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I was always taught to use a certain set of names for the length of a note, such as crotchet, minim, quaver, and so on. I'm aware though that those terms aren't used as much outside the UK, and that terms like "quarter note" and "half note" etc. are often preferred elsewhere.

Where did the British terms come from? Is there a historical reason why they differ from other English-speaking parts of the world?

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    Those terms are used in every English speaking country in the world except for Canada and the US
    – Neil Meyer
    Jul 10 at 19:58
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This direct quote from Wikipedia should answer your first question:

The British names go back at least to English renaissance music, and the terms of Latin origin had international currency at that time. Obviously, longa means 'long', and the rest rarely indicate relative shortness. Brave is from Latin bravis, 'short', minim is from minimus, 'very small', and quiver refers to the quivering effect of very fast notes. The elements sem-, dem- and hem- mean 'half' in Latin, French and Greek respectively, while quasi- means 'almost'. The chain semantic shift whereby notes which were originally perceived as short came progressively to be long notes is interesting both linguistically and musically. However, the crotchet is named after the shape of the note, from the Old French for a 'little hook', and it is possible to argue that the same is true of the minim, since the word is also used in palaeography to mean a vertical stroke in mediaeval handwriting.

For the second question, the first thing I found was this:

[the U.S. names] were originally translated from the German names for the notes because so many German composers immigrated to the United States in the 19th century.

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    I just have to add that neither of my sources show their sources and I am by no means an expert in history so... take with a grain of salt! Both explanations sound believable, though.
    – nonpop
    Oct 24 '13 at 23:19
  • The names are older than the renaissance, but I can't easily support that from here at work. Later, I hope. Oct 25 '13 at 15:18
  • German is not the only language with literal note names. The Afrikaans names we inherited from the dutch is also halfnoot, kwartnoot, heelnoot. Interesting that the Americans have changed the breve to a dubbel wholenote where the germanic languages still hold to the brewis or breve
    – Neil Meyer
    Jul 10 at 19:48
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Some corrections to "non pop"'s great answer: Breve from Latin brevis ("brief" not "brave"). "Minim" is from the Latin adjective "minima" ("smallest") which was originally a subcategory of the semibrevis called the "semibrevis minima" -- the name doesn't come from the stroke, since the earliest minimae did not have the stroke; it was a later invention. Why the crotchet has the name meaning "hook" but the first note with a hook (i.e., flag) is the quaver [not quiver] (8th note) comes from a shift that took place in the 15th c. where the notes minim or longer came to be written with hollow noteheads and all smaller notes shifted up a rhythmic level.

I've published a bit on the history of slowing down of music at https://www.academia.edu/243065/Changing_Musical_Time_at_the_Beginning_of_the_Renaissance_and_Today_

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Hemi, Demi and Semi are all synonyms, they all mean half. Semi-Quaver being half a quaver. A hemisphere being half a sphere. A hemi, Demi or semi being added to each division of two. I guess a semi-semi-quaver just sounds like you have a stutter.

The archiac meaning of the word crotchet is that it was a surgical device with a hook this makes sense as the hook the quaver has in modern notation was originally given to the crotchet. It probably has something to do with that.

Why the crotchets hook from which it's name is derived was given to the quaver but the name stayed unchanged is another good question.

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