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Talk:Time signature

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This article was the subject of a Wiki Education Foundation-supported course assignment, between 1 September 2020 and 18 December 2020. Further details are available on the course page. Student editor(s): ThomasBabaian.

Above undated message substituted from Template:Dashboard.wikiedu.org assignment by PrimeBOT (talk) 11:23, 17 January 2022 (UTC)

3/4 compound and complex just like 6
Sorry but all the online resources and wikipedia are kinda wrong because 3/4 isn't necessarily simple always. As this article mentioned there "In compound meter, subdivisions (which are what the upper number represents in these meters) of the beat are in three equal parts, so that a dotted note (half again longer than a regular note) becomes the beat." and it also says that "3 4 is a simple signature that represents three quarter notes (crotchets). It has a basic feel of (Bold denotes a stressed beat):

ONE two three (as in a waltz)" therefore according to that 3/4 can be a compound time signature, in the case of Waltz for instance. Because subdivisions are in three equal parts since the stressed is on every THREE notes therefore the "DOTTED NOTE BECOMES THE BEAT". And no, a measure with one beat isn't necessarily simple or not doable, take waltz as an example the main beat is on every 3 quarter notes. To say 3/4 is the simple version of 6/8 is extremely ignorant. What if the stress is on dotted half notes, every three notes in 3/4? It depends on the score and what the pulse is in that score. So I'm going to delete the word simple in "3 4 is a simple signature that represents". 6/4 for instance can be either compound or complex. If the beat is 6 quarter notes then it's simple if the beat is 2 dotted half notes it's s coumpound time signature. As simple as that. 3/4 isn't a device to make your time signature simple 6/8 doesn't automatically make your music complex either. What if I wanted to count that as a 6 beat measure. It's up to the piece and the composer.

Can someone clarify whether a time signature is complex or compound depends on the piece? — Preceding unsigned comment added by 70.27.243.174 (talk) 08:22, 9 October 2017 (UTC) I am adding a comment for my class. MusicAB (talk) 18:35, 11 September 2020 (UTC)

 * There are two issues here. One is misnotation. If a composer writes a piece in 3/2 but gives it a 6/4 time signature, that's just nonstandard; the standard is that any "6" signature means 2 groups of 3, not 3 groups of 2. Exceptions exist, both in Baroque and modern music, but there are counterexamples of just about every notation, since it evolved organically. That's not the same as undermining the definitions.
 * The other matter is that music in 3/4 is often heard as "one to a bar," which does have a similar feel to compound signatures. This is frequent enough that it does merit mention in the article. —Wahoofive (talk) 17:47, 12 September 2020 (UTC)

Ambiguous or misleading
The text currently first uses "common time" (a phrase which is a re-direct to this site) in the following: "(such as or 3 4) (read 'common time' and 'three four time', respectively)." This can easily be incorrectly understood to mean that common time is three four time. Please re-write this. Kdammers (talk) 11:30, 6 December 2012 (UTC)

measure
I know music is sort of axiomatic, but couldn't there be some way of making the lede clearer? The use of the word measure, while probably necessary, is, as currently used, opaque. It is a linked word, but the text of the link pretty much goes back to the first part of the article's sentence. What makes a measure a measure?Kdammers (talk) 11:41, 6 December 2012 (UTC)
 * I'm not sure I understand your complaint. The linked word goes to an explanation: "a segment of time defined by a given number of beats, each of which are assigned a particular note value". This only "goes back to the first part of the article's sentence" in the sense that it says a time signature "specif[ies] how many" beats there are and "which note value constitutes one beat". This seems to be the very pinnacle of simplicity and clarity. What part do you find difficult to follow? Does it need to say that "measure" and "bar" (the actual title of the article linked from the word "measure") are synonymous?—Jerome Kohl (talk) 17:41, 6 December 2012 (UTC)

 * The lead undeniably needs work. I get what it is saying, but to non-musicians or even aspiring musicians who have not had any formal training it think it would be rather difficult to understand. I am also somewhat surprised that the article takes quite some time to getting around to mentioning that 4/4 is by far the most common, as indicated by the fact that is also called common time. To a general audience think this is a very basic fact that could significantly contribute to their understanding of what a time signature is and why they so often hear musicians counting to four or clicking the drumsticks four times. Beeblebrox (talk) 21:31, 17 December 2012 (UTC)

 * The text is written for musicians as far as I can tell. My problem seems to be that it is like talking to me in Chinese when I don't know that language or telling me about colors if I'm blind.  Saying that a measure is a bar does not tell me what a measure is.Kdammers (talk) 03:34, 31 December 2012 (UTC)

reference needed or not for time signatures ?
Why do we need a reference when the value of the time signature of a piece of music is mentioned ? Wikipedia does rely on the competence of their contributors or not ? It is not difficult for a musician to know that Radiohead's 15 step is in 5/4. It belongs to the category of informations that are neither opinions neither private or "published" in any way by any entity, it is just a scientific reality. If we agree on this, I will contribute by adding a few other examples in popular music. — Preceding unsigned comment added by Laurent S. Vleminckx (talk • contribs) 21:51, 27 January 2013 (UTC)
 * I think you might want to have a look at Talk:List of musical works in unusual time signatures (and especially the archives of that page) before proceeding to do what you are proposing. Wikipedia emphatically does not (and cannot) rely on the competence of their contributors, especially because there are such things as vandalism and hoaxes.—Jerome Kohl (talk) 23:26, 27 January 2013 (UTC)
 * To put it another way, right or wrong you would be violating the principle of no original research. Beeblebrox (talk) 23:40, 27 January 2013 (UTC)

Thank you for your answer. As a new contributor, I will not engage myself in any troubles and respect your point of view, but if we stick to the principles of no original research, I would tell you that the meaning of my first message was precisely that I consider finding the time signature of a piece of music as a "routine calculation" as it is defined in the Wikipedia rules. I would go even further and say that it is not the fact that a score says that a song is n-tempered that makes the song actually be n-tempered : the song itself provides that information, and the source is thus a secondary source. Though, I must admit that it is true for the "n-tempered" thing but not for the time signature : a song can be attributed a few different time signatures but only one "n-tempered" number provided that n is a prime number (well here we are totally on original research :) ). That leads us to the question of the pertinence of the time signature (I would say that the only real important fact about a song would be the n-tempered number with n being a prime number) but I guess Wikipedia is not the place for such a debate. And thank you Beeblebrox for telling me that Wikipedia does not rely on the competence on the editors, I didn't know it, but I find it really sad. First because then editors are reduced to "sources finders". I thought that some editor having some knowledge in some fields could use that knowledge to enhance the community ? Besides, more generally, how can you guarantee that the journalist or whoever wrote the "source" (often citing another source) provides a more accurate information than the one directly computed by a Wikipedia contributor ?

Well, I guess there are other pages on Wikipedia where to talk about the pertinence of its principles and I guess that I should stick to the rules. Right now I have got some scorebooks so I will contribute with the few songs concerned. Laurent S. Vleminckx (talk) 00:52, 28 January 2013 (UTC)

 * Welcome to the weird and wacky world of Wikipedia. I hope the frustration wears off quickly, and you find the environment congenial. The principle of "no original research" is one of the things that newcomers to Wikipedia have the greatest difficulty getting used to. It is made necessary by the "open" editing used, which requires no credentials at all of editors. Regarding some of the other questions you raise, you should have a look at the introductory paragraphs for the List of musical works in unusual time signatures and the articles Quintuple meter and Septuple meter for discussions of the "pertinence of time signatures" and their possible problematic relationship to the actual meter of any given piece of music. Keep in mind that the subject of the present article is the former rather than the latter, though their inter-relationship is of course part of the discussion.—Jerome Kohl (talk) 01:34, 28 January 2013 (UTC)

Musical parameters in infobox
Musical parameters in infobox I'm not sure of the best place to ask this, but I've started a discussion over at Template_talk:Infobox_song, which essentially asks whether tempo and key should have their own parameters. — Preceding unsigned comment added by 86.175.220.114 (talk) 19:35, 15 March 2013 (UTC)

Stress and meter
There's an odd signature and timestamp in that section.--95.116.231.215 (talk) 18:42, 4 April 2013 (UTC)
 * The result of vandalism, or an extremely clumsy edit. I have reverted it. Thanks for pointing this out.—Jerome Kohl (talk) 19:00, 4 April 2013 (UTC)

"Irrational" time signatures
I am going to insist on restoring my previous edits in the section on "irrational" meters. I added a link to dyadic rationals, which is an article about the type of fractions that are most commonly used for time signatures. I also pointed out that so-called "irrational" time signatures are in fact precisely those which are not dyadic rationals, so that "non-dyadic" would be the mathematically proper way to refer to these fractions (rather than "irrational", which is exactly contrary to the mathematical usage of this term). Note that I did not attempt to rename the section, remove the "irrational" terminology, or otherwise conceal the fact that "irrational" is commonly used in this mathematically incorrect sense. My edits were entirely relevant, appropriate, and in the spirit of Wikipedia, and I object in the strongest possible terms to their reversion, especially given the presumptuous, spurious grounds cited by the user responsible ("mathematical details not helpful" -- as if that user's lack of interest in mathematics were a legitimate reason for deleting a pertinent cross-reference!). 24.127.68.167 (talk) 10:51, 24 May 2013 (UTC)
 * This is an article about music. Whether the terminology used in music corresponds to similar terminology in other fields isn't really relevant (rational is also used in fields like psychology and economics as well, as you can see from the disambiguation page, and the page called irrational isn't even about math), and anyway dyadic rationals include numbers like 3475/65536, which isn't much help in understanding time signatures. I just don't see how including a link to this mathematical topic could assist readers in understanding even this obscure subtopic of musical time signatures. —Wahoofive (talk) 17:59, 24 May 2013 (UTC)
 * Instead of assuming that you are in a position to know why every reader of this article is reading it, why not defer to other editors who did, in fact, happen to come upon the article as readers? There are plenty of people who are interested in the mathematics of music, and they may appreciate the link simply for the background information it provides -- not necessarily because it "helps them understand" the meaning of a particular time signature. This is an encyclopedia, not a pedagogical text. (Though, frankly, I would have appreciated the terminological note even in a pedagogical text.)
 * (It should also be obvious that the use of "irrational" in this context -- where it directly conflicts with the mathematical usage -- is much more problematic than the uses of this term in fields like psychology, where the meaning is simply unrelated. A note calling attention to the conflict serves to prevent confusion and is entirely appropriate.) 24.127.68.167 (talk) 04:58, 25 May 2013 (UTC)

 * I agree. Unless there is solid evidence for good quality sources that the term 'irrational' is in general use in music we should not use terminology that conflicts with that used in mathematics. Martin Hogbin (talk) 10:10, 20 July 2014 (UTC)
 * I take it that you have not actually read the section of this article that deals with this subject. Whether or not the term is in "general use" may be debated but, as far as I am aware, there is no alternative term that may be substituted for this type of time signature. One thing that has not been pointed out here is that this use of the word "irrational" is consistent with the common musical term "irrational rhythm" which, as you will discover if you follow the link to the pertinent section of that article, uses the word in a sense borrowed from ancient Greek prosody, rather than the quite different one from mathematics. You will also find that some of the alternative terms for these rhythms (ordinarily expressed by whole-number ratios) are also in conflict with mathematics, from which they are borrowed, but with misunderstandings and other distortions, piled on top of some etymological confusions perpetrated by mathematicians. Nevertheless, just as 5 + 5 = 9 in music, so has "irrational" come to signify "relationships expressed as numeric ratios".—Jerome Kohl (talk) 20:13, 20 July 2014 (UTC)

 * Agree with Jerome, as usual, but I find it amusing that your definition of "irrational" in music is exactly what mathematicians would call "rational" — the exact opposite. Still, the less said about this topic the better. It's beyond obscure. I bet 99.9% of musicians have never encountered a non-dyadic time signature. —Wahoofive (talk) 18:25, 21 July 2014 (UTC)

 * Not my definition! Not my definition!! I wash my hands of this whole sorry business. The expression has been around a lot longer than I have, and of course there is that irony. What is interesting, though, when you check the senses in the OED (which lists them in historical order) is that the prosodic sense predates the mathematical one, though in poetic analysis there is not the definition of numeric proportions—this only gets added in by the mathematically inept musicians when they borrowed the non-mathematical sense of the term. (We should keep in mind that music is an art, not a science, despite the best efforts of musicology to make it one.)—Jerome Kohl (talk) 19:02, 21 July 2014 (UTC)

Alla Breve, cut time
The second example in the table mentions '2/2' on the left and in the text, but the rightmost column still shows a '4/4' example? — Preceding unsigned comment added by Sejtam (talk • contribs) 23:25, 12 August 2013 (UTC)

East versus West
I'm just some person when talking music, a complete layman, but I recently found myself in a forum where Time Signatures became relevant, yet we were still left with some unresolved issues.

