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Source for formula definitions: Basic Atonal Theory by John Rahn. Hyacinth
Would you care to elaborate more on the (dis)connection between set theory and musical set theory? For instance, why tuple, rather than sequence? Feel free to respond in the article itself, as hopefully I/others can write a better introduction to "musical set theory" and the current introduction can become a section which clearly explains the differences. Hyacinth 23:35, 14 Oct 2004 (UTC)
I hope its OK to respond to you here. I replaced "sequence" with "tuple" for two reasons--one is that a sequence is normally taken to be a function from the integers to some range of objects, and hence the primary meaning is infinite sequence. The second is that "sequence" has a specific musical meaning. An n-tuple is what you'd ordinarily call n things in order with possible repetition. From a more CS point of view, you might call it a list.
Having said all that, I'm not sure what I could add to the musical set theory page to make things better.
Is this approach to atonal music theory actually known as "musical set theory" outside of Wikipedia? The references given don't seem to use the term, and the Google links I get all seem to refer to Wikipedia. If I restrict the Google search to .edu sites, I get 11 hits, with most of them links to the Java applet. If I'm correct and this article's title is unconventional, then I propose to find a title more in line with current usage; I don't think it is a good idea for Wikipedia to invent terms or to popularize terms used only at the fringes. The article [1] uses "pitch-class set analysis"; this term is more prevalent on .edu sites, but I'm not sure that it is exactly the same as what our article talks about. AxelBoldt 18:57, 1 Nov 2004 (UTC)
I agree with [[User::AxelBoldt|Axelboldt]]. A lot of this article seems like original research, and this is borne out by the fact that there is no history of the term, or the field. That doesn't mean that what is written on the page is wrong, but in my view this article is not wikipedia-worthy in with its current title and focus. Zargulon 17:37, 13 September 2005 (UTC)
Ok, it's certainly not original research. I am still a bit concerned about its notability though. My main problem is, it doesn't seem to have had a very long history or a very large following. In any case even if it is notable, it would be nice to say something about the circumstances and period in which it originated. Zargulon 08:44, 14 September 2005 (UTC)
It's worth pointing out that the set-theoretic approach to musical analysis has been applied to tonal music, and to folk and classical musics from other parts of the world. I wouldn't like us to give the impression that this technique only applies to something called "atonal music". Personally I think the title matches the content nicely, and I suspect a bit of history would dispel the notion that this subject was somehow quirky or obscure. I'm not qualified to add that, but I've just added links to a print journal (Perspectives of New Music) and an online one (Music Theory Online) where you can see plenty of evidence of ongoing activity in this field. Ornette 11:15, 15 September 2005 (UTC)
I have professional qualifications in both Mathematics and Music, and see myself as being well-read in both areas. I have never come across musical set theory or atonal theory or what is described in this article under any other name. That fact in itself doesn't make them obscure; but 'go read a book' is for a teacher to say to a student.. in this context it is inappropriate, and might be construed as insulting. I am certainly not qualified to talk about the title.. if you guys say that's what it's called, I believe you. But there simply needs to be more background. Lots of mathematicians think they can address questions about music, and it is not too hard for them to get a couple of publications. But people will be suspicious that this is an ephemeral topic, and I strongly urge you, as someone to whom you might expect this article to strongly appeal, to give more context. Zargulon 11:38, 15 September 2005 (UTC)
OK I understand.. you are saying that reading the article is enough to persuade someone that the topic is not somehow quirky or obscure.. right? I disagree.. that is exactly what I thought after reading it. It needs to have a paragraph like "m.s.t. arose in the 19xx's when X Y and Z wanted to address the question Q. It was a natural devlopment from the mature disciplines of A and B". (That is what I mean by background or context). Otherwise it seems like m.s.t. appeared out of nowhere, and will likely be gone by tomorrow (ephemeral). Another significant problem with this article is it doesn't make explicit how practical musicians or composers are enlightened from m.s.t.: the vast majority of the article is a very solid and clear description of the mathematics of m.s.t., and there are only a couple of vague throwaway sentences to suggest that m.s.t. has any repercussions in practice. Does it have any, and what are they? Zargulon 23:37, 15 September 2005 (UTC)
It sounds like the nub of all of this is that we need an historical note to (a) explain how it fits into the history of C20th music and musicology, and (b) to demonstrate the subject's credentials (although IMHO the latter is bogus).
