Talk:List of set classes

Vectors
 Prime Forms and Vectors of Pitch-Class Sets  The following is a table of all pitch-classes sets as cataloged by Allen Forte. Complementary sets are aligned in the same row. The above was at Set (music). Hyacinth (talk) 03:56, 17 January 2014 (UTC)

Information
Presumably it should be decided if we want to show information such as inversions (a minor chord would be 3-11A while a major chord would be 3-11B), if Z-related hexachords should be listed next to each other, etc. Hyacinth (talk) 05:16, 18 January 2014 (UTC)

Shouldn't there be a 3-9a (0,2,7) and a 3-9b (0,5,7)? That one's not symmetrical. (4/13/2016) — Preceding unsigned comment added by 71.255.46.134 (talk) 02:28, 14 April 2016 (UTC)

Ditto 4-6a (0,1,2,7) and 4-6b (0,5,6,7)? — Preceding unsigned comment added by 71.255.46.134 (talk) 14:50, 14 April 2016 (UTC)


 * Keep in mind sets are presented in normal form (most compact). (0,2,7) is not symmetric but 0,7,14 and 5,0,7 are.
 * If one inverts 0,2,7 to 0,t,5 and then transposes up 2 semitones, one reaches 2,0,7 ((0,2,7)). Hyacinth (talk) 02:47, 19 February 2017 (UTC)

Do not use T and E for 10 and 11. It is only conventional because music theory is white. Instead, use the the duodecimal digits A and B because it is more universal and less Anglo. (Sorry, I do not know the proper way to edit a wiki page.)

Audio examples incorrect.
I think I have found an inconsistency in the audio examples for a couple of the pitch-class sets in this list. Certainly I have not played all of them, but if I find one inconsistency, I am a bit wary about trusting any of the examples - I have at other times found errors in Wikipedia's coverage of music theory, and don't think this is one of Wikipedia's most solidly reliable topics.

I don't understand the pitch-class set notation, so cannot really say exactly what is incorrect. But the essence of what I have just found, and wish to note here, is that two of the audio examples for two purportedly different sets sound exactly the same. If you do a DOS-command-line comparison (with "FC" and the switch "/b"), only three bytes in the two files differ, although how significant that is, I can't say.

The two pitch-class sets in question are these:

Forte no. 3-3A; Prime form [0,1,4]; audio file-name 3-3A_trichord on C.mid

Forte no. 3-4A; Prime form [0,1,5]; audio file-name 3-4A_trichord on C.mid

Yet both of these sound the same, consisting of the notes C, Db, E - which cannot be correct, since these are two different sets.

I'm at a disadvantage in not really understanding the notation for these sets, so don't know which one is wrong - or maybe both are. And, even if I knew exactly, I don't have the capacity to produce corrected audio files for them.

I think it would be a good idea if some person knowledgeable in this area could go over all the audio files and check whether they are correct; it would not surprise me if some others were also incorrect. M.J.E. (talk) 14:17, 16 June 2017 (UTC)
 * Yes, 3-4A should be (C, D♭, F) and is currently the same as 3-3A. Looks like Hyacinth is taking a break or otherwise not currently active (hope they're ok?) so I'll have a go at wrangling a MIDI file (it's been a while!) and upload a new file for it. — Jon (talk) 21:43, 23 June 2024 (UTC)
 * FYI, unfortunately Hyacinth died late last year. They made a great contribution and will be missed 😢 Jon (talk) 02:00, 11 July 2024 (UTC)

Set 5-31A
The table lists 5-31A (Prime form [0,1,3,6,9]) as being a dominant minor 9th chord. Unless I'm mixing up my chord names, though, that seems impossible: There's only one half-step, between 0 and 1, and 0 is clearly not the root of a dominant 7th chord here, since we have no 4, 7, or 10. I'm assuming this is supposed to be a diminished 7 b9, but I figured I'd asked here first in case I was misreading something. 12tonevideos (talk) 09:54, 8 January 2018 (UTC)

Set [0134679]
The set [0134679] seems to be missing. The online PC set calculator refers to it as 7-31, which is marked to be [0134689] here. Here's a linked to the calculator:

https://www.mta.ca/pc-set/calculator/pc_calculate.html#

176.72.39.220 (talk) 23:19, 4 December 2018 (UTC)

[0,1,3,4,6,8,9] is now missing
Using the same calculator as the previous correction, 7-32A should be [0,1,3,4,6,8,9], but is listed as [0,1,3,4,5,7,9], which is a duplicate of 7-26A.

