# Talk:Pitch class

## Couple of things

1. Not sure what value is the "music set theory" paragraph adding. My understanding is that integer notation is not specific to "music set theory", but common to atonal music theory.
2. "In both cases, when "pitch class" is used, the use of "pitch" indicates a specific frequency or note and does not include its octaves." Can someone please clarify?
1. "both cases" - which cases?
2. Can I have an example please?
3. "Using integer notation, two pitches, x and y, are in the same pitch class only if for some integer n:
${\displaystyle x=12n+y}$"
Wait a second, using that notation x and y are in the same pitch class only if x=y. In other words, using that notation there is no such thing as x=3 and y=15, they are both 3.
4. To summarise, IMHO only the first paragraph is a keeper, the rest is confusing or redundant. One thing that perhaps is missing is whether or not Eb and D# are in the same pitch class - perhaps part of the clarification of point 2 above.
1. I joined the first two paragraphs.
2. A pitch class is a pitch and all its octaves. A pitch is one octave of a pitch class.
1. "Both cases" referred to the seperation of the first two paragraphs (as if pitch classes in set theory and general music theory where different, a possibility that I left open with my original addition). In other words, "In both music theory and set theory" is what it meant.
2. I have no example at this point.
3. Please consult Rahn, John (1980). Basic Atonal Theory. ISBN 0028731603.
Hyacinth 11:51, 26 December 2005 (UTC)

Your edit ([1] "Hope it's ok") was good. Hyacinth 11:53, 26 December 2005 (UTC)

I've added a bunch of stuff deriving from the fact that "pitch class" is hardly an idea belonging only to 12 equal temperament. It is instead an important notion in the great majority of tuning systems. I also seem to have put my foot in it by removing the ugly notation ip(n,m) and replacing it with what mathematicians use, |n-m|. Then I noticed that it seems this has been done before, and reverted. Why??? I think standard mathematical notation should be used unless there is a compelling reason not to, and in this case the standard math notation is clearly far superior. Gene Ward Smith 22:33, 4 January 2006 (UTC)

It was good to see the nonstandard and unexplained "ip" notation go, hopefully forever.
However, I would go even further. In my opinion, in the intro section, everything from "In a system where [...]" to the end should be moved to integer notation, if not modular arithmetic, because that's what is being explained. Even further, the paragraph starting "If one only knows [...]" is IMHO redundant, because it's an alternative definition that is completely equivalent. Actually, I think that even the first definition is dubious, and I'm not at all convinced of the contrary by the argument above ("Please consult Rahn, John" - which incidentally was the same argument for reinstating the ip notation), not having access to the source - and as I said on other occasions, an external reference should not be a pre-requisite for reading/understanding a WP article. As I mentioned above, if we use integer notation to represent x and y, their difference cannot be 12 or higher.
Also, I still don't understand if the classes labelled Eb and D# are in fact the same pitch class.
Finally, I am still not convinced by the answer 2 above. Anyway, if that is indeed the case, I propose changing the sentence to "In the context of pitch classes, a "pitch" indicates a specific frequency or note, chosen from an arbitrary octave. For example [...]". PizzaMargherita 07:54, 5 January 2006 (UTC)

## Reworking the entries

I've substantially reworked this material as well as the pitch-class space entry. My goal is to bring some of the music-theoretical entries in line with current thinking by professionals in the field. I hope people won't be offended.

The major changes are 1. emphasizing that chroma is an attribute of pitches; 2. mentioning that pitch classes result from identifying octave-related pitches, and 3. explaining that pitch class space is a quotient space. I emphasize the continuous perspective, since not all music confines itself to a discrete lattice.

