Chromatic circle: Difference between revisions
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Yilloslime (talk | contribs) WP:BOLDly removing pitch constellation stuff. This doesn't belong in this article and isn't notable enough for a stand alone article. Plus the tone is not encyclopedic. Likely original research. |
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==Comparison with circle of fifths== |
==Comparison with circle of fifths== |
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A key difference between the chromatic circle and the [[circle of fifths]] is that the former is truly a continuous space: every point on the circle corresponds to a conceivable [[pitch class]], and every conceivable pitch class corresponds to a point on the circle. By contrast, the circle of fifths is fundamentally a ''discrete'' structure, and there is no obvious way to assign pitch classes to each of its points. |
A key difference between the chromatic circle and the [[circle of fifths]] is that the former is truly a continuous space: every point on the circle corresponds to a conceivable [[pitch class]], and every conceivable pitch class corresponds to a point on the circle. By contrast, the circle of fifths is fundamentally a ''discrete'' structure, and there is no obvious way to assign pitch classes to each of its points. |
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==Pitch constellation== |
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[[File:pitch constellation chromatic.svg|thumb|400px|Pitch constellations for twelve-tone equal temperament showing all twelve chromatic pitches]] |
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A '''pitch constellation''' is a graphical representation of [[Pitch (music)|pitches]] used to describe [[musical scale]]s, [[Musical mode|modes]], [[Chord (music)|chords]] or other groupings of [[Pitch (music)|pitches]] within an octave range.<ref name="Slonimsky">{{Citation |
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| last = Slonimsky |
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| first = Nicolas |
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| year = 1947 |
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| title = Thesaurus of Scales and Melodic Patterns |
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| publisher = Music Sales America |
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| isbn = 0-8256-1449-X |
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}}{{page needed|date=September 2012}}.</ref><ref name="Burns">{{Citation |
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| last = Burns |
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| first = Edward M. |
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| year = 1999 |
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| title = Intervals, Scales, and Tuning. The Psychology of Music. |
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| publisher = Academic Press |
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| isbn = 0-12-213564-4 |
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}}{{page needed|date=September 2012}}.</ref><ref>{{Citation |
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| last = Lerdahl |
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| first = Fred |
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| year = 2001 |
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| title = Tonal Pitch Space |
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| publisher = Oxford University Press |
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| isbn = 0-19-505834-8 |
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}}{{page needed|date=September 2012}}.</ref> It consists of a circle with markings along the circumference or lines from the center which indicate pitches. Most pitch constellations use a subset of pitches chosen from the [[chromatic scale]]. In this case the points on the circle are spaced like markings on an analog clock or gauge where each tick mark represents a step. |
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===Scales and modes=== |
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The pitch constellation provides an easy way to identify certain patterns and similarities between [[Harmony|harmonic]] structures. |
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For example, in twelve-tone equal temperament: |
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* A [[major scale]] consists of a circle with markings at 0 (or 12), 2, 4, 5, 7, 9 and 11 o'clock. |
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* A [[minor scale]] consists of a circle with markings at 0 (or 12), 2, 3, 5, 7, 8 and 10 o'clock. |
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[[File:Pitch constellation degrees.svg|400px|Pitch constellations showing major and minor scales in degrees]] |
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The diagrams above show the two scales marked with [[Degree (music)|"scale degrees"]]. It can be observed that the [[Tonic (music)|tonic]], second, fourth, and fifth are shared, while the minor scale [[Flat (music)|flattens]] the third, sixth and seventh notes relative to the major scale.<ref name="Glaser">{{Citation |
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| last = Glaser |
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| first = Matt |
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| year = 1999 |
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| title = Ear Training for Instrumentalists (Audio CD) |
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| publisher = Homespun |
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| isbn = 0-634-00385-2 |
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}}{{page needed|date=September 2012}}.</ref> Another observation is that the minor scale's constellation is the same as the major scale, but rotated +90 degrees. |
