refer to different concepts in the discussion than in the diagram? If so, someone more mathematically more adept than me will need to step in to make things right.

--Beatrice57 (talk) 19:28, 24 August 2020 (UTC)

Continuous definition?
The statistics section mentions, without introduction, the harmonic means of several continuous probability distributions. There should be a definition of the continuous case somewhere. I suppose it would be something like integral(f(x))/integral(f(x)/x), where f(x) is a PDF that supplies the weights and the integrals are both taken over the whole domain. Elias (talk) 12:05, 27 November 2020 (UTC)

title: Talk:Harmonic mean


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text:
Omura's whale

Omura's whale or the dwarf fin whale (Balaenoptera omurai) is a species of rorqual about which very little is known. Before its formal description, it was referred to as a small, dwarf or pygmy form of Bryde's whale by various sources. The common name and specific epithet commemorate Japanese cetologist Hideo Omura.

The scientific description of this whale was made in Nature in 2003 by three Japanese scientists. They determined the existence of the species by analysing the morphology and mitochondrial DNA of nine individuals – eight caught by Japanese research vessels in the late 1970s in the Indo-Pacific and an adult female collected in 1998 from Tsunoshima, an island in the Sea of Japan. Later, abundant genetic evidence confirmed Omura's whale as a valid species and revealed it to be an early offshoot from the rorqual lineage, diverging much earlier than Bryde's and sei whales. It is perhaps more closely related to its larger relative, the blue whale.

In the third edition of Mammal Species of the World, the "species" is relegated to being a synonym of Balaenoptera edeni. However, the authors note that this is subject to a revision of the genus. The database ITIS lists this as a valid taxon, noting a caveat on the disputed systematics of this species, Balaenoptera edeni and Balaenoptera brydei.

Taxonomy
The six specimens obtained in the Solomon Sea in 1976 were only noted to be smaller at sexual maturity than the "ordinary" Bryde's whales caught off New Zealand, whereas the two caught near the Cocos-Keeling Islands in 1978 were not differentiated from the 118 other "ordinary" Bryde's whales taken in the eastern Indian Ocean, south of Java. As a result of allozyme analysis, their distinctive baleen and small size at physical maturity compared to Bryde's whale, and photographs obtained of the harvested whales (showing their fin whale-like coloration), Shiro Wada and Kenichi Numachi (1991) decided that these eight individuals represented members of a new species of baleen whale. However, due to the lack of a detailed osteological study and the absence of "conclusive data", the International Whaling Commission decided to consider them only as a regionally distinct group of "small-form Bryde's whale". Despite this declaration, the specific status of the Solomon Sea specimens was supported by a mitochondrial DNA study done by Hideyoshi Yoshida and Hidehiro Kato (1999).

The identity of these eight specimens was finally resolved in 1998 when an unidentified whale, which had died after colliding with a fishing boat in the Sea of Japan and was towed to Tsunoshima, was examined by Tadasu Yamada, Chief of the Division of Mammals and Birds at the National Science Museum, Tokyo. This specimen closely resembled the individuals caught in the 1970s in external appearance and allowed a complete osteological examination of the putative new species to be conducted. As a result of external morphology, osteology, and mitochondrial DNA analysis of two of the harvested whales and the Tsunoshima specimen, Wada, Masayuki Oishi, and Yamada described Balaenoptera omurai in the 20 November 2003 issue of the journal Nature. In honour of the people of Tsunoshima, who helped remove the flesh from the type specimen, it was given the Japanese vernacular name of Tsunoshima kujira (English: Horn Island whale).

Holotype and paratypes
The holotype is an 11.03 m adult female, NSMT-M32505 (National Science Museum, Tokyo), which stranded at Tsunoshima (34.35083°N, 130.88583°W) in the southern Sea of Japan on 11 September 1998. It includes a complete skeleton, both complete rows of baleen plates, and frozen pieces of muscle, blubber, and kidney collected by T. K. Yamada, M. Oishi, T. Kuramochi, E. Jibiki, and S. Fujioka. The type locality is the Sea of Japan, which may not be representative of the species’ typical range. The paratypes include the eight specimens (five females and three males), NRIFSF1-8 (National Research Institute of Far Seas Fisheries, Fisheries Research Agency, Shizuoka), collected by Japanese research vessels in the Indo-Pacific in the late 1970s. The longest baleen plate (NRIFSF6 includes 18 more baleen plates), an earplug, and a piece of the sixth thoracic vertebra with associated epiphysis were collected from each individual.

