Talk:Circle of fifths

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no c-flat[edit]

Why is there no C-flat in the picture of the circle of fifths on this page?

--Phil Kirlin 15:43, 14 November 2005 (UTC)

Probably for the same reason there is no E-sharp, A-sharp, G-sharp, etc. Having them would clutter the diagram. Probably anyone who knows what C-flat is knows that it's enharmonic to B. Merphant 05:34, 15 November 2005 (UTC)

Any of the rules of spelling notes are there to make reading music easier. B-natural and C-flat sound exactly the same. They are the same tone, but there are reasons to spell it one way or the other.

  For instance,  The C Major Scale is spelled like this:
  There is one of each of the letters (with the last being the same as the first)
  The G-flat scale is spelled like this
  There is one of each of the letters (with the last being the same as the first)

Scales need one of each letter so that they will read nicely on the staff. It won't do to have a Bb and a B-natural in the same scale. It would make for difficult reading.

While C-Flat is the enharmonic equivalent to B-natural, it definitely has it's place in the circle. As you travel counter-clockwise from the top of the circle, you encounter keys with 1 flat, then 2 flats, then 3 and so on. C-flat is the key with seven flats. That is one flat for each of the seven notes A through G.

The Circle of Fifths is a wonderful tool with no exceptions. It is invaluable to any student of 12 tone music


Alternate version[edit]

Here's an alternate version of the article:

A term used in music theory to relate the greek modes.

being a circle it can be travelled around either clockwise or counter clockwise, one direction resulting in fourths.

B E A D G C F Bb Eb Ab Db Gb (B)

(note that this is written with flats. the enharmonic sharp equivalents are just as valid)

left at an alternative page name by at 00:31, 22 Jan 2005; archived here. Noel (talk) 14:54, 3 Feb 2005 (UTC)

Major scale articles[edit]

Recently, I completed the list of Wikipedia articles for major keys, which include:

  • The scale (with the arpeggio notes strongly bolded, namely, 1-2-3-4-5-6-7-8)
  • The key signature
  • The relative minor
  • The Circle of fifths template
  • A category, Category:Musical keys

Anything else to include in each key article?? Georgia guy 00:53, 1 Jun 2005 (UTC)

WHY do we need a separate article on every key? —Wahoofive (talk) 15:58, 1 Jun 2005 (UTC)
There should be an article on every frequently used key, such as C major, D minor, etc. Someone perusing Wikipedia articles on composers is likely to wonder "What's so special about that trumpet sonata being in E-flat?" or "Is there a reason there are more violin concertos in D major than there are piano concertos in that key?" As for the less frequently used keys, such as G# minor, it would be just for the sake of completeness. Dmetric 19:18, 1 Jun 2005 (UTC)
I still don't get it. If you go to the E-flat major article and it says exactly the same thing as every other key, that doesn't help answer what's special about a concerto in E-flat. It seems to me that readers will learn much more by reading more comprehensive articles such as Key (music), Scale (music), and Key signature. —Wahoofive (talk) 20:28, 1 Jun 2005 (UTC)
That's the sort of thing we need to add to the article on E-flat major. User:Georgia guy has taken care of writing a lot of the repetitive verbiage, now you and I ought to fill in what's special about each key that it doesn't have in common with all the other keys. Right now, the articles on C major and B-flat major are better representatives of the potential of these articles. Dmetric 20:51, 1 Jun 2005 (UTC)
Well, good luck. I don't see much potential there myself. So some keys are easier to play in for some instruments, and occur statistically more often in some composers. Are you planning a comprehensive list of all pieces written in D major? —Wahoofive (talk) 21:56, 1 Jun 2005 (UTC)
Thanks. Those comprehensive lists have already been compiled, and musicologists agree on some points, debate others. Dmetric 17:14, 2 Jun 2005 (UTC)

All keys are equal. There is nothing special about E-flat major. It may be made special by context, but this is still only in relation to another key. Hyacinth 21:03, 2 Jun 2005 (UTC)

I'm also skeptical about the key articles. At least so far, they look to me like repositories for trivia. Trivia collecting is fun for Wikipedia editors, but it's not helpful for our readers, who deserve systematic coverage.
For instance, I think it is of interest to know what keys Haydn liked to write symphonies in, but this information should go in an article about Haydn's symphonies, and not scattered through the various articles about keys. Likewise, the fact that violin concertos are often in D is a fact about the violin, not about D, because it follows from the structure of the violin. (There are many violin concertos in G, A, and E, too.)
As the saying goes, scholars should try to "carve nature at the joints." You put the information in a place where, juxtaposed with other information, it will give you a pattern and not be arbitrary. Opus33 05:32, 6 Jun 2005 (UTC)

Ear's perception of major keys[edit]

Does anyone have an opinion on the ear's perception of certain major keys?? For example, some keys might be percieved as happy an pleasant; others as sleepy; others as boring; etc. Georgia guy 8 July 2005 00:04 (UTC)

In my personal experience, that doesn't have to do with keys, but rather with scales and modes. Uttaddmb 00:18, 23 July 2005 (UTC)
See perfect pitch and equal temperament. Hyacinth 08:16, 23 July 2005 (UTC)


Perhaps it's just me, but in the section talking about moving from C to G, my browser is showing that you move the F to F? not F#. In fact all through the article where there should be a # sign there is just a question mark.

Is this something fancy the author was trying to do that didn't work?

Someone recently changed all the "#" signs to unicode sharp signs. I think this is in keeping with some Wikipedia style guide, but it seems to be messing with your browser. Tymoczko 21:56, 22 May 2006 (UTC)
I agree. The flat signs do not show up either on my computer. I get these little boxes (like when your computer does not support a certain language format) where the symbols should be. Dachshund2k3 23:55, 20 March 2007 (UTC)

I updated the article to make use of the {{music}} template. This should improve display of Unicode sharps and flats. See Template:Music for details.--Dbolton 02:45, 20 July 2007 (UTC)

Chromatically-altered Circle of Fifths[edit]

Has anybody ever heard of this term? I was taught this concept in a 20th-century harmony class at University, and it specifically focused on the music of Alexander Scriabin. Basically, it's a progression that follows an "altered" circle-of-fifths passage, usually by using a bV-of-something (i.e. A-Db-G-C). I found it very fascinating, as it turns out to be a defining feature of MUCH of Scriabin's music (piano-music especially), and has helped me on MANY occassions to make sense of much of his music (the "middle" period-music, for the most part). Is there another name for this type of progression? Should it be included in this article? thoughts? --Crabbyass 05:27, 23 August 2006 (UTC)

Example for an easy to use table[edit]

Look in the Categorie for more Informations...

Spacer.gif Kreis4.svg Kreis1.svg Kreis5.svg Spacer.gif


Kreis3b.svg Kreis6.svg
Kreis6b.svg Kreis3.svg
Spacer.gif Kreis2b.svg Kreis4k5b.svg Kreis7.svg Spacer.gif

used for German Wikibooks (Gitarre)

Layman's terms[edit]

Not only is the phrase "layman's terms" kind of condescending, but it's hilarious that that section includes this passage:

The frequencies of two notes that are a perfect fifth apart differ by a ratio of approximately 3:2 = 1.5. A ratio of exactly 1.5 sounds best, and this explains why a perfect fifth sounds consonant, though for mathematical reasons it is not possible to get the circle of fifths to 'join up' (i.e. return to the original pitch after going round the circle) unless a close approximation to a perfect fifth is used, namely 2 to the power 7/12 = 1.498.

Way to simplify it for the average person!! Anyone would understand the circle of fifths after reading that. —Wahoofive (talk) 02:51, 21 March 2007 (UTC)

I just tried to improve this section a little. It still reads awkwardly though. Pfly (talk) 07:25, 22 November 2007 (UTC)
And, for what it's worth, the original addition of the "In Layman's Terms" section can be seen here. Looks like someone was just trying to explain what a fifth is. Pfly (talk) 07:31, 22 November 2007 (UTC)

This article is an example of writing "in media res" -- jumping into the subject without properly introducing it. It neglects to explain the creation of musical scales by the subdivision of a plucked string, and how this relates to consonant and dissonant intervals. If you don't understand this, then the reason a "perfect" perfect fifth doesn't work makes no sense. This article needs a careful revision that will enable the "layman" to understand it. WilliamSommerwerck (talk) 13:46, 28 November 2012 (UTC)

Why doesn't a "perfect" perfect fifth work? __ Just plain Bill (talk) 17:20, 28 November 2012 (UTC)


01-June-2007: Some of the most unprintable characters are the unicodes for sharps/flats: they have been reverted to '#'/'b' several times over the past year in other music-key articles. For more precise sharp/flat symbols, the superscripted pound/bee characters could be used (see below: Symbols for sharps/flats). However, there is no need to persecute '#'/'b' since they work well for all 12 notes, even "Eb"/"Ab", and more importantly, they create chords/notes that are searchable by search-engines. Guitar players have complete sets of chord notations that are easy to write, read, and are SEARCHABLE by search-engines: for example, look at these brilliant notations for D-sharp diminished, minor/major, 7th chords: D#dim D#sus D#sus4 D#m D#m7 D#maj D#maj7. There is little need for the obsession over unicode sharp/flat symbols. Although music-miming computer languages have become trendy (such as C# coding, F# coding and Db), the notes/chords are still searchable within music articles (hunting G#, Bb, C#, Eb, etc.). Try to avoid those get-a-life unicode characters for sharps/flats; they just aren't needed in the modern age. -Wikid77 10:00, 1 June 2007 (UTC)

Symbols for sharps/flats[edit]

01-June-2007: Over the past 2 years, the unprintable sharp/flat unicodes have been reverted in articles to use simple # / b; however, superscripted codes seem more accurate and universal. For more precise coding, that shows on most PC screens, use superscripts:

  • For flats (b), use "<sup>b</sup>"
  • For sharp (#), use "<sup>#</sup>"
  • Double-sharp, use "<sup>##</sup>"
  • Double-flats, use "<sup>bb</sup>"

Results: F#, C#, Bb, Eb, C##, Gbb. In general, the simple "# / b" characters are close enough, such as F# or Bb, especially considering the tedious coding of superscript expressions. However, because Wiki articles feed other sources, worldwide, avoid unicodes for sharp/flat or use words (such as: F-sharp, A-flat or C-natural). The more precise superscripted forms support music elitists as well as the vast general public on an amazing variety of PCs or Wifi Internet devices. -Wikid77 10:00, 1 June 2007 (UTC)

In general good, except:
(1) double-sharps don't look like that
(2) this discussion would be better placed at Wikipedia:Manual of style (music)
Wahoofive (talk) 20:46, 1 June 2007 (UTC)


Why does this article have absoloutly nothing on how to use the Circle of Fiths. After reading it over it says (much like many of Wikipedia's music theory articles) that this is what X is in several different ways, while avoiding any mention of how to actually use it, or any mention of ways it has been used with given examples.

