Talk:Circle of fifths

Circle or cycle?

A recent modification by user 71.17.203.128 (what a strange name!), already reverted, raised once again the question of "circle" vs "cycle," already discussed on several occasions in this talk page. There are reasons to prefer "circle" which may be more common in English – the situation would be the opposite, say, in French. But I do believe that adding a word about this in the article could prevent further discussions. "Cycle" is used at least three times in the article, but not all three cases may be justifiable. Something should be done about this. Whaddya think? — Hucbald.SaintAmand (talk) 08:17, 22 June 2024 (UTC)

Cycle might make more sense, as IP user suggested, but lots of things in music don't make sense, e.g. a third is twice as big as a second. I've never heard any musician say "cycle of fifths." Is there a source which does? —Wahoofive (talk) 15:27, 22 June 2024 (UTC)

@Wahoofive, I don't mean that we should replace "circle" by "cycle", merely that we should say that "cycle" is a possible variant. This should close the discussion. Sources are not lacking:

1. Our article itself uses "cycle" three times:
   1. "It was known in [A]ntiquity that a cycle of twelve fifths was almost exactly seven octaves" – cycle is preferred here because the idea is to stress the origin of the (Pythagorean) comma, i.e. that the series does not close on itself as a circle.
   2. "Cycle' appears in a quotation from A. Whitall in The Oxford Companion to Music (see note 17), which I was unable to check.
   3. It also appears about a quotation from R. Scruton, The Ring of Truth: The Wisdom of Wagner's Ring of the Nibelung (see note 20), but the truth is that Scruton never writes "cycle or fifths" and only once "circle of fifths" (p. 174, about circles of third progressions replacing it in Schubert).
2. In my own electronic library, I find "cycle of fifths" many times, among others in
   1. E. Amiot (2016), Music Through Fourier Space.
   2. W. Apel (1950), Harvard Dictionary of Music (s.v. "Chinese Music").
   3. Th. Christensen (2019), Stories of Tonality in the Age of François-Joseph Fétis.
   4. S. Clark (2011), "On the Imagination of Tone in Schubert's Liedesend [etc.]", in The Oxford Handbook of Neo-Riemannian Music Theories.
   5. D. Conklin & S. Weisser, "Pattern and Antipattern Discovery in Ethiopian Bagana Song", in Computational Analysis.
   6. E. Kurth (1991), Selected Writings.
   7. L. Meyer (1973), Explaining Music.
   8. N. Newton (2019), "Chromatic Linear Progressions in Popular Music", in The Routledge Companion to Popular Music Analysis.
   9. C. Sachs (1943), The Rise of Music in the Ancient World.
   10. N. Waltham-Smith (2019), "Sequence", The Oxford Handbook of Critical Concepts in Music Theory.
       Etc.
3. Scholar Google gives about 1990 replies for "cycle of fifths", as against about 9130 for "circle of fifths" (i.e. a relation of about 20% for the first). In French, about 414 results for "cycle des quintes", as against 60 only for "cercle des quintes" (80% for the first – the situation is the reverse from that in English).

Hucbald.SaintAmand (talk) 11:40, 23 June 2024 (UTC)

I should have noted, in addition, that the French article corresponding to this one on FR.WP is "Cycle des quintes"; this may be a typically French usage, other European languages prefer "circle" or its equivalent. — Hucbald.SaintAmand (talk) 16:05, 24 June 2024 (UTC)

Comment: The normal name is "circle of fifths"; no problem in also using the word "cycle" appropriately as well in the article. But I do not think it is an improvement to start one of those ridiculous WPrambles, listing all sorts of "alternative names" in other languages, scripts, obscure/deprecated/wrong Unicode characters that might have something to do with it etc etc etc... Meanwhile this comment strikes me as bizarre: "cycle is preferred here because the idea is to stress the origin of the (Pythagorean) comma, i.e. that the series does not close on itself as a circle." But the meaning of "cycle" is (at least in its mathematical sense) precisely that of a closed cycle, otherwise it would not be a cycle. The CoF is isomorphic to the cyclic group of order 12, whose multiplication table can be used to convert semitones to CoF positions, for example. So arguably "cycle of fifths" might have been a better term, just as "cheap at twice the price" would have been better than the usual expression. Imaginatorium (talk) 07:03, 2 July 2024 (UTC)

We are not "starting" a ridiculous ramble (it started long ago), we are trying to close it. Could we agree that while the main term is "circle", "cycle" is a possible variant, and say so somewhere in the article? – Hucbald.SaintAmand (talk) 11:38, 2 July 2024 (UTC)

I'm fine with this. —Wahoofive (talk) 15:25, 2 July 2024 (UTC)

I added five words at the beginning of the article, which (I hope) should close the discussion. But feel free to formulate this otherwise. I don't think references are needed, the list of sources above will remain available. — Hucbald.SaintAmand (talk) 21:16, 2 July 2024 (UTC)

Circle of major and minor 7th chords?