Osteology
Omura's whale has several unique skeletal features that distinguish it from its congeners, namely B. brydei and B. edeni. In B. omurai and B. brydei, the posterior end of the ascending process of the maxilla widens to become squarish, whereas in B. edeni, it is slender and round throughout its length. In B. omurai, this widened posterior portion conceals the premaxilla, which disappears below the maxilla and nasal and does not reach the frontal, whereas in both B. brydei and B. edeni, the premaxilla reaches the frontal. The parietals flare laterally in dorsal view in B. omurai and the Indo-Pacific form of B. brydei, but are invisible in dorsal view in B. edeni and the North Pacific form of B. brydei. B. omurai has two small foramina "along the suture between the parietal and squamosal in the posterior wall of the temporal fossa", which both B. brydei and B. edeni lack. B. omurai has an oblique ridge on the dorsal side of the maxilla near the base of the rostrum, which is absent in both B. brydei and B. edeni. Unlike B. edeni, the alisphenoid is separate from the squamosal in B. omurai. The head of the first rib is not bifurcated in B. omurai, unlike B. brydei and B. edeni.

Omura's whale has a total of 53 vertebrae, including seven cervical (the standard number among mammals), 13 thoracic, 12 lumbar, and 21 caudal. Like all members of its genus, it has only four digits on the manus of each pectoral fin (the third digit is missing). The phalangeal formula is: I-5, II-7, IV-6, V-3.

External appearance
Its appearance resembles the larger fin whale (thus the alternate common names of dwarf fin whale and little fin whale), both having a dark gray left lower jaw, and on the right side a white mandible patch, a white blaze, a dark eye stripe, a white inter-stripe wash, as well as a white chevron on the back, pectoral fins with a white anterior border and inner surface, and flukes with a white ventral surface and black margins. Like fin whales, it also exhibits a white left gape and a dark right gape, a reversal of the asymmetrical pigmentation on the lower jaw. It has a very falcate dorsal fin with a leading edge that gradually slopes into the back, halfway in shape between the more gradual slope of the fin whale and the more acute angle of Bryde's and sei whales. Its dorsal fin is also proportionally smaller and less upright than these other species. It typically has a single prominent median ridge on the rostrum, but can have faint lateral ridges, which are more pronounced in calves. Bryde's whale, on the other hand, has three prominent ridges on the rostrum. It has 45 to 95 ventral grooves that extend past the umbilicus. The type specimen (NSMT-M32505) had 203-208 pairs of baleen plates that were "short and broad with uncurled, stiff, grayish-white fringes", while NRIFSF6 had an estimated 181–190 on the right side – fewer than any other species in its genus. Other specimens of Omura's whale had between 204 and 246 pairs of baleen plates. Like the fin whale, NSMT-M32505 exhibited asymmetrical coloration in its baleen, as well: on the right side, the front third are yellowish-white, the intermediate 100 plates are bi-colored (dark on the outer side and yellowish-white on the inner side), and the remaining plates in the back were all black, while on the left side, the majority are bi-colored with the remaining back plates being all black like the right side. The average length and width for the nine specimens was 26 by, the smallest length-to-breadth quotient (1.22) for any species in its genus.

Omura's whale seen off New Caledonia, the Solomon Islands, West Sumatra, and East Kalimantan showed extensive scarring from cookiecutter shark bites, indicating they had ventured into deep waters; whereas those off Madagascar did not exhibit them.

Morphometrics
The rostrum is flat and V-shaped with the head occupying about a quarter of the body length. The pectoral fins are short but slender, being from about 13 to 15 per cent of the body length. The dorsal fin ranges between 7.5 and 20 cm (3 to 8 inches) in height and 18 to 60 cm (7 to 23.6 inches) in length for specimens 3.9 to 7.15 m (12.8 to 23.4 ft) in length, and is placed about 57 per cent of the body length back from the tip of the rostrum. It is 61.5 per cent of the body length from the tip of the lower jaw to the umbilicus, whereas it is about 63 per cent from the tip of the lower jaw to the end of the ventral grooves. The flukes are about a quarter to a fifth of the body length in width.