The question arose when talking about the soundtrack to the film Akira. It's posted in its entirety on a youtube channel, and aroused one of the better discussions there. Almost. Some may recall the original film, but most will remember the re-dub, which came with an extensive commentary. In it, most seem to have memory of the composer interviewing about his motivation to use time signatures based on those found in ancient Japanese culture, and of those who saw it, some say this was something like 10/16 or 11/16. Does that... exist?

I noticed no mention of it, specifically, or anything regionally specific, but am wondering if the article in its current state may simply be referring to Western and/or contemporary music, or does it include such a wide description as to cover all musical possibilities?

I suppose the former would mean revision and addition, while the latter would mean... they were wrong!

WillJonassen (talk) 09:40, 29 August 2013 (UTC)

 * Certainly meter signatures (such as 10/16 and 11/16) are specific to the European notation system, and therefore would have been unknown in Japan until modern times (say, the early 17th century). However, it is entirely possible that decuple or undecuple meters may have existed, or perhaps aksak rhythms whose unequal beats can be transcribed into Western notation using meters like 10/16 or 11/16. Notated examples of Japanese music date back as far as the 13th century, though to what extent rhythmic information can be reliably extracted from this notation I do not know. According to the Music of Japan article, rhythms of traditional music are "based on the intervals of human breathing rather than mathematical timing", which suggests that so specific a transcription as 11/16 would have to be very speculative. I am not very well-informed about Japanese traditional music, though, and other editors may have better information to offer.—Jerome Kohl (talk) 15:31, 29 August 2013 (UTC)

 * It's still an incredible volume of information and knowledge which you've presented, and amazing how much such a response warrants continued learning. This is always great. I did some further research on my own, too, and though I lack the musical vocabulary necessary to carry on a full dialogue, the composition in question centered on the use of the Taiko drum ensemble juxtaposed with the human voice. The Taiko drum, of course, goes naturally with some serious energy. If it follows the breathing of anything, really, it would have to be a herd of stampeding horses, but that's just a side observation. Your mention of "human breathing" intervals does make a certain sort of sense in this context, in fact, as the goal of the film score had been to fully utilize the human vocal chord as an instrument unto itself. Check it out sometime, by the way - it's pure magic with or without the film. To your historical notes, though, those still remain interesting dates, as even the 17th century's point of development would place a new form's popular and cultural usage within a feudal state. A traditionally minded state. It could be a simple matter of untranslatable context, as well, being communicated to an English speaking audience which then led to a confusion. To finalize my own point and original question, still, relating to the aforementioned, it's good to know that there is a specific link to extra-cultural references of this sort. Music of Japan is exactly what I'll check out next - and in the future, I'll have a better grip on where to search for a starting point without assuming any one single article is all-inclusive, thanks to your example.

 * I appreciate the sincere effort, sincerely.

WillJonassen (talk) 16:55, 3 September 2013 (UTC)

 * You are very welcome. Just one cautionary note: the Taiko drum ensemble only first came into existence in 1951. While it does draw on some aspects of traditional Japanese music, what they play is a modern interpretation, and the movement has also been "globalized", which means that elements from other cultural streams have influenced performances by such groups.—Jerome Kohl (talk) 17:06, 3 September 2013 (UTC)

 * Well, that's just hilarious. I can only assume, then, that the speaker (Musical Director Shoji Yamashiro), who conducted the interview in truncated English, had only made a misstep in his attempt to describe what was a massive project with a very wide and original scope. I suppose that's why no such information could be found... Oh, life. You so and so...

WillJonassen (talk) 17:13, 3 September 2013 (UTC)

"Note values don't match with actual values"
In a request for clarification, User:70.88.200.134 states that "note values don't match with actual values" in this passage:
 * The lower numeral indicates the note value that represents one beat (the beat unit).
 * The upper numeral indicates how many such beats there are in a bar.
 * For instance, means two quarter-note (crotchet) beats per bar— means three eighth-note (quaver) beats per bar.

This request needs some clarification, I think. In what way does the /4 not match a quarter note (crotchet), and /8 not match an eighth note (quaver)? Or am I missing something here? Is "actual value" some sort of specialized technical term I don't understand?—Jerome Kohl (talk) 06:33, 13 January 2014 (UTC)
 * Perhaps the reference is to compound meters. —Wahoofive (talk) 00:03, 15 January 2014 (UTC)
 * That's what I thought at first, but if this is the case, the flag is placed in the wrong section: it is here, in the section on simple time signatures.—Jerome Kohl (talk) 01:02, 15 January 2014 (UTC)
 * Conversely, maybe it's a challenge to the concept of beat, which doesn't always line up with the time-signature value. There's plenty of music in 4/4 which is experienced as two beats to the bar, and plenty of 3/4 (or 3/8) music which is one beat to the bar, etc. —Wahoofive (talk) 23:19, 15 January 2014 (UTC)
 * Possibly, though neither "note values" nor "actual values" seem directly to signify the lower numeral in a time signature. I do wish User:70.88.200.134 would chime in here and explain what the problem is. Failing clarification, I think the flag ought to be simply removed.—Jerome Kohl (talk) 00:11, 16 January 2014 (UTC)
 * In other words, you want to flag the clarification flag for clarification :-) —Wahoofive (talk) 16:50, 16 January 2014 (UTC)
 * LOL! Yes, I guess that is exactly what I want to do!—Jerome Kohl (talk) 00:40, 17 January 2014 (UTC)

West Side Story
At long last, I have finally replaced the contrived example of "America" from West Side Story, which showed both 6/8 and 3/4 signatures at the beginning, with an example from Tchaikovsky. In fact, "America" is written just in 6/8. I've been meaning to fix this for years. —Wahoofive (talk) 19:28, 2 June 2014 (UTC)

Meter/time signature
Is this and WP:Engvar thing. In the UK, the numbers at the start of a line of music are generally called the 'time signature'. 'Meter' is used though. Martin Hogbin (talk) 10:05, 20 July 2014 (UTC)

Removal of (clarification needed) tag
Without reading this talk page first, I have removed the (clarification needed) tag from the section "Simple time signatures" because there is a perfect (in terms of function) explanation just two lines below. Jerome Kohl was on the right track by suggesting that the tag be removed. The editor who placed the tag in the first instance appears to be failing to understand the subject. We are advised to "Be Bold" and ... er .. I was. Any editor who wishes to revert my edit needs to come up with a very good rationale. In other words: Don't waste our time! CaesarsPalaceDude (talk) 02:58, 27 September 2014 (UTC)
 * Bless you, my son.—Jerome Kohl (talk) 05:40, 27 September 2014 (UTC)
 * I won't change it back, but the article, especially the lede, is still pretty much opaque to all but initiates. Maybe you musicians should sit down with a nonmusical person and have them try to understand the article.  I think you'll find out how confusing it is.Kdammers (talk) 14:08, 27 September 2014 (UTC)
 * Finally, someone who actually does not find this sentence perfectly clear has chimed in! Thank you for stepping forward. Please explain to the rest of us what is opaque about the words "numeral", "note value", or "beat", so that we can rewrite this sentence in a way that you can understand it. There are already links to the articles Note value and Beat (music), where explanations of those terms are given. Surely "numeral" cannot be unfamiliar. Does "lower numeral" not clearly indicate the number at the bottom of the signature? If w cannot get past this sentence, then there is no chance at all of dealing with the lede, which I admit looks cluttered and confusing, but mainly because of the many badly formatted time-signature examples in its last paragraph. Here again there are many links to terms that beginners might find unfamiliar, but very little that I can identify as "opaque to all but initiates". Could you please be more specific?—Jerome Kohl (talk) 22:24, 27 September 2014 (UTC)
 * I don't want to go on and on on this, but I'll try to make clear what isn't clear. beat and bar/measure are unclear ( a beat is something measured by measures, which measure beats - except when they don't because 2/2 time is or is not the same as 4/4 time) and C  [probably  totally unknown to anyt  nonmusicians including those who took  two or three years of piano) and 3/4 are still left sounding like they may the the same thing.  We exchanged comments on this a couple of years ago, but you don't see that the links don't help, and I don't see how they help.  I think the best way to handle this is as I mentioned before:  have one of the editors who understand this sit get together in real time with a nonmusician  or two and see where they struggle. (Trying to do this long-distance is at least as frustrating as for non-computer people trying to get help from geeks over the Internet.) And, yes, the visual impact dramatically raises the affective filter of a non-insider. Kdammers (talk) 00:47, 28 September 2014 (UTC)
 * If this muddled account is what it says in the article "Beat (music)" (and I admit I have not checked to verify this), then that article is sorely in need of correction. A beat is not "measured by measures". A beat is simply a periodic pulse, like the ticking of a clock. I admit this is proving difficult for a musician to discuss with a non-musician, especially when I cannot tap on the table-top between us to demonstrate the difference between a regular (periodic) pulse and a series of taps with different intervals of time between them. Does the "Beat (music)" article really make such a shambles of explaining this? I shall have to go and see but, if it does, the problem clearly must be there and not here. All that is at issue here is what symbol is assigned to this counting value (called a "beat"). It is not the purpose of this article to make an elaborate explanation of this basic element of musical rhythm.—Jerome Kohl (talk) 02:25, 28 September 2014 (UTC)
 * I have just checked the "Beat (music)" article, and indeed it is even worse than you suggested. It is one of the most badly written ledes I have seen on any subject. Although everything in it is perfectly true, it unnecessarily introduces at the outset a whole collection of abstruse terms that appear to serve no purpose other than to prevent readers from progressing any further. I think the most helpful thing we could do here is to remove the link to this infected and confusing article, until such time as it has been hospitalised and successfully treated. The concept of "beat" is one of the simplest and easiest-to-understand aspects in all of music theory. Only a gang of Wikipedia editors could succeed in making it incomprehensible.—Jerome Kohl (talk) 03:00, 28 September 2014 (UTC)
 * This article is about as clear as mud. Try writing it more clearly. 76.126.195.34 (talk) 04:45, 16 August 2016 (UTC)
 * Thank you for your constructive criticism. Given that there are nine major section, could we start at the beginning? The first sentence reads: "The time signature (also known as meter signature,[1] metre signature,[2] or measure signature[3]) is a notational convention used in Western musical notation to specify how many beats (pulses) are to be contained in each bar and which note value is to be given one beat." What do you find unclear about this (admittendly rather long) sentence?—Jerome Kohl (talk) 04:59, 16 August 2016 (UTC)