I tried to add a section called Circle of major and minor 7th chords to this page, but someone removed it because they felt it should be its own page. I don't think I discovered anything groundbreaking other than I have never seen the note sequence DFACEGB mentioned before anywhere in music theory discussions. Anyway, I created a separate wikipedia page named "Circle of major and minor 7th chords" with a graphic that represents this 24 note sequence. If anyone knows if this synthetic scale or note sequence already has a name, please let me know in the "Talk" on that page.

BTW, if everyone agrees that the "Circle of major and minor 7th chords" is a useful addition to the topic of the circle of fifths, I would like to add a link to it in the "Circle of fifths" page. --Bsenim (talk) 23:56, 20 December 2024 (UTC)

As far as I can tell, this is simply saying "stacking four 5ths yields two octaves plus a third". Those pitch classes will, of course, start outlining a chord. And the idea that was presented isn't even sticking to perfect fifths. So I don't think it's a notable music theory phenomenon, and it's not really accurate. - Special-T (talk) 18:02, 19 December 2024 (UTC)

It seems to me that what is shown really is a circle of 3ds, which has been termed the tonal omnibus. It is a succession of neo-Riemannian R relations (from major to relative minor) and L relations (from minor to major by a "leading tone change", e.g. raising E to F in A–C–E to A–C–F). It only indirectly concern the circle of fifths. — Hucbald.SaintAmand (talk) 09:50, 20 December 2024 (UTC)

The 3-4-3-4-3-4... semitone step sequence (F#/G♭, B♭, D♭, F, A♭, C, E♭, G, B♭, D, F, A, C, E, G, B, D, F#, A, C#, E, G#, B, D#, F#/G♭) actually exists in most circle of fifths charts that include the associated minor keys in an inner circle. Not sure if it's correct to say "it only indirectly concern the circle of fifths." This is why I don't claim to have discovered anything groundbreaking. I just feel it's useful to mention that the 3-4-3-4... semitone step sequence (F#/G♭, B♭, D♭, F, A♭, C, E♭, G, B♭, D, F, A, C, E, G, B, D, F#, A, C#, E, G#, B, D#, F#/G♭) embeds all major and minor 7th chords and includes the circle of fifths. Bsenim (talk) 00:17, 21 December 2024 (UTC)

But the 3–4–3–4... sequence is not inherent in the circle of fifths (you write yourself "most circle of fifth charts"), and what you added to the article, F#/G♭–b♭–D♭–f–A♭–c–E♭–g–B♭–d–F–a–C–e–G–b–D–f#–A–c#–E–g#–B–d#–F#/G♭ obviously is a circle of thirds. It might be interesting to describe the link between these two circles, but to pass for that by a succession of 7ths seems somewhat farfetched. — Hucbald.SaintAmand (talk) 09:42, 21 December 2024 (UTC)

Isn't the circle of fifths technically a circle of 7 semitones? Bsenim (talk) 11:39, 21 December 2024 (UTC)

Maybe the answer lies in the origins of the name diatonic scale. Maybe "di" is tied to the relationship with the 3-4 step sequence. I'm not a music theory expert, so hopefully someone that has a deeper understanding of the theory behind a diatonic scale can add to this discussion. I don't think it's a chance coincidence that the components of a diatonic scale are logically mapped to the 3-4-3-4... sequence. It's actually really cool that the number 7 is deeply tied to the diatonic scale which is built on top of the number 12. Mannheim Steamroller's Fresh Aire 7 is one of my favorite albums. Bsenim (talk) 11:54, 21 December 2024 (UTC)

For what I know of ancient Greek, dia (δια) means "through." So, "diatonic" means "through tones," which won't help us here. Diatonic scales were described and discussed in Mesopotamia about 2500 years BC. The reason probably is that these scales permit maximizing the number of consonances of 4ths and 5ths, so that it might be argued that the circle of 5ths is at the origin of the diatonic scale. 3ds probably had not the same role because they are less consonant (particularly the minor one). But all this is speculation: I'd gladly discuss it further, but it obviously is not for WP. — Hucbald.SaintAmand (talk) 18:28, 21 December 2024 (UTC)