Size
Omura's whale is among the smallest of the rorquals – only the two species of minke whale, the common and Antarctic, which reach 9.75 and 10.7 m in length, respectively, are smaller. Of the eight specimens taken during Japanese whaling in the Indo-Pacific, the five females ranged in length from 10.1 to 11.5 m, while the three males ranged from 9.6 to 10.0 m. The females ranged in age from perhaps only 9 years (the earplug was damaged or partially lost) for an 11.2 m individual to 29 years for the longest female, whereas the three males ranged from perhaps 21 years (another damaged or partially lost earplug) for the longest male to 38 years for one of the 9.6 m specimens. All were physically mature with the exception of the smallest female. Of individuals found stranded in Taiwan and Thailand between 1983 and 2004, five males ranged in length from 5.13 to 10 m, while two females were 4.3 and 5.95 m, respectively – a specimen of unknown sex that stranded in 1983 in Phuket Province, Thailand, was 7 m in length.

Of 16 "Bryde's whales" caught by hunters from Pamilacan between 1991 and 1993, 12 were measured. These cluster into two size categories, nine whales less than 10 m, and three 12 m or more. Later, 85% (24 of 28) of the identified skull specimens examined from the Bohol Sea were found to be Omura's whales, whereas only 15% (4 of 28) were what was tentatively called the Indo-Pacific form of Bryde's whale (B. brydei). The former size category may be primarily (if not entirely) Omura's whale, whereas the larger whales – one of unknown sex of 12 m and two females of 13 m – would be the larger, offshore form of Bryde's whale. Of those smaller whales, four males ranged from 6.7 to 9.8 m, four females ranged from 4.9 to 9.3 m, and one of unknown sex was 9.4 m.

Lone individuals seen off Madagascar were estimated to range between 8 and 12 m (26.2 to 39.4 ft), while calves were estimated to be between 3 and 5 m (9.8 to 16.4 ft).

The identity of three mature specimens (two females and a male) examined by biologist Graham Chittleborough in 1958 at a whaling station in Western Australia, which ranged in length from 10.6 to 11.74 m, is uncertain – they may refer to Omura's whale or the smaller form of Bryde's whale (B. edeni). These three individuals were noted to have very small baleen plates – about 22 cm (8.8 in) by 15 cm, about 22 cm (8.8 in) by 16 cm (6.3 in), and 23.5 by, respectively – with length-breadth quotients of 1.34 to 1.46, within the upper range (1.00-1.43) of the 9 specimens included in the formal description of Omura's whale, but also within the lower range of the Bryde's whale complex (1.2 to 1.33). The holotype of the smaller form of Bryde's whale (B. edeni) from Myanmar, though, had baleen plates of 30 by, or a length-breadth quotient of exactly 2.0 – within the typical range of the Bryde's whale complex (1.8 to 2.4).

Mating
Of the three females caught in the Solomon Sea, two were ovulating and lactating and one was resting (not lactating, ovulating, or pregnant), while one of the two females caught near the Cocos Islands was accompanied by a calf estimated to be about 3 m (9.8 ft) in length. Nothing is known of the duration of gestation and lactation, and little is known of the timing or extent of breeding seasons. In August 2005, a 3.2 m female neonate with fetal folds and folded dorsal fin and flukes stranded at Miyazaki, Miyazaki Prefecture, on the eastern coast of Kyushu. Off northwestern Madagascar, six different cow-calf pairs were seen: one in August 2011, three in November 2013, and two in August 2015, suggesting a protracted calving season. The three calves observed in November had bent dorsal fins (indicating that they were fairly young) but did not have fetal folds, while one of the calves sighted in August had an erect dorsal fin, indicating that it was older but still probably born that year. These calves were estimated to range in length from 3 to 5 m (9.8 to 16.4 ft).

Behavior and diet
Little is known of the behavior and diet of Omura's whale. Their blow is low and diffuse. After surfacing, the dorsal fin is usually not visible until after the head and splashguard have disappeared and they don't fluke when diving. They have been seen lunge feeding, defecating, and breaching off both Komodo National Park and northwestern Madagascar; they've also been seen rolling at the surface in apparent mating (the last of which allowed the identification of a male) off the former area. Off Madagascar, average group size was only 1.1 individuals (272 individuals in 247 groups), but loose aggregations of as many as a dozen whales could be seen. A total of thirteen cow-calf pairs were observed between 2011 and 2016, including a female first seen in an aggregation in 2012, then with a calf in 2013, and alone again in 2015 and 2017, showing that individuals can exhibit strong site fidelity.