Yet another undefined term: hypermeter
The word hypermeter is used without explanation or link. 69.247.14.176 (talk) 22:53, 16 December 2014 (UTC)
 * Right you are. I have added a link.—Jerome Kohl (talk) 23:04, 16 December 2014 (UTC)

Progressive rock
Probably 21/8 in Wakeman's Anne of Cleves is also worth mentioning. Fedorkov Dmitry (talk) 17:43, 30 March 2015 (UTC)
 * See List of musical works in unusual time signatures. But you'd need documentation. —Wahoofive (talk) 23:32, 30 March 2015 (UTC)

Order of explantion
In an edit note, Jerome Kohl says "(changed order of presentation for clarity (explain first what it is we are going to count before explaining the counting itself)--the editorial note already refers to this)"

Could someone please indicate where it explains how the counting will go? I'm not seeing it. What I see see is someone explaining the lower number before the upper number. This isn't how we read this, and it isn't how we say it. 7/8 time isn't called "eight under seven time", it's "seven eight". — Preceding unsigned comment added by HacksawPC (talk • contribs) 07:56, 6 April 2015 (UTC)

 * If you were explaining to someone what a "grove of trees" was, would you insist on explaining the word "grove" first because it comes first in the phrase? If someone wasn't clear what a "tree" was, that would be difficult. —Wahoofive (talk) 18:17, 6 April 2015 (UTC)

math
Maybe this is article not the appropriate place to answer this question, but I would think it would be clear to someone reading it (but it isn't to me). What if any difference is there between 2/2 and 4/4 (or 3/4 and 6/8 etc.) is there? 64.53.191.77 (talk) 03:27, 7 April 2015 (UTC)
 * Do not confuse time signatures with mathematics. If the differences between these pairs of meters is not made clear by the article, then the article needs improvement. Can you please re-read the relevant sections, and then point out what parts are unclear?—Jerome Kohl (talk) 04:45, 7 April 2015 (UTC)
 * I'd agree that the difference between 6/8 and 3/4 is made explicit in the text, but not so for 2/2 and 4/4; in fact, the example shown in the table for 2/2 time is notated with a 4/4 time signature, a situation certain to confuse readers. Are they interchangeable or not? (Hint: sometimes). I'd say most musicians are pretty hazy on the distinction between 2/2 and 4/4 in practice, although they could cite the textbook difference. —Wahoofive (talk) 17:29, 7 April 2015 (UTC)
 * I'll have to take a closer look, I guess, but I thought the text was fairly plain that the upper numeral indicates the number of beats in the bar, and 2 ≠ 4. In practice, I find that the main areas of ambiguity are not with these specific time signatures (at least, not in "classical" music), but rather with whether the symbols and  should really be interpreted as 4/4 and 2/2, respectively. This is particularly ambiguous in the case of eighteenth-century music, of course. Is it appropriate, though, to go into such detail here, or is the "textbook definition" really sufficient?—Jerome Kohl (talk) 23:06, 7 April 2015 (UTC)
 * Closer look taken, and I see what you mean about the 4/4 example illustrating the 2/2 block in the table. It is clearly a misplaced example, and should be moved to the 4/4 section. It looks like it may have been put there in confusion over the use of "duple" as a category encompassing all "even" meters, in opposition to "triple" for "odd" meters.—Jerome Kohl (talk) 23:11, 7 April 2015 (UTC)
 * Yes, this is the right article to address ambiguities in C and C-slash, although it could be relegated to a lower-down section. Even Brahms used C-slash to mean 4/2 sometimes, so it's not exclusively an early-music thing. As for the difference between 2/2 and 4/4, I guess the textbook difference will have to suffice, but it begs the question of why one would choose one rather than the other: what's the practical difference between them? The upper number doesn't necessarily indicate the number of beats in a bar, e.g. 6/8 frequently only has two beats, and it's not that hard to find pieces notated in 4/4 which are really two beats to the bar, but the determination of what is a beat is a slippery slope. —Wahoofive (talk) 20:53, 8 April 2015 (UTC)

 * A salutary example, thanks for that. It reminds me that on occasion Brahms could be an historist composer (like Bruckner). The notation obviously refers back to the 16th century sources of aspects of that particular Brahms work. Should we also bring in Brahms's liking for dotting notes across barlines?—Jerome Kohl (talk) 06:20, 9 April 2015 (UTC)
 * " and it's not that hard to find pieces notated in 4/4 which are really two beats to the bar," -- I have no idea what all this means. I thought that a beat is a stressed note, so 2/2 would be Stress, unstress {by the way, can it be unstress, Stress??} versus 4/4, which would be Stress, unstress, unstress, unstress; but the text I just quoted seems to contradict that idea.  {Does stress simply mean louder?} 76.126.195.34 (talk) 04:49, 16 August 2016 (UTC)
 * A beat is regarded as stressed in comparison to parts of the bar that are not on a beat. On the other hand, some beats may be stressed more than others. Stress does not necessarily mean louder, but that can be am important component. In 4/4 time, beat one is the primary stress, but beat three has secondary stress. Beats two and four are regarded as unstressed. Under some circumstances (a faster tempo, for example) the unstressed second and fourth beats can diminish in importance until they feel less like beats and more like fractions of beats, leaving just the first and third quarter notes of the bar seeming like "beats". There will still be the relative sense of strength: a stronger first beat, and a weaker, but still stressed, second beat. Clear now?—Jerome Kohl (talk) 05:11, 16 August 2016 (UTC)

Irrational meters
Since there was no example previously, I have added a theoretical one. It may not be ideal in its notation. Double sharp (talk) 15:04, 4 June 2016 (UTC)

No TS?
Does Western music always have a time signature? I can't hear a repeating stress pattern in a lot of music. Is that because there is no set time signature, or is that a problem in my hearing? 76.126.195.34 (talk) 04:53, 16 August 2016 (UTC)
 * The short answer is "No." There are pieces of music and even whole genres that are notated without a time signature, and are not meant to have a regular metrical structure see "free rhythm" Metre_(music). That said, this is not very common in Western music, so it is also possible that you are somehow not observing a recurring pattern that is actually present.—Jerome Kohl (talk) 05:21, 16 August 2016 (UTC)
 * Also, technically speaking, a time signature is a notational device, so it is sometimes omitted even when there is a regular meter, for example in church hymnals or lead sheets for pop music. —Wahoofive (talk) 17:05, 27 December 2016 (UTC)

timey-wimey nonlinear history

 * The first deliberate quintuple meter pieces were apparently published in Spain between 1516 and 1520, though other authorities reckon that the Delphic Hymns to Apollo (one by Athenaeus is entirely in quintuple meter, the other by Limenius predominantly so), carved on the exterior walls of the Athenian Treasury at Delphi in 128 BC, are probably earlier.

It does not surprise me that some authorities reckon that something written in 128 BC is probably older than something published in AD 1516. What is this sentence trying to say? That the Delphic examples were long not recognized as quintuple? That not everyone agrees that they are quintuple? —Tamfang (talk) 00:53, 26 February 2017 (UTC)

 * OK, I hold my hands up on this one. What you have left out is the "reliable source" quoted in footnote number 8 immediately following the year 1520: "Tim Emmons, Odd Meter Bass: Playing Odd Time Signatures Made Easy (Van Nuys: Alfred Publishing, 2008): 4." This source declares outright that quintuple meter only first appears in Spain in the 16th century. This citation is an incontrovertible "fact", and I could not resist pointing out the slightly earlier examples (also with a reliable source: Documents of Ancient Greek Music: The Extant Melodies and Fragments, edited and transcribed with commentary by Egert Pöhlmann and Martin L. West [Oxford: Clarendon Press, 2001]: 70–71 and 85) that make this claim seem rather foolish. This, by the way, is one example of why I personally dislike tucking away source citations in footnotes: they are too easy to overlook and, under Wikipedia's verifiability, not truth policy, such "reliable sources" should be made as visible as possible (without, of course, disrupting readability). Parenthetical referencing serves this purpose much better than footnotes.—Jerome Kohl (talk) 22:46, 26 February 2017 (UTC)

 * All very amusing, but it seems legitimate to say that these Spanish items are the first published examples and suggest at least a local practice of these irregular meters, whereas ancient inscriptions don't meet most people's definition of "publishing" and tell us little about musical practice in the ancient world. The question of inline citations is a red herring.
 * This sentence could easily be edited to reflect this, in part by removing "probably" near the end, removing "deliberately" near the beginning (were there accidental ones?), and removing "other authorities reckon that" from the middle. —Wahoofive (talk) 23:52, 27 February 2017 (UTC)

 * The question of what constitutes "published" is relevant here because, although the cited source (Emmons) does not actually say so, the 16th-century collection is the Cancionero de Palacio, which is a manuscript, not a printed book. As such, engraving musical compositions on the wall of a public place (in the case of the Delphic Hymns, a temple to Apollo) would just as plausibly constitute "publication" in the ancient world as the copying of a manucript in the 16th century. Keep in mind that, at the time the Cancionero was initially copied, music printing did not yet exist. I would suggest it makes better sense to strike everying from the Emmons source, on grounds of unreliability, and accept the Pöhlmann and West citation as authoritative.—Jerome Kohl (talk) 03:50, 28 February 2017 (UTC)

 * I think it's safe to say that both the Delphi and the Spanish examples are anomalous uses, as far as we know. I'd vote to keep the reference. I've rewritten the sentence to remove the word "first", although that whole paragraph is a list of everybody's favorite examples thrown together higgledy-piggledy. I was under the impression that "publishing" meant something different from "printing," but now that I look under Publishing in Harvard Dictionary, I see they're essentially synonymous. —Wahoofive (talk) 21:22, 1 March 2017 (UTC)

 * Just for fun, here's an example from Handel's opera Orlando, published in 1733:

 * —Wahoofive (talk) 21:53, 1 March 2017 (UTC)

 * Yes, this example is discussed in the article Quintuple meter. I gree that the Spanish examples may be anomalous (though, again, see the discussion of Spanish traditional music in the Quintuple meter article). The Delphic examples certainly are not anomalous, since the Cretic meter in which they are composed is reasonably common in ancient Greek poetry, and specifically in paeans (which both hymns are).—Jerome Kohl (talk) 23:53, 1 March 2017 (UTC)

Move discussion in progress
There is a move discussion in progress on Talk:Commontime (album) which affects this page. Please participate on that page and not in this talk page section. Thank you. —RMCD bot 21:33, 30 December 2017 (UTC)

Importance
This article is listed as being at mid-important. Clicking on the links, I couldn't find a definition of high/mid etc., but intuitively, I don't understand why this article, if I understand what it is about (the basic way the rhythm of music is shown in Western notation?), is not listed as being of high importance. Kdammers (talk) 09:05, 10 March 2018 (UTC)

Mission: Impossible Theme
I edited the page to say it was in 10/8, rather than 5/4, as this is how it is felt, grouping 3+3+2+2. It may be written in 5/4 with syncopation (I have seen the score), however this is misleading and the way beats are grouped in 5/4 is different from 10/8, which makes it very frustrating to play when written as such. The beats of 5/4 just don't line up with the way you feel the song, which is more important than how it is written; you can write 9/8, but it can felt in 3+3+3, or less commonly some permutation of 2+2+2+3, which has a very different feel (3 triplets vs. a slightly stilted, lengthened 4/4). Edit: wow I messed up, I was not given feedback that my section had been posted, rather given a conflicting edit error, and didn't tell me anything was posted, so I tried again. Nice. Torzod (talk) 02:27, 9 October 2019 (UTC)
 * On Wikipedia, what matters most are reliable sources. If you have one to cover this situation, it would make a welcome addition to the article.—Jerome Kohl (talk) 02:37, 9 October 2019 (UTC)

and
What does "and" mean in " one and two and three and"? 2600:6C67:1C00:300:7830:E76D:2A41:8E86 (talk) 04:09, 24 March 2022 (UTC)

 * It's just a way to pulse on the half-beats. The word has no denotative meaning. —Wahoofive (talk) 21:47, 24 March 2022 (UTC)

12/8 time
The 12/8 example given, 'The Irish Washerwoman' is a jig, and jigs are normally notated 6/8. Slides (type of Irish dance tune) are notated in 12/8, so I suggest using 'Star above the Garter' [ref: https://www.irishtune.info/tune/1829/] which is a 'slide'. squike (talk) 14:24, 9 September 2022 (UTC)

2/2 vs 4/4
What's confusing to me is that the examples for 2/2 and 4/4 are exactly the same thing but at different tempos. Is it that simple? If I change the tempo of a song, am I changing the time signature? If these concepts are really not decoupled, then I am disappointed by the opaqueness with which the topics are presented. The rhythm of a song has a certain character that does not change with tempo, just like the characteristic harmony of a major chord does not change by shifting the note frequencies. One would hope that time signature reflects this invariance.