How about something along these lines: "m.s.t. in its modern form was formalised in the late 1970s and early 1980s in an effort to provide an analytical technique that was appropriate for atonal music, although many of its ideas, including the basic notion of pitch-class sets, date back as far as Schoenberg's early formulations of serialism in the first decades of the twentieth century. It was in part a reaction against the dominant Schenkerian style of analysis, which was deeply rooted in tonal music, just as serialism was a reaction against tonality in the discipline of composition. It aimed to complement and illuminate the techniques used by serialist composers, and also inspired composers such as Boulez, Stockhausen, Babbit, Birtwistle, Ferneyhough and Wuorinen greatly to extend their techniques. Among its pioneers were Allen Forte, John Rahn, Joseph Straus, George Perle and Joel Lester, all of whom are now authors of books on the subject. Set-theoretical techniques have since been applied to tonal music, most influentially perhaps by George Perle."
As I said, I'm not really qualified to do this & don't have references at hand (and the above might well contain some inaccuracies), but maybe it's a start. Hyacinth, perhaps you can make some improvements...
Ornette 15:47, 30 September 2005 (UTC)
Is this article about music? That's my best guess. Jaberwocky6669 19:04, 21 September 2005 (UTC)
In response I quote: "Musical set theory is an atonal or post-tonal method of musical analysis" while "Musical analysis can be defined as a process attempting to answer the question 'how does this music work?'." That's what its for. Hyacinth 10:06, 15 December 2005 (UTC)
I apologize for my initial comment. I was originally on a crusade about simplifying Wikipedia articles because most seem to be geared towards readers that may already understand the content. Jaberwocky6669 03:16, 5 May 2006 (UTC)
A few questions:
Thanks. PizzaMargherita 19:09, 25 December 2005 (UTC)
I'm answering some of the same questions from different people more than once. I suggest everyone also read this talk page. Hyacinth 11:24, 26 December 2005 (UTC)
The term "musical set theory" is appropriate for this article. The founding books, mainly Forte's text, use the label "atonal theory" or "analysis of atonal music." The word "atonal" has been an issue in the past. It was first used to refer to the music of Schoenberg and his main students, at the time, Berg and Webern (The Second Viennese School of Music). Many looked at this term as a derogatory reference to the schools music, in particular Schoenberg. The word can literally be interpreted as "without tone," but is commonly used to mean "without tonal centricity." Schoenberg preferred the term pantonal over atonal. Being that his music was dodecaphonic, or 12-tone, "pantonal" describes the music as "all tonal systems." Although the word pantonal made more logical sense than atonal it was not adapted by society. Instead, "atonal" was used in a rather broad way throughout the 20th century. When atonal theory was developed the word took on new implications. Now, atonal theory is the theory of "sets" from cardinals 3 - 9. After the release of many fundamental atonal theory books, probably around 1980, the label "set theory" began to be used as a synonym to atonal theory. This label reflects the mathematical perspective that the theory takes on. Although the theory uses a mathematical approach, it is not math and does not use terminology that reflects a mathematical definition.
The terms "atonal theory" and "set theory (music)" do refer to the same music theory of cardinals 3 -9. The other labels used to describe the Second Viennese School of Music, like pantonal, dodecaphonic, or 12-tone, refer to a 12-tone system that examines the music, mainly, in cardinal 12. The use of the term "atonal" to refer to the 12-tone system simply indicates a lack of tonal center. The use of the word atonal to refer to theoretical works of Babbitt, Forte, Perle, Rahn etc.... indicates "atonal theory." 65.9.15.143 22:56, 28 December 2005 (UTC)
John Rahn in "Basic Atonal Theory" presents a slightly different view of the forms of sets that were established by Forte 7 years earlier. Taking into account Rahn's definitions of the forms of sets the 3 types would look like this: Say we have a series of notes like G F# C B A E A F, they are numerically represented as (7 6 0 11 9 4 5). This is an unordered form of the notes (1). Next we are going to put the notes in an ascending order: (7 9 11 0 4 5 6). From here we have to find the greatest interval between adjacent notes and put the second element in that interval as the first element of the set. 7 to 9 is an interval of 2 (i2), 9 to 11 is i2, 11 to 0 is i1, 0 to 4 is i4, <4,5>=i1, <5,6>=i1, <6, 7>=i1. <0,4>=i4 is the largest interval and so 4 is placed first: {4 5 6 7 9 11 0}. Before achieving the next form of