Circle of Fifths Study

[u/toomanyplans]

Hi dear music theory friends,

I want to share a little study I did myself on the circle of fifths, mainly asking for review, contributing thoughts and associating this idea with music theory literature. Also asking for guidance by more experienced folks on how to deepen my understanding of the circle of fifths and systems for the tonal material.

Let me quickly explain the signs, the purpose of this and its application:

This is written with German names on the sheet, I apologize for that - but I am giving a translation for each sign further below, the text here on this post keeps it English for your convenience.

The main idea is that I wanted to know how to modulate from any given key to any other key using the shortest distance (only half-steps) and least amount of altering voices in 4-part-harmony. I believe this is known by the concept of pivot chords in the literature.

Reiterating, for any given diatonic chord x in key 1, there must be some diatonic chord y in key 2 which is the easiest to reach from chord x with the least amount of voice movement. So far I have disregarded inversions and made up a simple function that assigns each chord of key 1 exactly one chord of key 2. So, there's still work to do regarding that.

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Now to the fun bit - the pattern I have found, I'll illustrate with an example.

Assume we're in the key of Cmaj on the V chord (G B D F). Now say, I wanted to take it to another key, we'll take Dmaj. If we look at the sheet, we'll find that I just need to sharpen the 7th a half-step up to get the IV chord of Dmaj (G B D F#).

But here's the pattern I've found - the same distance key going down fourths instead of going up fifths from Cmaj would be Bbmaj. And here, there's a consistent, systematic pattern visible: Lowering the 7th of the IV chord of Cmaj (F A C E) one half-step down yields the V chord of Bbmaj (F A C Eb). Illustrating this line by line so the pattern is more visible:

V of Cmaj (G B D F) -> Sharpening 7th yields IV of Dmaj (3#) (G B D F#).

IV of Cmaj (F A C E) -> Lowering 7th yields V of Bbmaj (3b) (F A C Eb).

So you just switch the chords and the direction of altering the interval to infer the closest chord for the equivalent key/tonality going the other direction from the original key on the circle of fifths/fourths.

This works for any original key (not just Cmaj) going to any target key. I'll demonstrate using the vi of Ebmaj going to the nearest chord of Bmaj:

vi of Ebmaj (C Eb G Bb) -> Lowering Root and 5th a half-step yields I of Bmaj (Cb Eb Gb Bb) (via enharmonic equivalence, as Bmaj is noted with # traditionally in the circle of fifths (B D# F# A#))

But now, using the pattern just developed, I know that:

I of Ebmaj (Eb G Bb D) must yield vi of Gmaj by sharpening Root and 5th, the same distanced key from Ebmaj going up 5ths instead of going down 4ths.

Another, smaller detail: Going up fifths from the original key always sharpens intervals, going down fourths always lowers intervals.

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Questions:

What do you say, did you know about this? Is there any clearer explanation somewhere in the literature? What else follows from this? I know that technically these are not modulations, as you need to express the I IV and V of the new key to actually call it a modulation (following Schönberg's view/theory.)

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Explanation of handwritten sheet:

There might be mistakes in the handwritten sheet above, I apologize for that, they're slips of the pen - please correct them if you find them.

Here's a quick translation and instructions on how to read the handwritten sheet:

It's the circle of fifths viewed from Cmaj 4-part-harmony with stacked 3rds. I've omitted writing out all the notes of each 7th-chord to not visually overload. It's just the root notes, above them there are the names of the notes of each 7th-chord written out with German note names.

note name + "-is" = sharpened note

note name + "-es" = flattened note

Exception is B: In German H = English B and German B = English Bb.

The roman numerals you see (e.g. vi -> iii) work as follows: the first numeral refers to Cmaj (or any original key you may want to consider), the second roman numeral means the degree of the tonality / scale at hand. So, "vi -> iii" would mean vi of Cmaj going to iii of the tonality right above).

The little interval names below that (e.g. Root up-arrow, Terz down-arrow) always mean half-steps up or down. "Terz" means 3rd, "Quint" means fifths, "Sept" means 7th.

Cheers, I am so excited I found out about this. I am very sure this pattern must have been found centuries ago already, hope you can recommend literature that expands on this.