Several people here assured that 2/2 and 4/4 are not the same thing when pretty much any layperson would hear them as closely related if not the same thing. Why not just say "2/2 is 4/4 at half the tempo", or "2/2 is 4/4 played for half as long"? It's no wonder people are confused by this article. 2603:6080:6001:E5BF:71C2:DAC0:C0DD:1CB1 (talk) 05:22, 13 September 2022 (UTC)
 * The real time signature is often different from the notated one by a factor of two. This is accepted in a number of scholastic works, but I'd have to search my computer to find them. Tony (talk)  00:54, 14 September 2022 (UTC)

Query
"There are various types of time signatures, depending on whether the music follows regular (or symmetrical) beat patterns"—what does symmetrical mean here? Tony (talk)  03:59, 27 September 2022 (UTC)

"Simple time signatures consist of two numerals". Well, not if they're C or cut C. Tony (talk)  04:44, 28 September 2022 (UTC)

Videos
The first two videos stop after about half a second on Safari for Mac. Tony (talk)  04:50, 28 September 2022 (UTC)

Intro
In the introductory section, there is this: "There are various types of time signatures, depending on whether the music follows regular (or symmetrical) beat patterns, including simple (e.g., 34 and 44), and compound (e.g., 98 and 128); or involves shifting beat patterns, including complex (e.g., 54 or 78), mixed (e.g., 58 & 38 or 68 & 34), additive (e.g., 3+2+38), fractional (e.g., 2+1⁄24), and irrational meters (e.g., 310 or 524)." All these undefined terms (except via the blue links) with examples that don't clarify do not belong in the introduction of ar article in simple English. What, for example, makes 12/8 compound, while 4/4 is simple? Kdammers (talk) 02:49, 10 December 2022 (UTC)
 * I don't understand your objection. All those terms have links to the corresponding places on the page which explain them in further detail. That's what an intro is for, to introduce the topic. —Wahoofive (talk) 01:05, 15 December 2022 (UTC)
 * Not the person who left the original message, but basically, the examples do not illustrate or enlighten but rather obfuscate what is meant to a reader not already familiar with the relevant terminology. I'm not particularly familiar with musical notation or music theory. It's like trying to figure out the meaning of a word by looking up its definition in a Polish dictionary: a whole bunch of words that I'm sure make perfect sense to someone who speaks the language, but I don't.
 * To draw a comparison: Lepidoptera (moths and butterflies) are an interest of mine. So let's say that you're interested in learning about them, including what kinds there are. If I were to try to explain some of it to you by saying, say:
 * "Lepidoptera are commonly divided into the monophyletic butterflies, which include the swallowtails (e.g. Papilio machaon and Parnassius apollo), metalmarks (e.g. Hamearis lucina), whites and yellows (e.g. Pieris brassicae), gossamer-winged butterflies (e.g. Phengaris arion) and brushfooted butterflies (e.g. Danaus genutia) and usually the skippers (e.g. Pyrgus oileus) and American moth-butterflies (genus Macrosoma), and the paraphyletic moths, which contain all other Lepidopterans, such as owlets (e.g. Noctua comes), tortrix moths (e.g. Grapholita delineana), geometers (e.g. Biston betularia) and many others".
 * That probably wouldn't help you in feeling like you've learnt something about them, or like you're now familiar with the relevant concepts in order to start learning about them, no matter how much sense the above might make to me. It certainly wouldn't help you understand the classification system—because it doesn't actually explain the system used, it just lists the labels used. And even if all those labels link to the relevant explanations, most of those still don't explain the system. They just explain that specific label, and the reader is left to try and build an idea of how the system works by piecing together information from various small sections of the system. AddWitty  NameHere  07:26, 27 January 2023 (UTC)
 * Okay, I've taken out the examples and simplified the sentence to emphasize the most important parts. —Wahoofive (talk) 16:52, 11 February 2023 (UTC)
 * Wouldn't "rhythmic value" be more precise than "note value"? Just as we talk of the leading tone, not the leading note (the latter needing to be invested with pitch, volume, etc). Tony (talk)  02:50, 12 February 2023 (UTC)
 * And isn't "beat" also "an auditory feature"? At the moment only (bar-rank) meter is so designated. It's very fiddly to be precise at the start of such an article, I concede. Tony (talk)  02:56, 12 February 2023 (UTC)

 * "Beat" is a very fuzzy concept, arguably more auditory than notative, but I don't think "rhythmic value" would improve anything. Note values can be dotted notes, or triplets, or whatever. I'm trying to find a way to make this layperson-accessible. —Wahoofive (talk) 00:16, 14 February 2023 (UTC)
 * Why is it "a very fuzzy concept"? Tony (talk)  01:28, 14 February 2023 (UTC)
 * Because it's subjective. Some 4/4 time signatures are really 2 beats per bar, or 8. Some 6/8s are 6 beats to the bar; some 3/4s are 1 to the bar: but there's no objective way to tell, and sometimes different performers will "feel" the same piece different ways, or at different tempi. It's not really "beat" itself that's subjective so much as the distinction between "beat" and "subdivision." Some 7/8 passages are felt as two or three unequal beats, but admitting the existence of unequal beats is too much for an intro. —Wahoofive (talk) 22:36, 14 February 2023 (UTC)
 * I don't agree with most of what you've just said. First off, you're now talking of bars/measures, not beats. Tony (talk)  00:51, 15 February 2023 (UTC)

 * For reference, the article Beat starts with "The beat is often defined as the rhythm listeners would tap their toes to ...though in practice this may be technically incorrect." That's fuzzy in my book. Harvard Dictionary (2nd) goes with "The temporal unit of a composition as indicated by the up-and-down movements, real, or imagined, of a conductor's hand." Plenty subjective. I brought up "bars" earlier since the lede says the TS describes how many beats are in a bar, but if we're not certain what a beat is, that isn't helpful.
 * Suppose we go with something like "A TS (aka...) is a notational device which defines the length of each bar in terms of a multiple of a particular note value." This skirts the whole "beat" thing at the start, although it has to be addressed somewhere in the article.
 * —Wahoofive (talk) 21:45, 17 February 2023 (UTC)
 * Yes, those supposedly reliable sources are pretty bad on beat. Let's get rid of the hated "in terms of" (what terms, please?). And CMOS says to use "that", not "which".
 * "A time signature is a notational device that defines the length of each bar as a given number of beats of a give note value." <b style="color:darkgreen">Tony</b> (talk)  00:00, 18 February 2023 (UTC)

 * It pleases me to learn that I am not the only h8r of "in terms of", but I'd accept this last example if not for the redundancy: defines the length of each bar in terms of a particular note value would be okay with me, though defines the length of each bar as a multiple of a particular note value is better. —Tamfang (talk) 01:59, 14 May 2023 (UTC)

 * I'm sorry you've chosen to re-introduce the misleading and vague "beats" terminology into the lede. I can't be bothered to get into a revert war, but the simple fact is that a 6/8 (or any other) time signature doesn't really say anything certain about the number of "beats" in the bar, but it definitively says how many eighth notes fit into each bar. And "beats of a particular note value" doesn't even mean anything. —Wahoofive (talk) 16:45, 22 April 2023 (UTC)
 * 6/8 has two beats—no fewer, no more. <b style="color:darkgreen">Tony</b> (talk)  02:03, 23 April 2023 (UTC)
 * I don't know what kind of musician you are, Tony, but as a conductor I can tell you I've often had to conduct 6/8 passages with 6 beats. I've often conducted 4/4 passages with 2 beats, or 8, depending on the tempo. But evidently no one has any idea what a "beat" is, as discussed above, so using such a poorly-defined term only muddies the lede. —Wahoofive (talk) 15:16, 23 April 2023 (UTC)
 * No, you don't know what kind of musician I am.
 * First, please note: no hyphen after -ly adverbs.
 * Can you give an example of a movement in 6/8 that is played so slowly that eighth notes are beats? There's something wrong, if so. Either it should have been notated in 3/8 or it should be played faster. Or is the composer such a klutz that it should have been written in 3/4 time? Plain fact is: the downbeat of each bar (measure) needs to be entrainable. Once the downbeats are more than four or five seconds apart (depending on the information density), metricity is lost at the bar rank. I'm interested in continuing this dialogue in good faith; perhaps I can learn from you. <b style="color:darkgreen">Tony</b> (talk)  08:32, 25 April 2023 (UTC)

I should know better than to get tempted into OR, but I'm only human. Here are some examples, starting with Beethoven:

Mozart:

Brahms:

Bach:

There's plenty more where those came from. Plus, evidently you're under the impression that a 3/8 time signature means 3 beats to the bar. Frequently (perhaps more often) it's only one.

