Talk:Set theory (music)

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Source for formula definitions: Basic Atonal Theory by John Rahn. Hyacinth

Set theory

  • From user talk:

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)

Also, you're edit looks great, and I don't want my above question (or the request above it) to appear as criticism in any way. Hyacinth 23:40, 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.

user:Gene Ward Smith

Musical set theory?

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)

I do not believe the article to be original research (cause/but I wrote it). The source was Basic Atonal Theory by John Rahn as you may have read above.
"Pitch-class set analysis" is not an acceptible title for this article as it refers only to musical analysis of pitch class sets, while theory includes more than analysis and "musical set theory" may consider sets besides pitch-class sets.
I would agree that the topic of this article is not refered to as "musical set theory", but is usually called "set theory" with the "musical" left off due to context. See titles such as:
  • Set theory for musicians;: An examination of those concepts in set theory applicable to music, and a comparison of selected writers in the musical application by Charles William Stetzer
  • Set Theory Objects: Abstractions for Computer-Aided Analysis and Composition of Serial and Atonal Music (European University Studies, No 36) by Peter Castine
For the policy on article titles, see Wikipedia:Naming conventions. Wikipedia:Naming conventions#Use common names of persons and things was already cited. One could argue that Wikipedia:Naming conventions#Be precise when necessary conflicts in this case.
Since no one has proposed an alternative title, I will though I do not support it: Atonal theory.
Hyacinth 00:45, 14 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)

What certainly isn't original research, the article or its title? What doesn't seem to have a long history or following, the subject or the title of this article? What should be described historically, the title or the subject? Hyacinth 10:01, 15 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 rollbacked (reverted) your additions since they are not set theory specific. Hyacinth 10:47, 15 September 2005 (UTC)
Musical set theory is preferable over Atonal theory in that set theory is one way of looking at atonal music, not the way, and as Ornette points out, set theory may be used to describe non-atonal music. I think that doing some reading would be enough to show anyone that the topic is not somehow quirky or obscure. Hyacinth 10:49, 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)

First, it is worth noting that this discussion was started by a question about the title not the topic.
I am uninterested in your qualifications.
Notice I didn't say "go read a book", I actually meant that one could start with the article itself. For instance, reading:
"Although musical set theory is often assumed to be the application of mathematical set theory to music, there is little coincidence between the terminology and even less between the methods of the two."
may have limited your concern about horrible mathematicians stealing a glance at music.
What kind of context? Or: What makes this topic seem "ephemeral" or original research? Hyacinth 22:06, 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)

What I was saying is before you ask me a question about the article, see if it is already answered in the article.
I would be thrilled if this, and every article in Wikipedia, had a nice paragraph explaining the origin of the topic and term, but many, if not a majority, if not most, do not. To help assuage your concern regarding the topics ephemerality I have added the missing publication dates to the further reading.
Again, regarding applications and repercussions, I think most articles lack this information. I will say that the average performer, being hostile to atonal music, is probably also hostile to atonal theories and feel they have no application. From my point of view the theory is most useful for composition and, as with any form of analysis, also useful for guiding the performance of a piece (for (a vague) example, the discovery that the last note of measure X marks the completion of the first set presented may indicate the end of the first formal section).
Hyacinth 11:22, 26 December 2005 (UTC)
Very interesting. Zargulon 17:27, 26 December 2005 (UTC)
To make a drive-by comment, since I have this watchlisted: the subject isn't that quirky or obscure. The article itself as it stands might be inaccessible to people who haven't studied it, but as for the subject, every music major beyond the second year at my alma mater would have known what was referred to by "musical set theory" and how to perform the basic operations (at least for long enough to pass a test or two). I don't know what to tell you beyond what Hyacinth has said. You don't to "go read a book" -- but at least look up the reputation of the book and the number of books published about the subject. A drive-by comment, because I don't intend to do any major work on this article myself in the immediate future. It could use more history rather than technique, but that's just a missing section: as if an article on, say, cryptography, were missing history and influence as well while describing techniques. A drive-by comment, because I don't intend to do any major work on this article myself in the immediate future; the text here already is good, it could just use some work to make it accessible to people who aren't theory nerds. Mindspillage (spill yours?) 22:49, 21 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)

