The Cycle of Fifths
Understanding the Cycle
We learned in our study of ratios and Pythaorean relationships that when we ascend and descend from the central note of C, we reach a note that appears to complete the circle:
When these fifths are tuned in perfect, Pythagorean 3:2 ratios, however, the F# and the Gb do not meet—they are separated by the Pythagorean comma, which too be precise is 23.46 cents (1.5 to the 12 power divided by 2 to the seventh power = 1.01364 or 23.46 cents.) Since a "cent" in musical acoustics is equivalent to an equally tempered semitone, the circle misses by rougly a quarter of a semitone.
The purpose of equal temperament is to narrow the size of the tuned perfect fifths (when tuning a piano or fitting frets onto a guitar) such that the circle does in fact close. This involves de-tuning a piano by 2 cents for every perfect fifth or fourth within an octave. While this may seem miniscule, it is both perceptable and, more importantly, effects the intonation of all the other intervals besides. This must be contrasted with string players, who tune their strings to perfectly resonant 3:2 fifths, or singers in a capella ensembles, who do in fact sing (or do their best to sing) in something approaching the ideal of just intonation.
However, this "small sin" of equal temperment is unavoidable when it comes to tuning pianos or fixed-fretted instruments if one wishes to play in all the keys at our disposal. This leads us directly to the cyce of fifths.
If one takes the notes drawn on the musical staff above and writes them out again in a circle, we can diagram this as the familiar Circle of Fifths:
You can see here that the notes of the cycle are arranged with the overtonal (3/2) fifths moving around to the right, while the reciprocal (2/3) fifths moving around to the left.
From this basic arrangement we can notice a great deal. First of all, if we take the notes of the circle of fifths and play around the overtonal side, we can see that as we play from C to C, G to G, D to D, A to A, etc, using only the natural notes, we again create the ancient modes:
- C to C = Ionian
- G to G = Mixolydian
- D to D = Dorian
- A to A = Aeolian
- E to E = Phrygian
If we move around the first two notes on reciprocal side, we notice first the appearance of the Lydian mode:
- F to F = Lydian
- Bb to Bb = Lydian (if we use only natural notes except for the Bb).
However, the cycle of fifths is really a representation of the tonal system, based on a cycle of scales that are always major, posessing a major third, a perfect fourth and a fifth and a major second, sixth and seventh above the tonic. Thus, all the modes that automatically result from going around the cycle of fifths using only the natural notes (the white notes of the piano), must be adjusted to create major scales as you move around the cycle. How is this to be accomplished?
Key Signatures and the Cycle of Fifths in major
Overtonal (Sharp) Side
Notice that when we shift up a fifth from C to G, the resultant scale of natural notes is the mixolydian mode. In order to transform this mode into the Ionian mode, i.e. the major scale, we must raise the seventh step of the scale:
When we move up to D, the resultant scale of natural notes is the Dorian mode. However, if we "keep" the F# that we just employed for changing the G mixolydian into G major, we note that, once again, we have a mixolydian mode, but starting on D:
If we move to A, but "keep" both the F# and the C# from the previous "adjustments", we find we once again create a major scale by taking A mixolydian and transforming it into A major by raising the leading tone:
The same process, then, can be applied to the remaining notes on the overtonal side: keep the previous sharps and the sharp to the seventh step of the scale, transforming mixolydian into major:
Reciprocal (flat) side
A similar process can be used on the flat or reciprocal side. When we go down a fifth to F, we notice that we then create the Lydian mode. In order for the Lydian mode to be transformed in the major scale, the fourth step must be lowered a half step so that we have the perfect fourth between the first and fourth degrees of the scale, and also that we have the half step between 3 and 4:
If we go down another fifth, to Bb (because of course F down to B is a tritone, so we must expand the interval by a half step and reach down to Bb to create the perfect fifth) we "keep" the previous flat (in this case, it is also the Bb) and we must then lower the fourth again the take away the tritone between Eb and A and create a perfect fourth between Eb and Ab:
In this way, we can continue to create major scales from Lydian scales by "keeping" the flats from the previous key and lowereing the fourth step of the scale:
The Cycle of Fifths and Key Signatures
It is clear then that a simple pattern emerges:
- When moving around the cycle of fifths to the right, the overtonal or sharp side, keep the accidental from the previous position and sharp the seventh step to create a half step between 7 and 8 in the scale.
- When moving around the cycle of fifths to the left, the reciprocal side, keep the accidental from the previous position and flat the fourth step to create a half step between 3 and 4 in the scale.
With all this information, the use of key signatures should be readily apparant. When one wishes to compose or perform music in a key other than C, but one desires it to be in the major mode, use a key signature that changes the necessary notes automatically, for the entire composition. The arrangement of key signartures around the cycle of fifths then is:
Relative Minor Keys
Once these major keys are established, there remains one last part of the system. Just as C major and A minor (C Ionian and A Aeolian) were perceived to be polar to each other, and therefore relatives due to the majorness or minorness of the tonic, dominant and subdominant triads, all the other major keys have minor relatives. In all cases:
the relative minor is down a third (a minor third) from the root of the relative major. The easiest way to find this is to simply go down the major scale a third until you reach the sixth.
Just as A is a minor third down from C, so to is E a minor third down from G, D a minor third down from F and so forth.
Here is the cycle of fifths with the relative minors inside the circle:
Thus, finally, we can see these relative major and minor keys as sharing the same key signature:
The simple rules for recognizing the key signatures should be obvious to you, now that we've gone through the logic for creating them:
- When presented with a sharp key signature, take the last sharp to the right and go up a half step. Why? Because we always raise the seventh step of the scale to create the major scale from the mixolydian, therefore the last sharp out is always the note that leads up to the tonic.
- When presented with a flat key signature, take the last flat to the right and go down four. Why? Because we always lower the fourth step of the scale to create the major mode from the lydian, therefor the last flat out is always the fourth note of the scale.
- Since this flatted fourth is also, by definition, a fifth below the tonic, the next key of the cycle is always the note that was just flattened in the previous key. Thus a shortcut with flat keys is take the second to the last flat our from the right, which is in fact the key of the signature.