OK, I didn't get what you meant by circle of thirds at first, but I think you're correct in saying the name of the 3-4-3-4 step sequence is the circle of thirds, but nobody has added a graphic to that page yet, or mentioned the mnemonic DFACEGB yet, or talked about the fact that 7 sequential steps around the circle of thirds are components to a diatonic scale, or spelled out the repeating pattern of notes I have been describing here. Thanks for pointing out the circle of thirds page! Bsenim (talk) 12:39, 21 December 2024 (UTC)

I didn't even realize that there was a circle of thirds page on Wikipedia. As shown there (or here above), an alternation of major and minor thirds (4–3–4–3...) produces a chromatic scale and continues with enharmonies. The "tonal omnibus" is slightly different: in order to limit the circle to the diatonic scale, it has to involve two successive minor thirds. I found only one paper discussing it, "Tonal and "Modal" Harmony," which means that after all the concept might not be very common (nor very useful). — Hucbald.SaintAmand (talk) 18:49, 21 December 2024 (UTC)

The circle of thirds is useful for jazz musicians. This one sequence of 24 notes that repeats is an excellent pattern to memorize and get under your fingers for improvisation. Bsenim (talk) 19:13, 21 December 2024 (UTC)

Cycle of 48 notes of the first four notes of major scales in fifths order

This cycle of 48 notes represents the first four notes of each major scale, following the order of keys around the circle of fifths.

I found the link between the circle of fifths and the circle of thirds. If you take the cycle of 24 notes from the circle of thirds (the 3-4-3-4... minor/major thirds step interval)

F#/G♭, B♭, D♭, F, A♭, C, E♭, G, B♭, D, F, A, C, E, G, B, D, F#, A, C#, E, G#, B, D#, F#/G♭

And you use the jump pattern 15263748... repeating around the circle of thirds. You have a 48 note cycle of modular arithmetic (modulo 12) that groups common tones together.

C, D, E, F, G, A, B, C, D, E, F#, G, A, B, C#, D, E, F#, G#, A, B, C#, D#, E, G♭, A♭, B♭, B, D♭, E♭, F, G♭, A♭, B♭, C, D♭, E♭, F, G, A♭, B♭, C, D, E♭, F, G, A, B♭

The most succinct way to describe this cycle without coining any new term is to say that this 48 note cycle is:

A cycle of 48 notes representing the first four notes of each major scale, following the order of keys around the circle of fifths.

or

Cycle of 48 notes of the first four notes of major scales in fifths order

This cycle of 48 notes is all the major scales in succession in the order of the circle of fifths.Bsenim (talk) 10:36, 23 December 2024 (UTC)

I feel this is worthy of a new page because it is not original research and only facts about the circle of fifths and the circle of thirds. Bsenim (talk) 10:44, 23 December 2024 (UTC)

I'd like to talk here about what the appropriate name should be for the new page. Bsenim (talk) 10:45, 23 December 2024 (UTC)

This cycle of 48 notes also contains 4 modes. For example starting with C Major, the next mode is D Dorian, followed by E Phrygian, followed by F Lydian and then the mode cycle repeats with G Major. Bsenim (talk) 13:43, 23 December 2024 (UTC)

Oh yeah, I forgot to mention that it's probably not a coincidence that the Ionian mode maps to the 3-4-3-4... minor/major thirds step sequence because the Ionian mode half steps are also a 3-4-3-4... step sequence. Hopefully an expert on modular arithmetic can tie these two 3/4 ratios together. Bsenim (talk) 15:59, 23 December 2024 (UTC)

You need to read the section on Wikipedia:No original research again. —Wahoofive (talk) 21:35, 23 December 2024 (UTC)

Thanks, I thought the rule was against invented/coined/discovered ideas. I didn't know that there was a policy against mathematical facts that can be easily proved. Bsenim (talk) 21:47, 23 December 2024 (UTC)

Have you ever wondered why there isn't a page about why 1 + 1 = 2? -- Jack of Oz ^{[pleasantries]} 22:12, 23 December 2024 (UTC)

valid point Bsenim (talk) 22:13, 23 December 2024 (UTC)

• "I found the link..." means that this is transparently OR. But there is almost nothing here; you have simply taken the circle of fifths and filled in the notes of the major scale from one to five; obviously this automatically includes major and (natural) minors scales of all keys, most of the modes etc etc. Imaginatorium (talk) 04:21, 24 December 2024 (UTC)

Quinticircles and quarticircles

Quinticircles (quintine circles) use 12 clavisignatures between the fourflat (F minor/A-flat major [#0000CC/DHB]) and the sevensharp (A-sharp minor/C-sharp major [#CC6600/DHO]).

Quarticircles (quartine circles) use 12 clavisignatures between the foursharp (C-sharp minor/E major [#33FF33/LHG]) and the sevenflat (A-flat minor/C-flat major [#CC0066/DHP]).