## Logical positivism

Can someone give a cite for the alleged influence of logical positivism on the choice of the name pitch class? Without a cite it should probably be removed. Gene Ward Smith 02:16, 21 April 2006 (UTC)

The origin of the term "pitch class" with Babbitt is well-known. The term first appears in print in Babbitt's "Twelve Tone Invariants as Compositional Determinants." The influence of positivism on Babbitt is clearly stated in Babbitt's "Words About Music," where he refers to Schoenberg, Schenker, and Carnap as his "Viennese Triangle." The logical positivist analysis of properties as extensions (sets) is well-known, and can be found in any history of positivism. Tymoczko 16:15, 21 April 2006 (UTC)

Tymoczko: While you're quite right about both the historical orgins of "pitch class" and the influence of logical positivism on Babbitt, it's nonetheless something of a stretch to claim that logical positivism is directly responsible for the term "pitch class". Rather, the term is a natural application of the mathematical terminology (see Set, Equivalence class, etc.) that Babbitt introduced into music theory. Thus, anyone with Babbitt's level of familiarity with mathematics would probably have invented the same term, irrespective of their philosophical orientation. --Komponisto 12:08, 7 July 2006 (UTC)

I disagree with this. The idea of representing a physical *property* as a class is not at all intrinsic to mathematics, but is instead a characteristic of logical positivism considered as a philosophical movement. (In general, mathematicians do not ask questions like: "how should we represent the concept 'whiteness' in a logically perfect language?") There is a difference between "the class of all white things" and "whiteness" as a property. For instance, you can count the members fo the class of all white things, whereas ordinary linguistic practices do not allow us to count the elements of "whiteness." Tymoczko 12:07, 9 July 2006 (UTC)

## continuous labelling for pitch-classes

Hi. Most published accounts of non-12 pitch-class that I've seen label pitch-classes with integers from 0 to n-1. You're *advocating* the continuous system, and you give good reasons, but I honestly can't see why you would delete a principal objection to it, viz this one which you should return to the article, please, or I will:

"It must be noted that some chords may be embedded in two different tuning systems; for example {0,4,8} in the mod 12 universe sounds identical to {0,5,10} in the mod 15 universe. The different names are appropriate because chords have different voice-leading and combinatorial properties depending on which universe they are embedded in; in that sense {0,4,8} mod 12 and {0,5,10} mod 15 are not the same entity despite sounding identical out of context."

Without this, you're giving the fact that the labels are different as only a disadvantage of the integer system, without the opposing point of view that this can be seen as perfectly justifiable. Or do you disagree that the two structures mentioned have different properties? I can explain more fully if you don't understand.

I also think that one of your stated disadvantages of the integer system--that it doesn't generalise well to universes with an infinite number of pitch-classes--is misguided, and I see you removed my remark to that effect. But really, who has ever suggested a pitch-class space with an infinite number of members?

Also, it should be made clear to the reader that the continuous labelling system is *your* personal suggestion, and not something that is in use in print. I'll leave it to you to figure out how to incorporate this into the discussion, rather than edit your entry myself.