the set it is necessary to set 4=0, that is to transpose the set down by 4 so that the first element is 0: {0 1 2 3 5 7 8}. This is called normal form, but is sometimes refered to as "real prime form" (2). Next, we have to take into account inversion. The easiest way to go about this is to write the set backwards (retrograde) and see if the intervals are smaller towards the left: (8 7 5 3 2 1 0), 8=0: (0 1 3 5 6 7 8). We can see that the intervals are smaller to the left in {0 1 2 3 5 7 8} than in (0 1 3 5 6 7 8). The set that is smaller to the left is the form known as "best normal order" also called "prime form" or "Forte prime form." This prime form is the way the table of Pc sets is classified. This is a more contemporary method to arriving at the 3 forms. An alternate method, proposed by Forte is called "cyclic permutations." If you didn't quite get this I'd be happy to do another example. 65.9.15.143 02:17, 29 December 2005 (UTC)
I do not see how set is used in here in a way which is not "as everybody knows them". Hyacinth 11:14, 27 December 2005 (UTC)
Sorry to jump in, but an ordered list of elements from a base set S (say, the pitch classes 0-11), with repetitions allowed, is indeed a set. In fact it's a map from some subset of the natural numbers to S, and a map is just a set defined a certain way. Musicologists don't often think about low-level constructions like this, but that doesn't mean their theories aren't rigorous. Mathematicians don't generally do their proofs from the standard axioms of set theory either; it would be too tedious, and they get on better working at a higher level, knowing that someone else is making sure the lower-level stuff works. FWIW I think it's appropriate here to keep it fairly informal in this article, since this is an entry on music, not mathematics. Ornette 08:22, 10 January 2005 (UTC)
I think this page basically needs to be reconstructed. Although it is accurate, much more needs to be said in order to achieve even the fundamental concepts of musical set theory. You cannot ignore interval vectors, similarity relations (Rp, Ro, R1, R2), inclusion relations, complementation, derivation, union, intersection, invarience theorems, subset theorems, VTICS, multiplicative operations, rotation etc.... At least present a few different methods for reaching prime form (i.e. Rahn and Forte methods). Perhaps the author(s) of this page need to read or re-read the basic atonal theory texts, like "Basic Atonal Theory" by Rahn, "The Structure of Atonal Theory" by Forte, and "Serial Composition and Atonality: An Introduction to the Music of Schoenberg, Berg, and Webern." If you wish to even scratch the surface of atonal theory you need to re-write most of the sections that already exist and include many additions, like those mentioned above. Since set theory has so many dimensions I suggest you summarize some sections and expand upon them in separate articles. For example, if you were to explore the set-complex it would take up too much space, so there should only be an introduction and a separate article. 04:26, 28 December 2005 (UTC)
I have made substantial revisions to the page. I can assure any skeptics that the concepts discussed here are central to much thinking about music, both tonal and atonal. Western music simply would not exist without the notions of transposition and inversion.
There were a fair amount of errors and mistakes in the article, which I have done my best to correct. The article now provides a basic introduction to the central ideas of musical set theory. However, as a previous commentator has indicated, there are plenty of concepts that are not discussed. The article could certainly be expanded, though my own view is that it should remain accessible to non-specialists, and that there is a danger of overloading people with too much information. Tymoczko 15:22, 22 February 2006 (UTC)
Stated like that, I agree. Zargulon 22:05, 30 March 2006 (UTC)
The section on "sums" does not say what a "sum" is! This is very frustrating. Please, somedebody, add an explanation of this concept before the article plunges into an example.
I don't think people should criticize this article for introducing a lot of technical concepts that are unfamiliar to most musicians. I think they should fix this article when these concepts are introduced without a clear explanation.
Judging from the discussion above, the article is now much better than it used to be. I hope the experts continue to improve it! I would like to learn more about this subject.
John Baez
14:27, 24 May 2006 (UTC)
Sums are also occasionally used in musical set theory, though theorists do not agree about their significance. George Perle provides the following example:
D | D♯ | E | F | F♯ | G | G♯ | ||||||
D | C♯ | C | B | A♯ | A | G♯ |
Thus in addition to being part of the interval-4 family, C-E is also a part of the sum-2 family (with G♯ equal to 0).