The six paratypes taken in the Solomon Sea in 1976 reportedly only had krill (Euphausia diomedeae) in their stomachs, while crustaceans and fish were found in a 7 m (23 ft) female from Japan. Individuals in Madagascar have been observed lunge feeding on the krill Pseudeuphausia latifrons.

Song
Omura's whale produce amplitude-modulated songs of 15-50 Hz with a peak frequency of 36.1 Hz and an average duration of 9.2 seconds. This is sometimes followed by a tonal call of 17 Hz and four seconds in duration. These songs are repeated every two to three minutes, sometimes for as long as thirteen hours. Songs have been recorded off northwest Madagascar year-round, with peak activity from late October to late January and again from late May to late June. Overlapping choruses of several singing individuals have been recorded throughout the year as well.

Movements
Four Omura's whales were satellite-tagged off northwest Madagascar in November 2016. The tags stayed on an average of 42 days (range: 30–58 days). The tagged whales traveled an average of 2,530 km (range: 2,148 to 3,181 km) but remained within a small coastal range of only 230 to 405 km (average: 283 km) off the northwest coast of the island. All whales traversed their entire individual ranges several times, spending most of their time on shelf waters and rarely venturing into deep waters.

Range

Omura's whale are found in the Indo-Pacific and the Atlantic Oceans, primarily in shelf waters between 35° N and 35° S, with the majority of records in the tropics (between 23° 26' N and 23° 26' S). Their range includes southern Japan (with strandings and entanglements recorded in March and from May to October; including the prefectures of Yamaguchi, Miyazaki, Kagawa, Mie, Shizuoka, and Chiba), South Korea (December to January, from the provinces of South Jeolla and South Gyeongsang), China (with strandings from November to February and in August; including the provinces of Zhejiang, Fujian, Guangdong, and Guangxi), Taiwan (with strandings from November to March and in May; there are also sightings from April to May and July to August), Hong Kong (March),  the Philippines (Manila Bay, Busuanga, and the Bohol Sea), Vietnam (Quảng Ngãi Province), Malaysia (Pahang), Thailand (with sightings and strandings from February, May to June, and November to December; including the provinces of Phang Nga, Phuket, Prachuap Khiri Khan, and Songkhla), Indonesia (with sightings and strandings from February, April to June, and from September to November; with records from West Sumatra, the southern Java Sea, Bali, Komodo, the Solor Archipelago, Seram, Raja Ampat Islands, and East Kalimantan), the Cocos (Keeling) Islands, the Andaman Islands (April), Australia (Western Australia from Exmouth to north of Darwin in the Northern Territory at about 9° 30' S - 10° S, 130° E), South Australia (January, Gulf St Vincent), Queensland (November to December, Port Douglas and Mission Beach), the Solomon Islands, New Caledonia, Sri Lanka (February), the Chagos Archipelago, Iran (September, Strait of Hormuz), Egypt (April, the northern Red Sea) northwestern Madagascar (12° 01' S to 19° 23' S, with sightings by researchers from August and October to December, by whale watch operators from April to December, and acoustically detected year-round), Mauritania (November, Trarza Region), Brazil (September, Ceará), and in the vicinity of the Saint Peter and Saint Paul Archipelago.

Sightings
In 1999 and 2000, an unidentified species of rorqual was repeatedly seen in the waters of Komodo National Park. They were small (most estimated to be only 7 to 10 m in length) with asymmetrical coloration similar to the fin whale, only had a single prominent ridge on the rostrum, and an extremely hooked dorsal fin. At first, they were tentatively identified as a "pygmy or regionally distinct" form of Bryde's whale, which was confirmed when one was photographed and biopsied in October 2000 and its tissue sample sent to the Southwest Fisheries Science Center in La Jolla, California. There, its DNA was analyzed and found to be a complete match with a "pygmy Bryde's" sample obtained from the Philippines – later, however, it was discovered samples from the Philippines corresponded to B. omurai and not B. edeni.

During marine mammal surveys conducted in the Solomon Islands in 2009 and 2010, what were "most likely" Omura's whales were sighted on three occasions. They were estimated to be 6 to 10 m in length and lacked obvious auxiliary ridges. The sightings were made very close to shore (within a few hundred meters), one on the west coast of Malaita in November 2009, one on the northwest coast of Guadalcanal in November 2010, and a pair observed feeding within the Sandfly Passage of the Florida Islands on 14 November 2010.