The 12 Visibone melochromies (musical colors) are DHB (#0000CC), DHV (#6600CC), DHM (#CC00CC), DHP (#CC0066), DHR (#CC0000), DHO (#CC6600), LHY (#FFFF33), LHS (#99FF33), LHG (#33FF33), LHT (#33FF99), LHC (#33FFFF) and LHA (#3399FF).

In a catalogue of 46,656 songs 23,328 are quinticircular and 23,328 quarticircular. It starts in the "ZZW" (quarticircular baritonal fourflat) and ends in the "ZZV" (quinticircular mesosopranine onesharp).

200.155.118.244 (talk) 11:20, 25 January 2025 (UTC)

??? — Hucbald.SaintAmand (talk) 20:30, 29 May 2025 (UTC)

Commentary: 46,656 divided different songs by 12 different tonalities (0000CC/DHB, 6600CC/DHV, CC00CC/DHM, CC0066/DHP, CC0000/DHR, CC6600/DHO, FFFF33/LHY, 99FF33/LHS, 33FF33/LHG, 33FF99/LHT, 33FFFF/LHC and 3399FF/LHA) (Eolians or Jonians) are 3,888 different artists (bands, choirs, orchestras or singers). The ZZW (46,652) is a quarticircular tetrabemollic baritonal song and the ZZV (46,651) a quinticircular monodiesic mesosopranine.

Observation: Baritone (E♭♭2-B♯4), tenor (B♭♭2-F♯♯5), contralto (F♭3-C♯♯6) and mesosoprano (C♭4-G♯♯6).

2804:18:1079:5D62:1408:7EFF:FE3C:526B (talk) 21:59, 29 May 2025 (UTC)

Is this AI ??? To me, "a quarticircular tetrabemollic baritonal song" and "a quinticircular monodiesic mesosopranine" sound like jokes. — Hucbald.SaintAmand (talk) 06:25, 30 May 2025 (UTC)

Examples of quarticircular* clavisignatural* cantional* codes:

Baritone: ZX8, ZX9, ZXA, ZXB (sevenflat), ZXC, ZXD (fiveflat), ZXE, ZXF, ZXG, ZXH, ZXI (sixflat) and ZXJ; ZZW, ZZX, ZZY, ZZZ*, 0, 1*, 2, 3, 4, 5, 6* and 7;

Tenor: ZXK, ZXL, ZXM, ZXN*, ZXO, ZXP*, ZXQ, ZXR, ZXS, ZXT, ZXU* and ZXV; 8, 9, A, B*, C, D*, E, F, G, H, I* and J;

Contralto: ZXW, ZXX, ZXY, ZXZ*, ZY0, ZY1*, ZY2, ZY3, ZY4, ZY5, ZY6* and ZY7; K, L, M, N*, O, P*, P, Q, R, S, T, U* and V;

Mesosoprano: ZY8, ZY9, ZYA, ZYB*, ZYC, ZYD*, ZYE, ZYF, ZYG, ZYH, ZYI* and ZYJ; W, X, Y, Z*, 10, 11*, 12, 13, 14, 15, 16* and 17.

Examples of quinticircular* clavisignatural cantional codes:

Baritone: ZYK, ZYL, ZYM, ZYN (fivesharp), ZYO, ZYP (sevensharp), ZYQ, ZYR, ZYS, ZYT, ZYU (sixsharp) and ZYV; 18, 19, 1A, 1B*, 1C, 1D*, 1E, 1F, 1G, 1H, 1I* and 1J;

Tenor: ZYW, ZYX, ZYY, ZYZ*, ZZO, ZZ1*, ZZ2, ZZ3, ZZ4, ZZ5, ZZ6* and ZZ7; 1K, 1L, 1M, 1N*, 1O, 1P*, 1Q, 1R, 1S, 1T, 1U* and 1V;

Contralto: ZZ8, ZZ9, ZZA, ZZB*, ZZC, ZZD*, ZZE, ZZF, ZZG, ZZH, ZZI* and ZZJ; 1W, 1X, 1Y, 1Z*, 20, 21*, 22, 23, 24, 25, 26* and 27;

Mesosoprano: ZZK, ZZL, ZZM, ZZN*, ZZO, ZZP*, ZZQ, ZZR, ZZS, ZZT, ZZU* and ZZV; 28, 29, 2A, 2B*, 2C, 2D*, 2E, 2F, 2G, 2H, 2I* and 2J.