Someone fix this, please. —Preceding unsigned comment added by (talk) 05:07, 11 April 2008 (UTC)

It does say: "Tonal music often modulates by moving between adjacent scales on the circle of fifths." And goes on to describe why modulation is easy between steps of the circle. It follows you could use it for modulation purposes. There could be something about secondary dominants I suppose. But is the circle of fifths all that useful for composing music? I always thought of it as a mainly theory-oriented way of showing scale and key relations. Pfly (talk) 05:35, 11 April 2008 (UTC)

One may find that this is a problem with music theory in general and not with Wikipedia. You may be looking for a good composition manual, which some people maintain does not exist. Hyacinth (talk) 02:11, 21 June 2008 (UTC)

To me, it looks like this article is in rough shape. I find the style opaque, difficult to understand, and I flatter myself that I already know a bit about the subject. I don't understand why most of the charts are here, in fact I have difficulty understanding them, full stop. The article needs more emphasis on presenting some clear basics and needs to lose some of the more esoteric stuff that belongs in more specialized articles, keeping links and adding more relevant "see also" items, of course.
Mention of equal temperament comes early and often. Why? It's still called the circle of fifths in other contexts. (yeah, syntonic comma, bla bla...) I propose to lose that as well.
As time permits, I intend to do a pretty drastic rewrite here. Soon as there's something to point at in my sandbox, I'll point. __Just plain Bill (talk) 05:23, 20 July 2008 (UTC)
Please do. I wonder if there is a verifiable source that could be cited as a basis for the {{Circle of fifths unrolled}} table. Are the symbols in it standard or original? Having them all be links is an interesting feature.
I have added an elementary reference that describes the CoF, but am not qualified to critique it for accuracy or completeness.
--Jtir (talk) 20:50, 20 July 2008 (UTC)

I for one am willing to call this section closed. In the last week and a half, the article has come a long way, going from scattered to cohesive. __Just plain Bill (talk) 01:19, 31 July 2008 (UTC)

OK, but it begins with a nearly incomprehensible lead sentence, lacks citations in many sections, has an embryonic history section, and needs more copyediting. You could, however, look into submitting it for review. --Jtir (talk) 17:42, 31 July 2008 (UTC)
I'll go along with everything but the "nearly incomprehensible lead sentence" part. If someone who reads at the fourth-grade level or above (say Stan Marsh & Kyle Broflovski are typical fourth-graders) has trouble understanding it, then we as a society are in deep swill-flavored yogurt. My previous comment was more of a withdrawal of the offer of doing it all by myself in my sandbox, than an imprimatur of good-articlehood. I'll stand by the assertion that recent collective efforts have brought this article into a promising state of coherence. Be well, __Just plain Bill (talk) 18:01, 31 July 2008 (UTC)
OK, the article is developing just fine in place. I apologize for giving such a general criticism of the lead, but I was trying to write a list of criticisms of the article as a whole and wanted to say something about the lead. I'll start a new section when I feel up to it. --Jtir (talk) 18:13, 31 July 2008 (UTC)
Mais non, m'sieu, mea maxima culpa... I assume good faith on your part, and have just tweaked the lead based on some of your recent edits and summaries. Must say, you and that other fella, as well as some other unindicted co-conspirators have been egging me on here with your good efforts. Cheers! __Just plain Bill (talk) 18:44, 31 July 2008 (UTC)

Heinichen's musical circle[edit]

While trying to source Johann David Heinichen as the origin of the CoF, I found an illustration from 1711 that shows it. In the article, the date of publication is wrong, the title seems to be incomplete, the origin is older, and Heinichen calls it a musicalischer circul.

--Jtir (talk) 21:37, 20 July 2008 (UTC)

The 1st and 2nd eds. had different titles:
  • '" Neu erfundene und gründliche Anweisung . . ." (1711); 2nd ed. äs "Der Generalbass in der Composition, oder Neu erfundene," etc. (1728).' (1905)
--Jtir (talk) 21:47, 20 July 2008 (UTC)
There seems to be a mistake in the 1711 illustration — "h dur" should be "h mol" (at the 9:00 position).
Here is a nicely drawn German language version of the modern CoF. (The "b"s should be "♭"s and the "#"s should be "♯"s, though.)
--Jtir (talk) 12:43, 21 July 2008 (UTC)
Circul is not German for circle, and it appears to be an abbreviation in the 1711 illustration.
Maybe it is an abbreviation of the Latin word circulus.
--Jtir (talk) 13:02, 21 July 2008 (UTC)

Nikolay Diletsky presented a circle of fifths in 1679. (Lester says "around 1670".)

Lester cites Jensen re Diletsky and has more about Heinichen.

--Jtir (talk) 13:48, 21 July 2008 (UTC)

Audio request[edit]

If an audio recording really would be helpful, please include specifics.--Dbolton (talk) 01:52, 23 July 2008 (UTC)

Audio example added to article. Hyacinth (talk) 02:38, 23 July 2008 (UTC)
Thanks for adding these. They work great. I am not so happy with the page layout, though, so I am about to try putting them in a gallery, but that could go wrong on displays other than mine. A problem is that the lead needs more text. I would have put Heinichen's musical circle there, if there were more text. --Jtir (talk) 18:28, 23 July 2008 (UTC)

The recent audio additions are great!--Dbolton (talk) 05:27, 24 July 2008 (UTC)

I was going to ask for two more audio examples with chord progressions that systematically step through the circle, and discovered that Hyacinth had already done something like that in the article on chord progressions (they sound great). Could something similar be added here? (with the notated music and possibly with minor chords too).
Would it be more clear to put the ascending sequence first, because it begins with a fifth? (and say in the caption that the sequence is going clockwise around the circle for people who don't read music). BTW, the basic circle in the lead was very helpful while writing this. --Jtir (talk) 13:57, 24 July 2008 (UTC)


Jtir, you have expressed some concerns about the formatting and length of the lead section. The article will appear differently to different readers depending on the width of their screen, and thus there may be more or less white space for different readers and you may not be able to fix it for everyone. See Wikipedia:Image use#Displayed image size. Hyacinth (talk) 20:20, 23 July 2008 (UTC)

You prompted me to try something new. My display is 1280x1024, so I can simulate a smaller display by simply resizing my browser window. I created an 800x600 test image with GIMP, which is the minimum display size in the policy you linked, and by placing my browser window over it, I can exactly resize to 800x600. The lead of this article looks fine, with the text occupying about the middle third and the images the right-third. The media gallery is displayed below that, so by vertical scrolling, everything is visible. (The width of the media gallery or the Heinichen circle could be reduced a bit to make them appear side-by-side.) Further down in the article there is an overly larger image (CoF unrolled), however, that forces horizontal scrolling too.
Thanks for adding the additional sentence to the lead. I know just enough to understand it. The lead sentence is another matter.
--Jtir (talk) 22:30, 23 July 2008 (UTC)
heh. I'm on a machine with 800x600 CRT right now. The wiki-bar on the left takes up about 20% of the width, with the lead dividing the rest about evenly between text and pics at first, with a lot of white space. That's OK, I don't mind scrolling vertically, but that unrolled diagram could use some shrinkage. Can't just thumbnail the template, or can you? __Just plain Bill (talk) 22:41, 23 July 2008 (UTC)
Thanks for validating my simulation. Your description (20%, etc) accurately describes what I see when I resize my browser window to 800x600. I resized to 550px and thumbnailed the image in the template per MOS:MUSIC#Images. Fits much better now. --Jtir (talk) 12:55, 24 July 2008 (UTC)

The circle used in instrument building[edit]

I just removed this [below] from the article. At the very least, it needs some introduction. I have no idea what instrument is being described in the top two paragraphs. My OR-sense is tingling. __Just plain Bill (talk) 21:49, 23 July 2008 (UTC)

E-A-D-G-C-F-A-D-G-C-F-B, arranged in 3 clusters of 4 strings to make the field of strings more readable.

Because of this tuning all five neighbouring strings form a harmonic pentatonic scale and all seven neighbouring strings form a major scale, available in every key. This allows a very easy fingerpicking technique without picking false notes, if the right key is chosen.