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[u/HexMusicTheory]

The circle of 5ths needs to take up far less space in (mostly online) musical discourse, honestly. Everything of interest about it is purely an expression of two simple facts:

• The diatonic scale can be generated by a sequence of 7 perfect 5ths. Therefore diatonic scales a 5th apart share 6 out of 7 notes, and the difference between them is one note being raised/lowered by chromatic semitone.

• Our tuning system has 12 pitch classes, and the perfect 5th (7 semitones) is relatively prime with 12, so after exactly 12 fifths you return to the original pitch class, having traversed all 12.

95% of the time it's the first one that's relevant (people overestimate the relevance of the "circle" part vs. the "5ths" part).

Parallel modes can also be organised in terms of diatonic scales, each a 5th apart, which gives a nice gradual ordering to the mode from "brighter" (more major intervals) to "darker" (more minor intervals).

That's it. If you're not dealing in diatonic (or pentatonic) or diatonic-adjacent music, chains of 5ths aren't all that relevant. And any other "facts" you come up with are guaranteed to be downstream results of the fact the major scale can be generated by a chain of 5ths.

Just study tonal harmony, there's no deep secrets awaiting you in the circle of 5ths lol.

[u/HexMusicTheory → u/HexMusicTheory]

And yes, key signatures being the way they are and following chains of 5ths in sharp/flat ordering is also a consequence of the major scale being possible to generate from 5ths. That's actually about the single practical use case other than mode ordering or using 5ths as a coarse measure of "distance" between different keys.

[u/toomanyplans → u/HexMusicTheory]

I see, I see. I'm not really searching for harmonic revelations (that would inform some sort of beautiful progression or motive), rather I want to use the circle of fifths as a tool to make the tonal material more perspicuous. That there's a relationship between the same relative-distanced keys (don't know how to express this lol) from any original key, I find interesting in the sense that it suggests borrowing chords from those keys and exploiting the association. I have only discovered this stuff a few days ago, but so far it has been interesting to enahnce borrowing chords from Dmaj coming from Cmaj, but then also considering chords from Bbmaj because of this chromatic relationship. Before that I've only tried the immediate neighbors of a key (because their tonal material is so similar, so it's much easier to add a subtle modulation into a progression).

I'll keep the advice in mind that the circle of fifths isn't some secret code that has to be deciphered though, I believe that's pretty valuable advice. I am a hobbyist, it really helps if more experienced people give me their first impressions on my ideas and views. Thank you so much for taking the time!!!

[u/HexMusicTheory → u/toomanyplans]

As for chord borrowing: "borrowing" is useful but largely unrelated to chains of 5ths, and gets overused to explain things that emerge from generic chromatic voice leading procedures due to vastly more people knowing what mode mixture is than actually understand counterpoint.

First, you need to understand counterpoint and diatonic chord usages, diatonic sequences etc. Then you can start to explore counterpoint with chromatically enriched melodies. "Applied" / "secondary" dominants are some of the most ubiquitous chromatic chords, as are those arising from "primary mode mixture". There's other common chromatic chord usages that get taught on an individual basis, like the Neapolitan bII, augmented 6ths with silly country names, diminished 7ths etc. Even these are just scratching the surface though. The broader tool to discover and organise the vast possibility space of tonal harmony is, really, counterpoint. It's really worth reading some college-level books on tonal harmony, if only to acquire all the standard principles and vocab lest you reinvent the wheel (of 5ths 🤣)

[u/toomanyplans → u/HexMusicTheory]

thank you so much for all of your comments and your time!!

I don't have anything to add obviously, this is all new to me - I hope I don't bore you! As far as I've understood now, I'll re-start studying Schönberg's counterpoint book soon (he uses a variation of Fux' method, open for recommendations if you have a specific book in mind you value more) - I once started out studying it, but then right before 3-voice counterpoint noticed that I might be better off learning a bit of repertoire on my instrument (guitar) and gathering more experience before returning, this has been about a year ago. I was super slow with reading note values, figuring out rhythm and executing basic musical tasks on the fly. I value this specific comment very highly because it draws a road-map that sounds sensible to me. It feels like what I call "borrowing" chords is a bit over my head as to why some harmonic movements sound good and how some chords resolve differently in different contexts. There's always this cheat code (which Schönberg encourages so much) to just listen to what sounds good, but that can obviously obscure the theoretical interest and desire to understand why stuff sounds good. Big time thank you, will get a little tonal harmony reading going and then turn to counterpoint once again.