I would suggest rather that you go back farther, to Howard Hanson, who pioneered a lot of the concepts. It is not a purely atonal theory, and in fact has nothing to do with atonality really. Hanson was before Forte, Rahn and the rest of them. Even farther back, the Schoenberg's introduction of serialism was an impetus. As for the name, mathematician came up with the phrase "set theory", mean something completely different by it, and most people looking for the phase will want an article on mathematics. Hence, Set theory (music) or Musical set theory. Gene Ward Smith 22:06, 17 June 2006 (UTC)
I found the above very interesting and helpful, Ornette. If it's all kosher (which I can't judge) it would really improve the article. By the way, does anyone understand what is meant in the introduction by musical phenomena.. is it some kind of technical term? In any case, phenomena aren't generally susceptible to proof, only explanation. It was probably those couple of sentences that gave me a less than admiring impression about this page. Zargulon 23:37, 14 December 2005 (UTC)
I concur that if what Ornette says is correct it should be included in a historical section. I also agree that "musical phenomena" is inappropriate and propose "musical events" instead. PizzaMargherita 21:13, 25 December 2005 (UTC)


Is this article about music? That's my best guess. Jaberwocky6669 19:04, 21 September 2005 (UTC)

I rest my case. Zargulon 19:34, 21 September 2005 (UTC)
Sometimes I feel that Wikipedia is used as a way for people to show other people how much knowledge they have. Jaberwocky6669 21:00, 21 September 2005 (UTC)
I think it's about music.. The question is, what in the world is the point of it? Pfalstad 05:45, 22 November 2005 (UTC)
Sometimes I feel people's participation in Wikipedia is primarily to illistrate their igorance. Rather than slyly insulting the contributors (none of whom get credit in the article) to this page why don't you just ask your questions? (kudos to Pfalstad) Hyacinth 09:08, 15 December 2005 (UTC)
Zargulon, what is your case? Hyacinth 09:44, 15 December 2005 (UTC)
I'm sorry, Hyacinth, I honestly can't remember.. after all it has been almost three months since I made that comment. Zargulon 14:06, 15 December 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)

Fair enough. Still not quite clear about whether or not musical phenomena is a technical term, since in general, phenomena tend not to be man-made, whereas music is man-made. Of course, one talks about phenomena in the art or music world to refer to spectacular personalities or developments, but I guess these are not what musical set theory is trying to explain. I put this point in "question" form in my comment above. Zargulon 14:06, 15 December 2005 (UTC)
Semantics. I don't know if it is a "technical term" nor do I remember if it was used by my source(s), but I assume so as I was concerned about that terminology. Apart from that I think phenomena is appropriate because you may be analysing human made music or precompositionally dealing with sounds. Hyacinth 11:14, 26 December 2005 (UTC)
It also appears to be common in discussing music, for example:
  • "In this chapter rhythm will be used as a general term, covering all instances of musical phenomena undefinable as sound; sound and rhythm being considered as the two primary elements of music." Cowell, Henry (1996). New Musical Resources. ISBN 0521499747.
  • "There remains the question that if all audible phenomena may be used in any succession or combination-in other words, if anything goes-are there any conceivable criteria of excellence, is any value judgment valid or possible at all?" Stein, Leon (1979). Structure and Style: The Study and Analysis of Musical Forms. ISBN 0874871646.
Hyacinth 11:31, 26 December 2005 (UTC)
Ok, it sounds like there is a good basis for the term musical phenomemon (although in the second example "audible phenomena" seems to me to refer to something more general; maybe I wouldn't feel that way if I was familiar with the context). I edited the intro to propose a different, in my opinion better, way of putting it. Reasons are:
MST is a method of analysis or composition, as it's said, and the composition side doesn't have an 'explanatory' purpose (whereas the analysis side obviously does) so 'explain' is redundant. I also think 'features' is better than 'phenomena', even though phenomena is technically correct, because it's a less non-standard use of 'features' than of 'phenomena', and as far as I can see using 'features' makes it easier to understand without changing the meaning. I feel the whole sentence now reads better. But you're the expert, so feel free to edit/revert. Zargulon 17:27, 26 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)

