Circle of Fifths Study
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.