Circle of fifths
As Quintenzirkel is referred to in the music theory a series of twelve spaced tempered fifths arranged tones, the last tone in the same pitch class has as the first and can be accordingly set equal to it. However, this equation is only possible due to an enharmonic mix-up that can be made anywhere. Returning to the beginning results in a “tour”, which is graphically represented as a circle ( Latin : circulus “circle”).
The circle of fifths does three things:
- In its most common presentation today, he assigns the parallel major - and minor keys so that type, number and order of the sign can be read.
- He established the principle of the related fifths for the keys (as well as their root tones and the chords built on them ), which says that two keys are more closely related the closer they are in the circle of fifths. In the 19th and 20th centuries, other types of clay relationship were added, sometimes contradicting.
- The most important diatonic scales in Western music (major, natural minor and the modal scales ) can be derived from the circle of fifths.
A circle of fourths arises if one assumes tempered fourths instead of fifths , which, however, does not make any difference to the circle of fifths for theory. Arnold Schönberg justifies this as follows: “If you go in one direction of the circle (C, G, D, A etc.), then that is the circle of fifths, or as I prefer to say: circle of fifths upwards, because it is those that build up above the starting point Fifths are. If you go in the opposite direction of the circle, you get C, F, B, Eb, etc., which some call the circle of fourths, but this has little sense, because C, G is fifth up or fourth down and C, F Quint down or fourth up. That's why I prefer to call the opposite direction the circle of fifths downwards. "
Since you can replace fifth steps up with fourth steps down, it is possible to take a tour of the circle of fifths within an octave:
Circle of fifths and spiral of fifths
The circle of fifths is not a system of order given by nature, such as the overtone series , but an artificial construction that is only conceivable on the basis of tempered moods and their possibility of enharmonic reinterpretation of tones. In a tuning with theoretically pure fifths, such as the Pythagorean tuning , no closed circle of fifths , but only a spiral of fifths can be realized: Twelve fifths ( cgdaeh fis cis gis dis ais eis his ) enclose an interval of about seven octaves; However, taking the fifths and octaves with their pure vibration speed ratios 3: 2 and 2: to 1, as is shown by calculation that the Abschlusston his the fifth row slightly by 23.5 cents (by almost a quarter semitone) is higher than that reached at the same time last c of the octave row. This difference, called the Pythagorean comma , prevents the equation of his and c as well as all other enharmonic tone pairs ( f sharp / gb, d sharp / es, a sharp / b , etc.) , also different by the Pythagorean comma . The sequence of fifths does not close to a circle, but can only be represented as a spiral. If one were to try to close the circle "by force" regardless of this, it would only be possible via a so-called wolf fifth .
The problem of closing the spiral of fifths to a circle without such jarring wolf fifths is solved in today's widespread equally tempered tuning by reducing each of the twelve fifths by 1/12 of the Pythagorean comma, so that with the twelfth fifth step you get exactly to the seventh Octave of the original note. The even distribution of the comma ensures that, apart from the octave, no interval sounds completely pure, but the imperfections remain so small that they hardly disturb.
The circle of fifths in the major-minor system
The graphic on the right shows the major keys on the outside of the circle , which are designated in German with capital letters.
The graphics can be seen that for a progression in the cycle of fifths with each key, a signed add occurs or disappears.
- With a clockwise movement (fifth upwards) there is an additional increase every time (expressed by an additional ♯ ( cross ) or a disappearing ♭ ).
- With a counter-clockwise movement (fifth down) an additional humiliation takes place every time (expressed by an additional ♭ or a disappearing ♯ ).
The necessary enharmonic mix-up is usually made between F sharp and G flat major in order to keep the number of required accidentals small. The reinterpretation could, however, also be carried out at any other point in the circle of fifths. In principle, every flat key could also be noted as a sharp key and every sharp key also as a flat key. There is a simple connection between enharmonic keys in terms of their accidentals:
- The sum of the accidentals of two enharmonic keys always results in 12. Example: B flat major (2 ♭ ), A sharp major (10 ♯ ): 2 + 10 = 12
|Sign :||7 ♭ + fes
||6 ♭ + ces
||5 ♭ + total
||4 ♭ + des
||3 ♭ + as
||2 ♭ + es
||1 ♭ b
||0 ♭ / ♯
||1 ♯ f sharp
||2 ♯ + c sharp
||3 ♯ + g sharp
||4 ♯ + dis
||5 ♯ + ais
||6 ♯ + ice
||7 ♯ + his
|Major keys:||Ces||Ges||Of||As||It||B.||F.||C.||G||D.||A.||E.||H||F sharp||Cis|
|Minor keys:||as||it||b||f||c||G||d||a||e||H||f sharp||cis||g sharp||dis||ais|
The major keys with sharps sign ( ♯ )
The accidentals are added successively in a fifth apart, with the sharp keys progressing upwards (clockwise). The order of the notes to be provided with a ♯ (F, C, G, D, A, E, H ...) can also be read in the circle of fifths due to the distance between fifths, if you start at F and continue clockwise. (The crosses ( ♯ ) increase tones by a semitone step):
- G major: F sharp
- D major: F sharp, C sharp
- A major: F sharp, C sharp, G sharp
- E major: F sharp, C sharp, G sharp, D flat
- B major: F sharp, C sharp, G sharp, D sharp, A sharp (pronounced: A sharp )
- F sharp major: F sharp, C sharp, G sharp, D sharp, A sharp, ice (pronounced: E- flat )
- C sharp major: F sharp, C sharp, G sharp, D sharp, A sharp, ice, His
Numerous sayings have been invented to help you remember the order of the major keys with sharps signs in the circle of fifths, for example:
- G eh D u A lter E sel H ol Fis che (Instead of the word “Esel”, “Emil” can be used to avoid the risk of confusion with the note “Es”.)