Accordions also commonly use the stradella bass system for the left hand buttons. It follows the circle of fifths.

comments on section called "Effect on diatonic function"[edit]

Although this section is above my level, I can still offer some comments:

  • The section makes good use of direct quotes and is well sourced. The last quote is a nice academic smackdown.
  • The section name is unhelpful because the term "diatonic function" is not linked (nor should it be as I recall). The section name is inconsistent with the first sentence, which unexpectedly pipes "harmonic functionality" to "diatonic function".
  • The section begins: "One theory regarding ..." This suggests the section is going to discuss several theories, but I don't see a clear structure following this scheme (e.g. "A second theory...").
  • "(the use and role of chords in harmony)" is a helpful summary, but there isn't really a summary sentence that would help a person unfamiliar with the topic understand what it is about. It seems to be about ways of explaining harmonic practice in terms of the CoF. That makes me wonder if composers actually consult a CoF while composing.
  • 'According to Goldman's Harmony in Western Music, ...' is redundant with the citation. Saying simply "Goldman says ..." would suffice.
  • "See: plagal cadence and authentic cadence." tells the reader what to do and offers no transition from the preceding sentence. It would be better to write a sentence using these words, possibly as a note.

--Jtir (talk) 23:08, 23 July 2008 (UTC)

Great comments. Hopefully I will work on addressing them in the near future.
To address what I can here, a trained composer would probably not literally consult a circle of fifths as they would have it memorized. Hyacinth (talk) 23:36, 23 July 2008 (UTC)
Largely above me as well, but rather than "composers consulting the circle" I believe it is saying that the circle, and the "distances" it implies, help explain why some cadences sound more "final" than others. Not sure how well a discussion of historical harmonic practice relates to this article.
As it now stands, this section commits the rhetorical crimes of assuming too much prior knowledge of the subject, and assuming the reader is already warmed up to the context. __Just plain Bill (talk) 23:41, 23 July 2008 (UTC)
Unless you are more specific there is not much that may be done. Hyacinth (talk) 02:53, 24 July 2008 (UTC)
Removed it entirely. It was pretty much a straight copy & paste of a section in Diatonic function, which is now linked in "See also here." __Just plain Bill (talk) 03:32, 24 July 2008 (UTC)
and put it back, as a subheading of "Structure and use" since I now see Hyacinth has put some work into getting this piece into shape. __Just plain Bill (talk) 03:56, 24 July 2008 (UTC)
Thanks for putting it back. That looks like a good place for it. --Jtir (talk) 12:19, 24 July 2008 (UTC)

What a lot of pictures...[edit]

Any support for putting the various renditions of the circle itself into a gallery of thumbnails? There's also Image:Circle fifths.svg which shows the "lose three sharps or add three flats" thing between minor and major, that's mentioned in the text. __Just plain Bill (talk) 04:00, 24 July 2008 (UTC)

Basically this article (and the lead in particular) is too short to accommodate so many images. I favor putting a single modern rendition in the lead so readers can quickly understand the subject graphically. Ideally the historic images would go in a section on the history where they could be more fully explained in their historic context. Since there is no such section, a gallery would be fine. However, this approach will likely leave white space in the lead. I like the audio clips with the "score". Since they are images, they also need to be considered when laying out the article.
As for modern renditions … This search at commons finds a few basic ones. Here are more elaborate German and Japanese versions that show the key signatures. I haven't found one in English that shows the key signatures, not even in that article, and this seems to be a serious omission. The German article on the circle of fifths might suggest some ideas for this article.
--Jtir (talk) 12:17, 24 July 2008 (UTC)

Obviously a modern basic image needs to go in the introduction. The historical images should ideally accompany the text describing them and do not belong in the lead. Hyacinth (talk) 16:06, 24 July 2008 (UTC)

I agree, that the two historical images are not helpful at the beginning. The short historical note may be. The historical pictures may be combined with the footnotes of this historical section.
I concider the small image Fifths.png to be superfluous. The information is better displayed in Circle-of-fifths.svg.
I do not agree, that a circle image should contain key signatures. The number of accidentals is important in this article. (But for the article key signatures a graphic like this could be made.)
Hexadecimal (talk) 16:13, 24 July 2008 (UTC)
Grüß Gott, and I actually prefer the small image over the colorful one. Simple, clean, and uncluttered with chartjunk. Easier to read at any legible zoom, with or without my spectacles on. ;-) Seriously, not everyone has excellent youthful eyes, and Wikipedia needs to maintain some level of accessibility in that respect. Be well, __Just plain Bill (talk) 17:32, 24 July 2008 (UTC)
Thanks for your comments, Hexadecimal. Hyacinth's version seems to be a good solution. (There would be a lot of white space in the lead without one of the historical images.) There is plenty of source material to expand the history section. If the history section is expanded, a summary can be put into the lead.
WRT the alternative versions of the modern CoF:
At least one editor finds this circle Circle-of-fifths.svg "gaudy". I don't much care for the graphic design either. Here is another version (with key sigs) that I prefer (I'm not so sure about the colors and arrows though — Bill?).
I'll leave it to the specialists to decide whether this article should have a circle with key sigs. The nicer circle images, IMO, had them, so those were the ones I cited. Further, I believe they help explain the otherwise arcane naming and sequencing of the key sigs.
The 800px table does not conform to MOS:MUSIC#Images, which is a guideline, true, but it overwhelms the article on my 1280x1024 display — essentiallly cutting the article in half. At 800x600 it forces horizontal scrolling. Further, it is very busy and difficult to interpret. Last, it is not clear whether it is original research or not. At least one editor, in addition to me, would like to know. BTW, I was wondering why you removed a {{fact}} tag directed at this table.
--Jtir (talk) 17:49, 24 July 2008 (UTC)
I like this deluxe one better than that one. The arrows get in the way, and the pink and green reminds me of a late 60's or early 70's rumpus room. I'll undertake to translate the deluxe one into English, and in the process, I'll probably use red for major and green for minor text, both pretty saturated, since that fits my twisted synaesthetic view of things. That said, I think it's best to keep things simple in the lead section. That little PNG is nice too, but has symmetry and alignment problems. __Just plain Bill (talk) 18:10, 24 July 2008 (UTC)

Here's an English version of the "deluxe" circle:

Caption it any way you like...

__Just plain Bill (talk) 05:26, 25 July 2008 (UTC)

Thanks, Bill. That's very nice.
Placing smaller circles (and ellipses) around the main circle nicely highlights the twelve positions. Was it your intent to emphasize the number of accidentals that way?
I like the way you shaded the clef. I now see that the other versions were missing it. Good catch.
The QzD aligns the staffs with the enharmonic key sigs differently. Can you see any significance to that alignment?
Using a natural sign for the key of C is logical and musical, but zero is a number too! :-) The viewer has to infer that there are no accidentals and infer what the key sig looks like.
The red/green color scheme could cause problems for viewers who are color blind.
--Jtir (talk) 13:33, 25 July 2008 (UTC)

Here it is as a PNG, with my thoughts on your feedback:

This file is a lot smaller
  • A lot of the circles I see have numbers, with line of the main circle separating major & minor, or separating major from the numbers of sharps/flats. I see the small circles as a way to give the numbers a bit of their own space, using the numbers as a radial link between the Major and minor key letters. I see the image as rays coming out through the circle, each ray being a cluster of info about number & placement of sharps/flats, and two of the relevant modes in each case. For that matter, I believe we could stand to lose all that ink in the main circle, de-emphasizing those numbers a bit, and still have a readable chart. I'll try that next.
  • Yeah, thanks; the world I live in is not very G-clef centric. ;-)
  • Not sure I see much of a difference there; still lined up pretty much radially... ??
  • Ah, the natural sign. I pondered for a while about that. What I hope that does is put the dimensions on the numbers in the circles. They're not just numbers, but numbers of sharps or flats. If the viewer doesn't know what C Major's key sig looks like, a. there are other ways for them to find out, and b. doing the mental work of the inference will help fix the info in their ever-changing brain. Musical and logical is what it's about here.
  • The whole image could just as well be B&W. I purposely gave the red & green a fairly dark value, so translations to grey-scale, whether mechanical or biological, would still render well. __Just plain Bill (talk) 14:15, 25 July 2008 (UTC)

Regarding your first point: the main circle is now 20% grey, and the little circles are gone. I like that a lot better. Comment? __Just plain Bill (talk) 14:41, 25 July 2008 (UTC) and by the way, I left off the enharmonic spellings except for the one at position 6, which suggests the rest. I figure that by the time someone is interested in wickedly sharp/flat keys, they'll be able to fill in the "missing" info for themselves, and it made for a crowded chart. __Just plain Bill (talk) 14:50, 25 July 2008 (UTC)