PLB527097 (talk) 16:09, 6 November 2019 (UTC)

A Recent and Unclear Edit by an Anonymous User
"This wiki entry needs to be fixed. It is not a list of pitch-class sets, but rather it is a list of set classes." has been inserted as line 2 of this article, in an edit last September, 2021. Disregarding the manner and formatting of this complaint, I wanted to at least mention that it was made. Pretty sure it's just someone not agreeing with established terminology, but it'd be hard to contact an anonymous user who's only made one edit: https://en.wikipedia.org/w/index.php?target=63.153.162.37&namespace=all&tagfilter=&start=2000-01-01&end=2022-01-30&limit=50&title=Special%3AContributions Has there ever been any reference to pitch-class sets as "set classes" as they've stated? In any literature?

PLB527097 (talk) 05:07, 31 January 2022 (UTC)

Requested move 6 September 2022

 * The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review after discussing it on the closer's talk page. No further edits should be made to this discussion. 

List of pitch-class sets → List of set classes – The anonymous editor from last year was correct, even if putting their comment in the body of the article was an odd way to voice their opinion. A pitch-class set is a group of specific pitch classes which does not necessarily include pitch-class zero. By contrast, a set class (which is shorthand for "pitch-class set class") is the most compact way of enumerating the intervallic relationships in a pitch-class set, regardless of transposition or inversion.

For example, [1,3,5] and [2,4,6] are DIFFERENT pitch-class sets. They contain different pitch-classes. But both correspond to set class [0,2,4]. This article is a list of those set classes. A list of all possible pitch-class sets would be substantially longer, including as it would every possible transposition and inversion for each set class.

This is the standard usage of the terms. One of the standard textbooks on the topic, Straus's "Introduction to Post-Tonal Theory" goes into a lot more detail. For an online reference, you can consult Moseley and Lavengood's chapter in Open Music Theory.

For the time being, I'm proposing moving this page to "List of set classes", and adding a redirect from "List of pitch-class sets" to that page. If someone were feeling ambitious, an ACTUAL list of pitch-class sets could be created, but I'm sure not volunteering. :)

PianoDan (talk) 02:47, 6 September 2022 (UTC)


 * Wouldn't the proposed name hide the list's connection to pitch/music and suggest a connection to Class (set theory) instead? IOW, the current title might help the article's discovery for interested readers, and the suggested title might mislead. -- Michael Bednarek (talk) 03:19, 6 September 2022 (UTC)


 * The existing title is essentially incorrect, which I find a much larger issue than possible confusion on math vs music. Since the old title will still redirect here, I don't see this as a huge issue. Though potentially something like "List of set classes (music)" could be better? It might be rather wordy, but I would still prefer that to the existing one.  Aza24  (talk)   04:04, 6 September 2022 (UTC)
 * I have a mathematician friend who specializes in set theory - I'll see what he thinks. PianoDan (talk) 04:27, 6 September 2022 (UTC)
 * OK, my mathematician friend says that while "set" and "class" are both important and common terms, "set class" is not, so there is unlikely to be a mathematical use for a list called "List of Set Classes". As such, I don't think the (Music) qualifier is necessary in the title. PianoDan (talk) 04:36, 7 September 2022 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

Set Class Complementation
Per user @Hyacinth waaaaay back when in an edit at 04:05, 17 January 2014: "Sets are listed next to their complements"

Some Notation
Unfortunately, much of the following is somewhat confusing without some notational aid. As such, I will attempt to borrow as much as possible from only Joseph Straus' Introduction to Post-Tonal Theory for consistency, citing where possible. For set complementation, Straus uses the prime symbol (e.g. the complement of the set $$A$$ is $$A^\prime$$) at least once. Here I'll only be using it specifically to get the literal complement of a set. If we again represent a set as $$A$$, we can let $$N_{orm}(A)$$ represent the normal form of that set. If $$A = [7463]$$ then $$N_{orm}(A)=[3467]$$. Let $$A[0]$$ represent accessing the $$0$$th item in the zero-indexed array, then using the standard $$T_{n}$$ notation we can convert any set to its zero-transposed normal form—hereafter referred to as znormal form—with $$Z_{norm}(A)=T_{12-N_{orm}(A)[0]}(N_{orm}(A))$$. Taking again $$A = [7463]$$ as an example, this results in $$Z_{norm}([7463])=T_{12-3}([3467])=T_{9}([3467])=[0134]$$. Standard inversion notation of $$I$$ is also used.