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In the following drawing all of the major/minor scales are drawn. Note that the constellation for all the major scales or all the minor scales are identical. The different scales are generated by rotating the note overlay. The notes that need to be [[Sharp (music)|sharpened]]/flattened can be easily identified. |
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{| class="wikitable" width=400px |
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|- |
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| |
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{{show |
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|Major and minor scales |
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|[[File:Pitch constellation major scales.svg|400px|Major scales in all keys]] |
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[[File:Pitch constellation minor scales.svg|400px|Minor scales in all keys]]}} |
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|} |
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Moreover, if we draw all seven [[Diatonic and chromatic|diatonic]] [[Musical mode|modes]] we can see them all as rotations of the [[Ionian mode]].<ref name="Slonimsky"/><ref>{{Citation |
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| last = Yamaguchi |
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| first = Masaya |
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| year = 2006 |
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| title = Symmetrical Scales for Jazz Improvisation |
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| publisher = Masaya Music |
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| isbn = 0-9676353-2-2 |
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}}{{page needed|date=September 2012}}.</ref> Note also the significance of the 6 o'clock point. This corresponds to a [[tritone]]. The modes including pitches a tritone from the tonic ([[Locrian mode|Locrian]] and [[Lydian mode|Lydian]]) are least used. The 5 o'clock and 7 o'clock pitches are also important points corresponding to a [[perfect fourth]] and [[perfect fifth]] respectively. The most used scales/modes - major ([[Ionian mode]]), minor ([[Aeolian mode]]) and [[Mixolydian mode|Mixolydian]] - include these pitches. |
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<!--this image is displayed wider than 400px for detail--> |
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[[File:Modes.svg|500px|Modern musical modes]] |
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[[Symmetric scale]]s have simple representations in this scheme. |
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<!--this image is displayed wider than 400px for detail--> |
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[[File:Symmetric.svg|500px|Symmetric scales (as defined by Slonimsky)]] |
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More exotic scales - such as the [[Pentatonic scale|pentatonic]], [[Blues scale|blues]] and [[Octatonic scale|octatonic]] - can also be drawn and related to the common scales. |
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[[File:Exotic.svg|400px|Less common scales]] |
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A more complete [[list of musical scales and modes]] |
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[[File:PitchConstellations.svg|700px|An assortment of pitch constellations]] |
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===Other overlays=== |
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In previous sections we saw how various overlays (scale degrees, semi-tone numbering, notes) can be used to notate the circumference of the constellation. Various other overlays can be laid around the constellation. For example: |
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* [[Interval (music)|Intervals]] |
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* [[Solfège]] |
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* [[Interval ratio|Pitch ratios]] (ratios of pitch frequencies) |
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<!--this image is displayed wider than 400px for detail--> |
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[[File:Overlays.svg|500px|Pitch constellation overlays (i.e. - the hour markings)]] |
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Note that once a pitch constellation has been determined, any number of overlays (notes, solfège, intervals, etc.) may be placed on top for analysis/comparison. Often generating one harmonic relationship from another is simply a matter of rotating the overlay or constellation or shifting one or two pitch locations. |
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===Chords=== |
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Similarities between [[Chord (music)|chords]] can also be observed as well as the significance of [[Augmentation (music)|augmented/diminished]] notes.<ref name="Burns"/><ref name="Glaser"/> |
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For [[Triad (music)|triads]] we have the following: |
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[[File:Pitch constellation triads.svg|400px|Triadic chords (key of C)]] |
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And for [[seventh chord]]s: |
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[[File:Sev chord.svg|400px|Seventh chords (key of C)]] |
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Some chords are related to symmetrical (augmented or diminished) chords by a change in one note, which is related to [[otonality and utonality]]: |
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[[File:Symmetrical otonal utonal chords.svg|400px|Relationship between symmetrical, otonal and utonal chords]] |
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===Circle of fifths=== |