Guess I should have been more careful about what I said WRT the beaded version. The small circles look like they lie above the CoF ring (main circle), giving the image a slightly 3D quality that I like. Without the beads that quality is lost and the ring becomes choppy dashes. Further, the beads give each cluster a central focus (as you noted). Graying the ring slightly might actually enhance the 3D effect, because it would seem to recede in the distance. (Using different line weights, as you did, also helps.) Maybe the 3D effect could be extended to the entire cluster.
I see your radial design now. Here is a circle that actually uses central rays and lacks a main circle (not to mention a few other refinements).
I like your use of color (sorry I didn't say so before), because it helps to distinguish and unify the modes.
--Jtir (talk) 15:28, 25 July 2008 (UTC)
Never be sorry, neither regret nor apology. Darn, sorry I started this in the imperative mood. I could go on in that vein, and I shouldn't. Yeah, distinguish and unify the modes is what those colors are meant to do. I'll spend a day or so tweaking the bubbles or beads to see if anything nifty shows up. Dedicated lay-disciple of the great Edward Tufte that I am, I believe that most non-data ink should be expunged, leaving only enough to serve as error-correcting redundancy, e.g. after generations of photocopying have had their way with the image. __Just plain Bill (talk) 16:34, 25 July 2008 (UTC)
I have invited a few editors to take a look, so we can get some more input.
I'm not sure what "ink" means when images are digital, and digital images do not degrade when copied, although rendering and resizing can degrade them. I was shocked that the PNG image had so much less contrast and looked so fuzzy in the article compared to the previous version. Not sure what to do about it (I couldn't find a way to get Inkscape to export a bitmap image with anti-aliasing disabled as a test.).
--Jtir (talk) 19:00, 25 July 2008 (UTC)
The previous version is thumbnail-sized to begin with, so it doesn't lose anything being decimated down to 180px wide, or whatever the default is. I'll sit back and watch for a while now; thanks for your involvement here. I mentioned the usage of "ink" in this context on your talk page. __Just plain Bill (talk) 21:06, 25 July 2008 (UTC)
Idea: use a double-gradient to "fade-out" and "fade-in" the circle as it passes "behind" the text. This might reduce the dashed effect, which is caused by the abrupt ends of the arcs, while not interfering with the text. I have some minor technical comments on the SVG. Should I put them here or there? --Jtir (talk) 11:19, 26 July 2008 (UTC)
Better on the Commons, I think. You can use the earlier SVG as a base for your edits; the only change was to lose the stroke paint on the bead circles, and expand the stroke width and reduce the grey value of the main circle (or was it alpha? either way works, since there's a white layer behind everything.) __Just plain Bill (talk) 19:52, 26 July 2008 (UTC)
Idea2: expand the grey circle to an annulus whose bounds do not intersect any text. A problem with the circle is that it conceptually intersects the text, causing visual confusion . The annulus is a feature of this one that I like. I was going to experiment, but noticed the SVG version has not been updated. --Jtir (talk) 16:55, 26 July 2008 (UTC)
I've been playing with some things, including something annular, and when they're fit to show, you'll see them. Meanwhile, and as a thank-you gift to all who helped just lately, check this out, and this. __Just plain Bill (talk) 17:12, 26 July 2008 (UTC)
Less choppy?

There's a first try at it with the bubbles and main circle shaded differently. I still like the unadorned "choppy" circle better. Anything beyond the simplest presentation starts saying things like "look at what a skilful artist I think I am, and all-round clever fellow!" Believe it or not, some people find that annoying, and pay less attention to the content as a result.

It's uploaded as a PNG; the parent SVG will be about four times as big if I turn the fonts into paths so they render equally (instead of as Arial) on all machines. I've used Garamond for the letters and slashes, and NWC15 (Noteworthy Composer's font) for the sharps, flats, and natural sign. __Just plain Bill (talk) 05:15, 27 July 2008 (UTC)

Thanks, Bill. That is a very interesting use of gradients. The shaded bubbles are less obtrusive than the circles, yet still provide contrast. However, they tend to over-emphasize the numbers. I believe the number of accidentals is the least important feature of this diagram and don't really understand why they are needed when the key sigs show that info. Do musicians really count accidentals: "Bill, this is in the key of 3 sharps, but I can only play the white keys." :-) The shaded ring looks continuous, however the 3D shading does not convey useful information (Tufte speaks! :-)). I prefer to think of this as a 2.5D graphic, which is 2D information stacked in layers with the most important info in the top layer and the least important receding into the background (somewhat like a stack of transparencies).
Good points WRT the fonts (the Garamond looks great) — I'm fine with using the PNG version operationally.
--Jtir (talk) 10:23, 27 July 2008 (UTC)

I might not call those numbers the most important feature, but they are certainly not the least important part of it. Depending on how I'm looking at the chart at any given moment, and what for, I'd even promote them a bit over the major or minor key names. Number of sharps or flats comes up all the time. "This piece is in 2 sharps, J; is it D major, B minor, A mixy, E dorian, or what?"

The key sigs have been called redundant elsewhere, and the editor that said that is not wrong. Where redundancy exists, which element is the redundant one? Philosophical musings aside, the data in this chart is massively redundant, and it all serves to help the viewer understand the particulars of the effects of a handful of simple rules. Simple doesn't always mean easy.

By the way, accidentals, strictly speaking, are sharps or flats or naturals that occur outside the key signature. Piece in two sharps may use accidentals to sharp some G's, for example, effectively straying into A major for a while, before coming back home to D major. Similarly, an A major piece in three sharps might use accidental naturals on G's or C's to give it an A minor feeling for a while. It happens. Accidentals also get used here and there to make chromatic runs, which have their own set of effects. best, __Just plain Bill (talk) 10:52, 27 July 2008 (UTC)

Thanks for the usage info. You've convinced me. Why are the numbers in a ring between the note/key rings, instead of in a ring closer to the key sigs? Moving them out would put related info closer together and allow the circle to be an unimpeded (though discreet) separator between the mode rings. Your earliest design avoided the problem of interference between the ring and the text by using two rings, neither of which intersects any text. These could be interpreted as a schematic annulus or as two separators. --Jtir (talk) 14:06, 27 July 2008 (UTC)
The CoF's I've seen mostly have
  • just note names going around, with C on top, or
  • note names (or keys) on the outside, with numbers, sometimes accompanied by sharp/flat signs, on the inside, or
  • major keys, numbers, and minor keys, from the outside in, sometimes showing key signatures outside the circle as well, for lagniappe.
This being an encyclopedia, I'm not about to put myself in the business of inventing something new to report on a well-established presentation of musical theory. Many an old engineer has said to a junior one, "I can't understand this drawing." meaning approximately, "You'll have a better chance of getting through to me if you use the conventional graphical language I already understand, instead of trying to invent a new one."
Musical notation may not be as old as writing itself, but by now it is pretty well-established as its own (international, to a fair extent) language, with symbols and syntax that have been reasonably stable for a few centuries now. That said, the minor/numbers/major/signature layout works for me, putting the number of "signs of chromatic alteration" next to the names of both the major and minor keys. __Just plain Bill (talk) 00:36, 28 July 2008 (UTC)
OK. I believe I know why the PNG thumbnail looks so fuzzy compared to the SVG thumbnail (easily compared above; I moved the PNG to the left) — the PNG image has been antialiased by Inkscape during bitmap export and it has been resized. The SVG image has been rendered directly on the WP server at the thumbnail resolution. How would you rate the relative image quality of the two thumbnails above? --Jtir (talk) 12:35, 28 July 2008 (UTC)

These versions reduce the size of the "Major" and "Minor" labels and vary the way the circles are laid out. --Jtir (talk) 23:48, 28 July 2008 (UTC)

Nice work, I like the porthole one. I tweaked it by re-centering the "Minor" label and moving it up a bit. At the larger text size, the upper-case "M" unbalanced things when it was centered perfectly, so I had skootched it a bit to the right. Stroke width and size of the "circle" also got tweaked. It might now be an ellipse, slightly wider than tall. Interestingly, you'll find that some things, the key signatures especially, are not exactly lined up with their dots. The alignment dots in the hidden layer were precisely placed at (r•cosθ, r•sinθ) and everything started out centered on its dot, then got fudged around til things looked right. I think it was Edgar Degas who said painting is the art of deception. ;-) __Just plain Bill (talk) 23:55, 28 July 2008 (UTC)
Just resized the thumbs in the gallery to match the article, and enlarged the major/minor labels in the tweaked one. Going live with that now... __Just plain Bill (talk) 14:19, 29 July 2008 (UTC)

I don't have much to add here, just that encoding information in a red vs. green distinction is problematic for people with red-green color blindness. — Gwalla | Talk 05:14, 29 July 2008 (UTC)

