Circle of fifths
Contents |
Definition
The circle of fifths is a circular diagram of all chromatic tones by perfect fifths.
Circle of fifths
Major scales in order of accidentals
It is possible to construct a major scale on every tone, and different accidentals are needed to induce the proper order of steps: whole, whole, half in both tetrachords (4 tone scale part). Only the scale of c major yields this without use of accidentals. Both c d e f and g a b c share this structure: 1 1 ½. These tetrachords are both connected to the fundamental c, the first has it as lowest, the second as highest note. The remaining interval f g is a consequence of this tetrachordal construction, and in itself a whole step (major second). The total scale thus shows 1 1 ½ 1 1 1 ½.
When constructing this on different tones, we can set the scales we find into the following order by number of accidentals, those we find having sharps:
And those we find having flats:
If we allow for the use of double sharps and flats, more scales are possible, but for practical reasons we limited the list of scales to those employing a maximum of 7 accidentals.
Minor scales in order of accidentals
It is also possible to construct a similar series when studying minor. In order to obtain similar accidentals, we use the minor-parallel scale, or aeolian mode, which can be found on the VI of any major scale. VI of c being the a we find the following order by number of accidentals, with sharps:
And another series yet, employing flats instead, starting with the parallel of f, d:
Key signatures in order of accidentals
The order of appearance of sharps is fixed as: f# → c# → g# → d# → a# → e# → b#, the example shows the correction notation of the key-signatures with sharps on two clefs:
The order of appearance of flats is also fixed and the exact reverse as to the use of basic tones: b♭ → e♭ → a♭ → d♭ → g♭ → c♭ → f♭, the example shows the correction notation of the key-signatures with flats on two clefs:
Note that both orders are based upon this diatonical series of fifths, either f c g d a e b or the reverse b e a d g c f.
Geometry of the full circle of fifths
We can now summarize this structure and knowledge in the circle of fifths.
The image of the circle of fifths summarizes and visually represents the observations on major and minor scales and keys and the key signatures with number and order of sharps and flats, but it also allows a further geometrical look into the intervals, and their place within this representation.
Here are some relevant observations which help in understanding and memorizing this visual representation of "musical geometry":
- There are the enharmonic equivalences of the major keys C♭=B (7 flats equals 5 sharps enharmonically), G♭=F# (6 flats equals 6 sharps enharmonically), D♭=C# (5 flats equals 7 sharps enharmonically);
- There are the enharmonic equivalences of the minor keys a♭=g# (7 flats equals 5 sharps enharmonically), e♭=d# (6 flats equals 6 sharps enharmonically), and b♭=a# (5 flats equals 7 sharps enharmonically);
- The enharmonic equivalences are true for the tempered tuning system alone, but this is the contemporary tuning system which we actually use in this basic music theory;
- The circle of fifths is not valid in other tuning systems;
- In the outer black ring the major keys are represented in white capitals;
- In the inner white ring the minor keys are represented in black minuscules;
- In the inner gray circle all possible interconnections between points are shown;
- Outside the rings, the number of accidentals is shown, from zero up to seven;
- On the top and bottom left and right are red accidentals, to show the left side contains flats, and the right side contains sharps;
- The fact that the starting point C - a is represented on top, is arbitrary but customary;
- Intervals can also be visualized in this geometrical representation, as the innermost gray circle contains all interconnections between all 2 points, these represent the following intervals:
- neighbours are a fifth (or its inversion the fourth) apart - completing this as a pattern one gets a dodecagon or the full circle of fifths;
- neighbours to neighbours, skipping one in between, are a major second apart (or its inversion the minor seventh) - completing this as a pattern one gets a hexagon or half the circle as a whole tone scale (acoustically there are only two such scales);
- skipping two one finds a minor third (or its inversion the major sixth) - completing this as a pattern one gets a square or a diminished seventh chord (acoustically there are only 3 such chords);
- skipping three one finds a major third (or its inversion the minor sixth) - completing this as a pattern one gets an equilateral triangle or an augmented triad (acoustically there are only 4 such triads);
- opposites are in a tritone (augmented fourth or its inversion diminished fifth) relationship (acoustically there are only 6 such intervals);
- neigbours to opposites are a minor second apart (or its inversion the major seventh) - completing this as a pattern one gets the full circle as a star representing the chromatic scale (acoustically there is only 1 such scale);
- A still fuller circle of fifths is possible when wrapping the full sequence of flats, naturals and sharps of basic tones in order onto itself, 21 tones wrapped in the form of a 12 tone circle, with only g, d and a having no enharmonic equivalents with a single accidental:
- f♭ - c♭ - g♭ - d♭ - a♭ - e♭ - b♭ - f - c - g - d - a - e - b - f# - c# - g# - d# - a# - e# - b#;
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- doing this however would lead to needlessly encumbering the image with highly unusual keys and scales, as well as to overflooding the eye with too much information, but one can and should attempt to imagine this overcomplete image nevertheless.
Making sure to check and gradually understand fully all of these steps towards, and observations on, the circle of fifths, will enable one to memorize it as more than a mere photographic image, but rather as a dynamic "Gestalt" implied by and underlying the tone-structure of the tempered tuning system.