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Beginning with a pitch constellation of a chromatic scale, the notes of a [[circle of fifths]] can be easily generated. Starting at C and moving across the circle and then one tick clockwise a line is drawn with an arrow indicating the direction moved. Continuing from that point (across the circle and one tick clockwise) all points are connected. Moving through this pattern the notes of the circle of fifths can be determined (C, G, D, A ...). |
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[[File:Pitch constellation fifths.svg|400px|Generating notes for a circle of fifths from a pitch constellation of a chromatic scale]] |
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[[File:Pythagorean_tuning_geometric.svg|thumb|right|A broken circle of fifths, using just fifths on a chromatic circle. (Starting point and ending point are at right, with a gap.)]] |
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One can also depict non-tempered intervals on a chromatic circle, which allows one to depict [[Comma (music)|commas]] (small intervals), particularly [[comma pump]]s. For example, using a sequence of twelve just fifths (3:2 ratio) does not quite return to the starting point (the size of the gap is the [[Pythagorean comma]]), resulting in a "broken" circle of fifths. |
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===Technical note=== |
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The ratio of the frequencies between two pitches in the constellation can be determined as follows.<ref>{{Citation |
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| last = Josephs |
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| first = Jess L. |
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| year = 1967 |
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| title = The Physics of Musical Sound |
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| publisher = Van Nostrand Company |
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| isbn = |
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}}{{page needed|date=September 2012}}.</ref> Take the length of the arc (measured clockwise) between the two points and divide by the circumference of the circle. The frequency ratio is two raised to this power. For example, for a standard perfect fifth ('''P5''', which is located at 7 o'clock relative to the tonic '''T''') the frequency ratio is: |
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<math>{\text{f}_\text{P5} \over \text{f}_\text{T}} = 2 ^ {( 7 / 12 )} \approx 1.49821 \approx {3 \over 2}</math> |
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==References== |
==References== |
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Revision as of 05:30, 26 April 2024
 | It has been suggested that this article should be split into a new article titled Pitch constellation. (discuss) (March 2024) |
The chromatic circle is a clock diagram for displaying relationships among the equal-tempered pitch classes making up a given equal temperament tuning's chromatic scale on a circle.
Explanation
If one starts on any equal-tempered pitch and repeatedly ascends by the musical interval of a semitone, one will eventually land on a pitch with the same pitch class as the initial one, having passed through all the other equal-tempered chromatic pitch classes in between. Since the space is circular, it is also possible to descend by semitone.
The chromatic circle is useful because it represents melodic distance, which is often correlated with physical distance on musical instruments. For instance, assuming 12-tone equal temperament, to move from any C on a keyboard to the nearest E, one must move up four semitones, corresponding to four clockwise steps on the chromatic circle. One can also move down by eight semitones, corresponding to eight counterclockwise steps on the pitch class circle.
Larger motions (or in pitch space) can be represented in pitch class space by paths that "wrap around" the chromatic circle one or more times.
For any positive integer N, one can represent all of the equal-tempered pitch classes of N-tone equal temperament by the cyclic group of order N, or equivalently, the residue classes modulo twelve, Z/NZ. For example, in twelve-tone equal temperament, the group has four generators, which can be identified with the ascending and descending semitones and the ascending and descending perfect fifths. In other tunings, such as 31 equal temperament, many more generators are possible.
The semitonal generator gives rise to the chromatic circle, while the perfect fourth and perfect fifth give rise to the circle of fifths.
Comparison with circle of fifths
A key difference between the chromatic circle and the circle of fifths is that the former is truly a continuous space: every point on the circle corresponds to a conceivable pitch class, and every conceivable pitch class corresponds to a point on the circle. By contrast, the circle of fifths is fundamentally a discrete structure, and there is no obvious way to assign pitch classes to each of its points.
References
- ^ "Prelude to Musical Geometry", p.364, Brian J. McCartin, The College Mathematics Journal, Vol. 29, No. 5 (Nov., 1998), pp. 354-370. (abstract) (JSTOR)
Further reading
- Brower, Candace (2000), "A Cognitive Theory of Musical Meaning", Journal of Music Theory, 44 (2), Duke University Press: 323–379, doi:10.2307/3090681, JSTOR 3090681.
- Kučinskas, Darius (2005), "Symmetry in creative work of Mikalojus Konstantinas Čiurlionis" (PDF), Menotyra, 38 (1): 42–46.
- Olson, Harry F. (1967), Music, Physics and Engineering, Dover Publications, ISBN 0-486-21769-8
External links
- Notenscheibe web application - pitch constellations of scales, triads, intervals and the circle of fifths, with basic audio
- On-line app illustrating pitch constellations
- ScaleTapper - IPhone app which utilizes pitch constellations.
- PDF of musical scales