In October 2015, an international team of scientists, led by Salvatore Cerchio of the New England Aquarium and Woods Hole Oceanographic Institution, released the first images and field observations of the species from a population off northwestern Madagascar. Forty-four sightings of Omura's whale were made between 2011 and 2014, with the majority in 2013 (thirteen) and 2014 (twenty-five). Forty-two were made off Nosy Iranja and the Ampasindava Peninsula, while only two occurred off Nosy Be. They were observed in open shelf waters that averaged 31 m (100 ft) deep (range: 4 to 202 m, or 13.1 to 662 ft) and were never seen in deep waters off the shelf break or in shallower coastal waters or embayments.

In 2018, Pierre Laboute and Philippe Borsa published a paper on sightings of Omura's whale made off Nosy Be in 1991–1995, 1998, and 2000, including photographs of an aggregation of a dozen whales seen skim and lunge feeding in 40 to 80 m (130 to 260 ft) of water in November 1994 at 13° 26' S, 48° 05' E.

In 2017, the first confirmed live sightings of Omura's whale were made off Taiwan and Sri Lanka.

On December 31, 2023, tourists on a charter boat off Phuket's coast in the Andaman Sea captured a pair of Omura's whales with a drone, one of which was an exceptionally rare all-white Omura whale.

Artisanal whaling

As early as the late 19th century, the natives of Lila, Bohol, began hunting whales in the Bohol Sea. By the turn of the century, this had spread to nearby Pamilacan Island and later to Sagay, Camiguin. At Pamilacan, whales were caught as early as January and as late as June, but most were taken in April and May. When a whale was spotted from shore, between 10 and 20 pump boats (boats with motors originally used for pumps) were launched in chase. When within range, a "hookman" jumped onto the whale's head and stabbed it with a 35-cm-long, 22.5-cm-wide stainless steel hook attached to a heavy line of 30 m with a 2 m bamboo spar buoy at the end of it. At Camiguin, they harpooned it with a toggle-headed grommet harpoon with a 2.5 m wooden shaft similar to the "dolphin irons" used by American whalemen in the mid-19th century, which in turn was either attached to a 140 m rope with a plastic fishing float or a 400 m rope with a plastic float or oil drum at the end. After an hour or more, the whale tired, and men took turns cutting it until it died. The boats usually towed the carcass to Lila for butchering, where it was sold and the meat eaten raw or cooked. Pamilacan hunters alone caught 10 to 20 per year, whereas at Camiguin, they caught them only sporadically.

When Maria Louella L. Dolar and colleagues (1994) examined photographs or baleen of the whales caught in the Bohol Sea between 1991 and 1993, they identified them as Bryde's whales; this was supported by osteological examinations of skulls collected from the same sites by Dolar, William F. Perrin, and others (1996), who suggested they were a "small form" of Bryde's whale. The specimens were deposited in the collections of the Marine Research Laboratory of Silliman University in Dumaguete, Negros Oriental, Philippines, where they were examined by T. K. Yamada and co-workers (2008). They discovered that 24 of the skulls were actually from Omura's whales, whereas only four were from the Indo-Pacific Bryde's whale (B. brydei). This was supported by genetic studies, which found that, based on comparison of published phylogenies, the small "Bryde's whales" from the Philippines correspond to Omura's whale.

A monograph from 1923 describes three skulls of balaenopterids taken by native whalers in Indonesia. Later examination of photographs of those skulls by one of the scientists (Tadasu K. Yamada) that had formally described Omura's whale showed that two of them, one from Bangsri, Java, and another from Lamakera, Solor, belonged to B. omurai. There is also a photograph (taken between 1915 and 1944) of a whale caught by the villagers of Lamakera that "strongly resembles a young Omura's whale in size and shape".

Commercial whaling
Among the small "Bryde's whale" caught 40 mi off Shionomisaki, Wakayama Prefecture, in June 1976, were two sexually mature females of 7.9 m (25.9 ft) and 8.5 m (27.9 ft), both of which are thought to likely be Omura's whale.

Scientific whaling
Of the eight individuals taken by Japanese scientific whaling in the 1970s, six were processed aboard the factory ship Tonan Maru No. 2 in the Solomon Sea (9°49'-10°17'S, 157°29'-157°56'E) on 24 October 1976, and two were processed aboard the factory ship Nisshin Maru No. 3 near the Cocos Islands (-10.85°N, 97.03333°W and -10.88333°N, 94.48333°W, respectively) on 15 and 17 November 1978.