Thanks 132.206.141.143 03:48, 23 May 2006 (UTC)Andy

Thanks for writing. First, note that I did *not* invent the continuous labeling system. As noted in the article, it's built in to the MIDI spec. Furthermore, it has been used (in print, in scholarly music theoretical articles) by Cliff Callender, Jay Rahn, myself, and others. Second, with regard to the deleted sentence: adapting it a bit, you suggest that {0, 4, 8} (mod 12) and {0, 8, 16} (mod 24), have different voice leading properties, but this is misleading. Any voice leading in the mod 12 system is equivalent to a voice leading in the mod 24 system. For instance, take the voice leading (0, 4, 8)->(0, 4, 7) which moves the G# down by semitone. This is equivalent to the mod 24 voice leading (0, 8, 16)->(0, 8, 14), which moves pitch class 16 by 2 quarter tones = 1 equal tempered semitone. So I disagree with the suggestion that chords *inevitably* have different voice leading properties. Third, many, many people have proposed pitch class systems with infinite numbers of pitch classes -- consider a standard "unconformed Tonnetz" as represented by a 2D lattice with perfect fifths in one direction and major thirds in the other. These are infinite pitch class systems, and they do not lend themselves to integer labeling. Finally, it seems to me that the tone should strive to be relatively neutral -- the article can present a variety of systems for naming pitch classes, as it currently does, without being particularly ideological. There are advantages and disadvantages to every approach. The article was beginning to take on too argumentative a tone, with its different paragraphs attempting to refute each other. Tymoczko 14:17, 23 May 2006 (UTC)
Hi again. One basis for saying that the integer-labelling system is the standard is a recent discussion on the SMT list, where the integer system is described as one of 4 pitch-class dogma. The author there, who I assume is you, proposed the continuous system as an alternative to the standard system, and he met resistance from the community of scholars there. There have been many publications that label pitch-classes in xenotemperaments with integers - you know them, I'm sure.
Now, your voice-leading example is a real stretch. Bear with me for a second. Think of Cohn flips in 12-tet, where one voice moves by semitone to form another member of the same set-class. Harmonic triads, pentatonic collections and diatonic collections support this voice-leading connection in 12-tet. Do you think that the important thing about the definition is that the voice-leading move is by exactly 1/12th of an octave? No, clearly the important thing is that it's the minimum motion within the space: one voice moves to an adjacent pitch-class in pitch-class space. The sensible generalization is the one that follows the line of integer representation. Or do you think that Cohn flips are a priori impossible in any equal temperament that is not a multiple of 12? In 13-tet there is no possible voice-leading move of distance "1" in your continuous representation; yet I would want there to be Cohn flips possible in 13-tet (assuming there *are* any - I have no clue whether there are or not). Wouldn't you? That's what I meant by different voice-leading possibilities. The different combinatorial possibilities (of the same sonority embedded in different universes) are even more obvious.
And as for your mention of the midi spec, midi note numbering has to do with pitch, not pitch-class. Midi note numbers aren't gathered into classes, they're just 7-bit numbers (plus a decimal part, if the tuning spec is implemented) that a device can interpret as it sees fit (although note 60 should be mapped to middle C according to the standard). There's nothing in the midi spec to identify note 60 with note 72, etc., and no cyclic aspect to midi note space. I don't think a mention of midi belongs in an article about pitch-class space, though I don't care that much about it.
I agree with your Tonnetz example with infinite pitch-classes, I hadn't thought of that as possibly relating to pitch-classes but clearly it can be construed that way. (I still think it'd be crazy to label the pitch-classes with irrational real numbers though, in this case - I'd just use letter-names, accidentals, and a symbol to indicate the number of comma shifts up or down from the "home" row.)
I understand the advantages to the continuous system and I'm glad you've added it to the article. But the article should also include the justification for different labellings for the same sonority when embedded in different universes. Why should the article say something is a disadvantage, when from another point of view it makes perfect sense? NPOV dictates it should go back! You can put it back in a way that doesn't seem argumentative to you, if you prefer to do it yourself.
Regards -Andy 132.206.141.143 19:17, 24 May 2006 (UTC)andy
I'm having a bit of trouble understanding your second paragraph. The following points seem pretty uncontroversial to me: 1) Some equal temperaments, like 24tet, contain others, like 12tet. Anything I can play in 12tet can also be played in 24tet. 2) Whether it is interesting to move by "minimum motion within the space" is entirely a function of the size of the temperament: in 1200tet, "minimal motion within the space" is uninteresting, because it is imperceptible. 3) Finally, it is clearly a difficulty (not an insurmountable one, but a difficulty) that a system labels notes only by way of some supposed division of the octave: suppose you're listening to a 12tet piece, mentally analyzing its chords and voice leadings, when suddenly, 10 minutes in, there's a single quarter tone. On your view, you have to go back and rework the entire analysis: what you thought were (interesting) "minimum motions within the space" turn out to be completely different things, uninteresting "non-minimal" motions within the space; the chords you were hearing had "entirely different combinatorial and voice leading properties" from what you supposed they had. Etc. All of this seems self-evidently problematic to me.
Straw man. What you refer to as "my view" says absolutely nothing about what to do in the example you made up. "My view" is that contexts in which it's useful to use a scale-step-based labelling system are more numerous than contexts in which it's more useful to use a continuous system interpolating from 12-tone equal temperament. I'd agree that the example you made up is one of the latter, though --Andy
BTW, I do agree that it is often useful to measure in terms of "scale steps" -- here I think we're dealing with two separate metrics on circular pitch class space at the same time. So, given your example of 13tet, we have two metrics: a scale-dependent metric according to which the 13 notes are 1 step apart from each other, and a scale-independent metric according to which they're each 12/13 of a circle apart from each other. But this is a subtle matter that deserves its own article ("scale step") on Wikipedia. Perhaps the best way to resolve this dispute is to write an article on scale steps and to link this article to that one.
I don't see it as a subtle side-issue--the scale-step issue is the obvious reason to relabel pitch-classes, when you change the universe they're embedded in. See my Cohn-flip example above. When you're in a situation with an unchanging underlying scale system (which is, I grant you, not 100% of the time, but is surely a large percent of the time for a large percent of people likely to ever need to label pcs), you would like your pitch-class labels (and interval metric) to match scale steps --Andy
The step from continuous pitch numbering to continuous pitch class numbering seems trivial to me.
My point is that the article makes it seem like the MIDI numbering has anything to do with pitch-*classes*. The mention of "base 128" is a real red herring. No number above 128 can be represented, and there aren't 128 different digit-symbols, so what kind of "base 128" is this? Internally it'll get represented as a 7-bit binary number. (Plus however many bits for the representation of the decimal part.) It would be accurate to say that it's a number whose integral part is between 0 and 127, and it has a decimal part with such-and-such an accuracy... but that has nothing to do with pitch-class. There are no pitch-classes implied, because there's no cyclic aspect to MIDI numbering. And quite apart from that, it's not true that the MIDI spec forces devices to interpret note number 61 as having a pitch one twelfth of an octave above note number 60. --Andy
The business about using irrational numbers seems merely notational: we use symbols like "pi" and "log_2 3" to refer to irrational numbers without difficulty. The important thing is having a clear conceptual framework for understanding music. Finally, my memory of the SMT discussion is somewhat different from yours, perhaps because I got a number of private emails indicating agreement with my view, and I know of at least one author who has adopted the continuous labeling system in a recent paper.
I would use it too, in the right context! My point was really that scale-step based labelling is more natural than the article currently allows for. You know, the article would seem fine to me now if you took out the "base 128" bit (maybe the whole MIDI bit), and made it clear that having different labellings for the same sonority embedded in different universes is a feature, not a bug, in many contexts. --Andy
Perhaps we should continue this over email. Tymoczko 13:55, 25 May 2006 (UTC)
Made a slight change on June 8th 2006: moved the mention of midi note mapping earlier in the article where it fits better, and changed the bit about "base 128" to something more sensible (see above). --Andy