(also works: G erda D enkt A n E in H OHES Fis Since lighter at the other saying F-Dur (is mistaken because of fish) with F # major.)
The major keys with B flat sign ( ♭ )
The movement of the keys as well as the addition of the accidental in the interval between fifths, which force a lowering of the notes, takes place in the ♭ keys downwards :
Here, too, the order of the notes to be provided with a ♭ (H, E, A, D, G, C, F, ...) can be read off based on the distance between fifths in the circle of fifths, if you start at B and continue counterclockwise.
- F major: B
- B flat major: B flat
- E flat major: B, E flat, A flat
- A flat major: B, E flat, A flat, D flat
- D flat major: B, E flat, A flat, D flat major, D flat major
- G flat major: B, E flat, A flat, D flat, G flat, Ces
- C flat major: B, E flat, A flat, D flat, G flat, C flat, Fez
Numerous sayings have been invented to help you remember the order of the major keys with a flat key in the circle of fifths, for example:
- F innovative B rötchen There sen As se Des Ges angvereins
The minor keys
The minor keys are assigned to the major keys as parallel keys in the circle of fifths. (Parallel keys have the same accidentals.) If you know the accidentals for the major keys, you can easily determine the accidentals of a certain minor key if you know that the parallel major key is always a minor third (= three semitones) higher. For example, G minor has the same accidentals as B flat major, which is a minor third higher.
Keys with more than six accidentals
The series of sharp and flat keys could be continued as desired, with the sharp keys, for example: C sharp major with seven ♯ , G sharp major with eight ♯ to ice major with eleven ♯ etc. But that's not just about the notation very confusing; further increases in already heightened tones only lead to tones that were already there in the sound image.
In His major (twelve ♯ ), in addition to the 6 ♯ already known from F sharp major, a “His” (sounding like C), a “Fisis” (double increase of F, sounding like G), a “Cisis” ( sounding like D), a “Gisis” (sounding like A), a “Disis” (sounding like E) and an “Aisis” (sounding like H). However, since His major does not differ from C major (with equal tuning ), such a notation would make little sense.
Therefore, it utilizes the phenomenon of aural equality different designated tones to enharmonic : instead of the increasing complexity of sharp keys to use the corresponding, equal sounding ♭ -Tonarten, as instead Gis major (8 ♯ ) As-Dur (4 ♭ ). Or, to stay with the example of the key “ice major”: Instead of eleven ♯ , F major only needs a single ♭ .
Keys with more than six accidentals are almost never used when it comes to choosing the basic key of a piece of music. Bach's use of C sharp major (7 ♯ ) in the Well-Tempered Clavier is rare. However, in the course of a piece of music it can be useful to use keys with many accidentals. So you would z. For example, with a brief modulation from E major (4 ♯ ) to the upper mediante G sharp major (8 ♯ ), keep the notation with crosses in order to clarify the harmonic relationship. Only when a longer stay in the new key is sought is it common to remove the old accidentals and replace them with the simpler notation of the enharmonic key (in this case A flat major with 4 ♭ ).
Application of the circle of fifths to modal keys
The modal scales that emerged from the church tones (modes), which also include the scales belonging to major ( Ionic ) and natural minor ( Aeolian ), can be derived from the circle of fifths. To do this select seven each neighboring circle of fifths sounds and sorts them then as to that they in Sekundabstand consecutive. For example, to get all scales with the root C, you apply this procedure to all seven-tone circle-fifths that contain the C. The result is shown in the following table:
|Circle of fifths
rearranged to the scale
|CGDAEH Fis||CDE Fis GAH||C- Lydian||1 ♯|
|FCGDAEH||CDEFGAH||C- Ionic / major||-|
|BFCGDAE||CDEFGAB||C- Mixolydian||1 ♭|
|There BFCGDA||CD Es FGAB||c- Doric||2 ♭|
|As it BFCGD||CD Es FG As B||C- Aeolian / minor||3 ♭|
|The As It BFCG||C Des Es FG As B||c- Phrygian||4 ♭|
|Ges Des Ace Es BFC||C Des Es F Ges As B||c- Locrian||5 ♭|
In the table "same", that is listed on the same tone-based keys, the major sex keys with a large C, the minor sex are marked with a small c. The circle representation on the right shows parallel keys: The keys on a certain radius have the same accidentals.