The tone row to Alban Berg's Lyric Suite, , is a series of six dyads, all sum 11. If the row is rotated and retrograded, so it runs , the dyads are all sum 6.
C | G | D | D♯ | A♯ | E♯ | |||||
B | E | A | G♯ | C♯ | F♯ |
hello, I'm posting the message below at various pages. I believe that now Wikipedia needs the help of real music experts. Recently some articles are being edited according to an agenda, a sort of plan that is willing to delete almost 30 years of history of popular music and mislead the connections and differences between genres. This is the message (I apologize in advance if my language may appear as German-Spanish inluenced):
For historical information see the Grove Dictionary of Music and Musicians including the articles Set and Analysis. For instance:
"Aspects of set theory entered the theory of musical composition with J.M. Hauer’s theory of tropes (1925), and are evident in the writings of René Leibowitz, Josef Rufer, George Perle, George Rochberg (1955, 1959) and Pierre Boulez (1964, chap.2; 1966, part ii). The proper formulation of a set theory of music was the work of Milton Babbitt (1955, 1960, 1961, 1972), Donald Martino, David Lewin and John Rothgeb (JMT, iii–v, x, xi). But while Babbitt’s work, using particularly the mathematical concept of the group, dealt with harmony and with the functions of melodic and rhythmic configurations in 12-note music, and also with the interaction of components over longer spans of time, it belonged to the realm of compositional theory rather than analysis."
From the History section of IAN D. BENT/ANTHONY POPLE: 'Analysis', Grove Music Online ed. L. Macy (Accessed [30 June 2006]), < http://www.grovemusic.com>
-- Amazzing5 20:23, 30 June 2006 (UTC)
Routine WP:NOR notification. Note: Citing sources and avoiding original research are inextricably linked: to demonstrate that you are not presenting original research, you must cite reliable sources that provide information directly related to the topic of the article, and that directly support the information as it is presented. Please do not assume readers have prior knowledge of subject matter. Semitransgenic ( talk) 13:07, 26 August 2008 (UTC)
Why, where, how? Hyacinth ( talk) 13:01, 30 December 2009 (UTC)
Mathematics does speak of 'ordered sets', and musical 'ordered sets', as they are, are covered. —Preceding unsigned comment added by 41.185.115.52 ( talk) 11:34, 17 January 2011 (UTC)
As a professional musician with no credentials other than playing, writing, arranging, being leader and member of ensembles too numerous and eclectic to remember entirely and having an utter devotion to the notion of music being art of the highest order and having the most profound ability to affect its audience of any art form- especially in its immediacy- I would love to see academia be less selfish and grandiose in its scope. Granted, science has long since left the layman in the intellectual dust and we are better off for it inasmuch to a large degree. However, the beauty of music is not in its analysis, nor are we more enlightened in our experience because of it. If I have to analyze every piece of music to "get it" then I will probably never enjoy it like some other inexplicably simple musics that make the pain and joy of life more appreciable. And, before anyone says it- I adore Babbit, Cage, Gloria Coates and a lot of other cats you have never heard of. But when I've finished this rant, I will pick up my horn and play whatever comes to mind, for no other reason than that I'm urged to. That's all I'll write now, I've got other things to do. Sammyflow ( talk) 21:43, 9 February 2011 (UTC)
I have initiated a formal RM action to move Musical scale to Scale (music). Contributions and comments would be very welcome; decisions of this kind could affect the choice of title for many music theory articles.
Noetica Tea? 00:12, 21 June 2012 (UTC)
Unless I misunderstood something, the top figure in both examples in the first image file should be 8 and not 4. Signed: Basemetal (write to me here) 06:43, 16 December 2012 (UTC)
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Currently, the article begins: "Musical set theory provides concepts for categorizing musical objects and describing their relationships. Many of the notions were first elaborated by Howard Hanson (1960) in connection with tonal music," I looked at the Hanson book. It says nothing about set theory that I could find. I did a search on an online version of the book for "set " and got exactly one hit that was not about set theory. If the idea is that the categories and their relationships that Hanson defines are similar to concepts in set theory then I agree but whether I agree or not doesn't matter, IMO the reference clearly doesn't support that opening sentence, and inferring something from another reference rather than using that reference to directly support your point is wp:OR. If no one responds with an explanation then I'm either going to either edit the intro or put an wp:OR tag on this article. -- MadScientistX11 ( talk) 23:33, 25 January 2018 (UTC)
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Source for formula definitions: Basic Atonal Theory by John Rahn. Hyacinth
Would you care to elaborate more on the (dis)connection between set theory and musical set theory? For instance, why tuple, rather than sequence? Feel free to respond in the article itself, as hopefully I/others can write a better introduction to "musical set theory" and the current introduction can become a section which clearly explains the differences. Hyacinth 23:35, 14 Oct 2004 (UTC)
I hope its OK to respond to you here. I replaced "sequence" with "tuple" for two reasons--one is that a sequence is normally taken to be a function from the integers to some range of objects, and hence the primary meaning is infinite sequence. The second is that "sequence" has a specific musical meaning. An n-tuple is what you'd ordinarily call n things in order with possible repetition. From a more CS point of view, you might call it a list.