## Just a minor note

Does the note "b-double sharp" (or E##) actually exist? Wouldn't that only exist in C## keys or higher (i.e. with more sharps)? I would be shocked if I could find a composer using either (E## or B##) correctly. Sorry to nitpick Diddydoobop 10:57, 13 November 2007 (UTC)

It just occurred to me that the same issue arises with Cbb and Fbb. If no one has any objections I will remove these from the article. Diddydoobop 11:00, 13 November 2007 (UTC)

Why, where, how, and what needs additional citations in this article? Hyacinth (talk) 07:37, 20 March 2010 (UTC)

Tag removed. Hyacinth (talk) 16:52, 16 September 2010 (UTC)

## Merge: Integer notation

Seems like this highly related topics, or topic and subtopic, would be more clear all in one article. Hyacinth (talk) 07:35, 20 March 2010 (UTC)

Concur. 100%. I had no idea the article "Integer notation" existed. It clearly declares itself at the beginning to be a method of notation strictly for application to musical pitch classes, and it is small enough to be absorbed easily into this one.—Jerome Kohl (talk) 19:50, 20 March 2010 (UTC)
Quick merge done. Hyacinth (talk) 04:09, 29 March 2010 (UTC)

## "Other ways" table

The current table is shown on the right, a two column version in the center, and a three column version on the left.