Clavisignatural semitonarchy* (legend above):

Fourflat (F minor/A-flat major): W, 8 and K;

Threesharp (F-sharp minor/A major): X, 9 and L;

Twoflat (G minor/B-flat major): Y, A and M;

Sevenflat (A-flat minor/C-flat major) [quarticircle] and fivesharp (G-sharp minor/B major) [quinticircle]: Z, B and N;

Zeroaccident or zeroalteration (A minor/C major): 0, C and O;

Fiveflat (B-flat minor/D-flat major) [quarticircle] and sevensharp (A-sharp minor/C-sharp major) [quinticircle]: 1, D and P;

Twosharp (B minor/D major): 2, E and Q;

Threeflat (C minor/E-flat major): 3, F and R;

Foursharp (C-sharp minor/E major): 4, G and S;

Oneflat (D minor/F major): 5, H and T;

Sixflat (E-flat minor/G-flat major) [quarticircle] and sixsharp (D-sharp minor/F-sharp major) [quinticircle]: 6, I and U;

Onesharp (E minor/G major): 7, J and V.

200.155.125.114 (talk) 10:26, 30 May 2025 (UTC)

I am confused. Is that nomenclature Lusitanian, or something else? I am familiar with sol-fa as used in many Romance languages, along with the letters (CGDAE and so on) used in my native English idiom. A link explaining the basis of your terminology would be useful here. cheers, Just plain Bill (talk) 16:46, 30 May 2025 (UTC)

The 15 mentioned names of the 15 clavisignatures (key signatures) above are organized semitonarchically (in semitonal order). The 36 mentioned unitarian alphanumeric digits (alphanumeric HTU system [hundred, ten and unit]) follow this mentioned tonal sequence above. I do not confounded the seven notes (C, D, E, F, G, A and B) with my commentary above. "Quarticircle" means "circle of fourths" and "quinticircle" "circle of fifths".

2804:18:107C:3F6F:DC18:ABFF:FE5B:A1BD (talk) 20:26, 30 May 2025 (UTC)

The whole "ZXK, ZXL, ZXM..." style is opaque to me. In what context is that a standard notation? Just plain Bill (talk) 20:56, 30 May 2025 (UTC)

What's the point of renaming "key signatures" as "clavisignatures", or "semitonal order" as "semitonachically", etc.? Are you trying to invent a new language? Ain't you satisfyed with English? – Hucbald.SaintAmand (talk) 21:13, 30 May 2025 (UTC)

Explication: I am a logatomatist (creator of words of the none) and a protologist (creator of first new words) because I am a verbophile (logophile). The mentioned alphanumeric tridigital codes I use for possibly appreciate until 3,888 different artists (1,944 quarticircular and 1,944 quinticircular [bands, choirs, orchestras or singers]) [46,656 divided different songs {23,328 quarticircular and 23,328 quinticircular} by 12 different tonalities]. "46,655" is "ZZZ" alphanumerically and "0" "0", exemplarly.

I am a passionated man about the 216 Visiosteic (Visibone Anglocentric Color Code - Visiosteum) colors, as 0000CC/DHB, 6600CC/DHV, CC00CC/DHM, CC0066/DHP, CC0000/DHR, CC6600/DHO, FFFF33/LHY, 99FF33/LHS, 33FF33/LHG, 33FF99/LHT, 33FFFF/LHC and 3399FF/LHA, exemplarly, who I consider they as melochromies (musical colors). These sequencial colors represent the notes geesharp/ayflat, geedoublesharp/ay/beedoubleflat, aysharp/beeflat, aydoublesharp/bee/ceeflat, beesharp/cee/deedoubleflat, ceesharp/deeflat, ceedoublesharp/dee/edoubleflat, deesharp/eflat, deedoublesharp/e/efflat, esharp/ef/geedoubleflat, efsharp/geeflat and efdoublesharp/gee/aydoubleflat. They also represent the 12 unitarian groups who are W-8-K, X-9-L, Y-A-M, Z-B-N, 0-C-O, 1-D-P, 2-E-Q, 3-F-R, 4-G-S, 5-H-T, 6-I-U and 7-J-V. The 36 digits are the 36 units in the HTU (hundred, ten and unit) system. Exemplarly, "ZZZ" is composite by the hundred "Z", by the ten "Z" and by the unit "Z". "Semitonarchy" means "semitonal order" in my second commentary.

Read my commentary carefully over my exemplar tridigital codes of each one of the four cited chorophonies (choral voices) in my second.

200.155.125.114 (talk) 01:13, 31 May 2025 (UTC)

Proposal to restore 3D visualization (Umbilic Torus) with clarified caption

@Squandermania: @Hucbald.SaintAmand:

I would like to discuss the recent removal of the animated visualization of the Circle of Fifths on the Umbilic Torus.