Bycatch
Three of the seven records from Japan involve bycaught individuals, including a 9.2 m (30.1 ft) male in Sagami Bay in October 2003, a 10.05 m (33 ft) female in Tokyo Bay in May 2004, and a 6.3 m (20.7 ft) female near Ise Bay in March 2012. Both records from South Korea were individuals taken as bycatch, including a 6.3 m (20.7 ft) female off Geoje in January 2004 and a 6.4 m (21 ft) male off Goheung in December 2006. A 4.4 m (14.4 ft) male calf was also caught in small-mesh herring seine nets in the Gulf of Thailand, Songkhla Province, in May 2011, while a 7.5 m (24.6 ft) whale (thought to likely be an Omura's whale) was caught in fishing gear off Negombo, Sri Lanka, in August 1985. A live whale seen off Sri Lanka in 2017 also had a scar on its rostrum suggestive of an entanglement with fishing gear.

Ship strikes
Two individuals of Omura's whale were victims of ship strike, including the holotype of the species, an 11 m (36.1 ft) female found off Tsunoshima in the Sea of Japan in September 1998, and a whale brought into Manila Bay on the bow of a ship.

Conservation
Omura's whale is listed on Appendix II of the Convention on the Conservation of Migratory Species of Wild Animals. It is listed on Appendix II as it has an unfavourable conservation status or would benefit significantly from international co-operation organised by tailored agreements.

In addition, Omura's whale is covered by the Memorandum of Understanding for the Conservation of Cetaceans and Their Habitats in the Pacific Islands Region (Pacific Cetaceans MOU).

title: Omura's whale


text:
Talk:Omura's whale

confirmation of the validity of B. omurai
The validity of Balaenoptera omurai has been confirmed by DNA analysis (Sasaki et. al., 2006). According to their DNA analysis, B. omurai is more derived than the blue whale, but more primitive than the sei and Bryde's whales. Additionally, B. edeni and B. brydei are closer to each other than either is to the sei whale or B. omurai. Therefore, this article must be updated and title of this article should be changed to Omura's Whale. This will update the Wikipedia article Bryde's Whale

Sasaki, T., Nikaido, M., Wada, S., Yamada, T.K., Cao, Y., Hasegawa, M., and Okada, N. 2006. Balaenoptera omurai is a newly discovered baleen whale that represents an ancient evolutionary lineage. Molecular Phylogenetics and Evolution 41(1): 40–52. —Preceding unsigned comment added by 68.4.61.237 (talk) 03:12, 12 September 2009 (UTC)
 * I agree, per above, and Jefferson et al (2008), which labels the species Omura's whale. The page will be moved as agreed. OldBabyBlue (talk) 03:45, 20 February 2011 (UTC)

Map and images needed
Yeah, I don't know how to make a map on here, so anyone feel free to make one. Only include its actual distribution though. We could include small blue areas where several strandings/catches have occurred (southern Japan, the west coast of Taiwan, the Bohol Sea). A star could be placed where the type specimen was found and perhaps an asterisk where confirmed sightings have occurred (Komodo National Park, southern New Caledonia). The only map on WikiCommons is outdated and inaccurate, showing it in places it has never been confirmed to occur (most of Indonesia, etc.). Images may be harder to come by. Perhaps the best we could do would be to find a free image of a museum specimen. Any help would be much appreciated. SHFW70 (talk) 01:11, 16 September 2013 (UTC)

Size comparison chart needed
Someone with inkscape and more patience than me needs to upload an Omura's whale size comparison chart. All you need to do is take the Bryde's whale size comparison chart, remove the lateral ridge, give it a more hooked dorsal fin, and scale the 1.8 m diver to a 11.5 m whale. SHFW70 (talk) 19:43, 18 September 2013 (UTC)

First confirmed wild sighting of _B.omurai_
http://www.whoi.edu/news-release/OmurasWhales Figured I'd post this here, since it's not currently documented in the article itself. As of October 22nd of this year, we have the first confirmed sightings of live, wild Omura's whales. Raptormimus456 (talk) 00:51, 28 October 2015 (UTC)
 * That's not true. They had been sighted off Komodo in 1999 and 2000 and off Madagascar in 1994. The authors were just lazy and didn't bother do enough research. OM2003 (talk) 19:22, 20 May 2018 (UTC)