I vowed to get to the bottom of this article and actually understand it

A few questions:

  1. It would be nice to state where the expression "music set theory" was originated (i.e. in this very article) and how/why. I'm in favour of renaming the article to the proposed "Pitch-class set theory" (because it would be less made-up) and perhaps mention that the concepts of this theory can be generalised to other kind of musical object or events other than pitch-classes (with examples).
  2. Why are "assumptions of atonal theory" in this article and not in the article about atonality?
  3. Is "music set theory" a special case of atonal music theory or not? Ornette seems to disagree...
  4. The definition of set ("a collection of any musical materials or qualities") is so vague that could mean anything without explaining what "musical materials" and "musical qualities" mean. Some examples wouldn't harm either.
  5. My understanding is there are three types of (musical) sets: "sets", "ordered sets", and "unordered sets". I'm missing the difference between "sets" and "unordered sets". Is it repetitions?
  6. "Sets may be simultaneities or successions." - How about "The elements of a set may be simultanous or successive events, {or both|but not both}."
  7. The link to domain is rather generic and does not help understanding. Ditto for type.

Thanks. PizzaMargherita 19:09, 25 December 2005 (UTC)

First I would like to ask that you keep in mind that many of your questions are unresolved. I would avoid phrasing questions and suggestions here and on other talk pages as if they had been.
  1. The phrase "musical set theory" originates quite commonly when needed to distinguish between "true" set theory. Per your complaints and the more recent ambiguity on Wikipedia:Disambiguation#Topic_page_naming, I moved this article.
  2. Given that your question directly below this one ("Is "music set theory" a special case of atonal music theory or not") this question seems unanswerable. Secondly, your question seems to have two parts: 1. Is "assumption" an appropriate term here? and 2. If so, why are these not at atonal music? At this point the best answer I have is that the "assumptions" of a theory and what that theory hopes to describe are different things.
  3. Yes. I think that the body of theories around twelve tone music provide an example of an atonal theory that is not set theory (and historically precede it). Other examples would include Hisama's analysis of String Quartet (Crawford-Seeger). This is theory regarding atonal music, but it is not set theory.
  4. I clarified with "any musical materials or qualities (such as pitch classes or rhythms)". Its supposed to be vague because one could work with any aspect of music.
  5. Good question. I don't know. Perhaps the stated difference ({} vs ()) is simply a variation in notation for what is actually the same thing. Since I contributed to this article before I cited sources I can't immediatley go back to the source.
  6. I clarified with "Sets may be presented as simultaneities or successions" before I realized your clarification is better. I left of "or both" because that is implied.
  7. I assume that's Domain (mathematics) and type points to Equivalence class.
Lastly, I would like to suggest that you acquaint yourself with one or more of the references listed in the article. That would allow you to answer some of your own questions and improve the article independently of other's answers.
Hyacinth 11:10, 26 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)