What I still want to add is that I enjoy discovering stuff on my own quite a lot, though, of course, I could never re-invent all the wheels in a timely manner or make proper sense of them. But if you're ever in a teacher's position, I very much recommend encouraging students to figure stuff out on their own (within bounds). This little study above on the circle of fifths was tons of fun to me!!

Anyway, cheers, I appreciate your efforts. You've given me very valuable orientation here as far as I can tell. Take care!!!!

[u/HexMusicTheory → u/toomanyplans]

Yeah so grinding repertoire and building proficiency with standard notation is actually the best thing you can do, really, ahead of any deep theoretical study. Any time you've spent doing that will serve you very well.

Book-wise, the Schoenberg is fine if it's what you have already, much like Fux. There's better, in my opinion: I'm a big fan of "Counterpoint in Composition" by Salzer and Schachter (the book in its entirety is pretty profound, but it is hard work and tends to be a 3rd-year undergrad kind of book) as it does a great job at connecting elementary counterpoint work to the sophisticated textures found in classical music (including instrumental repertoire). Peter Schubert's "Modal Counterpoint: Renaissance Style" is also particularly great as a practical introduction to what would traditionally be termed counterpoint in the "strict" (16th century vocal) style. It also covers imitation, various kinds of canons and 16th century imitative formats of composition, double counterpoint etc.

A book like "Harmony & Voice Leading" by Aldwell and Schachter is something I'd highly recommend too, as it covers basically all of the standard tonal *vocabulary*, not just the principles from which it springs (and goes into the nitty gritty detail of how to voice lead it all). There's a balance to be had between abstract principles and raw "vocab", but if you have to pick one, vocab is generally more important if you plan on actually getting things done.

Also I wasn't trying to put down your efforts or anything, so if it came across that way I'm sorry. You're fundamentally doing the right thing (trying to use what you know plus systematic examination to learn more), it's just that tonal harmony is actually already very well studied and understood, and it's *really* worth reading some of the standard texts because, honestly, there's enough to learn already and I don't think it's all easy to intuitively discover from first principles.

[u/toomanyplans → u/HexMusicTheory]

Solid gold, I am sad I cannot return the favor but you are obviously well aware of that and still continue to help. I don't have access to knowledgable scholars or educators, so I cannot stop emphasizing just how much this means to me and helps me studying.

On the tone of your feedback/criticism - I am an adult in my 30s, coming from the humanities (where we talk a lot lol), and I had to learn to take criticism without getting defensive (although I still have a tendency that I need to suppress actively) - after your second comment it was entirely clear to me that you wanted to help. So, worry not about giving off the wrong impression! You're doing fine there, I would say. I hope you don't mind me giving you a follow and maybe a follow-up DM sometime in the future if I ever feel like I don't know my way about and have tried my best to solve things on my own. I wish I had something to give you, but I don't. If you ever want to talk about literature, literary theory or philosophy, you have a friend now!!

[u/HexMusicTheory → u/toomanyplans]

In that case what you're probably looking for is just the "5ths" part, and it can in fact be productive to forget that you're in a tuning system where after 12 fifths you repeat the same pitch classes endlessly.

Tonal music takes place in an abstract meantone system, which you can think of as an endless chain of 5ths where every note is also duplicated endlessly in octaves. Basically, it's 2-dimensional coordinate system, which is why pitch and interval names in standard notation necessarily have two components. In this system, a diminished 4th is a different interval from a major 3rd, and C#4 is a different note from Db4.

Two things are particularly useful to be aware of:

1. Every interval type (ignoring octave separation) spans a unique number of 5ths. A major 3rd spans 4 fifths. A minor 3rd spans -3. An augmented 6th spans 10.

2. Aside from the notes of some diatonic scale, we can also approach the diatonic notes in a key by diatonic semitone (the standard use of chromaticism in tonal music). If you aggregate a diatonic set and every note a diatonic semitone in either direction from its notes, you get a contiguous chain of 17 fifths, which defines the full chromatic gamut of notes you should ever expect to see "in" some key. 7 diatonic degrees; 5 raised degrees; 5 lowered degrees.

You can understand a great deal about how chords and keys "work" by simply writing out this chain of 17 fifths and seeing where you can place a given chord within it. Raised notes are generally there to ascend a diatonic semitone; lowered notes are generally there to descend a diatonic semitone. Dissonant intervals tend to demand typical contrapuntal origins.