Some Examples
As best as I can tell from inference of the set classes and their relations—all being in their znormal form according to Rahn's approach—any two sets $$A$$ and $$B$$ related by $$B=Z_{norm}(A') \Leftrightarrow A=Z_{norm}(B')$$ should match up side-by-side with one another. This is the case for most set classes found in the article, whether the set class inverts to itself or not. Initially, I thought that an inverted set class would have a prime complement, and a prime set class would have an inverted complement, but this isn't always the case either. By my estimation there are 51 set classes which invert to themselves, 115 of them which have an inverted complement, and 14 whose complement matches the inversion of the starting set class.

Four arbitrarily chosen examples of correctly associated set classes are:


 * $$A_1 = [0127]$$ or 4-6, which inverts to itself, and thus its znormal form is always its prime form. $$A_1^\prime = [0123456789te] - [0127] = [345689te]$$. The normal form of this is unchanged as $$N_{orm}([345689te])=[345689te]$$, and thus $$T_9$$ of this is then $$Z_{norm}(A_1^\prime)=[01235678]$$. This matches up with 8-6, as expected from the table. Its znormal form is also always its prime form.


 * $$A_2 = [012457]$$ or 6-z11A, which has an inversion whose znormal form does not match its prime form. Applying the same process, we get $$A_2^\prime = [3689te]$$, then $$N_{orm}(A_2^\prime) = [3689te]$$ and thus $$T_9$$ of this is then $$Z_{norm}(A_2^\prime) = [035678]$$. This corresponds with 6-z40B as expected from the table. Note we went from 6-z11A (whose znormal form matches its prime form) to 6-z40B (whose znormal form does not match its prime form). In other words, the ending tacked onto the Forte number went A → B.


 * $$A_3 = [034]$$ or 3-3B, which is an example of a set class which is inverted relative to the prime form. As before, $$A_3^\prime = [1256789te]$$, then $$N_{orm}(A_3^\prime) = [56789te12]$$ and thus $$T_7$$ of this is $$Z_{norm}(A_3^\prime) = [012345689]$$. This matches up with 9-3A in the table, meaning the inversion relative to prime form flipped again, from B → A rather than A → B.


 * Lastly, $$A_4 = [023457]$$ or 6-8, which again inverts to itself but is also self-complementary: $$A_4^\prime = [1689te]$$, then $$N_{orm}(A_4^\prime) = [689te1]$$ and thus $$T_6$$ of this is $$Z_{norm}(A_4^\prime) = [023457]$$. This is the exact set we started with, 6-8.

Two arbitrarily chosen examples of incorrectly associated set classes are:


 * $$A_5 = [013]$$ or 3-2A. $$A_5^\prime = [2456789te]$$, then $$N_{orm}(A_5^\prime) = [2456789te]$$ (thus already in normal form), so $$T_{10}$$ of this is $$Z_{norm}(A_5^\prime) = [023456789]$$. This is in 9-2B rather than the correspondingly listed 9-2A.


 * $$A_6 = [02347]$$ or 5-11A. $$A_6^\prime = [15689te]$$, then $$N_{orm}(A_6^\prime) = [5689te1]$$, so $$T_{7}$$ of this is $$Z_{norm}(A_6^\prime) = [0134568]$$. This is in 7-11A rather than the corresponding 7-11B, meaning the inversion relative to prime form did not flip, resulting in A → A rather than A → B for the ending of the Forte number.

Summing Up
Not all of the set classes in the article correspond with what I would expect, which I hope is relatively clear with the enumeration of examples above combined with some explanation of the notation. I'm not entirely sure what should be done, since many theorists consider the inversion of a set class to be an invariance relation to begin with. It's also possible that I have misinterpreted the intention of the listing of set class complements and therefore the inconsistencies I've found aren't actually inconsistencies.

Beyond this, there are hexachords whose znormal complement have the same prime form as the original set class, but whose complements are in the inversion of the originating set class. Should those hexachords then be duplicated to the other column to indicate the complementary relationship? Should it only be the set classes which aren't self-complementary? I'm not entirely sure what the answers should be.

I've calculated the znormal form of the complement of each set class and the results are listed below.