Determination of the fifth
The circle of fifths can also be used to determine the size of the width of the fifths. If one assigns its position in the circle of fifths to each tone of a tone group, the greatest distance between two tones can easily be read in fifths.
- Example A major triad: A - E - B - F sharp - C sharp → The chord has a fifth width of 4 QB.
Compositions with special reference to the circle of fifths
The realization of the circle of fifths with the help of tempered moods and the possibility of unrestricted use of all keys was reflected in compositions which exploit the possibilities of transposition and modulation in a special way.
- Johann Sebastian Bach : The Well-Tempered Clavier (Part I, 1722). The pairs of movements, consisting of the prelude and fugue , go through all keys and are arranged in ascending chromatic order, with the minor key of the same name always following the respective major key.
- Georg Andreas Sorge : Keyboard exercise from 24 Praeludia through the entire Circulum Modorum (1730). The sequence of the pieces corresponds to the circle of fifths.
- Johann Sebastian Bach : The Well-Tempered Clavier (Part II, 1740/42). Key sequence as in Part I.
- Frédéric Chopin : 24 Preludes op. 28 (1836/39). Arrangement of the individual pieces according to the circle of fifths upwards, with each major key being followed by the parallel minor key.
- Dmitri Shostakovich : 24 preludes for piano solo op. 34 (1932/33). Arrangement of the keys as with Chopin.
- Paul Hindemith : Ludus tonalis (1942). Although this work has no direct reference to the circle of fifths, it deserves mention here because it follows a system of tone relationships developed by Hindemith himself, which, in his opinion, should take the place of the traditional circle of fifths. The Ludus tonalis sees itself as a modern counterpart to Bach's Well-Tempered Clavier .
- Dmitri Schostakowitsch : 24 Preludes and Fugues, Op. 87 (1950/51). Key sequence as with Chopin.
The following compositions modulate through all keys of the circle of fifths.
- Johann Mattheson : Exemplary organist rehearsal (1719, 10th test piece)
- Johann David Heinichen : Fantasia through all keys in the general bass in the composition (1728). The work was ascribed to JS Bach for a long time (cf. BWV Anh. 179).
- Ludwig van Beethoven : Deux Preludes par tous les 12. Tons majeurs pour le Fortepiano ou l'Orgue op.39 (1789, published 1803)
- Frédéric Chopin : Nocturne op.37, no.2 . Shortly before the end there is a three-bar passage in which the basic harmony notes in the bass run through all twelve stations of the circle of fifths (downwards).
Examples from rock / pop music: At the end of the 1970s, the French percussionist Pierre Moerlen played some pieces that moved through the circle of fifths, among others. a. with his brother Benoît and the Oldfield siblings.
- Mike Oldfield : Incantations Part Four from the album Incantations (1978)
- Pierre Moerlen's Gong : Downwind from the album Downwind (1979)
- Sally Oldfield : Mandala from the album Celebration (1980)
Synesthetic representations of the circle of fifths
Of synesthetic associations were between synesthesia and music produced. Accordingly, the levels of the circle of fifths have been associated with color ideas in many ways, although the color assignments can differ from case to case. Here are some examples:
- Gregory Barnett: Tonal organization in seventeenth-century music theory. In: Thomas Christensen (Ed.): The Cambridge History of Western Music Theory (The Cambridge History of Music). Cambridge University Press, Cambridge 2002, doi: 10.1017 / CHOL9780521623711.015 .
- Willibald Gurlitt , Hans Heinrich Eggebrecht (Ed.): Riemann Music Lexicon. Material part. 12th, completely revised edition. B. Schott's Sons, Mainz 1967.
- Marc Honegger, Günther Massenkeil (ed.): The great lexicon of music. In 8 volumes. Updated special edition. Herder, Freiburg (Breisgau) a. a. 1987, ISBN 3-451-20948-9 .
- Joel Lester: Between Modes and Keys. German Theory 1592-1802. Pendragon Press, Hillsdale, (New York) 1989, ISBN 978-0-918728-77-7 .
- Diether de la Motte : Harmony . 16th edition. Bärenreiter, Kassel u. a. 2011, ISBN 978-3-7618-2115-2 .
- Arnold Schönberg: Harmony . 3rd, increased and improved edition. Universal Edition, Vienna 1922.
- The circle of fifths as software for the PC
- The circle of fifths on lehrklaenge.de
- Exercise on recognizing keys on lehrklaenge.de
- The circle of fifths and related ideas on musikanalyse.net
- Newly invented and thorough instructions ... to complete learning of the general bass , p. 261. See Lester 1989, p. 108–111.
- Arnold Schönberg: Harmony . 3rd, increased and improved edition. 1922, p. 187 .
- Willibald Gurlitt , Hans Heinrich Eggebrecht (Ed.): Riemann Music Lexicon. Material part . 12th, completely revised edition. 1967, p. 775 .
- Beethoven, Preludes op. 39 : Sheet music and audio files in the International Music Score Library Project