Pitch class
Pitch
class
Tonal counterparts
0 C (also BTemplate:Music, DTemplate:Music)
1 [[C♯ (musical note)|CTemplate:Music]], [[D♭ (musical note)|DTemplate:Music]] (also BTemplate:Music)
2 D (also CTemplate:Music, ETemplate:Music)
3 [[D♯ (musical note)|DTemplate:Music]], [[E♭ (musical note)|ETemplate:Music]] (also FTemplate:Music)
4 E (also DTemplate:Music, FTemplate:Music)
5 F (also ETemplate:Music, GTemplate:Music)
6 [[F♯ (musical note)|FTemplate:Music]], [[G♭ (musical note)|GTemplate:Music]] (also ETemplate:Music)
7 G (also FTemplate:Music, ATemplate:Music)
8 [[G♯ (musical note)|GTemplate:Music]], [[A♭ (musical note)|ATemplate:Music]]
9 A (also GTemplate:Music, BTemplate:Music)
10, t or A [[A♯ (musical note)|ATemplate:Music]], [[B♭ (musical note)|BTemplate:Music]] (also CTemplate:Music)
11, e or B B (also ATemplate:Music, CTemplate:Music)

Pitch class
Pitch
class
Tonal counterparts
0 C also BTemplate:Music, DTemplate:Music
1 [[C♯ (musical note)|CTemplate:Music]], [[D♭ (musical note)|DTemplate:Music]] also BTemplate:Music
2 D also CTemplate:Music, ETemplate:Music
3 [[D♯ (musical note)|DTemplate:Music]], [[E♭ (musical note)|ETemplate:Music]] also FTemplate:Music
4 E also DTemplate:Music, FTemplate:Music
5 F also ETemplate:Music, GTemplate:Music
6 [[F♯ (musical note)|FTemplate:Music]], [[G♭ (musical note)|GTemplate:Music]] also ETemplate:Music
7 G also FTemplate:Music, ATemplate:Music
8 [[G♯ (musical note)|GTemplate:Music]], [[A♭ (musical note)|ATemplate:Music]]
9 A also GTemplate:Music, BTemplate:Music
10, t or A [[A♯ (musical note)|ATemplate:Music]], [[B♭ (musical note)|BTemplate:Music]] also CTemplate:Music
11, e or B B also ATemplate:Music, CTemplate:Music

Pitch class
Pitch
class
Tonal counterparts
0 BTemplate:Music C DTemplate:Music
1 BTemplate:Music [[C♯ (musical note)|CTemplate:Music]] [[D♭ (musical note)|DTemplate:Music]]
2 CTemplate:Music D ETemplate:Music
3 [[E♭ (musical note)|ETemplate:Music]] [[D♯ (musical note)|DTemplate:Music]] FTemplate:Music
4 DTemplate:Music E FTemplate:Music
5 ETemplate:Music F GTemplate:Music
6 ETemplate:Music [[F♯ (musical note)|FTemplate:Music]] [[G♭ (musical note)|GTemplate:Music]]
7 FTemplate:Music G ATemplate:Music
8 [[G♯ (musical note)|GTemplate:Music]] [[A♭ (musical note)|ATemplate:Music]]
9 GTemplate:Music A BTemplate:Music
10, t or A [[B♭ (musical note)|BTemplate:Music]] [[A♯ (musical note)|ATemplate:Music]] CTemplate:Music
11, e or B ATemplate:Music B CTemplate:Music