Results
These are, I believe, all currently incorrectly listed complements. First up is the set class corresponding to what I believe is the correct complementary form, then after the colon is a brief working out of the znormal of the complement. The hexachords whose complements are in the set class of their inversion:
 * 3-2A → 9-2B: $$Z_{norm}([013]^\prime) = Z_{norm}([2456789te]) = T_{10}[2456789te] = [023456789]$$
 * 3-2B → 9-2A: $$Z_{norm}([023]^\prime) = Z_{norm}([1456789te]) = T_{8}[456789te1] = [012345679]$$
 * 5-11A → 7-11A: $$Z_{norm}([02347]^\prime) = Z_{norm}([15689te]) = T_{7}[5689te1] = [0134568]$$
 * 5-11B → 7-11B: $$Z_{norm}([03457]^\prime) = Z_{norm}([12689te]) = T_{6}[689te12] = [0234578]$$
 * 5-z18A → 7-z18B: $$Z_{norm}([01457]^\prime) = Z_{norm}([23689te]) = T_{6}[689te23] = [0234589]$$
 * 5-z18B → 7-z18A: $$Z_{norm}([02367]^\prime) = Z_{norm}([14589te]) = T_{8}[4589te1] = [0145679]$$
 * 6-z10A → 6-z39A: $$Z_{norm}([013457]^\prime) = Z_{norm}([2689te]) = T_{6}[689te2] = [023458]$$
 * 6-z10B → 6-z39B: $$Z_{norm}([023467]^\prime) = Z_{norm}([1589te]) = T_{7}[589te1] = [034568]$$
 * 6-2A → 6-2B: $$Z_{norm}([012346]^\prime) = Z_{norm}([5789te]) = T_{7}[5789te] = [023456]$$
 * 6-5A → 6-5B: $$Z_{norm}([012367]^\prime) = Z_{norm}([4589te]) = T_{8}[4589te] = [014567]$$
 * 6-9A → 6-9B: $$Z_{norm}([012357]^\prime) = Z_{norm}([4689te]) = T_{8}[4689te] = [024567]$$
 * 6-15A → 6-15B: $$Z_{norm}([012458]^\prime) = Z_{norm}([3679te]) = T_{9}[3679te] = [034678]$$
 * 6-16A → 6-16B: $$Z_{norm}([014568]^\prime) = Z_{norm}([2379te]) = T_{5}[79te23] = [023478]$$
 * 6-18A → 6-18B: $$Z_{norm}([012578]^\prime) = Z_{norm}([3469te]) = T_{9}[3469te] = [013678]$$
 * 6-21A → 6-21B: $$Z_{norm}([023468]^\prime) = Z_{norm}([1579te]) = T_{7}[579te1] = [024568]$$
 * 6-22A → 6-22B: $$Z_{norm}([012468]^\prime) = Z_{norm}([3579te]) = T_{9}[3579te] = [024678]$$
 * 6-27A → 6-27B: $$Z_{norm}([013469]^\prime) = Z_{norm}([2578te]) = T_{7}[578te2] = [023569]$$
 * 6-30A → 6-30B: $$Z_{norm}([013679]^\prime) = Z_{norm}([2458te]) = T_{10}[2458te] = [023689]$$
 * 6-31A → 6-31B: $$Z_{norm}([014579]^\prime) = Z_{norm}([2368te]) = T_{6}[68te23] = [024589]$$
 * 6-33A → 6-33B: $$Z_{norm}([023579]^\prime) = Z_{norm}([1468te]) = T_{8}[468te1] = [024679]$$
 * 6-34A → 6-34B: $$Z_{norm}([013579]^\prime) = Z_{norm}([2468te]) = T_{10}[2468te] = [024689]$$

Updating incorrect pitch class sets
I've found a few more mistakes in the MIDI audio samples, and made a start at uploading corrected sample files to Commons (3-4A, 3-4B, 3-5A, 3-5B). There is currently a bug T368364 in the Commons transcoding back end that is not rendering the audio files (Ogg, MP3), so the audio for these will not play. Please be patient, hopefully it will be fixed soon :) Jon (talk) 08:51, 25 June 2024 (UTC)
 * Update: these are now working — Jon (talk) 20:47, 26 June 2024 (UTC)
 * I've now gone through all the trichord and tetrachord audio samples, and found that in addition to 3-4A as identified by User:M.J.E. above, the samples for 3-4B, 3-5A, 3-5B, 4-3, 4-17, 4-19B, and 4-27A were also incorrect. I've uploaded new MIDI files to Commons, and the trichords now play correctly; the tetrachords may take a few hours for the Commons bots to generate the audio. Once that's done, can someone please check my working (e.g. User:Boppennoppy seems to know what he's talking about!) When I find the strength to carry on, I'll go through the rest of the table. — Jon (talk) 22:21, 26 June 2024 (UTC)