I agree with the feedback that the previous caption ("Major 7th progression") was confusing; it made the image seem like a specific chord chart rather than a structural diagram. However, I propose restoring the image with a new caption and a specific placement.

The 3D visualization offers a topological insight that the 2D circle cannot, serving a crucial role for readers interested in the mathematical structure of harmony:

1. The "Strip" is the Subject: The article defines the circle as a sequence of perfect fifths. Topologically, this sequence forms a continuous path. The animation visualizes the Circle of Fifths as a single continuous manifold (an Umbilic Torus), showing how the "sharp" side connects to the "flat" side through the geometric twist.

2. The Moving Chord is a "Cursor": The moving rectangular plane is not intended to show a random progression, but acts as a visual cursor (or tangent frame). Without this movement, the static torus looks like an abstract shape. The movement allows the eye to trace the specific path of Harmonic Resolution (V → I). (Note: The cursor moves in the direction of the Circle of Fourths (C → F → Bb...), capturing the natural gravitational pull of Western harmony).

3. Pedagogical Depth: While the standard 2D diagram serves the novice well, Wikipedia also serves readers looking for deeper structural connections. This visualization bridges the gap between music theory and geometric topology. Removing it denies advanced readers (such as those interested in the mathematical properties of the chromatic scale) the ability to visualize the continuity of the system.

Proposal:

A visualization of the Circle of Fifths on a toroidal surface. The moving plane acts as a cursor, tracing the path of Harmonic Resolution (C → F → Bb...) to illustrate the geometric continuity between keys.

— Preceding unsigned comment added by Jimishol (talk • contribs) 10:29, 9 February 2026 (UTC)

I suggest placing the image in the "Modulation and chord progression" section (as it directly visualizes the chromatic closure), or any other section you deem appropriate, with this strictly geometric caption:

"A visualization of the Circle of Fifths on a toroidal surface. The moving plane acts as a cursor, tracing the path of Harmonic Resolution (C → F → Bb...) to illustrate the geometric continuity between keys."

Does this distinction between the "musical content" (the chord) and the "geometric function" (the cursor) resolve the issue of clarity? Jimishol (talk) 08:45, 6 February 2026 (UTC)

What the image represents remains unclear. A point moving around the circle of fifths would also show "the chromatic closure" – if "chromatic closure" means where the succession involves enharmony. A static image does not stress a direction (descending 5ths) above the other. One could argue that descending 5ths are the privileged direction in tonal music, but to prove this is not the purpose of the article.

Here, four points move at the same time, representing the four notes of a chord. But these notes do not simultaneously reach the point of enharmonic change, which in turn raises the question of tuning. It already was the case in John Bull's Fantasia supra Ut re mi fa sol la, in the Fitzwilliam Virginal Book, where an F♯ chord is changed into a G♭ one, leading the modern editors to believe that the piece was an early example of equal temperament.

Toroïdal Tonnetz

So, the example probably shows more than what the article discusses. The image of the toroïdal Tonnetz, hereby, even if perhaps not much easier to understand, may show as much (the circle of fiths can be seen along the blue lines).

But this is my opinion, and others may see all this otherwise. — Hucbald.SaintAmand (talk) 11:12, 10 February 2026 (UTC)

• I do not honestly think that this animation helps the article. Several points: firstly, unless an animation really makes something immediately obvious, it obstructs working out what is going on. I really can't see what the "cursor" represents, and I cannot immediately find a way to stop it moving. I cannot see that anyone could understand what the cursor is doing unless they already understood the answer. More generally, why this particular arrangement? You have noticed that 12=3x4, in effect showing that all 12 notes are generated by moving an augmented major chord up a semitone, in four positions. What does this illuminate? If I can call your torus a 3x4 shape (four triangles), there are three more obvious diagrams: 4x3 (three quadrilateral; the three diminished sevenths), 6x2 (two hexagons; the two whole tone scales), and 2x6 (a six-rung Möbius strip; the six augmented fourths). How do any of these illustrate more or less about anything relevant? Imaginatorium (talk) 15:34, 10 February 2026 (UTC)

  @Imaginatorium: @Hucbald.SaintAmand:

  Thank you for the detailed feedback. I would like to address the geometric necessity of this specific visualization and how it directly illuminates the existing text of the article regarding the "spiral" nature of fifths.

  1. To Imaginatorium: "Why this particular arrangement (3x4)?"

  You asked why we use the Major 7th (3x4) rather than Diminished (4x3) or Whole Tone (6x2).