Which is superior? Hyacinth (talk) 04:09, 29 March 2010 (UTC)

## Integer notation

I propose the following paragraph or something to the likes of it be added into the Integer notation section, mainly because it's fucking hard to figure it out! 93.103.93.103 (talk) 11:18, 15 January 2012 (UTC)

```The following Python code produces a pitch class integer (aligned with A = 0, ..., G♯ = 11) given a frequency:
```
```def frequency_to_pitch_class(freq):
return round(((log2(freq / 440) * 12) % 12) + 12) % 12
```
How does an explanation that relies on a programming language that only a tiny fraction of readers can possibly understand help with a difficult-to-understand explanation? I would say the problem must have to do with the wording, which evidently needs revision for clarity. Alternatively, an ordinary mathematical formulation would have a better chance of reaching more readers.—Jerome Kohl (talk) 22:23, 15 January 2012 (UTC)
Mathematical formulation, I agree! Then again, round, log and mod (either prefix or infix) are functions in math too, so it doesn't really make much difference to the initiated. I'm not an editor nor I strive to be one (so I won't do it:-), but I'd really like to see it included in the article as I think it might help someone. Thanks. 93.103.93.103 (talk) 16:59, 16 January 2012 (UTC)
It seems to me that the first step is to scrutinize the description, in order to see whether there may be ways of clarifying the English. Once that is done, a mathematical formula could be added. I am neither a mathematician nor a programmer (though I do have some background in both areas—never heard of Python, though), so I am not really qualified to do this, but I will look at the English.—Jerome Kohl (talk) 20:31, 16 January 2012 (UTC)
Well, having examined the offending passage, I can see two problematic things: first, it succeeds in making a very simple concept sound complicated and, second, it does so in part through the use of just the sort of mathematical explanations being called for here! Because I already know what is being described I can follow the convoluted explanations, but I doubt whether any beginner could do so. I think what is really needed is a simple explanation in plain English, with the more complicated mathematical formulas given only afterward (if at all). I'll see what I can come up with.—Jerome Kohl (talk) 23:31, 16 January 2012 (UTC)
Lol, I too understand the 'Integer notation' section quite well, and having in common only the mathematical background, it didn't help me not one bit devise the formula above. That's what I think is missing, but you're free to rewrite the section as you please (though I though simpler explanations are preferred on simple.wikipedia.org). Anyway, thanks. Regards. 93.103.93.103 (talk) 15:57, 20 January 2012 (UTC)

## File:Pitch class on C.png

Ideally you would arpeggiate all of the unison octaves, then sound them together. The same should be done with the single octave example. Thanks in anticipation. Arrond wolf (talk) 09:11, 12 February 2012

Why? Hyacinth (talk) 00:51, 13 February 2012 (UTC)

I wasn't referring to the image file. I was referring to the attatched general midi example of the image. If the image were arpeggiated along with the midi file being altered, it would simply illustrate the idea and it's practical application better. For the same reason that when you take piano I or accompanied a vocalist warm up with a piano that you would sound <If major>

```1-3-5-3-1, Triad.
```

With the general midi, it is nearly impossible to distinguish one octave from the next when only sounded in unison without prior arpeggiation of the tones. It would simply illustrate the concept better if the audio example aurally informed the inquiring mind (if continuing to use general midi-I don't know if wikipedia supports wave files) if it were charted C1 - C2 - C3, cont. ascending, then descending back to C1 then sound them simultaneously. As for the image, the dynamic marking seems moot. I don't care to create the file as I described (both the audio and the notation) if that is agreed upon. If It were a well tuned grand piano or an electric piano with an authentic tone, it might not sound so muttered or vague. That's why. — Preceding unsigned comment added by Arrond wolf (talkcontribs) 11:52, 19 February 2012 (UTC)