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title: Talk:Omura's whale


text:
Ada Gospels

The Ada Gospels (Trier, Stadtbibliothek, Codex 22) is a late eighth century or early ninth century Carolingian gospel book in the Stadtbibliothek, Trier, Germany. The manuscript contains a dedication to Charlemagne's sister Ada, from where it gets its name. The manuscript is written on vellum in Carolingian minuscule. It measures 14.5 by 9.625 inches (36.8 by 24.5 cm). The Ada Gospels are one of a group of manuscript illuminations by a circle of scriptoria that represent what modern scholars call the "Ada School". Other products of the Ada School include the Soissons Gospels, Harley Golden Gospels, Godescalc Evangelistary and the Lorsch Gospels; ten manuscripts in total are usually recognised.

The manuscript is illuminated. Its illuminations include an elaborate initial page for the Gospel of Matthew and portraits of Matthew, Mark and Luke. The illuminations show Insular, Italian and Byzantine influences. The Evangelist portraits show a firm grasp of Classical style typical of the Carolingian Renaissance.

In 1499 the codex was given a rich sculpted gold binding that includes the Late Antique Eagle Cameo displaying the family of Emperor Constantine.

title: Ada Gospels


text:
Friedrich Gulda

Friedrich Gulda (16 May 1930 – 27 January 2000) was an Austrian pianist and composer who worked in both the classical and jazz fields.

Early life and career
Born in Vienna the son of a teacher, Gulda began learning to play the piano at age 7 with Felix Pazofsky at the Wiener Volkskonservatorium. In 1942, he entered the Vienna Music Academy, where he studied piano and musical theory under Bruno Seidlhofer and Joseph Marx.

During World War II as teenagers, Gulda and his friend Joe Zawinul would perform forbidden music, including jazz, in violation of the government's prohibition of playing of such music.

Gulda won first prize at the Geneva International Music Competition in 1946. Initially, the jury preferred the Belgian pianist Lode Backx, but when the final vote was taken, Gulda was the winner. One of the jurors, Eileen Joyce, who favoured Backx, stormed out and claimed the other jurors were unfairly influenced by Gulda's supporters. Gulda began to play concerts worldwide. He made his Carnegie Hall debut in 1950. Together with Jörg Demus and Paul Badura-Skoda, Gulda formed what became known as the "Viennese troika".

Career as classical pianist
Although most renowned for his Mozart and Beethoven interpretations, Gulda also performed the music of J. S. Bach (often on clavichord), Schubert, Chopin, Schumann, Debussy and Ravel. His recordings of Bach's The Well-Tempered Clavier are well regarded, but Gulda performed very few other pieces by Bach and recorded even fewer. In the late 1960s Gulda recorded the complete Beethoven sonatas. He continued to perform classical works throughout his life, composing cadenzas for two Mozart concertos, which he famously recorded with his former pupil Claudio Abbado, although he sometimes conducted from the keyboard himself. A notable feature of his Mozart recordings were his own improvisations.

Phillips Records included Gulda in its Great Pianists of the 20th Century CD box set, which came out in 1999. His piano students included Martha Argerich, who called him "my most important influence," and the conductor Claudio Abbado.

Jazz, free music and composition
In 1956, Gulda performed and recorded at Birdland in New York City and at the Newport Jazz Festival. He organized the International Competition for Modern Jazz in 1966, and he established the International Musikforum, a school for students who wanted to learn improvisation, in Ossiach, Austria, in 1968. From the 1950s on Gulda cultivated a professional interest in jazz, writing songs and free improvisation or open music improvisations. He also recorded as a vocalist under the pseudonym "Albert Golowin", fooling music critics for years until it was realized that Gulda and Golowin were the same person. He played instrumental pieces, at times combining jazz, free music, and classical music in his concerts. He once said:

"There can be no guarantee that I will become a great jazz musician, but at least I shall know that I am doing the right thing. I don't want to fall into the routine of the modern concert pianist's life, nor do I want to ride the cheap triumphs of the Baroque bandwagon."

In jazz, he found "the rhythmic drive, the risk, the absolute contrast to the pale, academic approach I had been taught." He also took up playing the baritone saxophone.

In the 1960s, Gulda wrote a Prelude and Fugue with a theme suggesting swing. Keith Emerson liked Gulda's Fugue so much, that he often performed it in Emerson, Lake & Palmer concerts in the 1970s, and a studio version was also issued on Emerson, Lake & Palmer's The Return of the Manticore.

