Circle of Fifths

Now that I’ve written a basic introduction into major and minor keys (https://functionalanalysis.blog/2017/11/20/major-keys/ and https://functionalanalysis.blog/2017/11/20/minor-keys/ ) let’s talk a bit about how keys are related to one another.

 

Most music that can be described as functionally tonal, where it is useful to talk about keys and functional harmonic chordal relationships, doesn’t stay in one key for the whole piece unless it is quite short. It will usually come back to the same key it started in, but depending on the era, composer, and length, it will briefly refer to, highlight, or even be in[*] other keys. When we look at music to analyze it to better play or understand it, we often need to know what these keys are for tuning, understanding chord relationships, and larger structures. One graphic that describes relationships between keys is the circle of fifths:

circle-of-5ths w bass.jpg

To understand this, let me explain key signatures. Key signatures are shown at the left of each staff of music, and tell the person reading it which accidentals happen every time without needing to be indicated. The following shows a key signature of three sharps, or A major.

k331-ph-1-clean-copy

An A major scale has three sharps (C#, F#, and G#) to make the wwhwwwh interval pattern. So it is easier to indicate in the key signature that we expect every F, C, and G to be sharped unless otherwise noted for a whole piece in A major. So, playing in C major (all white keys on the piano) there are no accidentals/black keys in the major scale, so the key signature is no sharps or flats, and if the music called for a black key-chromatic note, it would be indicated with a sharp or a flat. Likewise, D minor needs a B♭ to complete the whwwhww pattern, it’s key signature shows one flat on B. The minor key signatures match up with the natural minor scale.

You can work out the key signature to any given tonic pitch by constructing the major or natural minor scale and then figuring out which flats or sharps you used, or (as usually preferred by teachers and conductors) you just memorize all of them. So when you see a piece of music with one flat you know: F major or D minor.

When writing a key signature, the order of flats and sharps written in a key signature is the same as the order they are added as one goes around in the circle of fifths. So while A♭ major has A♭, B♭, D♭, and E♭, we write them in the order B♭, E♭, A♭, and D♭ because one flat is B♭, two is B♭ and E♭, three is B♭ E♭ and A♭, etc. Same with sharps; while A major has three sharps, C#, F#, and G#, they are listed F#, C#, and G#.

The order of sharps as they go around the circle is F, C, G, D, A, E, B.

The order of flats is the reverse: B, E, A, D, G, F, C.

From just looking at the key signature, the way to determine a major sharp key is to look at the last printed sharp and go up one half step. For example six sharps is, F#, C#, G#, D#, A#, and E#. One half step up from E# is F#. This note will be your tonic.

For flats, the second to last printed flat is the tonic; with 5 flats you have B♭E♭A♭D♭G♭, so D♭ is the tonic.

Minor keys signatures are often learned in connection to their relative major. Relative major and minor keys share the same key signature. To find a relative minor of any given major key, go down a minor third, which is another way of saying to find the note corresponding to the solfège syllable la or scale degree 6. A minor third down from F is D, so F major and D minor share a key signature. Likewise a minor third up from a minor tonic or the solfège me in minor keys will tell you the relative major of that key. B minor has 2 sharps (F# and C#), and its scale degree 3 is D, so D major is its relative major, also having two sharps.

Another key relationship is parallel keys. Two parallel keys share the same tonic: C major and C minor. A parallel minor key will have 3 more flats than its major: C major has 0 flats and C minor has 3. On the sharp side, this means three fewer sharps: F# major has 6 sharps, F# minor has three. This is to lower scale degrees 3, 6, and 7: mi to me, la to le, and ti to te.

While a key is often defined by what notes it is using, in functional tonality how the notes act is almost as important as what they are. By looking at or listening to how the notes are behaving, you can determine whether you are in the major or minor key given by the key signature, or if the key of the music does not match the key signature written on the page.[†] The note that the piece ends on is usually tonic. Tonic is also usually (but not always) somewhere near the beginning, either the first note or the first down beat if there is a rhythmic pick-up. Check for the ti do half step. If you see a note that is consistently raised with a sharp or natural, it might be the leading tone, ti. Follow ti up a half step to find do/tonic. If the piece is in minor, it will often have ti chromatically emphasized with a sharp (or natural on the flat side). Minor keys have to use an accidental to get ti and la, so if the piece you’re looking at has more accidentals, it might be in minor. Look/listen for resting points in the music (cadences) and see what notes and chords are emphasized there.

Try practicing figuring out key signatures by looking at whatever music you’re working on. Or here’s some examples for major: http://musictheoryexamplesbywomen.com/example_categories/major/ and minor: http://musictheoryexamplesbywomen.com/example_categories/minor/ . I use youtube to search for recordings, just make sure you get the op. number and movement right!

 

Music that changes key (modulates) or emphasizes other keys often goes first to closely related keys. Closely related keys are defined as keys that are within one sharp or flat on the circle of fifths, neighbors/touching on the above diagram. So D major’s (2 sharps) closely related keys are one sharp more with 3 sharps (A major and F# minor), and one sharp less with one sharp (G major and E minor), as well as the relative minor (B minor). Other relationships can be described using fancier terminology, but closely related keys are the most common in most types of tonal music. You can use the same techniques as finding the original key of a piece by looking for accidentals that might emphasize notes or chords as more important for a given phrase or chunk of music that looks/sounds like it may have modulated. Trust your ear! If it sounds like a different note is being emphasized as tonic, you’re probably right.

———-

[*] depending on your terminology, hearing, and overall structural biases

[†] This happens when the music has changed keys halfway thru (called modulating), or if there’s a reference to a mode of some kind, or other various reasons the composer may have used a different key signature. Sometimes, composers don’t use signatures at all, but the music can still sound like its in a key, so it’s good to know what to look for.

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