Having said all that, I'm not sure what I could add to the musical set theory page to make things better.
Is this approach to atonal music theory actually known as "musical set theory" outside of Wikipedia? The references given don't seem to use the term, and the Google links I get all seem to refer to Wikipedia. If I restrict the Google search to .edu sites, I get 11 hits, with most of them links to the Java applet. If I'm correct and this article's title is unconventional, then I propose to find a title more in line with current usage; I don't think it is a good idea for Wikipedia to invent terms or to popularize terms used only at the fringes. The article [1] uses "pitch-class set analysis"; this term is more prevalent on .edu sites, but I'm not sure that it is exactly the same as what our article talks about. AxelBoldt 18:57, 1 Nov 2004 (UTC)
I agree with [[User::AxelBoldt|Axelboldt]]. A lot of this article seems like original research, and this is borne out by the fact that there is no history of the term, or the field. That doesn't mean that what is written on the page is wrong, but in my view this article is not wikipedia-worthy in with its current title and focus. Zargulon 17:37, 13 September 2005 (UTC)
Ok, it's certainly not original research. I am still a bit concerned about its notability though. My main problem is, it doesn't seem to have had a very long history or a very large following. In any case even if it is notable, it would be nice to say something about the circumstances and period in which it originated. Zargulon 08:44, 14 September 2005 (UTC)
It's worth pointing out that the set-theoretic approach to musical analysis has been applied to tonal music, and to folk and classical musics from other parts of the world. I wouldn't like us to give the impression that this technique only applies to something called "atonal music". Personally I think the title matches the content nicely, and I suspect a bit of history would dispel the notion that this subject was somehow quirky or obscure. I'm not qualified to add that, but I've just added links to a print journal (Perspectives of New Music) and an online one (Music Theory Online) where you can see plenty of evidence of ongoing activity in this field. Ornette 11:15, 15 September 2005 (UTC)
I have professional qualifications in both Mathematics and Music, and see myself as being well-read in both areas. I have never come across musical set theory or atonal theory or what is described in this article under any other name. That fact in itself doesn't make them obscure; but 'go read a book' is for a teacher to say to a student.. in this context it is inappropriate, and might be construed as insulting. I am certainly not qualified to talk about the title.. if you guys say that's what it's called, I believe you. But there simply needs to be more background. Lots of mathematicians think they can address questions about music, and it is not too hard for them to get a couple of publications. But people will be suspicious that this is an ephemeral topic, and I strongly urge you, as someone to whom you might expect this article to strongly appeal, to give more context. Zargulon 11:38, 15 September 2005 (UTC)
OK I understand.. you are saying that reading the article is enough to persuade someone that the topic is not somehow quirky or obscure.. right? I disagree.. that is exactly what I thought after reading it. It needs to have a paragraph like "m.s.t. arose in the 19xx's when X Y and Z wanted to address the question Q. It was a natural devlopment from the mature disciplines of A and B". (That is what I mean by background or context). Otherwise it seems like m.s.t. appeared out of nowhere, and will likely be gone by tomorrow (ephemeral). Another significant problem with this article is it doesn't make explicit how practical musicians or composers are enlightened from m.s.t.: the vast majority of the article is a very solid and clear description of the mathematics of m.s.t., and there are only a couple of vague throwaway sentences to suggest that m.s.t. has any repercussions in practice. Does it have any, and what are they? Zargulon 23:37, 15 September 2005 (UTC)
It sounds like the nub of all of this is that we need an historical note to (a) explain how it fits into the history of C20th music and musicology, and (b) to demonstrate the subject's credentials (although IMHO the latter is bogus).