Bach:

Beethoven:

In some cases it's dependent on what tempo the performer (or conductor) chooses, but that only underscores that beat is a hopelessly confusing term. The time signature is an unreliable guide to the number of beats, but it DOES definitely determine how many note values fill up a bar, and for accuracy we should stick to the latter. I encourage you to expand the section I created on "Beat and subdivision" if you want to explain that most often certain time signature are associated with certain numbers of beats, but I'd strongly advise against including the as-yet-undefined term beat in the lede. —Wahoofive (talk) 16:49, 26 April 2023 (UTC)

 * Thanks for these examples; but none of them implies eighth note as beat/pulse (note that some theorists distinguish between beat and pulse: see Hasty p. 12). Are all of the metronome indications by the composers? The Brahms is to be played at a fast pace. Caplin refers to (here I paraphrase) notated time signature and real time signature (usually half a bar); many conductors beat twice to every notated bar.
 * "I'd strongly advise against including the as-yet-undefined term beat in the lede"—I'm pretty sure the link to "beat" was only shortly after where I relocated it.
 * "evidently you're under the impression that a 3/8 time signature means 3 beats to the bar."—Nothing of the sort: I suppose one could find a performance of music in 3/8 so slow that listeners will hear it as three beats, but it's unlikely.
 * The first example (Beethoven): this is the best example of a passage where the could be six pulses per bar, but I believe it would be ponderous played that way. And wasn't Beethoven's metronome faulty at one stage?
 * The third Beethoven example: it's too fast at 112 eighth notes per minute not to feel three pulses in the bar.
 * I'm unconvinced. But your query about "note value" ... I'll return to explore a better way of expressing it. <b style="color:darkgreen">Tony</b> (talk)  12:25, 27 April 2023 (UTC)
 * Thank you for giving this some more consideration. All the pieces have opus numbers in the image descriptions if you want to listen to recordings and see what you think (although we are still talking OR). The Mozart MM marking is certainly editorial (although it's frequently played slower than 76); not sure about the Beethoven ones. Can a beat be as slow as, say, MM30? I don't think so, but as a conductor I find there are practical limitations to such a slow gesture. Absent an agreed definition of beat we can't say definitively. —Wahoofive (talk) 03:15, 28 April 2023 (UTC)
 * Justin London's book Hearing in Time (2012, 2nd ed.) surveys the lit. on maximum duration you can entrain for. Entrainment, of course, is essential for beat (and for meter), otherwise there's not much point. Without consulting it I'm pretty sure 4–5 s is the limit. But scholars have also shown that the density of information and the background/skills of the listener might alter this. And here's my RO: if the content is repeated within beat-lengths or bar-lengths, it might extend that viable duration. But that can't go into the article, naturally. <b style="color:darkgreen">Tony</b> (talk)  05:23, 28 April 2023 (UTC)

Wahoofive is correct. It is absolutely normal in the Classical era to find 3/8 so slow that it has to be treated as three beats. The slow movement of Beethoven's Third Piano Concerto is an obvious (if extreme) example: I should like to hear how a proposed one-beat-per-bar performance is going to deal with all the 128th notes. In Charles Rosen's book Beethoven's Piano Sonatas: A Short Companion he makes the point that "The last movement of the "Emperor" Concerto, as well, is misrepresented in most editions: the tempo should not be Allegro but the more deliberate Allegro non tanto, and the dance movement sometimes requires six emphatic beats per bar and not simply a more graceful two". Seeing a 6/8 time signature tells you nothing about whether it is six beats per bar, two beats per bar, or even one beat per bar (in Borodin's Polovtsian Dances). Double sharp (talk) 10:16, 31 July 2023 (UTC)
 * I strongly disagree. Wahoofive is confusing (i) the conductor's need to signal, and (ii) actual beats from the listener's perspective, which can be as far apart as entrainment allows. The examples he has provided above (take the first four) don't convince me at all that eighth notes are "beats". <b style="color:darkgreen">Tony</b> (talk)  08:41, 1 August 2023 (UTC)

 * We'll never resolve this until the article Beat has a comprehensible and less subjective definition. —Wahoofive (talk) 15:38, 1 August 2023 (UTC)
 * I don't mind the lead of Beat, except for the vague "or the numbers a musician counts while performing", and the unreferenced "In popular use, beat can refer to a variety of related concepts, including pulse, tempo, meter, specific rhythms, and groove." <b style="color:darkgreen">Tony</b> (talk)  03:40, 2 August 2023 (UTC)
 * I provided a source specifically talking about a piece in 6/8 meter (the finale of Beethoven's Emperor Concerto) and remarking that sometimes it has six beats in a bar and sometimes it has two. And it's not even a slow piece, the marking is Allegro non tanto. You have given no argument from sources, only a personal lack of conviction. And as for such lack of conviction, it's firstly OR, and secondly – regarding the tempo of the first piece, the 64th notes that appear later should lead to obvious conclusions about the tempo. (As should the 128th and almost-256th notes in the 3/8 slow movement of Beethoven's Third Piano Concerto.) Not to mention that the Emperor Concerto example shows that one can hear six beats to a 6/8 bar even at a rather fast tempo. Double sharp (talk) 08:56, 3 August 2023 (UTC)
 * So let me guess: your theory is not RO?
 * I don't have a score for Beethoven's third piano concerto. You need to present evidence. <b style="color:darkgreen">Tony</b> (talk)  10:58, 3 August 2023 (UTC)
 * What is "RO"? If you mean "OR", I quoted a source (Rosen). You didn't.
 * It somewhat beggars belief that you are talking about classical music and don't know about IMSLP, but here's a link to the 3/8 slow movement of Beethoven's third piano concerto. Now how is anyone supposed to play all those 128th notes if it was meant to be one beat per bar?
 * And here's another source, from David Montgomery's Franz Schubert's Music in Performance, distinguishing possible interpretations of the time signature 3/4: "The finale of D784, then, is taken at three to the measure, and the scherzo of D845 in one." He goes on to distinguish slower "type A" and faster "type B" movements in various time signatures, in which type A move by a shorter beat and type B by a longer one, e.g. 2/4 in two (e.g. D804 finale) vs in one (e.g. D845 finale), 6/8 in two (e.g. D664 finale) vs in one (e.g. D810 finale), 3/4 in three (e.g. D850 scherzo) vs in one (e.g. D812 scherzo), and 3/8 in three (e.g. D574 slow movement) vs in one (D537 finale). All these examples are straight from Montgomery. Since there are too many of them, I feel free not to link to them all. Just search the catalogue numbers with "IMSLP".
 * And here is yet another source, regarding the 6/8 seventh variation of the Brahms Op. 56 variations: "Variation VII is felt principally in six beats to a bar". And the author in fact argues earlier for why it should be six, not two: because only with six beats to the bar does Brahms' tempo marking Grazioso make sense.
 * The whole idea that the time signature tells you how many beats there are in a bar is one of those lies to children we get told to start off our journeys into music. I was told it too as a child. But it simply isn't so, as the sources (and in some cases, like the Largo e mesto from Beethoven Op. 10 No. 3, common sense from seeing 64th notes) prove. Double sharp (talk) 16:05, 3 August 2023 (UTC)
 * So you don't mind calling eighth notes the "beats" in a slow 3/4 measure containing eight 8th-notes? BTW I'm OK with the current wording in Beats and subdivisions: "Conversely, at slow tempos, the beat might even be a smaller note value than the one enumerated by the time signature." – though I wouldn't mind "very slow". And I'm OK with the current wording in Time_signature: "Most often, 6/8 is felt as two beats, each being a dotted quarter note (crotchet), and each containing subdivisions of three eighth notes (quavers). It is felt as one two three four five six ...". ("The one and a two and a" will confuse the reader without better formatting.) <b style="color:darkgreen">Tony</b> (talk)  04:43, 4 August 2023 (UTC)
 * Why not? A slow 2/4 might really have four eighth-note beats, and a slow 3/4 six eighth-note beats. See the Classical-era sources quoted here.
 * Considering Rosen's example from the Emperor Concerto, probably "at slow tempos" should rather be "especially at slower tempos". Other than that, the wording seems fine to me. Double sharp (talk) 07:06, 4 August 2023 (UTC)

 * I'm totally on board with saying that 6/8 is 2-beats-per-bar and 3/8 is one-per-bar the vast majority of the time; my only objection is to using the word "beat" in the lede as if it were part of the meaning of the time signature itself. Double Sharp is right that that's a simple definition for beginners. Are you familiar with Lies My Music Teacher Told Me? —Wahoofive (talk) 00:54, 6 August 2023 (UTC)

lower numbers in groups of threes -?
(Still struggling to understand this article.)  "the note values specified by the bottom number are grouped into threes,"-- How are the numbers 4 and 8 grouped into threes? Kdammers (talk) 04:52, 1 March 2023 (UTC)
 * It's not the numbers being grouped, it's the corresponding note values. Quarter notes and eighth notes (represented by the numbers 4 and 8) can easily be grouped in threes. I welcome editing the sentence to make this clearer. —Wahoofive (talk) 17:11, 1 March 2023 (UTC)
 * Okeh, note values. But I am still at a loss.  The article says "are" grouped, whereas you write "can easily be."  What is this talking about?  I have no idea whatsoever.  So what if the note values are or can be grouped into threes.  What has that to do with a given time signature?  For example, 4/4 time, as I understand it, has four beats within a given time, marked by the bars.  where does three come in? Kdammers (talk) 21:11, 1 March 2023 (UTC)
 * Indeed 4/4 cannot group quarter-note beats into threes (at least, not unless there's some kind of cross-rhythm going on), but it is not compound. The article says that only for compound time signatures, like 6/8, 9/8, and 12/8. Notice that in these cases the upper number is divisible by 3. Double sharp (talk) 06:26, 15 September 2023 (UTC)

"The time signature is a notational device representing the meter, which is an auditory feature of the music." (my italics) ... What else would meter be? <b style="color:darkgreen">Tony</b> (talk)  05:08, 2 March 2023 (UTC)

"Simple time signatures consist of two numerals, one stacked above the other:" – that belies what appears above concerning the time symbol for common time. And don't compound time signatures also consist of two stacked numerals? Why is this implied as exclusive to simple time? <b style="color:darkgreen">Tony</b> (talk)  05:08, 2 March 2023 (UTC)
 * Okay, I've addressed some of these comments. More edits to come. —Wahoofive (talk) 04:58, 5 March 2023 (UTC)

Time signature notation in ancient Greece
Early in the description of complex signatures is this reference to the Delphic hymns to Apollo, argued about above. But this article is about the notational device of the time signature, not Metre (music). Was there any kind of time signature in the notation back then? If not, it really needs to be omitted here. —Wahoofive (talk) 17:37, 7 March 2023 (UTC)
 * Sensible. <b style="color:darkgreen">Tony</b> (talk)  05:26, 28 April 2023 (UTC)

Under number of 16
Is 16 a usually-used under number of a time signature? Χιονάκι (talk) 15:44, 1 August 2023 (UTC)
 * Occasionally you see it. The composer might be asked whether they could have made do with the simpler "8". <b style="color:darkgreen">Tony</b> (talk)  03:41, 2 August 2023 (UTC)
 * To be fair, x/16 meters sometimes have an advantage when it comes to beaming. If we compare 6/8 with 6/16 in the case where they're both duple, then you can't beam the rhythm     according to the beat, but you could beam its equivalent      that way. Double sharp (talk) 12:40, 23 August 2023 (UTC)
 * They were more common in Baroque times. For example, in WTC II the F major fugue is in 6/16 and the C-sharp minor fugue is in 12/16. In List of musical works in unusual time signatures, 6/16, 9/16, and 12/16 are considered "usual" (I suppose because they are the natural compound-time counterparts of 2/8, 3/8, and 4/8). Double sharp (talk) 08:53, 3 August 2023 (UTC)
 * Then why not also mention the 16, since it is common in known pieces of Baroque music? Χιονάκι (talk) 12:20, 6 August 2023 (UTC)
 * I personally would not object to it, for the reason of consistency with the "unusual time signatures" list. Though I have misgivings about its definition (surely 12/4 can't be more common than 5/4), it is at least sourced and one I recall. Double sharp (talk) 16:35, 6 August 2023 (UTC)
 * In List of musical works in unusual time signatures he does not consider the time signatures 6/16, 9/16 and 12/16 unusual. Χιονάκι (talk) 22:04, 6 August 2023 (UTC)
 * Yes, that's what I said. Double sharp (talk) 06:48, 9 August 2023 (UTC)

title: Talk:Time signature


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Simplex

In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. For example,
 * a 0-dimensional simplex is a point,
 * a 1-dimensional simplex is a line segment,
 * a 2-dimensional simplex is a triangle,
 * a 3-dimensional simplex is a tetrahedron, and
 * a 4-dimensional simplex is a 5-cell.

Specifically, a $k$-simplex is a $k$-dimensional polytope that is the convex hull of its $k + 1$ vertices. More formally, suppose the $k + 1$ points $$u_0, \dots, u_k$$ are affinely independent, which means that the $k$ vectors $$u_1 - u_0,\dots, u_k-u_0$$ are linearly independent. Then, the simplex determined by them is the set of points $$ C = \left\{\theta_0 u_0 + \dots +\theta_k u_k ~\Bigg|~ \sum_{i=0}^{k} \theta_i=1 \mbox{ and } \theta_i \ge 0 \mbox{ for } i = 0, \dots, k\right\}.$$

A regular simplex is a simplex that is also a regular polytope. A regular $k$-simplex may be constructed from a regular $(k − 1)$-simplex by connecting a new vertex to all original vertices by the common edge length.