I think of the color as lagniappe, much of the encoding being done by their placement in/outside the porthole frame, with labels nearby. I'll read a bit more and sit with this for a while-- red for major and green for minor fits my own modal view, but open to suggestion for colors more easily distinguished by dichromats. __Just plain Bill (talk) 14:19, 29 July 2008 (UTC)
That's true. It's like how red and green traffic lights are distinguished by position. And I'm not sure if the distinction as it appears in this image is not noticeable to dichromats; they look to be about the same darkness to me, but I'd have to pass it through a red-green color blindness simulating filter to be sure (as I myself am not color blind). — Gwalla | Talk 17:24, 29 July 2008 (UTC)
Gwalla, I was wondering about the colors too.
Bill, thanks for tweaking the text — that looks much nicer. While experimenting with the circles, I realized that the text was irregularly positioned wrt the circles, but since I don't know anything about typesetting, I am happy have you do that.
IMO, the "bullseye" more effectively reveals the radial design, which seems to pop out at me.
Has anyone else noticed that effect? (I believe that the circles somehow counter the irregular sizes and shapes of the text.) --Jtir (talk) 19:12, 29 July 2008 (UTC)
The rays do pop out of the bullseye design, but I have to make an effort to see a unifying circle in the composition there. In the porthole design, I can see it either way easily. __Just plain Bill (talk) 21:37, 29 July 2008 (UTC)
The two circles tend to fuse into an annulus, but they also guide me around the circle. --Jtir (talk) 18:03, 30 July 2008 (UTC)
The two circles tend to fuse into an annulus, if you have sharp eyes, are close to the screen, and make the effort. They disappear if I stand back 6'~7' from the screen, while the composition of the necklace and porthole versions stays readable a lot further than that. From across the room, the bullseye one starts to resemble a dart board without much of a middle ring. A very pale grey fill between circles would help that, but then you're getting back to a porthole presentation. __Just plain Bill (talk) 19:02, 30 July 2008 (UTC)
OK, the circles disappear and the rays are much more prominent than in the others. However the (minor) text at the center tends to fuse into an inner circle around the top inside. At the bottom, the enharmonic equivalents become heavy clumps (I believe Gwalla noticed the same — not sure what to do about it). Further, the missing key sig at the top disrupts the circle there, and the minor label becomes a light green wart. This distance viewing is more revealing than I expected — are you doing your 6' viewing with the 183px res. images? (I'm not sure why Hyacinth reduced them. This isn't an article page and we should be evaluating them at the article page resolution.) "grey fill": The bullseye design is intended to eliminate visual interference between the text and other graphic elements. "dart board" <g> Would you like me to add a small disk at the center? --Jtir (talk) 19:53, 30 July 2008 (UTC)
I'm OK with the 183x183 gallery, and yes that's what I was looking at. It fits three images on the 800x600 screen I can get to most easily in the evening shift. Music is all about warts and blivets, and how they can be consonant in the context of a quasi-ideal system, if I may wax philosophical for a tick. The word "Major" sort of fills in for that missing key sig; keys with lots of blivwidgets in them (more than 3 flats or 4 sharps) always seemed kind of clumpy to me anyway, so I don't have a problem with the lower part of the chart. In musical reality, it is what it is. Consensus works OK for me; you've pretty much seen my 2 bits by now... __Just plain Bill (talk) 20:30, 30 July 2008 (UTC)
Well, one more bit: I'm not sure I'd be happy with a perfect featureless circular arrangement. As it stands, the various lumpinesses and other asymmetries provide topographical cues to what's where. There may be idealists who would like to claim otherwise, but all keys are not equivalent. D major and B flat major both fall easily under the hand on a violin in standard tuning, and manage to be different, possibly based on the resonances of the box, its air cavity, and the open strings. Likewise E minor and G minor, one sharp, the other flat. __Just plain Bill (talk) 21:16, 30 July 2008 (UTC)
--- and yet another bit: At most zooms, I see the "Minor" label as forming a supporting bowl-like structure around C, with F and G being the limbs of the cup. Fits quite well with my notion of the basic three-chord harmony with the I chord supported by the IV and the V. That said, I could just as easily favor the bullseye arrangement, with the following issues:
  • The "Major" and "Minor" text could stand to be enlarged a tad, maybe not as much as the porthole one, but still...
  • Those same labels sit outside the annulus, separated from their referents.
Six, or a half dozen? Hard time choosing... __Just plain Bill (talk) 18:59, 31 July 2008 (UTC)
I think the "pale" presentation is the best in terms of organization, as it avoids clutter at the bottom and clearly groups everything. The only problems are that the text is much smaller, the bold sans-serif font tends to blur into indistinct black blobs, and it's too, well, pale. Some more contrast would make the dividing negative-space "spokes" more obvious. (Also, salmon next to fluorescent green is kind of hideous). The spiral presentation probably shouldn't be grouped as an alternative, because it doesn't illustrate quite the same thing (although elements of it could potentially be worked in). — Gwalla | Talk 22:07, 31 July 2008 (UTC)

Thinking about this a bit more, it seems like the number of sharps and flats is redundant with the key signatures. The paired numbers (for keys that can be spelled with sharps or flats) also get a bit crowded down at the bottom and obscure the groupings. — Gwalla | Talk 01:58, 30 July 2008 (UTC)

Bill reports above that musicians commonly refer to the number of sharp or flat signs: "This piece is in 3 flats." Would you say something like that? I agree that the enharmonic notes and key sigs disrupt the regularity. Can you suggest another way of displaying (or not) them? There could be variants of this CoF that are less than deluxe — maybe "light" and "standard" versions. :-) Bill has put the significant features (e.g. the key sigs) in separate Inkscape layers, so they can be easily hidden or removed. There are a lot of CoF variants on the web: (# of signs in key sig inc. zero)(# of signs, zero)(key sigs, arrows)(basic, arrows)(harp)(clock)(mandala)(interactive) … --Jtir (talk) 18:24, 30 July 2008 (UTC)
In the full deluxe presentation, with a little visual effort, following the sharp keys clockwise or the flat keys widdershins, one may see the beginnings of a spiral there. __Just plain Bill (talk) 19:41, 30 July 2008 (UTC)
While I've got an edit window open, let me restate the "three flats" thing here in context. It's pretty easy to tell that a piece is in three flats at a single glance from a fair distance. It takes a bit more scrutiny to tell how much time that piece spends in F dorian or A flat mixolydian or what, or whether the composer got all ambiguous about their tonality.
Music and ambiguity are by no means strangers. I think it was while plowing through Douglas Hofstadter's ponderous brainy book that I saw mention of the pianist (Glenn Gould?) who played some J.S. Bach piece so evenly that it was impossible to tell by listening what meter it was in, or more to the point, that the listener could mentally impose various meters on it. Herr Bach may even have intended it so. <g> __Just plain Bill (talk) 21:16, 30 July 2008 (UTC)


Someone capable of it could also make a representation of the circle of fifths on the chromatic circle as a star dodecagon as in "Prelude to Musical Geometry", p.364, Brian J. McCartin, The College Mathematics Journal, Vol. 29, No. 5 (Nov., 1998), pp. 354-370. Hyacinth (talk) 01:35, 29 July 2008 (UTC)

Does this quick sketch work? __Just plain Bill (talk) 04:34, 29 July 2008 (UTC)
Yes, like that! Hyacinth (talk) 04:52, 29 July 2008 (UTC)
Thank-you both. The paper sounds interesting and the image is informative (i.e. I didn't know that :-)).
The full paper doesn't seem to be online, so here is an abstract and the JSTOR page.
BTW, here is a star dodecagon in Kepler's Harmony of the World (1619).
--Jtir (talk) 18:50, 29 July 2008 (UTC)
McCartin (1998) in a figure to the immediate right of the star dodecagon, illustrates the creation of the diatonic scale from a chain of seven perfect fifths. Hyacinth (talk) 19:54, 29 July 2008 (UTC)
Not sure what this would look like. Does it feature a circle? --Jtir (talk) 15:16, 30 July 2008 (UTC)

Removed: In coordinate system[edit]

Pitch classes in a coordinate system

In order to understand and memorize the relations between pitch classes easier, it is possible to put all the keys into a Cartesian coordinate system according to their signatures[citation needed].

In the resulting graph, the X coordinate represents the number of minor key signatures, treating the number of flats as a negative (as it makes a note negative). The Y coordinate represents the major key signature count the same way. C is therefore in (-3;0) as C minor has 3 flats and C major a 0 (zero).

When minor and major keys are paired into single dots, various relations become visible more clearly from the graph:

  • Minor and major key signature amounts are always separated by 3 units.
    • Examples: if we know that C major has 0, we get that C minor would have 0 + 3 = 3 flats. If G minor has 2 flats (-2), G major would have -2 + 3 = 1 sharp.
  • The amount of signatures in a sharp/flat key is always 7 more than the same note key without sharp/flat tonic.
    • Examples: if C major has 0, C sharp major would have 7 sharps. If C minor has -3, C sharp minor would be -3 + 7 = 4 sharps.
  • The relative key is also easy to find. If a relative is needed of the C major, which stands on y = 0, we look for a key that stands on 0 of the minor axis. In another example, the relative of B minor (x = 2) is D major (y = 2).
  • The order of signature placement is the same as the key order on the axes. If the key goes down becoming flat, flats are placed from B downwards. Respectively, sharps are placed from F upwards.
  • It is also clear that all of the keys tend to recur every 12 units because of the chromatic scale consisting of 12 notes. Therefore a major key with 120 sharp signatures would remain C major.

The above was removed as it appears to be textbook like original research. Hyacinth (talk) 16:01, 24 July 2008 (UTC)

I agree, that the above section was absolutely not helpful. It´s so strange and complicated, that I can not even guess, if it makes any sense.
Hexadecimal (talk) 16:24, 24 July 2008 (UTC)


Do anyone agree that the entire article could benefit from context? Not just the "Effect on diatonic function" section? Hyacinth (talk) 16:28, 24 July 2008 (UTC)

Yes, although moving that section gave it a bit more context, IMO. __Just plain Bill (talk) 17:33, 24 July 2008 (UTC) hit "save page" too soon here... with your recent edits and others, I think the article is coming nicely into shape, and has plenty of context for the likes of me. Regarding that, I think it's at a point where a lot of other eyes will be helpful. __Just plain Bill (talk) 17:55, 24 July 2008 (UTC)

Request regarding Circle_progression_in_major.png[edit]

Hyacinth, would it be possible to harmonize this example using root position chords? I think it's important to also see and hear the descending fifth/ascending fourth motion in the bass. Thanks- --Blehfu (talk) 15:38, 26 July 2008 (UTC)

Done. Hyacinth (talk) 00:52, 27 July 2008 (UTC)

Deleted: unrolled circle with Pythagorean comma shown[edit]

I just removed it from the article for the following reasons:

  • I have not seen these symbols used elsewhere.
  • As a result, I find the image difficult to decipher.
  • The background color and other chartjunk gets in the way of my comprehension of this graphic.
  • Sometimes the only computer available to me has an 800x600 pixel CRT display. The "unrolled" graphic doesn't all fit on that screen, requiring sideways scrolling to view "the big picture."
  • This info is more relevant to Pythagorean comma, which now appears in the "see also" section here.