  • Symmetry vs. Tonality: The 4x3 and 6x2 partitions are perfectly symmetrical, making them effectively atonal (no defined root).
  • The Atom of the Key: The Circle of Fifths is a map of Keys. The Major 7th chord (3x4) is the fundamental asymmetric structure that defines a Key (Root + Mode + Leading Tone).
  • Topological Efficiency: Unlike many "Toroidal Tonnetz" diagrams that use "phantom notes" (duplicating C in multiple locations), this 3x4 arrangement maps the 12 notes of 12ET onto a 3D surface with zero duplication. It is a strict bijection.

  2. To Hucbald: "Why not a moving point?"

  • Line vs. Surface: A single point moving along the circle traces a spiral (a 1D line). It shows the sequence, but it does not show the manifold.
  • Visualizing the Twist: To see the topology of the Umbilic Torus, the reader needs to see the orientation of the surface. The moving rectangular plane acts as a "tangent frame." It reveals how the surface twists to connect the sharp keys to the flat keys—something a dimensionless point cannot show.

  3. The "Infinite Helix" vs. "Closed Torus"

  The article correctly notes that "In just intonation the sequence of fifths can therefore be visualized as a spiral, not a circle."

  • The Visualization's Purpose: This animation demonstrates exactly how 12ET allows that infinite helix to close into a finite, continuous loop.
  • Without this 3D view, the concept of "closing the spiral" is abstract. The animation makes the geometry of the 12ET closure explicit, showing the "functional succession" (ascending fourths) described in the text.

  Summary

  The animation is not arbitrary art. It uses the Major 7th to preserve Tonality, and it uses movement to reveal the topological "twist" that allows the Circle of Fifths to exist as a closed loop in 12ET. ~~~~

  Jimishol (talk) 23:58, 10 February 2026 (UTC)

  I will reply in two parts, in reverse order: the second part of the answer is that this is fairly clearly all Original Research, and does not belong on WP; the end. The first part is more interesting, because dismissing contributions as OR is not very constructive.

  Why this arrangement? You said: "... Major 7th (3x4)". But by 3x4, I mean that each of your triangles (actually deltoids on the umbilical torus, although there is no obvious distinction between laying this out on a any particular type of torus) represents an augmented triad (eg CEG#); how is this a "Major 7th"? Actually it is precisely as atonal as all the other arrangements, which consist of the elements of Z12 generated by a divisor of 12 (2, 3, 4, or 6; representing semitone, whole tone, minor 3rd, major 3rd, augmented 4th). (OK, I managed to stop the animation...) I see that the "cursor" is a rectangle marking the sides of two successive triangles. Sure, but because of the obvious symmetry, you could equally show the complement of this diagram, which has 3 loops on a 4-sided torus; each loop is a dim7 chord, and the same "cursor" represents the "complement" of a Maj7, i.e. minor 3rd, major 3rd, minor 3rd. So you have not shown that there is anything special about the particular "3x4" arrangement. This is all remarkably similar to the last OR contribution I looked at - immediately above, named "Cycle of 48 notes of the first four notes of major scales in fifths order".

  Your claim, "This animation demonstrates exactly how 12ET allows that infinite helix to close into a finite, continuous loop." is I think simply false. The animation shows the closed "circle of fifths"; the fact that this is incompatible with just intonation is not illustrated anywhere; the fact that a fifth is supposed to be a 3:2 ratio rather than 2^(19/12) makes no difference to the diagram. You could just as easily have the 4x4 diagram illustrating something similar in 16TET, not really related to diatonic harmony.

  In summary, I do not think this animation illuminates anything. It does not take something complicated and show it more simply; rather like the German Tonnetz diagrams, it is better at impressing the student with the teacher's obscure knowledge than actually providing any explanation. Imaginatorium (talk) 08:40, 11 February 2026 (UTC)

  One problem that I see in all this is a confusion between successions of notes, of chords, and of keys. This problem existed as early as the images of the circle themselves, in the 17th and early 18th centuries. The Circle of fifths article begins saying that "the circle is a way of organizing pitches as a sequence of perfect fifths" and adds that "this order places the most closely related key signatures adjacent to one another" – "pitches," "fifths," and "key signatures!"

  What this article says of chord progressions is not (or not exactly) the same as what the article chord progression says and, in both, some theories of chord progressions remain unmentioned (see for instance Tymoczko, "Root Motion, Function, Scale-Degree," or Hedges and Rohrmeier, Exploring Rameau and Beyond: A Corpus Study of Root Progression Theories). We might perhaps better begin sorting that out... — Hucbald.SaintAmand (talk) 10:10, 11 February 2026 (UTC)

  Final Statement regarding WP:OI and the suppression of geometric visualization

  @Imaginatorium: @Hucbald.SaintAmand:

  I am writing this as a final statement on the matter. I will not be restoring the image myself, but I feel compelled to correct the record regarding the misapplication of Wikipedia policy that has taken place here.