The standard simplex or probability simplex is the $(k − 1)$-dimensional simplex whose vertices are the $k$ standard unit vectors in $$\mathbf{R}^k$$, or in other words $$\left\{x \in \mathbf{R}^{k} : x_0 + \dots + x_{k-1} = 1, x_i \ge 0 \text{ for } i = 0, \dots, k-1 \right\}.$$

In topology and combinatorics, it is common to "glue together" simplices to form a simplicial complex. The associated combinatorial structure is called an abstract simplicial complex, in which context the word "simplex" simply means any finite set of vertices.

History
The concept of a simplex was known to William Kingdon Clifford, who wrote about these shapes in 1886 but called them "prime confines". Henri Poincaré, writing about algebraic topology in 1900, called them "generalized tetrahedra". In 1902 Pieter Hendrik Schoute described the concept first with the Latin superlative simplicissimum ("simplest") and then with the same Latin adjective in the normal form simplex ("simple").

The regular simplex family is the first of three regular polytope families, labeled by Donald Coxeter as $α_{n}$, the other two being the cross-polytope family, labeled as $β_{n}$, and the hypercubes, labeled as $γ_{n}$. A fourth family, the tessellation of $n$-dimensional space by infinitely many hypercubes, he labeled as $δ_{n}$.

Elements
The convex hull of any nonempty subset of the $n + 1$ points that define an $n$-simplex is called a face of the simplex. Faces are simplices themselves. In particular, the convex hull of a subset of size $m + 1$ (of the $n + 1$ defining points) is an $m$-simplex, called an $m$-face of the $n$-simplex. The 0-faces (i.e., the defining points themselves as sets of size 1) are called the vertices (singular: vertex), the 1-faces are called the edges, the ($n − 1$)-faces are called the facets, and the sole $n$-face is the whole $n$-simplex itself. In general, the number of $m$-faces is equal to the binomial coefficient $$\tbinom{n+1}{m+1}$$. Consequently, the number of $m$-faces of an $n$-simplex may be found in column ($m + 1$) of row ($n + 1$) of Pascal's triangle. A simplex $A$ is a coface of a simplex $B$ if $B$ is a face of $A$. Face and facet can have different meanings when describing types of simplices in a simplicial complex.

The extended f-vector for an $n$-simplex can be computed by $(1,1)^{n+1}$, like the coefficients of polynomial products. For example, a 7-simplex is (1,1)8 = (1,2,1)4 = (1,4,6,4,1)2 = (1,8,28,56,70,56,28,8,1).

The number of 1-faces (edges) of the $n$-simplex is the $n$-th triangle number, the number of 2-faces of the $n$-simplex is the $(n − 1)$th tetrahedron number, the number of 3-faces of the $n$-simplex is the $(n − 2)$th 5-cell number, and so on.

An $n$-simplex is the polytope with the fewest vertices that requires $n$ dimensions. Consider a line segment AB as a shape in a 1-dimensional space (the 1-dimensional space is the line in which the segment lies). One can place a new point $n$ somewhere off the line. The new shape, triangle ABC, requires two dimensions; it cannot fit in the original 1-dimensional space. The triangle is the 2-simplex, a simple shape that requires two dimensions. Consider a triangle ABC, a shape in a 2-dimensional space (the plane in which the triangle resides). One can place a new point $C$ somewhere off the plane. The new shape, tetrahedron ABCD, requires three dimensions; it cannot fit in the original 2-dimensional space. The tetrahedron is the 3-simplex, a simple shape that requires three dimensions. Consider tetrahedron ABCD, a shape in a 3-dimensional space (the 3-space in which the tetrahedron lies). One can place a new point $D$ somewhere outside the 3-space. The new shape ABCDE, called a 5-cell, requires four dimensions and is called the 4-simplex; it cannot fit in the original 3-dimensional space. (It also cannot be visualized easily.) This idea can be generalized, that is, adding a single new point outside the currently occupied space, which requires going to the next higher dimension to hold the new shape. This idea can also be worked backward: the line segment we started with is a simple shape that requires a 1-dimensional space to hold it; the line segment is the 1-simplex. The line segment itself was formed by starting with a single point in 0-dimensional space (this initial point is the 0-simplex) and adding a second point, which required the increase to 1-dimensional space.

More formally, an $Δ^{n}$-simplex can be constructed as a join (∨ operator) of an $E$-simplex and a point, $(n + 1)$. An $$-simplex can be constructed as a join of an $n$-simplex and an $m$-simplex. The two simplices are oriented to be completely normal from each other, with translation in a direction orthogonal to both of them. A 1-simplex is the join of two points: $(m + n + 1)$. A general 2-simplex (scalene triangle) is the join of three points: $∨ = 2 ⋅$. An isosceles triangle is the join of a 1-simplex and a point: $∨ ∨$. An equilateral triangle is 3 ⋅ or {3}. A general 3-simplex is the join of 4 points: $\{ \} ∨$. A 3-simplex with mirror symmetry can be expressed as the join of an edge and two points: $∨ ∨ ∨$. A 3-simplex with triangular symmetry can be expressed as the join of an equilateral triangle and 1 point: $\{ \} ∨ ∨$ or $3.∨$. A regular tetrahedron is $\{3\}∨$ or $n$ and so on.

In some conventions, the empty set is defined to be a (−1)-simplex. The definition of the simplex above still makes sense if $4 ⋅$. This convention is more common in applications to algebraic topology (such as simplicial homology) than to the study of polytopes.

Symmetric graphs of regular simplices
These Petrie polygons (skew orthogonal projections) show all the vertices of the regular simplex on a circle, and all vertex pairs connected by edges.

Standard simplex
The standard $\{3,3\}$-simplex (or unit $n$-simplex) is the subset of $n = −1$ given by
 * $$\Delta^n = \left\{(t_0,\dots,t_n)\in\mathbf{R}^{n+1} ~\Bigg|~ \sum_{i = 0}^n t_i = 1 \text{ and } t_i \ge 0 \text{ for } i = 0, \ldots, n\right\}$$.

The simplex $R^{3}$ lies in the affine hyperplane obtained by removing the restriction $R^{n+1}$ in the above definition.

The $Δ^{n}$ vertices of the standard $n$-simplex are the points $t_{i} ≥ 0$, where

A standard simplex is an example of a 0/1-polytope, with all coordinates as 0 or 1. It can also be seen one facet of a regular $n + 1$-orthoplex.

There is a canonical map from the standard $n$-simplex to an arbitrary $n$-simplex with vertices ($e_{i} ∈ R^{n+1}$, ..., $e_{0} = (1, 0, 0, ..., 0),$) given by
 * $$(t_0,\ldots,t_n) \mapsto \sum_{i = 0}^n t_i v_i$$

The coefficients $e_{1} = (0, 1, 0, ..., 0),$ are called the barycentric coordinates of a point in the $n$-simplex. Such a general simplex is often called an affine $n$-simplex, to emphasize that the canonical map is an affine transformation. It is also sometimes called an oriented affine $n$-simplex to emphasize that the canonical map may be orientation preserving or reversing.

More generally, there is a canonical map from the standard $$(n-1)$$-simplex (with $n$ vertices) onto any polytope with $n$ vertices, given by the same equation (modifying indexing):
 * $$(t_1,\ldots,t_n) \mapsto \sum_{i = 1}^n t_i v_i$$

These are known as generalized barycentric coordinates, and express every polytope as the image of a simplex: $$\Delta^{n-1} \twoheadrightarrow P.$$

A commonly used function from $e_{n} = (0, 0, 0, ..., 1)$ to the interior of the standard $$(n-1)$$-simplex is the softmax function, or normalized exponential function; this generalizes the standard logistic function.

Examples

 * Δ0 is the point $(n + 1)$ in $v_{0}$.
 * Δ1 is the line segment joining $v_{n}$ and $t_{i}$ in $R^{n}$.
 * Δ2 is the equilateral triangle with vertices $1$, $R^{1}$ and $(1, 0)$ in $(0, 1)$.
 * Δ3 is the regular tetrahedron with vertices $R^{2}$, $(1, 0, 0)$, $(0, 1, 0)$ and $(0, 0, 1)$ in $R^{3}$.
 * Δ4 is the regular 5-cell with vertices $(1, 0, 0, 0)$, $(0, 1, 0, 0)$, $(0, 0, 1, 0)$, $(0, 0, 0, 1)$ and $R^{4}$ in $(1, 0, 0, 0, 0)$.

Increasing coordinates
An alternative coordinate system is given by taking the indefinite sum:

\begin{align} s_0 &= 0\\ s_1 &= s_0 + t_0 = t_0\\ s_2 &= s_1 + t_1 = t_0 + t_1\\ s_3 &= s_2 + t_2 = t_0 + t_1 + t_2\\ &\;\;\vdots\\ s_n &= s_{n-1} + t_{n-1} = t_0 + t_1 + \cdots + t_{n-1}\\ s_{n+1} &= s_n + t_n = t_0 + t_1 + \cdots + t_n = 1 \end{align} $$ This yields the alternative presentation by order, namely as nondecreasing $n$-tuples between 0 and 1:
 * $$\Delta_*^n = \left\{(s_1,\ldots,s_n)\in\mathbf{R}^n\mid 0 = s_0 \leq s_1 \leq s_2 \leq \dots \leq s_n \leq s_{n+1} = 1 \right\}. $$

Geometrically, this is an $n$-dimensional subset of $$\mathbf{R}^n$$ (maximal dimension, codimension 0) rather than of $$\mathbf{R}^{n+1}$$ (codimension 1). The facets, which on the standard simplex correspond to one coordinate vanishing, $$t_i=0,$$ here correspond to successive coordinates being equal, $$s_i=s_{i+1},$$ while the interior corresponds to the inequalities becoming strict (increasing sequences).

A key distinction between these presentations is the behavior under permuting coordinates – the standard simplex is stabilized by permuting coordinates, while permuting elements of the "ordered simplex" do not leave it invariant, as permuting an ordered sequence generally makes it unordered. Indeed, the ordered simplex is a (closed) fundamental domain for the action of the symmetric group on the $n$-cube, meaning that the orbit of the ordered simplex under the $n$! elements of the symmetric group divides the $n$-cube into $$n!$$ mostly disjoint simplices (disjoint except for boundaries), showing that this simplex has volume $(0, 1, 0, 0, 0)$. Alternatively, the volume can be computed by an iterated integral, whose successive integrands are 1, $n$, $(0, 0, 1, 0, 0)$, $(0, 0, 0, 1, 0)$, ..., $(0, 0, 0, 0, 1)$.

A further property of this presentation is that it uses the order but not addition, and thus can be defined in any dimension over any ordered set, and for example can be used to define an infinite-dimensional simplex without issues of convergence of sums.

Projection onto the standard simplex
Especially in numerical applications of probability theory a projection onto the standard simplex is of interest. Given $$ (p_i)_i$$ with possibly negative entries, the closest point $$\left(t_i\right)_i$$ on the simplex has coordinates
 * $$t_i= \max\{p_i+\Delta\, ,0\},$$

where $$\Delta$$ is chosen such that $\sum_i\max\{p_i+\Delta\, ,0\}=1.$

$$\Delta$$ can be easily calculated from sorting $R^{5}$. The sorting approach takes $$O( n \log n)$$ complexity, which can be improved to $1/n!$ complexity via median-finding algorithms. Projecting onto the simplex is computationally similar to projecting onto the $$\ell_1$$ ball.