In addition, Gulda composed "Variations on The Doors' 'Light My Fire'" (aka 'Variationen über "Light My Fire" (von Jim Morrison)') for solo piano, and released it in 1971 on Track 11 (LP disc 1, side 2, track 1) of "The Long Road To Freedom (Ein musikalisches Selbstporträt in Form eines Lehrgangs)". An earlier instrumental rock-style piano/bass/drums trio version (sans any of the complex Gulda composed and improvised variations...) of 'Light My Fire' can also be found on Gulda's album As You Like It (1970), an album that also includes standards such as "'Round Midnight" and "What Is This Thing Called Love?", as well as Gulda's classic "Blues For H.G. (dedicated to Hans Georg Brunner-Schwer)."

From the late 1960s through the 1980s – while continuing his straight-ahead swing and bop-based jazz (often in European Jazz big bands, which he often organized yearly) performances and recordings, and his classical performances and recordings, he also performed and/or recorded (often using a custom electrically amplified clavichord, percussion instruments, and a bass recorder wooden flute) with a wide range of musicians involved in Free improvisation, including: Cecil Taylor, Barre Phillips, Ursula Anders, John Surman, Albert Mangelsdorff, Stu Martin, and Fritz Pauer. Gulda spoke of a fascination with the boundaries in music, believing all music to have worth, regardless of how society judged it. He believed experiments in what he called 'free music' were wonderful musical experiences, even if nobody else believed it was music. One such experiment was a performance in which he and Ursula Anders would both improvise whilst nude and shouting about being mad.

In the late 1970s and 1980s, Gulda was involved in yearly music festivals, such as the Münchner Klaviersommer – where musical guests coming to perform over the years with him included Herbie Hancock, Joe Zawinul, and Chick Corea.

In 1980, he wrote his Concerto for Cello and Wind Orchestra, which has been called "as moving as it is lighthearted", in five movements "involving jazz, a minuet, rock, a smidgen of polka, a march and a cadenza with two spots where a star cellist must improvise."

In 1982, Gulda teamed up with jazz pianist Chick Corea, who was between the breakup of Return to Forever and the formation of his Elektric Band. Issued on The Meeting (Philips, 1984), Gulda and Corea communicate in lengthy improvisations mixing jazz ("Some Day My Prince Will Come" and the lesser known, adapted by Miles Davis song "Put Your Foot Out") and classical music (Brahms' "Wiegenlied" ["Cradle song"]).

Gulda and Corea continued their musical relationship and recorded Mozart's Double Piano Concerto with the Concertgebouw Orchestra with Nikolaus Harnoncourt (conductor). They also played jazz piano duets of Gulda's "Fantasy For Two Pianos" and Corea's "Ping Pong For Two Pianos".

In the late 1980s and 1990s, organist/MIDI keyboardist Barbara Dennerlein also studied with and performed with Gulda.

These unorthodox practices along with sometimes refusing to follow clothing conventions (he was notoriously described as resembling, in one South German concert, "a Serbian pimp" ) or announce the program of his concerts in advance, earned him the nickname "terrorist pianist". In 1988, he cancelled a performance after officials of the Salzburg Festival objected to his including jazz musician Joe Zawinul on the program. When the Vienna Music Academy awarded him its Beethoven Ring in recognition of his performances, he accepted it but then later reconsidered and returned it. To promote a concert in 1999, he announced his own death in a press release so that the concert at the Vienna Konzerthaus could serve as a resurrection party.

Gulda died of heart failure at the age of 69 on 27 January 2000 at his home in Weissenbach, Austria. He is buried in the cemetery of Steinbach am Attersee, Austria. He gave instructions for there to be no obituary.

Personal life
Gulda was married twice, firstly to actress Paola Loew (1956–1966) with whom he had two sons, David Wolfgang and Paul, and secondly to Yuko Wakiyama (1967–1973) with whom he had another son, Rico. Both Paul and Rico became accomplished pianists. In 1975 Gulda began a relationship with the oratorio singer Ursula Anders which lasted until his death.

In 2007 a documentary film for television was made about his life, ''So what?! – Friedrich Gulda''.

Decorations and awards

 * 1959: Austrian Cross of Honour for Science and Art
 * 1989: Honorary Ring of Vienna

title: Friedrich Gulda