I only just recently discovered that Hyacinth is no longer with us 😢 so I am now determined to fix up this article, and correct the audio samples, in their honour. It currently takes about a day for Commons to generate the audio samples from uploaded MIDI files. For reference and/or if anyone else feels like helping, I am generating the MIDI files from Lilypond using something like this: — Jon (talk) 02:10, 11 July 2024 (UTC)


 * Sad to see Hyacinth gone as well. I'm thinking this list would work a lot better in one long table, rather than the two next to each other. A single table would allow the individual parameters to actually be sortable, particularly helpful for the Carer numbers. We can disable the sortability for parameters where it would be unhelpful (like audio, for instance).  Aza24  (talk)   20:39, 15 July 2024 (UTC)
 * I'd support this, it would make it easier to maintain as well. The only obvious disadvantage is not having the complements alongside each other, but with a bit of magic pokery we could have a column for the complements, which link to visible anchors within the table. — Jon (talk) 21:58, 15 July 2024 (UTC)
 * e.g.
 * {| class="wikitable"

! Forte no. !! Prime form !! Interval vector !! Carter no. !! Audio !! Possible spacings !! Complement
 * - style="background:#F5F5F5;"
 *  || [0,1,3] || <1,1,1,0,0,0> || 12 || || ... || 9-2B
 * colspan=7 ... | ...
 *  || [0,2,3,4,5,6,7,8,9] || <7,7,7,6,6,3> || ||  || ... || 3-2A
 * }
 * Nice mockup! Yeah, that would work great. It'd be nice to get some references in eventually, particularly for the "possible spacings" column.
 * Another thought: In my experience, the Prime Form is used more often to identify individual set classes than the Forte number(s). If this assertion is even correct, I wonder if the Forte numbers should be moved after the Interval vector column. –  Aza24  (talk)   22:39, 15 July 2024 (UTC)
 * I'm also unclear why some are in bold and others are not. Jon (talk) 03:58, 16 July 2024 (UTC)
 * Well it appears that sets with the same interval vectors have only one (one of the compliments) bolded, presumably to identity the more common name? I'm not entirely sure either, although there is at least this pattern.  Aza24  (talk)   23:35, 16 July 2024 (UTC)
 * The Rahn spelling of a “prime form” is often described (even in Rahn’s own book) as that version of a set class that is “most packed to the left”.  This is not correct.  The Rahn spelling is the version that is most dispersed from the right.  By contrast, the Forte spelling is the version that is most packed to the left within the smallest span.  Sometimes, these spellings are all the same; but in many cases, they are not.
 * The Wikipedia list uses Rahn spellings with Forte numbers.  For sets with a distinct inverse (i.e. unsymmetric sets), out of all possible versions (including inversions), the version that is most dispersed from the right gets an “A” designation (and bold-face).  Then the version of the inverse of the “A” set that is most dispersed from the right gets a (light-face) “B” designation.
 * To see the difference between the Rahn spelling and the version that is most packed to the left, consider (Forte number) 5-25A.  Adjacency intervals are shown here as subscripts between respective pitch-class numerals.  The “Brightness” of the version (defined as the cumulative sum of all pitch-class numerals in the particular spelling of the set) is shown in a superscript.
 * The Rahn spelling of 5-25A, i.e. the version that is most dispersed from the right (which is the same as the Forte spelling—in this case), is:
 * [0221325384](18)
 * The version of 5-25A that is most packed to the left (note the placement of the largest adjacency interval) is:
 * [01123364T2](20)
 * For reference, the darkest version of 5-25A (i.e. that version with the smallest Brightness) is the same as the Rahn (and Forte) spelling—in this case.
 * As a matter of interest, the brightest version of 5-25A is:
 * [03347291T2](29).  Although, as seen below, the brightest version of the inverse, 5-25B, is even brighter—as might be expected.
 * For 5-25B, i.e. the inverse of 5-25A, the Rahn (and Forte) spelling is:
 * [0332516284](22).
 * The version that is most packed to the left (again, note the position of the largest adjacency interval) is:
 * [01123473T2](21).
 * The darkest version is:
 * [0221325493](19).
 * And the brightest version (of all) is:
 * [04437291T2](30) Boppennoppy (talk) 19:26, 17 July 2024 (UTC)
 * And the brightest version (of all) is:
 * [04437291T2](30) Boppennoppy (talk) 19:26, 17 July 2024 (UTC)