__Just plain Bill (talk) 00:17, 27 July 2008 (UTC)

More pictures[edit]

Here are some images I made for comparison. I can't figure out at this moment how to even describe them (ascending vs descending, fourth vs fifth), and they all seem equally of value. The top two are currently in the introduction to the article. Hyacinth (talk) 01:38, 27 July 2008 (UTC)

Thanks for bringing this up. I had a similar problem when trying to explain this to a friend.
IMO, the simplest way to illustrate the CoF is to start at the lowest musical pitch possible and ascend in pitch by fifths until a harmonic of the starting pitch is reached. If I understand Pythagorean comma correctly, this way would span seven octaves.
A second way, which is what I believe you are doing, is to constrain the pitches to one octave (or two?), ascend from a starting pitch by fifths until the pitch falls outside the octave, replace that pitch with an equivalent pitch that falls within the octave, and continue ascending.
BTW, the article talks about ascending by fifths on a piano, but it never says that doing this will require the full width of the keyboard.
--Jtir (talk) 13:36, 27 July 2008 (UTC)
Thanks for the additional pictures and annotation. We might want to indicate in the caption whether the movement is clockwise or counterclockwise around the CoF (maybe CW or CCW to save space).
The description of the first picture puzzles me though. ISTM that it is "1 octave, fourths, ascending".
If editors feel that it would be useful to include all of these, it might be possible to put some of them into a {{Dropimage}}:
Traversing the circle of fifths
I haven't used these before now, so I don't know what could go wrong.
--Jtir (talk) 20:02, 27 July 2008 (UTC)
I added all of these in a dropimage, but made no other changes except for removing the asterisks around "multi octave, fifths, descending". In the context of the article, it might be better to put "multi octave, fifths, ascending" first. --Jtir (talk) 16:27, 31 July 2008 (UTC)

CoF slide rules[edit]

  • "You can buy linear or circular slide rules based on the circle of fifths; they give you the key signatures, the principal chords and so on." [1]
  • TEACHING AID FOR DEMONSTRATING THE HARMONIC SYSTEM (US Patent: 3791254. Issued: Feb 1974) (linear design) [2]
  • Music magic circular calculator (US Patent: 4074607. Issued: February 21, 1978) [3]
  • Chord display device for fretted string instrument (US Patent: 4503748. Issued: Mar 12, 1985) (circular design) [4]
  • The Chord Wheel: The Ultimate Tool for All Musicians (Hal Leonard. 2000) (color) [5]

--Jtir (talk) 19:00, 30 July 2008 (UTC)

Introduction: Accidental sign[edit]

Was including "accidental" or "accidental " useful? It was not strictly necessary. I'm not sure, however, why one would refer to musical keyboard when removing the term. Also, accidentals do not necessarily refer to black keys. Hyacinth (talk) 03:42, 1 August 2008 (UTC)

See also Talk:Accidental (music)#Inflections vs accidentals. Hyacinth (talk) 04:01, 1 August 2008 (UTC)

That mention of black keys in my recent edit summary came in the context of a discussion that Jtir and I had regarding what collective term to use for those signs as a category. I maintain that since there are so few types of them, that "sharps and flats" is adequately compact and universally understandable, natural signs being understood to be fully included in the group. (Varying degrees of membership in a category may be found in George Lakoff's book, Women, Fire, and Dangerous Things: What Categories Reveal About the Mind, University of Chicago Press, 1987, ISBN 0-226-46804-6, which I consider relevant to a deeper discussion of "do we even need to find a collective term for sharps, flats, and naturals?")
Since I enjoy playing with language, I'd be happy to find that there is such a term. "Accidentals", as I've said elsewhere, I believe should be reserved for their application outside a key signature, based on my own experience of that usage, Wikipedia's own Accidental (music) article, and lots of dictionary cites, including the paper OED sitting in the next room. The closest things I've seen so far are:
  • Signs of chromatic alteration -- accurate, understandable, but unwieldy
  • Accidental signs -- accurate if you think about it, but casual readers might miss the "collective group" sense of it
  • Inflections -- whose sole merit is that it's a different word. Most musically literate people, I believe, will go "Huh!?! What's that supposed to mean?" when they see it.
Again, I'd be happy to find such a term, but also surprised. I haven't seen such a thing yet, and I've been looking. For me, "sharps and flats" (with naturals understood as full members) or, maybe not quite so aptly, "black keys" do the job. __Just plain Bill (talk) 13:20, 1 August 2008 (UTC)
I hope that if I check Talk:Natural sign I won't find a discussion about how the term "natural" is inappropriate given that they are often used as "accident"als, and are thus far less natural than the original pitches! (on second thought, I hope I do!) Hyacinth (talk) 20:54, 1 August 2008 (UTC)

comments on the lead[edit]

Comments refer to this version of the article:

  • "The lead should be able to stand alone as a concise overview of the article." (WP:LEAD)
  • Why is it a circle and why is it called the Circle of fifths?
  • Why in the adjacent diagram do the letters go C-G-D-… and how does anyone remember them?
  • Who invented the CoF and why?
  • The lead seems to be giving three definitions of the term: theoretical, graphical, genetic. ISTM, that it would be better to go from concrete to abstract: graphical, genetic, theoretical.
  • The graphical description implies that the arrangement is based on the number of accidentals. IIUC, the arrangement is based on the sequence of fifths, and the number of accidentals is a consequence, because the CoF determines the tonic of a scale and that scale determines the number of accidentals. (I had long wondered why the accidentals are added to the key sigs in the order that they are. The CoF helps explain that order and the lead could say so.)
  • The genetic def describes the impression given when traversing by fourths. What is the impression given when traversing by fifths?
  • There is no indication as to why a reader should "See cadence" (this needs to be formed into a sentence.)
  • The lead does not say that the CoF can be used to construct and explain chord progressions and modulations. (Not sure about the usage here: e.g. "construct modulations"?)
  • The lead does not mention the history (The history section is only two sentences, but the history should still be mentioned in the lead for perspective and context.)

--Jtir (talk) 19:15, 3 August 2008 (UTC)

  • "Why in the adjacent diagram do the letters go C-G-D-… and how does anyone remember them?"
Excellent question. Not sure it needs to be addressed at any great length in the lead, but my answer is "after you've used them a lot, they come naturally." Playing a stringed instrument tuned in fifths or fourths helps give a tactile/visual structure to the memory. One of the night guys at my evening job is a beginning guitarist, and is asking exactly that question. Last month he looked at this article, and found it incomprehensible. I've pointed him at it again, in hope that it may help. Another day/evening person there, a low brass player, said she just put up a poster of the circle where she could see it often, and let it become familar that way. I'm thinking the guitarist may wind up using one of our pics as screen wallpaper for the same purpose.
I'll be ruminating on your comments... __Just plain Bill (talk) 21:26, 3 August 2008 (UTC)

It's easy to remember the order because it is called the circle of fifths, and so you just have to count if you forget. If you are on A, for instance, you count up to E.

          = Fifth
1 2 3 4 5

Hyacinth (talk) 22:51, 13 August 2008 (UTC)

What happens at B?
1 2 3 4 5
--Jtir (talk) 15:33, 14 August 2008 (UTC)

On the piano there's a visual cue -- all the perfect fifths are either white-white or black-black keys, EXCEPT at the crossroads surrounding C - - the perfect fifth is B-F# and/or Bb-F. it's a really important thing to learn for a keyboardist, that weird spot. if you are a teacher you can use all kinds of metaphors, i'm fond of 'going from the field into the forest' [crossing over from wh to bl] or somesuch. i'm sure there is a great poem in all this.

right, and, yup, you basically just have to memorize it, and usually you do that in connection with your instrument. But it doesn't exist in a vacuum - - if you are playing basic western/eurocentric melodic music, the interval of a fifth is extremely common, so they are 'under your fingers' already - -- then you just group them in a more logical order. Magnolia63 (talk) 01:44, 15 August 2008 (UTC)

'going from the field into the forest' <g> You sound like a really good teacher.
Could you take a look at the section In lay terms? (It uses the piano to explain the CoF.)
ISTM, that it could be more specific about where to start on the keyboard and say what you did above about the white and black keys. IIUC, one loop around the CoF is seven octaves, so how do you play it without running out of keys? Could the number of keys on a piano be explained, in part, by the CoF? There are several sheet music examples in the section. Are they realistic examples of what a piano student might practice?
--Jtir (talk) 15:01, 15 August 2008 (UTC)
If you start on the bottom C on the piano (C1), seven octaves will bring you to the top C (C8). This will work starting on A, B{{music|b}], B, and C for pianos with the standard 88-key range. Double sharp (talk) 08:41, 8 February 2015 (UTC)

--This is a way of remembering:

Clockwise Kindly (0#) Give Dad An Electric Blanket, Father Christmas! CGDAEB,F#C#
Anticlockwise 'Cos (0♭) Former Blanket Exploded And Dad Got Cold. CFB♭E♭A♭D♭G♭C♭

The comma divides the signatures with # from those without - CGDAEB, F#C#. Going anticlockwise all the signatures after f have a flat ♭. Wodorabe (talk) 16:36, 7 April 2017 (UTC)

Removed: Simpler terms[edit]

Saying the same thing in simpler terms: there are twelve "major keys," which means that the sequence "do-re-mi-fa-so-la-ti-do" can be played or heard starting on twelve different pitches or tones. There are only twelve, because there are only twelve different tones (or half steps) before the tones would repeat again. (This is seen easily on a piano: pick any note, and play every black and white key, counting to twelve, and you'll see that the 13th note is the beginning note again.)
The circle of fifths is a functional way to organize and understand these twelve major keys. It can also be thought of as a representation of musical principles that naturally exist (the sound of the Circle of Fifths is common in much of Western music.) Instead of listing those twelve keys in simple sequential order (i.e., C, C#, D, D#, E, F, F#, G, G#, A, A#, B), the sequence is altered so that the interval of a fifth (in one direction) or a fourth (in the other direction) is between each key (i.e., C, G, D, A, E, B, F#/Gb, C#/Db, Eb, Ab, Bb, F, and back to C - - or in reverse). When the keys are arranged thus, musicians can access and use information much more quickly and clearly than they might if they keys were in random or simple sequential order.