Corner of cube
Finally, a simple variant is to replace "summing to 1" with "summing to at most 1"; this raises the dimension by 1, so to simplify notation, the indexing changes:
 * $$\Delta_c^n = \left\{(t_1,\ldots,t_n)\in\mathbf{R}^n ~\Bigg|~ \sum_{i = 1}^n t_i \leq 1 \text{ and } t_i \ge 0 \text{ for all } i \right\}.$$

This yields an $x$-simplex as a corner of the $n$-cube, and is a standard orthogonal simplex. This is the simplex used in the simplex method, which is based at the origin, and locally models a vertex on a polytope with $n$ facets.

Cartesian coordinates for a regular $n$-dimensional simplex in Rn
One way to write down a regular $n$-simplex in $x^{2}/2$ is to choose two points to be the first two vertices, choose a third point to make an equilateral triangle, choose a fourth point to make a regular tetrahedron, and so on. Each step requires satisfying equations that ensure that each newly chosen vertex, together with the previously chosen vertices, forms a regular simplex. There are several sets of equations that can be written down and used for this purpose. These include the equality of all the distances between vertices; the equality of all the distances from vertices to the center of the simplex; the fact that the angle subtended through the new vertex by any two previously chosen vertices is $$\pi/3$$; and the fact that the angle subtended through the center of the simplex by any two vertices is $$\arccos(-1/n)$$.

It is also possible to directly write down a particular regular $n$-simplex in $x^{3}/3!$ which can then be translated, rotated, and scaled as desired. One way to do this is as follows. Denote the basis vectors of $x^{n}/n!$ by $p_{i}$ through $O(n)$. Begin with the standard $R^{n}$-simplex which is the convex hull of the basis vectors. By adding an additional vertex, these become a face of a regular $n$-simplex. The additional vertex must lie on the line perpendicular to the barycenter of the standard simplex, so it has the form $R^{n}$ for some real number $n$. Since the squared distance between two basis vectors is 2, in order for the additional vertex to form a regular $α$-simplex, the squared distance between it and any of the basis vectors must also be 2. This yields a quadratic equation for $n$. Solving this equation shows that there are two choices for the additional vertex:
 * $$\frac{1}{n} \left(1 \pm \sqrt{n + 1} \right) \cdot (1, \dots, 1).$$

Either of these, together with the standard basis vectors, yields a regular $α$-simplex.

The above regular $n$-simplex is not centered on the origin. It can be translated to the origin by subtracting the mean of its vertices. By rescaling, it can be given unit side length. This results in the simplex whose vertices are:
 * $$\frac{1}{\sqrt{2}}\mathbf{e}_i - \frac{1}{n\sqrt{2}}\bigg(1 \pm \frac{1}{\sqrt{n + 1}}\bigg) \cdot (1, \dots, 1),$$

for $$1 \le i \le n$$, and
 * $$\pm\frac{1}{\sqrt{2(n + 1)}} \cdot (1, \dots, 1).$$

Note that there are two sets of vertices described here. One set uses $$+$$ in each calculation. The other set uses $$-$$ in each calculation.

This simplex is inscribed in a hypersphere of radius $$\sqrt{n/(2(n + 1))}$$.

A different rescaling produces a simplex that is inscribed in a unit hypersphere. When this is done, its vertices are
 * $$\sqrt{1 + n^{-1}}\cdot\mathbf{e}_i - n^{-3/2}(\sqrt{n + 1} \pm 1) \cdot (1, \dots, 1),$$

where $$1 \le i \le n$$, and
 * $$\pm n^{-1/2} \cdot (1, \dots, 1).$$

The side length of this simplex is $\sqrt{2(n + 1)/n}$.

A highly symmetric way to construct a regular $n$-simplex is to use a representation of the cyclic group $R^{n}$ by orthogonal matrices. This is an $e_{1}$ orthogonal matrix $n$ such that $e_{n}$ is the identity matrix, but no lower power of $Q$ is. Applying powers of this matrix to an appropriate vector $(n − 1)$ will produce the vertices of a regular $Q$-simplex. To carry this out, first observe that for any orthogonal matrix $n$, there is a choice of basis in which $Q$ is a block diagonal matrix
 * $$Q = \operatorname{diag}(Q_1, Q_2, \dots, Q_k),$$

where each $(α/n, ..., α/n)$ is orthogonal and either $Z_{n+1}$ or $n × n$. In order for $Q$ to have order $Q^{n+1} = I$, all of these matrices must have order dividing $v$. Therefore each $Q_{i}$ is either a $2 × 2$ matrix whose only entry is $1 × 1$ or, if $Q$ is odd, $n + 1$; or it is a $n + 1$ matrix of the form
 * $$\begin{pmatrix}

\cos \frac{2\pi\omega_i}{n + 1} & -\sin \frac{2\pi\omega_i}{n + 1} \\ \sin \frac{2\pi\omega_i}{n + 1} & \cos \frac{2\pi\omega_i}{n + 1} \end{pmatrix},$$ where each $Q_{i}$ is an integer between zero and $n$ inclusive. A sufficient condition for the orbit of a point to be a regular simplex is that the matrices $1 × 1$ form a basis for the non-trivial irreducible real representations of $1$, and the vector being rotated is not stabilized by any of them.

In practical terms, for $n$ even this means that every matrix $−1$ is $2 × 2$, there is an equality of sets
 * $$\{\omega_1, n + 1 - \omega_1, \dots, \omega_{n/2}, n + 1 - \omega_{n/2}\} = \{1, \dots, n\},$$

and, for every $ω_{i}$, the entries of $Q_{i}$ upon which $Z_{n+1}$ acts are not both zero. For example, when $Q_{i}$, one possible matrix is
 * $$\begin{pmatrix}

\cos(2\pi/5) & -\sin(2\pi/5) & 0 & 0 \\ \sin(2\pi/5) & \cos(2\pi/5) & 0 & 0 \\ 0 & 0 & \cos(4\pi/5) & -\sin(4\pi/5) \\ 0 & 0 & \sin(4\pi/5) & \cos(4\pi/5) \end{pmatrix}.$$ Applying this to the vector $2 × 2$ results in the simplex whose vertices are

\begin{pmatrix} 1 \\ 0 \\ 1 \\ 0 \end{pmatrix}, \begin{pmatrix} \cos(2\pi/5) \\ \sin(2\pi/5) \\ \cos(4\pi/5) \\ \sin(4\pi/5) \end{pmatrix}, \begin{pmatrix} \cos(4\pi/5) \\ \sin(4\pi/5) \\ \cos(8\pi/5) \\ \sin(8\pi/5) \end{pmatrix}, \begin{pmatrix} \cos(6\pi/5) \\ \sin(6\pi/5) \\ \cos(2\pi/5) \\ \sin(2\pi/5) \end{pmatrix}, \begin{pmatrix} \cos(8\pi/5) \\ \sin(8\pi/5) \\ \cos(6\pi/5) \\ \sin(6\pi/5) \end{pmatrix}, $$ each of which has distance √5 from the others. When $n$ is odd, the condition means that exactly one of the diagonal blocks is $Q_{i}$, equal to $v$, and acts upon a non-zero entry of $Q_{i}$; while the remaining diagonal blocks, say $n = 4$, are $(1, 0, 1, 0)$, there is an equality of sets
 * $$\left\{\omega_1, -\omega_1, \dots, \omega_{(n-1)/2}, -\omega_{n-1)/2}\right\} = \left\{1, \dots, (n-1)/2, (n+3)/2, \dots, n \right\},$$

and each diagonal block acts upon a pair of entries of $1 × 1$ which are not both zero. So, for example, when $−1$, the matrix can be
 * $$\begin{pmatrix}

0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & -1 \\ \end{pmatrix}.$$ For the vector $v$, the resulting simplex has vertices

\begin{pmatrix} 1 \\ 0 \\ 1/\surd2 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \\ -1/\surd2 \end{pmatrix}, \begin{pmatrix} -1 \\ 0 \\ 1/\surd2 \end{pmatrix}, \begin{pmatrix} 0 \\ -1 \\ -1/\surd2 \end{pmatrix}, $$ each of which has distance 2 from the others.

Volume
The volume of an $n$-simplex in $n$-dimensional space with vertices $Q_{1}, ..., Q_{(n − 1) / 2}$ is

\mathrm{Volume} = \frac{1}{n!} \left|\det \begin{pmatrix} v_1-v_0 && v_2-v_0 && \cdots && v_n-v_0 \end{pmatrix}\right| $$

where each column of the $2 × 2$ determinant is a vector that points from vertex $v$ to another vertex $n = 3$. This formula is particularly useful when $$v_0$$ is the origin.

The expression

\mathrm{Volume} = \frac{1}{n!} \det\left[ \begin{pmatrix} v_1^\text{T}-v_0^\text{T} \\ v_2^\text{T}-v_0^\text{T} \\ \vdots \\ v_n^\text{T}-v_0^\text{T} \end{pmatrix} \begin{pmatrix} v_1-v_0 & v_2-v_0 & \cdots & v_n-v_0 \end{pmatrix} \right]^{1/2} $$ employs a Gram determinant and works even when the $n$-simplex's vertices are in a Euclidean space with more than $n$ dimensions, e.g., a triangle in $$\mathbf{R}^3$$.

A more symmetric way to compute the volume of an $n$-simplex in $$\mathbf{R}^n$$ is

\mathrm{Volume} = {1\over n!} \left|\det \begin{pmatrix} v_0 & v_1 & \cdots & v_n \\ 1  & 1   & \cdots & 1 \end{pmatrix}\right|. $$

Another common way of computing the volume of the simplex is via the Cayley–Menger determinant, which works even when the n-simplex's vertices are in a Euclidean space with more than n dimensions.

Without the $(1, 0, 1/\sqrt{2})$ it is the formula for the volume of an $n$-parallelotope. This can be understood as follows: Assume that $n$ is an $P$-parallelotope constructed on a basis $$(v_0, e_1, \ldots, e_n)$$ of $$\mathbf{R}^n$$. Given a permutation $$\sigma$$ of $$\{1,2,\ldots, n\}$$, call a list of vertices $$v_0,\ v_1, \ldots, v_n$$ a $n$-path if
 * $$v_1 = v_0 + e_{\sigma(1)},\ v_2 = v_1 + e_{\sigma(2)},\ldots, v_n = v_{n-1}+e_{\sigma(n)}$$

(so there are $(v_{0}, ..., v_{n})$ $n$-paths and $$v_n$$ does not depend on the permutation). The following assertions hold:

If $n$ is the unit $P$-hypercube, then the union of the $n$-simplexes formed by the convex hull of each $n$-path is $n$, and these simplexes are congruent and pairwise non-overlapping. In particular, the volume of such a simplex is
 * $$ \frac{\operatorname{Vol}(P)}{n!} = \frac 1 {n!}.$$

If $P$ is a general parallelotope, the same assertions hold except that it is no longer true, in dimension > 2, that the simplexes need to be pairwise congruent; yet their volumes remain equal, because the $n$-parallelotope is the image of the unit $n$-hypercube by the linear isomorphism that sends the canonical basis of $$\mathbf{R}^n$$ to $$e_1,\ldots, e_n$$. As previously, this implies that the volume of a simplex coming from a $i$-path is:
 * $$ \frac{\operatorname{Vol}(P)}{n!} = \frac{\det(e_1, \ldots, e_n)}{n!}.$$

Conversely, given an $n$-simplex $$(v_0,\ v_1,\ v_2,\ldots v_n)$$ of $$\mathbf R^n$$, it can be supposed that the vectors $$e_1 = v_1-v_0,\ e_2 = v_2-v_1,\ldots e_n=v_n-v_{n-1}$$ form a basis of $$\mathbf R^n$$. Considering the parallelotope constructed from $$v_0$$ and $$e_1,\ldots, e_n$$, one sees that the previous formula is valid for every simplex.