How about something along these lines: "m.s.t. in its modern form was formalised in the late 1970s and early 1980s in an effort to provide an analytical technique that was appropriate for atonal music, although many of its ideas, including the basic notion of pitch-class sets, date back as far as Schoenberg's early formulations of serialism in the first decades of the twentieth century. It was in part a reaction against the dominant Schenkerian style of analysis, which was deeply rooted in tonal music, just as serialism was a reaction against tonality in the discipline of composition. It aimed to complement and illuminate the techniques used by serialist composers, and also inspired composers such as Boulez, Stockhausen, Babbit, Birtwistle, Ferneyhough and Wuorinen greatly to extend their techniques. Among its pioneers were Allen Forte, John Rahn, Joseph Straus, George Perle and Joel Lester, all of whom are now authors of books on the subject. Set-theoretical techniques have since been applied to tonal music, most influentially perhaps by George Perle."
As I said, I'm not really qualified to do this & don't have references at hand (and the above might well contain some inaccuracies), but maybe it's a start. Hyacinth, perhaps you can make some improvements...
Ornette 15:47, 30 September 2005 (UTC)
Is this article about music? That's my best guess. Jaberwocky6669 19:04, 21 September 2005 (UTC)
In response I quote: "Musical set theory is an atonal or post-tonal method of musical analysis" while "Musical analysis can be defined as a process attempting to answer the question 'how does this music work?'." That's what its for. Hyacinth 10:06, 15 December 2005 (UTC)
I apologize for my initial comment. I was originally on a crusade about simplifying Wikipedia articles because most seem to be geared towards readers that may already understand the content. Jaberwocky6669 03:16, 5 May 2006 (UTC)
A few questions:
Thanks. PizzaMargherita 19:09, 25 December 2005 (UTC)
I'm answering some of the same questions from different people more than once. I suggest everyone also read this talk page. Hyacinth 11:24, 26 December 2005 (UTC)
The term "musical set theory" is appropriate for this article. The founding books, mainly Forte's text, use the label "atonal theory" or "analysis of atonal music." The word "atonal" has been an issue in the past. It was first used to refer to the music of Schoenberg and his main students, at the time, Berg and Webern (The Second Viennese School of Music). Many looked at this term as a derogatory reference to the schools music, in particular Schoenberg. The word can literally be interpreted as "without tone," but is commonly used to mean "without tonal centricity." Schoenberg preferred the term pantonal over atonal. Being that his music was dodecaphonic, or 12-tone, "pantonal" describes the music as "all tonal systems." Although the word pantonal made more logical sense than atonal it was not adapted by society. Instead, "atonal" was used in a rather broad way throughout the 20th century. When atonal theory was developed the word took on new implications. Now, atonal theory is the theory of "sets" from cardinals 3 - 9. After the release of many fundamental atonal theory books, probably around 1980, the label "set theory" began to be used as a synonym to atonal theory. This label reflects the mathematical perspective that the theory takes on. Although the theory uses a mathematical approach, it is not math and does not use terminology that reflects a mathematical definition.
The terms "atonal theory" and "set theory (music)" do refer to the same music theory of cardinals 3 -9. The other labels used to describe the Second Viennese School of Music, like pantonal, dodecaphonic, or 12-tone, refer to a 12-tone system that examines the music, mainly, in cardinal 12. The use of the term "atonal" to refer to the 12-tone system simply indicates a lack of tonal center. The use of the word atonal to refer to theoretical works of Babbitt, Forte, Perle, Rahn etc.... indicates "atonal theory." 65.9.15.143 22:56, 28 December 2005 (UTC)
John Rahn in "Basic Atonal Theory" presents a slightly different view of the forms of sets that were established by Forte 7 years earlier. Taking into account Rahn's definitions of the forms of sets the 3 types would look like this: Say we have a series of notes like G F# C B A E A F, they are numerically represented as (7 6 0 11 9 4 5). This is an unordered form of the notes (1). Next we are going to put the notes in an ascending order: (7 9 11 0 4 5 6). From here we have to find the greatest interval between adjacent notes and put the second element in that interval as the first element of the set. 7 to 9 is an interval of 2 (i2), 9 to 11 is i2, 11 to 0 is i1, 0 to 4 is i4, <4,5>=i1, <5,6>=i1, <6, 7>=i1. <0,4>=i4 is the largest interval and so 4 is placed first: {4 5 6 7 9 11 0}. Before achieving the next form of