I removed the above because I do not see how is it simpler to introduce more than eight new technical terms (major, do, etc.). Hyacinth (talk) 22:50, 13 August 2008 (UTC)

Hi, I'm a newbie wikieditor, I'm a piano teacher, and I recently told a student to look up "Circle of Fifths" on wiki cuz he was having trouble conceptualizing it. So today I thought I'd look and see...and after nearly 40 years of playing the piano, I had trouble making heads or tails of this article. It seems to be written for advanced music theorists. I know it would scare away my poor student.

Thus, my attempt (writing the above text) was to provide something for even a beginning musician to grab onto.

I think maybe your objection to the new "technical terms" would only apply to someone who has never had a 5th grade music class or somehow successfully avoided the Sound of Music. Yes, I grant you, they are musical terms....but how can we talk about music without musical terms? The terms I used are far more common and accessible than those first two paragraphs. (honest question: How do we parse whether a wiki reader really has never heard of do re mi??)

I realize now this conversation has been going on for a long time. I feel committed to joining the team here and working on this article to make it accessible. I think there should be a starting paragraph with the simplest possible terms. For example -- using the simpler term "visual" instead of the immediately complex "geographic" in the first sentence would improve things immensely imho. OK - I am going to attempt a first pass at that kind of edit. let's keep talking.

Magnolia63 (talk) 02:53, 14 August 2008 (UTC) further thought...the terms "major" and "minor" are right there on the diagram. Magnolia63 (talk) 03:18, 14 August 2008 (UTC)

What about people who have never had "a fifth grade music class", or what I assume they taught when you where in fifth grade? Hyacinth (talk) 05:20, 14 August 2008 (UTC)
Not here to validate or refute anyone's assumptions. Elsewhere, Hyacinth, you've mentioned that Wikipedia has or had a policy about being understandable by a lay reader. I've been looking here and there for particular guidance about that. Got any further pointers?
Pretty sure we don't need to assume the audience is a man from Mars who needs every little thing laid out in exquisite agonizing detail, and it's plain that we're not writing for folks with advanced music degrees either. Put more bluntly: How far should we dumb this down, according to Wikipedia policy? (It ain't "My Little Golden Book of Songs and How to Sing Them" here.)
Forgive me if that seemed harsh. I want to reach the ones who are looking for info here, in a way they understand, in a way they can use effectively, which means not turning them off with a lot of densely packed jargon. But... by the same token, I don't believe we should be in the business of spoon-feeding the audience at an elementary level. That's one thing the hyperlinks are for. Comment, anyone? __Just plain Bill (talk) 12:33, 14 August 2008 (UTC)
Wikipedia:Explain jargon is the closest I can find. Hyacinth (talk) 00:04, 16 August 2008 (UTC)
Wikipedia:Make technical articles accessible. Hyacinth (talk) 00:08, 16 August 2008 (UTC)

Hyacinth - - yes I was making a huge leap re the 5th grade music class - - I assumed someone who was looking up CoF would cross-correlate with either "Having seen the Sound of Music" [age, gender, class race marker] and/or "Having Had some basic music instruction, ie 5th grade music class curriculum." [privilege marker] Yet, I think that might be a valid marker/benchmark, though it is worth imagining: OK who would access this article? who is the Lowest Common Denom.?

it's a really good question about 'how much to dumb down.' I use wiki alot for reference and most articles give a really solid ground/context before leaping into the technical info. With music it's a little confusing - - do we try to put the 5th grade music class on the CoF page, or somewhere else with links...or do we say in the very first sentence: "this is a somewhat advanced musical concept. Please refer to these links, x,y,z, for background." ?? (is that ever done??) Magnolia63 (talk) 01:53, 15 August 2008 (UTC)

Introductory music theory books give a good idea what the prerequisites are:
Miller, Michael. The Complete Idiot's Guide to Music Theory, 2nd ed. Indianapolis, IN: Alpha, 2005. ISBN 1592574378.
Aikin, Jim. A Player's Guide to Chords & Harmony: Music Theory for Real-World Musicians. Backbeat music essentials. San Francisco: Backbeat Books, 2004.
  • Chapter 11 (of 21): Key Signatures and the Circle of Fifths.
Pilhofer, Michael, and Holly Day. Music Theory for Dummies. Hoboken, NJ: Wiley, 2007.
Magnolia63, when and how is the CoF first taught to music students?
--Jtir (talk) 14:03, 14 August 2008 (UTC)

Hey people, this article is really shaping up. I like the new edits to my edits. go team. I'll have more to contribute another day...been chewing on this quite a bit...

to answer Jtir: it depends on the teacher and their curriculum. Some [many] never teach it. Some teachers don't use or understand it. Some curricula (the famous kid books like Schaum) put it right in Book I. (but sometimes it's too soon and overwhelming)

I have been engaged in developing a curriculum which provides a broad base of knowledge so that my students can branch out in a direction that interests them: classical, pop, folk, blues, or jazz. If you are going to play jazz (imho) you must understand the CoF. So I start even the youngest kids on the CoF on the piano, which btw has a beautiful logic: if you start on middle C and move "up" or to the right, you get to F# (across the 'clock' from C) - - and then if you go left from C down, you finish the circle to Db. So tidy. [oh wait i think this is already on the wiki page isnt it?] Anyhow -- you don't need to know what it "is" in its vastness to memorize and play it...lights will pop on later. I like the idea of locking it in in the very early development far it's working.

also - to addrss the "clock" question: I just expanded that idea b/c the previous editor had used that term -- but again my students find it helpful. We even say -- "What's One Sharp O'Clock?" So yeh clockface is a fine term but i'm not attached. Magnolia63 (talk) 01:33, 15 August 2008 (UTC)

Thanks for the background. What topics do you consider prerequisites for understanding the CoF? (e.g. notes, scales, intervals, key sigs, etc.) ISTM, that the level of the lead should start by assuming such prereqs. and then become more technical, a sequence not yet fully apparent.
The clock/clockface analogy is used here ("… like the numbers on the face of a clock.") and in Music Theory for Dummies (p. 136). It also meshes with descriptions of the clockwise and counterclockwise traversals of the circle. Here is a literal clock.
--Jtir (talk) 15:43, 15 August 2008 (UTC)

What do "up" and "down" mean?[edit]

Another "man from Mars" question, perhaps... do we need to explain here (or give a link to an explanation of) what "higher" and "lower" mean in terms of pitch? The difference isn't always clear-cut. __Just plain Bill (talk) 14:39, 14 August 2008 (UTC)

Just read the first few sentences of Tritone paradox and came away not knowing what it actually is — perfect example of not stating the obvious. --Jtir (talk) 15:27, 14 August 2008 (UTC)
Talk pages are aimed at a different audience than articles. First clause of first sentence: "The tritone paradox is an auditory illusion..." Some people hear a given pair of Shepard tones as ascending, others hear it as descending. Some can hear it both ways, but not simultaneously. Interesting to listen to the externally linked stuff there. __Just plain Bill (talk) 16:15, 14 August 2008 (UTC)
Thanks for the explanation. I was critiquing the article to which you linked, not what you wrote. The clause you quote states the genus, but a definition requires a differentia too — the lead of the article does not tell a reader how this illusion differs from other such illusions by saying what it actually sounds like, which may be why you had to. I commented at Talk:Tritone paradox. --Jtir (talk) 13:51, 15 August 2008 (UTC)
Yeah, sometimes it takes me a while... I realized you were critiquing the article about a day after I wrote that "aimed at different audiences thing" in haste. I've responded over there, you see. __Just plain Bill (talk) 15:57, 15 August 2008 (UTC)

linking clockwise[edit]

The question was asked: Do we really need to link cloc directions here?[6]

A: WP:CONTEXT is the guideline.

The previous version included the explanatory phrase "to the right". Linking is more concise. Linking both clockwise and counterclockwise could be considered overlinking, however.