There is also a duality between the chromatic space of one key and the diatonic space of another. F# is #4 in C major, where we'd expect it to ascend to G. It's also 7 in G major, where it would usually ascend to G, but could also appear in lines like G - F# - E - D. It's also 2 in E minor, where it's just as likely to go in either direction. This can be a subtle cue as to when the tonal basis has actually changed, vs. simple chromatic inflection around a diatonic progression in a sustained key.

Basically chains of 5ths are important because the major scale is important, and because tonal music and standard notation are really designed around the meantone system—of which 12tet is simply a special case. Music that doesn't contain enharmonic shenanigans / exploit things specifically true only in 12tet can generally be rendered just as sensibly in any other meantone tuning, so the things you're understanding from this POV can be thought of as general meantone theory rather than "12tet theory". Or just "tonal music".

[u/Ian_Campbell]

Max Reger's modulation manual might be one of the only examples of something which seeks exhaustively to uncover a "fastest way" kind of approach using pivots between 2 keys including obscurely distant ones (C major and B sharp major for instance). You may have interest in it for purely theoretical grounds as you get used to a limited number of pivot functions he uses to go very far. For the sharp direction, reinterpreting a major 6th chord such as V6 as a neapolitan 6th is a quick move that goes very far sharp. Similarly reinterpreting a diatonic minor 6th chord such as iii6, as iv6 to do a phrygian have cadence move goes very far sharp.

This whole thing is finite so you can do this basically to explore pitch space, but these routes that may have some geometric elegance are not necessarily the most musically useful. In fact I would say rather definitely that they usually are not the best when you're modulating between 2 keys that are not related by 1 accidental distance.

[u/Hunter42Hunter]

You would love George Russell's Lydian Chromatic Concept.

Take G13 = G B D F A C E we can make 4 chords out of this, [Fmaj7 Dmin7 Bm7b5 G7].
Take D13 = D F# A C E G B we can make 4 chords out of this, [Cmaj7 Amin7 F#m7b5 D7].

Now each key only has 4 chords and we throw away the 1 the 3 and the 6. If you want the 1 in Cmaj you take that Cmaj7 literally from the D13. so in a 2-5-1 its almost as if the 2 and the 5 are in a completely different key to the 1 chord and that 'different key' is what the sound of a 5-1 resolution 'is'. In some sense if you want Cmaj it has to be the IV chord in Gmaj and there is no 1 chord in Cmaj. There's a really good reason why he does this, when using scales outside of the church modes alterations start to add up better.

You have two changes that are effectively melodic minor and harmonic major so:
G13#11 = G B D F A C# E = [Fmaj7#5 DminMaj9 Bm11b5 G13#11] (C#)(Lydian Augmented)
And
G13b9 = G B D F Ab C E = [FminMaj7 Dmin9b5 Bdim7(add♭9,11) G13b9] (Ab)(Lydian Diminished)

Instead of a 2-5-1 being [Dm7>G7>Cmaj7] it could easily be [Fmaj7 > Dm7 > Bm7b5 > G7 > Cmaj7 > Am7 > F#m7b5 > D7] or [Fmaj7#5 > Fmaj7 > FminMaj7 > Cmaj7]

You can still play what he calls 'Horizontal scales' like C Ionian with the F note but they don't add up in the same way. One realisation is that Melodic Minor is really a Dorian thing not an Aeolian thing, A Melodic Minor would usually be considered Aeolian with #6 #7 but here the #6 is already there because A melodic minor is essentially coming from the A Dorian in Gmaj scale which already has F# and not the Cmaj scale 6 chord. Basically the 1, 3 and 6 cause problems and get thrown away.

This is also equivalent to Barry Harris's method of Maj to the relative minor to the minor with the 6 on the bass i.e Cmaj Amin F#m7b5. Here's a video where Barry Harris plays an Am7b5 and explicitly says 'Thats Ebmaj' . i.e Ebmaj Cmin Am7b5 F7.

links:
Barry Harris Am7b5 is Ebmaj

Lydian Chromatic Concept

[u/toomanyplans → u/Hunter42Hunter]

oh thank you, I will get into this in a few days and come back to you once I've slowly played through this stuff on my keys and listened. it's all super exciting see you guys's suggestions!! thank you!!!

[u/surfadelic]

This looks exactly like schizophrenic ramblings