the set it is necessary to set 4=0, that is to transpose the set down by 4 so that the first element is 0: {0 1 2 3 5 7 8}. This is called normal form, but is sometimes refered to as "real prime form" (2). Next, we have to take into account inversion. The easiest way to go about this is to write the set backwards (retrograde) and see if the intervals are smaller towards the left: (8 7 5 3 2 1 0), 8=0: (0 1 3 5 6 7 8). We can see that the intervals are smaller to the left in {0 1 2 3 5 7 8} than in (0 1 3 5 6 7 8). The set that is smaller to the left is the form known as "best normal order" also called "prime form" or "Forte prime form." This prime form is the way the table of Pc sets is classified. This is a more contemporary method to arriving at the 3 forms. An alternate method, proposed by Forte is called "cyclic permutations." If you didn't quite get this I'd be happy to do another example. 65.9.15.143 02:17, 29 December 2005 (UTC)
I do not see how set is used in here in a way which is not "as everybody knows them". Hyacinth 11:14, 27 December 2005 (UTC)
Sorry to jump in, but an ordered list of elements from a base set S (say, the pitch classes 0-11), with repetitions allowed, is indeed a set. In fact it's a map from some subset of the natural numbers to S, and a map is just a set defined a certain way. Musicologists don't often think about low-level constructions like this, but that doesn't mean their theories aren't rigorous. Mathematicians don't generally do their proofs from the standard axioms of set theory either; it would be too tedious, and they get on better working at a higher level, knowing that someone else is making sure the lower-level stuff works. FWIW I think it's appropriate here to keep it fairly informal in this article, since this is an entry on music, not mathematics. Ornette 08:22, 10 January 2005 (UTC)
I think this page basically needs to be reconstructed. Although it is accurate, much more needs to be said in order to achieve even the fundamental concepts of musical set theory. You cannot ignore interval vectors, similarity relations (Rp, Ro, R1, R2), inclusion relations, complementation, derivation, union, intersection, invarience theorems, subset theorems, VTICS, multiplicative operations, rotation etc.... At least present a few different methods for reaching prime form (i.e. Rahn and Forte methods). Perhaps the author(s) of this page need to read or re-read the basic atonal theory texts, like "Basic Atonal Theory" by Rahn, "The Structure of Atonal Theory" by Forte, and "Serial Composition and Atonality: An Introduction to the Music of Schoenberg, Berg, and Webern." If you wish to even scratch the surface of atonal theory you need to re-write most of the sections that already exist and include many additions, like those mentioned above. Since set theory has so many dimensions I suggest you summarize some sections and expand upon them in separate articles. For example, if you were to explore the set-complex it would take up too much space, so there should only be an introduction and a separate article. 04:26, 28 December 2005 (UTC)
I have made substantial revisions to the page. I can assure any skeptics that the concepts discussed here are central to much thinking about music, both tonal and atonal. Western music simply would not exist without the notions of transposition and inversion.
There were a fair amount of errors and mistakes in the article, which I have done my best to correct. The article now provides a basic introduction to the central ideas of musical set theory. However, as a previous commentator has indicated, there are plenty of concepts that are not discussed. The article could certainly be expanded, though my own view is that it should remain accessible to non-specialists, and that there is a danger of overloading people with too much information. Tymoczko 15:22, 22 February 2006 (UTC)
Stated like that, I agree. Zargulon 22:05, 30 March 2006 (UTC)
The section on "sums" does not say what a "sum" is! This is very frustrating. Please, somedebody, add an explanation of this concept before the article plunges into an example.
I don't think people should criticize this article for introducing a lot of technical concepts that are unfamiliar to most musicians. I think they should fix this article when these concepts are introduced without a clear explanation.
Judging from the discussion above, the article is now much better than it used to be. I hope the experts continue to improve it! I would like to learn more about this subject.
John Baez
14:27, 24 May 2006 (UTC)
Sums are also occasionally used in musical set theory, though theorists do not agree about their significance. George Perle provides the following example:
D | D♯ | E | F | F♯ | G | G♯ | ||||||
D | C♯ | C | B | A♯ | A | G♯ |
Thus in addition to being part of the interval-4 family, C-E is also a part of the sum-2 family (with G♯ equal to 0).