Should we link circle? :-)

--Jtir (talk) 18:52, 14 August 2008 (UTC)

I happen to think "clockwise" is a "plain English word," right at the top of the list of things not to link.
I suppose I can live with it being a "technical term," but isn't that a stretch?
If someone doesn't know what clockwise means, the key of G is mentioned in the next few words, and there's the image alongside to guide them. If a reader doesn't get it from that, either they're trying to read way above their level, or they don't really care to know. I don't think we should link "clockwise" any more than we should link "circle." __Just plain Bill (talk) 05:23, 15 August 2008 (UTC)

Circle of fifths vs fourths[edit]

It's called the circle of fifths! Hyacinth (talk) 07:26, 29 June 2009 (UTC)

Yes, but the article leads off with "In music theory, the circle of fifths (or circle of fourths)..." The article doesn't yet explain well anything to do with fourths. One should not assume that the reader is yet familiar with the fact that a fourth is the reverse of a fifth. It is certainly not intuitive to those not familiar with music notation that an interval has one name going up scale and a different name going down scale.
It is also not yet clearly stated that one can use the circle of fifths to go up scale or down scale both clockwise and counterclockwise, and what the results will be in each case.
I note some debate on the talk page about what level this article should be written at. IMO there is not one music theory article on wikipedia that is written at too low a level. OTOH, there is not one really good introductory article on music theory for those readers completely new to the subject. In other words, there is a lot of explaining yet to be done in this entire field; you aren't even close to oversimplifying too much. I dearly wish for an article that explains the whys and hows of music terminology written to a newcomer's level. It's not at all clear why intervals are called intervals and not factors or proportions. (Normally, one thinks of an interval as an additive distance value, not a multiplicative coefficient.) It's not at all clear what the 'fifths' in 'circle of fifths' refers to. (Bottles of whiskey and rye?) And that's not getting to any of the confusing paradoxes such as the fact that we customarily use a chromatic tuning to express diatonic and pentatonic scales. I note that it is even difficult to find out exactly which fractions are assumed to make up a diatonic or pentatonic scale. Instead, we find a multitude of sources giving patterns of 'whole' and 'half' notes without defining what they mean by 'whole' and 'half'. And no, that's not something that a beginner can safely assume means anything like half+half=whole arithmetically, because it's not. Dlw20070716 (talk) 18:20, 9 August 2011 (UTC)


In my opinion, "circle of fifths" is a misnomer for "cycle of fifths".Unfree (talk) 06:11, 30 August 2009 (UTC)

Very probably. Unfortunately, "circle of fifths" is the term found nearly universally in textbooks.—Jerome Kohl (talk) 19:06, 30 August 2009 (UTC)

"It is intended to be a guide for composition and now it has become essential to writing music"[edit]

I beg to differ. While it is no doubt helpful - to those who understand it - it is by no means essential in the process of writing music. How do I know? I cannot grasp the meaning of the circle - at all - yet I have written several successful and well-received music pieces, including a symphony. Nor am I alone in this regard. Therefore, can I change "essential" to "helpful" and expect the change to not result in an immediate revert? —Preceding unsigned comment added by RadicalTwo (talkcontribs)

I went ahead and rephrase the overstatement. They are plenty of styles of music that do not necessarily rely on the circle of fifths. Also it possible to understand the concepts of keys and chords without thinking of them in a circle as you mention. --dbolton (talk) 19:29, 23 October 2009 (UTC)

Circle of fifths image/mid file[edit]

Isn't the image and accompanying midi file for the circle of fifths being played clockwise near the top of the page incorrect? In both, the fourth note is given as an A sharp, but shouldn't it be an A? (talk) 05:31, 28 December 2010 (UTC)

The image is certainly incorrect, just as you say. I haven't listened to it but, if the accompanying sound file also has A instead of A, then it, too, is in error and should be corrected. Unfortunately, correcting these things is not within my realm of competence.—Jerome Kohl (talk) 18:17, 28 December 2010 (UTC)
Ah! I see this was the result of a recent edit by User:Hyacinth. Could we have a correction, please, Hyacinth?—Jerome Kohl (talk) 18:23, 28 December 2010 (UTC)
Thanks. Fixed. Hyacinth (talk) 08:09, 8 January 2011 (UTC)

discussion of a whole document does not merit a 'needs page number' quibble.[edit]

Considering the flurry of malicious edits and reversions ongoing here, I'm not about to enter the fray.

I'm going instead merely to mention a minor problem rather than to repair it [this is true also in part because I have no idea how to repair the problem without breaking something], and leave the repair of the problem to wiser and more trusted editors.

Anyway, in the History section, the "[page needed]" quibble at the end of the first sentence is inappropriate.

The paragraph author at this point is discussing the treatise (a self-standing document in most cases) as a whole, not some particular point within that treatise, and so has no page number to provide in response to the quibble at the indicated point.

Xanthian (talk) 07:43, 13 February 2012 (UTC)

Sometimes it is necessary to look more closely at a problem like this. It is not the named book that is being cited, but rather a secondary source. The note reads <ref name="Jensen">Jensen (1992)</ref>{{Page needed|date=May 2010}}, so the demand for a page reference is from Claudia Jensen's 1992 article in JAMS, rather than from Diletskii's Grammatika. I am frankly astounded that this request has not been dealt with for nearly two years, since JAMS is an extremely well-known journal readily accessible to many editors.—Jerome Kohl (talk) 17:45, 13 February 2012 (UTC)

general criticism[edit]

The article is very vague and confusing. The sentence, which is supposed to be a definition, is not even grammatical: "Since the term 'fifth' defines an interval or mathematical ratio which is the closest and most consonant non-octave interval, then the circle of fifths is a circle of closely related pitches or key tonalities." Saying that the circle of fifths is a circle of "closely related pitches" is vary vague. Don't you want to say that these pitches are precisely 5 tones apart?

Adam — Preceding unsigned comment added by (talk) 15:27, 25 May 2012 (UTC)

How is the sentence ungrammatical? To me, it seems to parse OK. In most general terms, two pitches making the interval of a perfect fifth are three tones plus a semitone apart, or seven semitones. How can we reduce the confusion here? __ Just plain Bill (talk) 17:31, 25 May 2012 (UTC)

7 sharps and flats[edit]

How does the circle not stop at 7 sharps and flats (quote from article)? Realistically, enharmonically, you start running into equivalent keys and there is no point to extend it beyond. — Preceding unsigned comment added by (talk) 03:21, 27 June 2012 (UTC)

I suspect that keys with double sharps and double flats are of more theoretical significance, not so much of any great practical use. There is also the fact that "enharmonic" does not necessarily mean "identical pitch" in tuning systems other than ET. But, as that section says, the circle (or spiral) does continue, starting over by adding Fdouble sharp or Bdouble flat and so on. I don't think I would enjoy sight reading a piece in such a cluster of flats or sharps, but perhaps they do exist out in the wild. Awaiting comment from more expert editors... __ Just plain Bill (talk) 13:33, 27 June 2012 (UTC)
There's more to music than key signatures. For example, pieces in C-sharp major routinely modulate to the dominant, G-sharp major. Of course they have to use double-sharps to notate those passages, but to the extent we're using the circle of fifths to understand key relationships (or even harmonic progressions), we have to allow for keys with double accidentals. Below is a passage from Bach's Well-Tempered Clavier: —Wahoofive (talk) 21:01, 27 June 2012 (UTC)

G-sharp major example.png


Well I looked hard at the picture of the Vicious Girdle of Fifths, trying again after forty unhappy years of "Not Getting It" ::: and realised after about 10 minutes that there was no C# major? Scrolling down the talk, I also find there is no C♭, (or similarly abstruse key-sig) because it "clutters the table up????" Duh??????????????????????? So there is no point in reading the talk, I just hate all music theory and all theorists because they can never explain what the f they are actually on about, namely Producing Beautiful Music Which Revivifies The Soul. I have always hated the circle, and WP does nothing in the slightest to modify my attitude, especially since "The Circle" has been populated by things which have apparently escaped from Ian M. Banks' vocabulary such as Pitch-class space, like a Raptor-class interstellar terraforming AI with a somewhat detached hive mind of its very very very very own. And, unless you are careful, you end up with a -Class planet-destroyer. Trumpets and drums to you, too. MinorProphet (talk) 12:42, 20 October 2015 (UTC)

Holy Cow! You are right about that diagram, of course, but the problem is much worse than you describe. If we are to believe what we see there, the key of D-flat major may have a signature of seven sharps, and the key of B major may be written with a signature of seven flats. Let us see if we can track down User:Just plain Bill, who created this diagram, and see if we can get it fixed. On the other hand, I doubt very much that music theorists are concerned with Producing Beautiful Music Which Revivifies The Soul, which is something they leave to performers. Thanks for the heads-up.—Jerome Kohl (talk) 16:01, 20 October 2015 (UTC)
I may be able to fit the proper nomenclature for those two keys into that end of the diagram. Might take a few days to find time... Just plain Bill (talk) 16:59, 20 October 2015 (UTC)
Had a cancellation, so was able to update the svg file sooner than I expected. Now it is shown in this article. For some reason the commons would not let me upload a new rev of the existing image, so it is a different file. Sorting out that cluster of issues will take some time, another day. Just plain Bill (talk) 17:41, 20 October 2015 (UTC)
Some theorists are also performers, so they might also want to PBMWRTS. ;-) —Wahoofive (talk) 20:43, 20 October 2015 (UTC)
Commons eventually showed the "upload new version" link, so disregard that bit above. Back to the same file name, newest rev shows the extra key names. Just plain Bill (talk) 02:54, 21 October 2015 (UTC)

what no music?[edit]

Here we are with a huge exposition and no composers' works cited. many people cannot think in musical notes. they see the words but don't get it. try this: Mike Oldfield, Incantations, easily discovered on youtube. the original, not anything that has been sample etc. the entire composition is based on the circle of fiths and the first few bars will make the Circle of Fifths into an understood reality. i don't know how to add a sample but perhaps one of you does. (talk) 16:22, 9 April 2016 (UTC)

Introduction: sharps and flats[edit]

for those who know the topic I guess this is okay, but I don't know what sharps and flats are, they are never defined in the Introduction, so I'm lost from the start — Preceding unsigned comment added by (talkcontribs) 16:16, 4 August 2018 (UTC)

Sharps and flats are linked in the "Definition" section, the first one after the table of contents. They are not mentioned in the introduction, which is meant to summarize the content of the article, not explain all the rudiments of the topic. Just plain Bill (talk) 17:50, 4 August 2018 (UTC)