1. I think the origin of the title still needs an explaination in the article, because as I understand it none of the books sourced uses the expression "Set Theory" in this new context. Also, your renaming doesn't address the fact that a lot of other articles are still linking here using the expression "musical set theory", whose origins IMO need to be explained.
2. I don't understand your answer, maybe my question wasn't clear, but I can't break it down much more than that. "Assumptions" seems appropriate, so I'm not questioning that. The question is: Given that MST is indeed a special case of atonal music theory, why are its assumptions (that I gather coincide with the ones of atonal music theory) in this article and not in the article about atonality?
4. When defining a formal theory we should not be vague.
5. You mean you no longer have access to the referenced books? If we want to present a theory we can't be sloppy like that about the very concept that underlies it. This should be clarified.
6. Ok, I'll change it then. I'll leave "or both" in, because what could be obvious for you may not be for someone else (me, in this case).
7. Domain (mathematics) refers to functions, whereas here we are talking about the set (in the proper sense) of all (musical) sets. Again we are being vague by allowing the "or other criteria" clause. I'll rephrase as I understand it, in respect of pre-existing mathematical terminology. I'll drop the link "types" because we already have a link to Equivalence class.
"I would like to suggest that you acquaint yourself with one or more of the references listed in the article. That would allow you to answer some of your own questions and improve the article independently of other's answers." - I have absolutely no intention to do that. I am merely trying to read this article. References should be read for further information, not as a precondition for reading an article.
My impression of this article so far is not great, and can be summarised as follows: "MST is a musical theory, except that nobody really calls it that. It redifines vaguely the term 'set' in complete contradiction with a concept that has been there for a few centuries. It allows you to do stuff with musical stuff. Somebody wrote a few books about concepts and methods that could be roughly put under this made-up MST umbrella. So read those books first if you want to understand this article." I hope this impression will change with your help. Thanks. PizzaMargherita 17:25, 27 December 2005 (UTC)
References may not be a "precondition for reading an article" but you are quite obviously also editing the article.
Once again, people do "call it that". I could refer you to ##Musical_set_theory.3F, but I will simply requote. "See titles such as:
  • Set theory for musicians;: An examination of those concepts in set theory applicable to music, and a comparison of selected writers in the musical application by Charles William Stetzer
  • Set Theory Objects: Abstractions for Computer-Aided Analysis and Composition of Serial and Atonal Music (European University Studies, No 36) by Peter Castine"
Hyacinth 10:46, 28 December 2005 (UTC)
Can't see the phrase "Musical set theory" anywhere in those titles.
I am editing the article either after having discussed and agreed the change with the "experts", or confident that they will revert a "bad" change shortly after without feeling the need to discuss it with us plebeans. PizzaMargherita 11:02, 28 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." 22:56, 28 December 2005 (UTC)

Thanks for that. So if I get it right, you are saying that "musical set theory" = "atonal theory" uses sets of integers between 0 and 12, and "set theory (music)" uses sets of integers between 3 and 9? So are other aspects of music and "music material" ruled out? And what kind of sets are we talking about? Would you be able to define them in both theories? Thanks. PizzaMargherita 23:13, 28 December 2005 (UTC)
No, I think there was a misunderstanding. It would go more like this: musical set theory = set theory (music), obviously. Musical set theory = atonal theory. Atonal theory ≠ 12-tone theory (although there is much overlap, and atonal theory is in a way a development of 12-tone theory). Both 12-tone theory and atonal theory operates in mod 12. Meaning that the integers range from 0 - 11... (12 = 0 (mod 12)). Atonal theory is based on sets from cardinals 3 - 9. Meaning a set with 3 elements [0 1 2] to one with 9 elements [0 1 2 4 5 6 8 9 10 ]. Allen Forte's "The Structure of Atonal Theory" is based around a table of all sets in prime form from cardinal 3 - 9. Meaning that any group of notes in cardinals 3 - 9 can be categorized into one of the sets in the table. I am going to answer your questions about sets below this post. 02:17, 29 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. 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)

This explains why you can't see how some people find this article (title and contents) quite questionable. I refer you to the concept of set as taught in elementary schools: a set cannot contain multiple copies of an element and the elements are not ordered. I suppose the so-called theorists that redefined this concept as they saw fit didn't pay much attention in class. PizzaMargherita 12:06, 27 December 2005 (UTC)
I refer you to the more common definition:
  • "a number of things of the same kind that belong or are used together" [2]
More importantly I think the better, and less ugly, way to clarify is to explain the difference between the term as used in music and as used in other areas. See: Set (disambiguation).
Hyacinth 13:49, 27 December 2005 (UTC)
The dictionary definition (which I find anyway an inappropriate foundation for a formal theory) still doesn't mention ordering. Also, it is implied that no repetitions are allowed. Unless of course you are comfortable with the following set of professions: doctor, musician, mathematician, lawyer, professor, doctor, doctor, clown. Even so, I'm afraid that would be only you and John Rahn, because anybody else would call it a list with repetitions, or a multiset, or a tuple, or something like that, but definitely not a set.
Just because somebody wakes up one day and redefines a concept that has been accepted and taught for centuries because he either didn't understand it, or he didn't have the creativity to come up with a different term, it doesn't mean that we have to call that a theory and make it a WP article. Especially if there is no agreement over the "fundamental concept" of such a "theory". PizzaMargherita 14:46, 27 December 2005 (UTC)
Stop talking about me. Mathematicians are the ones who took a word and redefined it. However, one doesn't find me complaining about it. Hyacinth 08:23, 28 December 2005 (UTC)
Stop using the word "you", I am deeply offended by that. If you must address me in this discussion (of which I don't see the need unless you wanted to attack me personally), please use "Mr. Pizza" and refer to me in the third person.
Now seriously, what mathematicians have done is to give a rigorous definition that doesn't conflict with common sense. I can't say the same of (musical) set "theorists", judging from the article. PizzaMargherita 10:19, 28 December 2005 (UTC)
Now, that edit (you don't -> one doesn't) was unfair, it broke my joke! I'll appeal to the Supreme Court. PizzaMargherita 10:55, 28 December 2005 (UTC)
I guess I feel that "sets and subsets" is preferable, without "Musical", since it's explained shortly thereafter what the relationship with the mathematical terms is. I'm not sure about invoking the dictionary, since subset is usually a technical term, even if set isn't, in my experience. Zargulon 15:16, 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)