The tone row to Alban Berg's Lyric Suite, , is a series of six dyads, all sum 11. If the row is rotated and retrograded, so it runs , the dyads are all sum 6.
C | G | D | D♯ | A♯ | E♯ | |||||
B | E | A | G♯ | C♯ | F♯ |
hello, I'm posting the message below at various pages. I believe that now Wikipedia needs the help of real music experts. Recently some articles are being edited according to an agenda, a sort of plan that is willing to delete almost 30 years of history of popular music and mislead the connections and differences between genres. This is the message (I apologize in advance if my language may appear as German-Spanish inluenced):
For historical information see the Grove Dictionary of Music and Musicians including the articles Set and Analysis. For instance:
"Aspects of set theory entered the theory of musical composition with J.M. Hauer’s theory of tropes (1925), and are evident in the writings of René Leibowitz, Josef Rufer, George Perle, George Rochberg (1955, 1959) and Pierre Boulez (1964, chap.2; 1966, part ii). The proper formulation of a set theory of music was the work of Milton Babbitt (1955, 1960, 1961, 1972), Donald Martino, David Lewin and John Rothgeb (JMT, iii–v, x, xi). But while Babbitt’s work, using particularly the mathematical concept of the group, dealt with harmony and with the functions of melodic and rhythmic configurations in 12-note music, and also with the interaction of components over longer spans of time, it belonged to the realm of compositional theory rather than analysis."
From the History section of IAN D. BENT/ANTHONY POPLE: 'Analysis', Grove Music Online ed. L. Macy (Accessed [30 June 2006]), < http://www.grovemusic.com>
-- Amazzing5 20:23, 30 June 2006 (UTC)
Routine WP:NOR notification. Note: Citing sources and avoiding original research are inextricably linked: to demonstrate that you are not presenting original research, you must cite reliable sources that provide information directly related to the topic of the article, and that directly support the information as it is presented. Please do not assume readers have prior knowledge of subject matter. Semitransgenic ( talk) 13:07, 26 August 2008 (UTC)
Why, where, how? Hyacinth ( talk) 13:01, 30 December 2009 (UTC)
Mathematics does speak of 'ordered sets', and musical 'ordered sets', as they are, are covered. —Preceding unsigned comment added by 41.185.115.52 ( talk) 11:34, 17 January 2011 (UTC)
As a professional musician with no credentials other than playing, writing, arranging, being leader and member of ensembles too numerous and eclectic to remember entirely and having an utter devotion to the notion of music being art of the highest order and having the most profound ability to affect its audience of any art form- especially in its immediacy- I would love to see academia be less selfish and grandiose in its scope. Granted, science has long since left the layman in the intellectual dust and we are better off for it inasmuch to a large degree. However, the beauty of music is not in its analysis, nor are we more enlightened in our experience because of it. If I have to analyze every piece of music to "get it" then I will probably never enjoy it like some other inexplicably simple musics that make the pain and joy of life more appreciable. And, before anyone says it- I adore Babbit, Cage, Gloria Coates and a lot of other cats you have never heard of. But when I've finished this rant, I will pick up my horn and play whatever comes to mind, for no other reason than that I'm urged to. That's all I'll write now, I've got other things to do. Sammyflow ( talk) 21:43, 9 February 2011 (UTC)
I have initiated a formal RM action to move Musical scale to Scale (music). Contributions and comments would be very welcome; decisions of this kind could affect the choice of title for many music theory articles.
Noetica Tea? 00:12, 21 June 2012 (UTC)
Unless I misunderstood something, the top figure in both examples in the first image file should be 8 and not 4. Signed: Basemetal (write to me here) 06:43, 16 December 2012 (UTC)
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Currently, the article begins: "Musical set theory provides concepts for categorizing musical objects and describing their relationships. Many of the notions were first elaborated by Howard Hanson (1960) in connection with tonal music," I looked at the Hanson book. It says nothing about set theory that I could find. I did a search on an online version of the book for "set " and got exactly one hit that was not about set theory. If the idea is that the categories and their relationships that Hanson defines are similar to concepts in set theory then I agree but whether I agree or not doesn't matter, IMO the reference clearly doesn't support that opening sentence, and inferring something from another reference rather than using that reference to directly support your point is wp:OR. If no one responds with an explanation then I'm either going to either edit the intro or put an wp:OR tag on this article. -- MadScientistX11 ( talk) 23:33, 25 January 2018 (UTC)