  1. The Abuse of WP:OR vs. WP:OI

  You have categorized this visualization as "Original Research" (OR). This is a fundamental misreading of WP:OI (Original Images), which states: "Original images created by a Wikipedian are not considered original research, so long as they do not illustrate or introduce unpublished ideas."

  • The Data is Neutral: The GIF merely plots the sequence of Perfect Fifths in 3D space. This is not a "theory"; it is a geometric plot of the interval data defined in the article.
  • The Topology is Standard: The fact that the sequence of fifths in 12ET forms a closed loop (a continuous curve on a torus) is established mathematical fact. The article's own main image explicitly equates 6 flats to 6 sharps (Gb=F#). My visualization introduces no new data; it merely projects this established enharmonic equivalence onto a continuous 3D curve to show how the closure happens.
  • The "Cursor" is a Tool: Using a Major 7th rectangle to track the movement is a design choice for visualization, identical to choosing a specific color scheme or projection for a map. It introduces no new musical data.

  2. The "Moving Goalposts" of Rejection

  The objection process has been inconsistent, suggesting a predetermined outcome rather than constructive editing:

  • Phase 1 (Archive 3): The objection was a lack of "accompanying text" or "encyclopedic value."
  • Phase 2 (Squandermania): When context was provided, the objection shifted to "This is cool but it's not really clear what it means."
  • Phase 3 (Current): When the geometric meaning (the cursor tracking the functional succession of fourths) was clarified, the objection shifted immediately to "Original Research."
  • If the goal was clarity, we would be discussing captions. Instead, the goal appears to be exclusion.

  3. The Image Fits Multiple Contexts

  This visualization resolves the "Circle or cycle?" debate discussed on this Talk page. It demonstrates visually that in 12ET, the "cycle" (sequence) geometrically becomes a "circle" (closed loop) through the topological twist. It supports existing text in multiple potential sections:

  • Definition: It visualizes the "sequence of perfect fifths" (which the text admits is generally, but not exclusively, shown as a circle).
  • Modulation: It visualizes the "functional succession" of ascending fourths described in the text.
  • Tonnetz: It directly visualizes the topology described in the Twentieth-century reinterpretation section, which explicitly states that the harmonic network is "mathematically isomorphic to a torus." My visualization provides the geometric realization of this isomorphism.

  4. Conclusion

  I have received correspondence from a reader (with a background in physics and mathematics) who encountered the visualization on this article before its removal. He explicitly stated that this specific model described the intuition connecting music theory to "Lie groups and supersymmetry" more elegantly than any previous object, inspiring him to start a project on GitHub based on it. By removing it, you are not protecting the article from "confusion"; you are preventing future readers from accessing that geometric reality.

  The file remains in the Wikimedia Commons. The responsibility to place it where it belongs—whether here to illustrate the Definition, in the Modulation section, or in the Tonnetz article—now rests with you. I have provided the geometry; it is up to the editors to decide if they want to show it.

  P.S.

  To clarify the distinction between Original Research (Text) and Original Imagery (Visualization), I offer the following example. The text below describes a novel pedagogical method (let's call it "Clock Arithmetic for Keys") that fits perfectly in the Relation with chromatic scale section. Because this specific method does not appear in standard bibliography, it would be classified as Original Research, and I have never asked to include it. I asked to include the GIF, which contains none of this novel methodology, but simply plots the standard intervals.

  A Novel Pedagogical Method (Clock Arithmetic):

  To recognize a key: Treat the clock as a chromatic scale where C is 00:00 and each hour is one semitone (C#=1, D=2, etc.). Every sharp (#) in the signature adds 7 hours to the clock, and every flat (b) adds 5 hours. For example, 3 sharps = 3 * 7 = 21 hours; on a 12-hour clock, 21:00 is 9:00 (A Major). To find the relative minor, look 3 hours earlier (6:00 is F# minor).

  To construct a signature: Find the hour of your Major key (e.g., E Major is 4:00). Add 12 to that hour until it is divisible by 7 (for sharps) or 5 (for flats). For E Major: 4 + 12 + 12 = 28; since 28 / 7 = 4, you need 4 sharps. To find which notes are sharp, start at F# (6:00) and add 7 hours for each subsequent note (6+7=1, 1+7=8, 8+7=3). For flats, start at Bb (10:00) and add 5 hours for each subsequent note.

  Confusing a novel textual theory (like the above) with a neutral geometric plot (the GIF) is the error in your judgment. ~~~~

  Jimishol (talk) 07:43, 17 February 2026 (UTC)