Finally, the formula at the beginning of this section is obtained by observing that
 * $$\det(v_1-v_0, v_2-v_0,\ldots, v_n-v_0) = \det(v_1-v_0, v_2-v_1,\ldots, v_n-v_{n-1}).$$

From this formula, it follows immediately that the volume under a standard $n$-simplex (i.e. between the origin and the simplex in $n × n$) is
 * $${1 \over (n+1)!}$$

The volume of a regular $n$-simplex with unit side length is
 * $$\frac{\sqrt{n+1}}{n!\sqrt{2^n}}$$

as can be seen by multiplying the previous formula by $v0$, to get the volume under the $P$-simplex as a function of its vertex distance $n$ from the origin, differentiating with respect to $n$, at $$x=1/\sqrt{2}$$ (where the $n$-simplex side length is 1), and normalizing by the length $$dx/\sqrt{n+1}$$ of the increment, $$(dx/(n+1),\ldots, dx/(n+1))$$, along the normal vector.

Dihedral angles of the regular n-simplex
Any two $vk$-dimensional faces of a regular $n$-dimensional simplex are themselves regular $1/n!$-dimensional simplices, and they have the same dihedral angle of $n!$.

This can be seen by noting that the center of the standard simplex is $\left(\frac{1}{n+1}, \dots, \frac{1}{n+1}\right)$, and the centers of its faces are coordinate permutations of $\left(0, \frac{1}{n}, \dots, \frac{1}{n}\right)$. Then, by symmetry, the vector pointing from $\left(\frac{1}{n+1}, \dots, \frac{1}{n+1}\right)$ to $\left(0, \frac{1}{n}, \dots, \frac{1}{n}\right)$  is perpendicular to the faces. So the vectors normal to the faces are permutations of $$(-n, 1, \dots, 1)$$, from which the dihedral angles are calculated.

Simplices with an "orthogonal corner"
An "orthogonal corner" means here that there is a vertex at which all adjacent edges are pairwise orthogonal. It immediately follows that all adjacent faces are pairwise orthogonal. Such simplices are generalizations of right triangles and for them there exists an $n$-dimensional version of the Pythagorean theorem:

The sum of the squared $R^{n+1}$-dimensional volumes of the facets adjacent to the orthogonal corner equals the squared $x^{n+1}$-dimensional volume of the facet opposite of the orthogonal corner.
 * $$ \sum_{k=1}^n |A_k|^2 = |A_0|^2 $$

where $$ A_1 \ldots A_n $$ are facets being pairwise orthogonal to each other but not orthogonal to $$A_0$$, which is the facet opposite the orthogonal corner.

For a 2-simplex, the theorem is the Pythagorean theorem for triangles with a right angle and for a 3-simplex it is de Gua's theorem for a tetrahedron with an orthogonal corner.

Relation to the (n + 1)-hypercube
The Hasse diagram of the face lattice of an $n$-simplex is isomorphic to the graph of the $(n − 1)$-hypercube's edges, with the hypercube's vertices mapping to each of the $n$-simplex's elements, including the entire simplex and the null polytope as the extreme points of the lattice (mapped to two opposite vertices on the hypercube). This fact may be used to efficiently enumerate the simplex's face lattice, since more general face lattice enumeration algorithms are more computationally expensive.

The $x$-simplex is also the vertex figure of the $(n − 1)$-hypercube. It is also the facet of the $cos^{−1}(1/n)$-orthoplex.

Topology
Topologically, an $x$-simplex is equivalent to an $n$-ball. Every $n$-simplex is an $n$-dimensional manifold with corners.

Probability
In probability theory, the points of the standard $n$-simplex in $(n − 1)$-space form the space of possible probability distributions on a finite set consisting of $(n − 1)$ possible outcomes. The correspondence is as follows: For each distribution described as an ordered $(n + 1)$-tuple of probabilities whose sum is (necessarily) 1, we associate the point of the simplex whose barycentric coordinates are precisely those probabilities. That is, the $n$th vertex of the simplex is assigned to have the $n$th probability of the $(n + 1)$-tuple as its barycentric coefficient. This correspondence is an affine homeomorphism.

Aitchison geometry
Aitchinson geometry is a natural way to construct an inner product space from the standard simplex $$\Delta^{D-1}$$. It defines the following operations on simplices and real numbers:

 * Perturbation (addition)

 * $$ x \oplus y = \left[\frac{x_1 y_1}{\sum_{i=1}^D x_i y_i},\frac{x_2 y_2}{\sum_{i=1}^D x_i y_i}, \dots, \frac{x_D y_D}{\sum_{i=1}^D x_i y_i}\right] \qquad \forall x, y \in \Delta^{D-1}

$$

 * Powering (scalar multiplication)

 * $$ \alpha \odot x = \left[\frac{x_1^\alpha}{\sum_{i=1}^D x_i^\alpha},\frac{x_2^\alpha}{\sum_{i=1}^D x_i^\alpha}, \ldots,\frac{x_D^\alpha}{\sum_{i=1}^D x_i^\alpha} \right] \qquad \forall x \in \Delta^{D-1}, \; \alpha \in \mathbb{R}

$$

 * Inner product

 * $$ \langle x, y \rangle = \frac{1}{2D}

\sum_{i=1}^D \sum_{j=1}^D \log \frac{x_i}{x_j} \log \frac{y_i}{y_j} \qquad \forall x, y \in \Delta^{D-1}$$

Compounds
Since all simplices are self-dual, they can form a series of compounds;
 * Two triangles form a hexagram {6/2}.
 * Two tetrahedra form a compound of two tetrahedra or stella octangula.
 * Two 5-cells form a compound of two 5-cells in four dimensions.

Algebraic topology
In algebraic topology, simplices are used as building blocks to construct an interesting class of topological spaces called simplicial complexes. These spaces are built from simplices glued together in a combinatorial fashion. Simplicial complexes are used to define a certain kind of homology called simplicial homology.

A finite set of $n$-simplexes embedded in an open subset of $(n + 1)$ is called an affine $n$-chain. The simplexes in a chain need not be unique; they may occur with multiplicity. Rather than using standard set notation to denote an affine chain, it is instead the standard practice to use plus signs to separate each member in the set. If some of the simplexes have the opposite orientation, these are prefixed by a minus sign. If some of the simplexes occur in the set more than once, these are prefixed with an integer count. Thus, an affine chain takes the symbolic form of a sum with integer coefficients.

Note that each facet of an $n$-simplex is an affine $(n + 1)$-simplex, and thus the boundary of an $n$-simplex is an affine $n + 1$-chain. Thus, if we denote one positively oriented affine simplex as
 * $$\sigma=[v_0,v_1,v_2,\ldots,v_n]$$

with the $$v_j$$ denoting the vertices, then the boundary $$\partial\sigma$$ of $n$ is the chain
 * $$\partial\sigma = \sum_{j=0}^n (-1)^j [v_0,\ldots,v_{j-1},v_{j+1},\ldots,v_n].$$

It follows from this expression, and the linearity of the boundary operator, that the boundary of the boundary of a simplex is zero:
 * $$\partial^2\sigma = \partial \left( \sum_{j=0}^n (-1)^j [v_0,\ldots,v_{j-1},v_{j+1},\ldots,v_n] \right) = 0. $$

Likewise, the boundary of the boundary of a chain is zero: $$ \partial ^2 \rho =0 $$.

More generally, a simplex (and a chain) can be embedded into a manifold by means of smooth, differentiable map $$f:\mathbf{R}^n \to M$$. In this case, both the summation convention for denoting the set, and the boundary operation commute with the embedding. That is,
 * $$f \left(\sum\nolimits_i a_i \sigma_i \right) = \sum\nolimits_i a_i f(\sigma_i)$$

where the $$a_i$$ are the integers denoting orientation and multiplicity. For the boundary operator $$\partial$$, one has:
 * $$\partial f(\rho) = f (\partial \rho)$$

where $n$ is a chain. The boundary operation commutes with the mapping because, in the end, the chain is defined as a set and little more, and the set operation always commutes with the map operation (by definition of a map).

A continuous map $$f: \sigma \to X$$ to a topological space $k$ is frequently referred to as a singular $k$-simplex. (A map is generally called "singular" if it fails to have some desirable property such as continuity and, in this case, the term is meant to reflect to the fact that the continuous map need not be an embedding.)

Algebraic geometry
Since classical algebraic geometry allows one to talk about polynomial equations but not inequalities, the algebraic standard n-simplex is commonly defined as the subset of affine $(n + 1)$-dimensional space, where all coordinates sum up to 1 (thus leaving out the inequality part). The algebraic description of this set is $$\Delta^n := \left\{x \in \mathbb{A}^{n+1} ~\Bigg|~ \sum_{i=1}^{n+1} x_i = 1\right\},$$ which equals the scheme-theoretic description $$\Delta_n(R) = \operatorname{Spec}(R[\Delta^n])$$ with $$R[\Delta^n] := R[x_1,\ldots,x_{n+1}]\left/\left(1-\sum x_i \right)\right.$$ the ring of regular functions on the algebraic $k$-simplex (for any ring $$R$$).

By using the same definitions as for the classical $k$-simplex, the $n$-simplices for different dimensions $n$ assemble into one simplicial object, while the rings $$R[\Delta^n]$$ assemble into one cosimplicial object $$R[\Delta^\bullet]$$ (in the category of schemes resp. rings, since the face and degeneracy maps are all polynomial).

The algebraic $σ$-simplices are used in higher K-theory and in the definition of higher Chow groups.

Applications

 * In statistics, simplices are sample spaces of compositional data and are also used in plotting quantities that sum to 1, such as proportions of subpopulations, as in a ternary plot.
 * In probability theory, a simplex space is often used to represent the space of probability distributions. The Dirichlet distribution, for instance, is defined on a simplex.
 * In industrial statistics, simplices arise in problem formulation and in algorithmic solution. In the design of bread, the producer must combine yeast, flour, water, sugar, etc. In such mixtures, only the relative proportions of ingredients matters: For an optimal bread mixture, if the flour is doubled then the yeast should be doubled. Such mixture problem are often formulated with normalized constraints, so that the nonnegative components sum to one, in which case the feasible region forms a simplex. The quality of the bread mixtures can be estimated using response surface methodology, and then a local maximum can be computed using a nonlinear programming method, such as sequential quadratic programming.
 * In operations research, linear programming problems can be solved by the simplex algorithm of George Dantzig.
 * In game theory, strategies can be represented as points within a simplex. This representation simplifies the analysis of mixed strategies.
 * In geometric design and computer graphics, many methods first perform simplicial triangulations of the domain and then fit interpolating polynomials to each simplex.
 * In chemistry, the hydrides of most elements in the p-block can resemble a simplex if one is to connect each atom. Neon does not react with hydrogen and as such is a point, fluorine bonds with one hydrogen atom and forms a line segment, oxygen bonds with two hydrogen atoms in a bent fashion resembling a triangle, nitrogen reacts to form a tetrahedron, and carbon forms a structure resembling a Schlegel diagram of the 5-cell. This trend continues for the heavier analogues of each element, as well as if the hydrogen atom is replaced by a halogen atom.
 * In some approaches to quantum gravity, such as Regge calculus and causal dynamical triangulations, simplices are used as building blocks of discretizations of spacetime; that is, to build simplicial manifolds.

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title: Simplex