Dear anonymous friend, I have another idea: why don't you help us get to grips with the basics first? (See unanswered questions above.) Thanks. PizzaMargherita 07:24, 28 December 2005 (UTC)
What you are trying to get to grips with is not the basics. You are getting into a very senseless debate between mathematicians and music theorists over the use of labels in atonal theory. It should be apparent that the music theory does not properly use terms, as they would be defined in a math book. Rather, they use the term as it appears in the dictionary, as a very broad word that can be re-interpreted by multiple fields. This is not an issue only in atonal theory. It is apparent in most music theory that uses a mathematical perspective, like "The Schillinger System of Music Composition" by Joseph Schillinger. Schillinger states that he is using a different notation than appropriate in a math text. This is also an assumption in atonal theory. You cannot assume that 4/2 means 4 divided by 2, you have to read the context of the theory. What you are debating is the terms as used in the reference books, and that are common to the theory. This article is accurate in its use of terms. So...I have another idea: why don't you come to grips with silly terminologies and actually try to learn the theory Pizza Man. 23:25, 28 December 2005 (UTC)

Substantial Revisions

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)

Even if your speculation Western music simply would not exist... were justified (which it isn't), the notions of transposition and inversion vastly predate the notion of musical set theory.. so I (as a semi-skeptic) am not assured by that! Do you really think that every time Bach transposed some chorale or inverted or augmented a theme in counterpoint, he was thinking of musical set theory? I certainly don't. But I think you did good work on the article. Can you put in something about the history of the development of the field? Zargulon 17:53, 8 March 2006 (UTC)
Probably this isn't what Tymoczko was thinking of, but my take on it is that when Bach was, say, transposing a theme up a fifth, he was doing something that can be neatly explained using a set-theoretic approach to pitch classes. It's fatuous to object that Bach wouldn't have given the same explanation were he to be asked what he was doing. A professional cricketer can't necessarily write down and solve the equation of motion for an object accelerating under gravity but he can still catch a ball much better than any physics professor. Doing something and analysing it require different skills and are done with different goals in mind. The fact is, a school of musical analysis exists that uses set theory to provide such analyses. These analyses may or may not be of interest to you; you might, for example, want to know what thought-processes Bach himself might have gone through while composing, in which case a set-theoretic analysis is probably not much help. I still don't get why the existance of a particular analytic method (among numerous others) would worry anyone. ornette 10:37, 27 March 2006 (UTC)
Neither do I.. I think you completely misunderstood me. The objection that you describe as fatuous is not one that anyone is making.. you are setting up a fatuous straw man. Equally I don't accept your analogy with the cricketer.. cricket would still exist without the notion of gravity, just as Western music, in my opinion (but not Tymoczkos) would still exist, perhaps exactly as it is now, without the notions of transposition or inversion. That is the point I was making. I hope it is clearer now. Zargulon 10:56, 27 March 2006 (UTC)
I really don't think there's much of an issue here. Composers like Bach, Mozart, Beethoven, etc., were unquestionably aware of the concepts "transposition" and "inversion." These theoretical terms (or their equivalents) are centuries old. In this sense, the sports analogy is misleading: while one can play baseball without knowing the laws of physics, one simply can't compose convincing classical-style music without a fair degree of training, training that will in one way or another involve the concepts of "transposition" and "inversion." Modern day musical set theory goes beyond this of course, and introduces concepts that would have been totally foreign to Bach and Mozart. But its basic notions would have been reasonably familiar. Tymoczko 16:37, 30 March 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)

John makes a good point. I think this section is confusing -- especially since it doesn't say what sums are, why they're important, or how to generalize to higher cardinalities. Are all trichords having the same sum equally important? I'm putting the excised material here. If someone wants to clarify this stuff and put it back in, all power to them. However, it strikes me that the topic is pretty marginal, and is better left to an article on Perle. Tymoczko 03:23, 26 May 2006 (UTC)
Symmetry isn't important in music? Hyacinth (talk) 13:04, 30 December 2009 (UTC)

Sums are also occasionally used in musical set theory, though theorists do not agree about their significance. George Perle provides the following example:

"C-E, D-F♯, E♭-G, are different instances of the same interval… the other kind of identity… has to do with axes of symmetry. C-E belongs to a family of symmetrically related dyads as follows:"
D D♯ E F F♯ G G♯
D C♯ C B A♯ A G♯
Axis pitches italicized, the axis is pitch class determined.

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.

Successive dyads from Lyric Suite tone row, all sum 11
C G D D♯ A♯ E♯
B E A G♯ C♯ F♯
Axis pitches italicized, the axis is dyad (interval 1) determined

Your help is needed

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):

I have a question for music fans: are you aware that electronic music is not electronic dance music? Did you ever realize that no university in the USA regards electronic music and the music for dancing as the same genre? Are you aware that this is not a music magazine? Are you aware of the meaning of the term encyclopedic? Are you aware of the difference between idiomatic expressions, slang and encyclopedic (formal) language? Are you aware that most of articles claiming to deal with "electronic music subgenres" are unsourced or grounded only on independent websites? Brian W 00:21, 15 June 2006 (UTC)


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]), <>

--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)

Is there and how [is] original research in this article? Hyacinth (talk) 13:02, 30 December 2009 (UTC) [[[User:Hyacinth|Hyacinth]] (talk) 07:03, 19 September 2010 (UTC)]


Why, where, how? Hyacinth (talk) 13:01, 30 December 2009 (UTC)

Removed. Hyacinth (talk) 07:03, 19 September 2010 (UTC)

Ordered sets

Mathematics does speak of 'ordered sets', and musical 'ordered sets', as they are, are covered. —Preceding unsigned comment added by (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)

Nice rant. Did you have any suggestions for improving this article?—Jerome Kohl (talk) 22:14, 9 February 2011 (UTC)
"I adore Babbit, Cage, Gloria Coates and a lot of other cats you have never heard of. " original hipster, this one is. (talk) 17:35, 21 September 2011 (UTC)

Requested move: "Musical scale" → "Scale (music)"

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.

NoeticaTea? 00:12, 21 June 2012 (UTC)

First image in the article

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)

On the face of it you are correct, though of course 8 is the inversion of 4, and therefore the smaller number is the interval class representing both intervals. The first question to ask here, however, is whether the diagram correctly reproduces the example in Schuijer's book, from which it is taken. If it is an accurate transcription, then the question must be directed to Schuijer; if not, then it should be corrected.—Jerome Kohl (talk) 07:53, 16 December 2012 (UTC)
Oops. You're right. Thanks. Since the numbers represent intervals between pitch classes then octaves don't count and the interval between pitch-class C and pitch-class G-sharp is 4. Probably then the example is correctly reproduced, but I'll let the creator of the image, Hyacinth, answer that. I also asked this question on the file talk page File talk:Z-relation Z17 example.png because I didn't know who was watching what. I've redirected people who'll read the question there to the answer on this talk page. Incidentally, I've learned since then that this was not an appropriate use of an image file talk page. Signed: Basemetal (write to me here) 08:46, 16 December 2012 (UTC)
See Interval class and Interval vector. Hyacinth (talk) 10:18, 16 December 2012 (UTC)