Lydian and Locrian in the Circle of Fifths
The relationship between Lydian and Locrian can be read as a natural consequence of the circle of fifths. If the modes are arranged along this circle, they no longer appear as isolated structures, but as points of a single movement: from a maximum openness toward ascending fifths to a maximum closure toward descending fifths. In this space, Lydian and Locrian occupy the two extreme poles of the system.
Diatonic Modes Ordered on the Circle of Fifths
The seven diatonic modes can be ordered along the circle of fifths from the brightest to the darkest. Taking C as the reference note, the order is as follows:
| Mode | Notes | Tones and Semitones | Variation |
|---|---|---|---|
| C Lydian | C - D - E - F# - G - A - B | T - T - T - S - T - T - S | — |
| C Ionian | C - D - E - F - G - A - B | T - T - S - T - T - T - S | F# → F |
| C Mixolydian | C - D - E - F - G - A - Bb | T - T - S - T - T - S - T | B → Bb |
| C Dorian | C - D - Eb - F - G - A - Bb | T - S - T - T - T - S - T | E → Eb |
| C Aeolian | C - D - Eb - F - G - Ab - Bb | T - S - T - T - S - T - T | A → Ab |
| C Phrygian | C - Db - Eb - F - G - Ab - Bb | S - T - T - T - S - T - T | D → Db |
| C Locrian | C - Db - Eb - F - Gb - Ab - Bb | S - T - T - S - T - T - T | G → Gb |
In this sequence, each mode arises from the previous one by lowering a single note by a semitone. The transition from the brightest mode to the darkest mode thus occurs step by step: F#, B, E, A, D, and G are progressively lowered, following the order of the circle of fifths.
| Mode | Position in the Modal Series | Interval Formula | Character |
|---|---|---|---|
| C Lydian | Bright extreme | 1 - 2 - 3 - #4 - 5 - 6 - 7 | Maximum openness |
| C Locrian | Dark extreme | 1 - b2 - b3 - 4 - b5 - b6 - b7 | Maximum closure |
Lydian and Locrian occupy the two extremes of the modal series. Lydian is the most open mode because it contains, relative to the tonic, a major second, major third, augmented fourth, perfect fifth, major sixth, and major seventh. Locrian, on the other hand, is the most closed mode because it contains a minor second, minor third, perfect fourth, diminished fifth, minor sixth, and minor seventh.
Ambivalence of the Tritone
| Mode | Characteristic Note | Interval Formula | Reading of the Tritone |
|---|---|---|---|
| C Lydian | F# | #4 | Tritone as augmented fourth |
| C Locrian | Gb | b5 | Tritone as diminished fifth |
The distance between C and F#/Gb is the same from a sonic standpoint: a tritone. In C Lydian it is written as #4, that is, augmented fourth, because it derives from the raising of the fourth. In C Locrian it is written as b5, that is, diminished fifth, because it derives from the lowering of the fifth. The tritone is therefore the same acoustic interval, but it takes on two different names and two different theoretical functions depending on the modal context.
Opposition and Contiguity
The apparent opposition between Lydian and Locrian, however, can be overcome with a minimal shift of the tonal center.
Having traversed the entire modal sequence on the same tonic, from C Lydian to C Locrian, six notes out of seven have been lowered. The only note that remains unchanged is the modal tonic, that is, C. At this point, by also lowering the fundamental by a semitone, the system does not simply close: it starts over a semitone lower.
| Mode | Notes | Tones and Semitones | Variation |
|---|---|---|---|
| C Locrian | C - Db - Eb - F - Gb - Ab - Bb | S - T - T - S - T - T - T | — |
| B Lydian | B - Db - Eb - F - Gb - Ab - Bb | T - T - T - S - T - T - S | C → B |
Starting from C Locrian, the darkest form of the modal system built on C, only the fundamental note remains to be lowered. If C descends to B, while all the other notes remain identical, the scale becomes B Lydian. In enharmonic notation, B - Db - Eb - F - Gb - Ab - Bb corresponds to B - C# - D# - E# - F# - G# - A#, that is, B Lydian. The transition from Locrian to Lydian thus occurs through the lowering of the modal tonic alone: having exhausted the cycle of alterations on the same fundamental, the system begins again with an analogous sequence, but transposed a semitone lower.
For this reason Lydian and Locrian are at the same time the opposite extremes of the modal cycle and two contiguous forms: they are extremely distant when observed as modes on the same tonic, but they become immediately connected if the tonal center is shifted down by a single semitone.
The system thus shows a double structure. On one hand, by ordering the modes on the same tonic, the transition from Lydian to Locrian occurs through a progressive darkening of the scale, note after note, semitone after semitone. On the other hand, Lydian can transform directly into Locrian without changing the sonic material, simply by lowering the fundamental and reassigning the same seven notes a new tonal function.
Opposite Halves in the Circle of Fifths
If one takes a note as center, for example C, and places it ideally at noon in the circle of fifths, Lydian represents the maximum expansion toward the right, that is, toward ascending fifths:
C - G - D - A - E - B - F#
Reordered in scalar form, these notes produce C Lydian:
C - D - E - F# - G - A - B
Locrian instead represents the maximum expansion toward the left, that is, toward descending fifths:
C - F - Bb - Eb - Ab - Db - Gb
Reordered in scalar form, these notes produce C Locrian:
C - Db - Eb - F - Gb - Ab - Bb
Lydian is therefore the most “rightward” mode of the circle of fifths, while Locrian is the most “leftward” mode. The former concentrates major or augmented intervals; the latter concentrates minor or diminished intervals.
Spiral Form
The relationship between Lydian and Locrian can also be imagined as a double spiral built around the origin of a Cartesian system. The tonic lies at point 0; from there the spiral proceeds by semitones, alternating movement below and above the center. With each half-circle, the radius increases by 1, that is, by one semitone.
The formula of the spiral is r = n, θ = nπ, where n indicates the progressive number of the half-circle. Every time θ increases by π, the spiral completes half a turn and the radius grows by one unit. The intersections with the x-axis thus produce a sequence of alternating chromatic distances relative to the tonic.
Table 1 — Lydian Spiral
| X-axis | Shift in Semitones from C | Note Obtained | Interval Function in the Scale |
|---|---|---|---|
| 0 | 0 | C | 1 |
| -1 | 1 semitone below | B | 7 |
| +2 | 2 semitones above | D | 2 |
| -3 | 3 semitones below | A | 6 |
| +4 | 4 semitones above | E | 3 |
| -5 | 5 semitones below | G | 5 |
| +6 | 6 semitones above | F# | #4 |
Table 2 — Locrian Spiral
| X-axis | Shift in Semitones from C | Note Obtained | Interval Function in the Scale |
|---|---|---|---|
| 0 | 0 | C | 1 |
| +1 | 1 semitone above | Db | b2 |
| -2 | 2 semitones below | Bb | b7 |
| +3 | 3 semitones above | Eb | b3 |
| -4 | 4 semitones below | Ab | b6 |
| +5 | 5 semitones above | F | 4 |
| -6 | 6 semitones below | Gb | b5 |
The two spirals are thus mirror images of each other. Lydian organizes the intervals in the direction of maximum openness: 2, 3, #4, 5, 6, 7. Locrian reverses the rotation and organizes the intervals in the direction of maximum closure: b2, b3, 4, b5, b6, b7.
The extreme point of both spirals is the tritone. In Lydian it is reached as +6 semitones, that is, #4; in Locrian it is reached as -6 semitones, that is, b5. From a graphical standpoint, it is therefore possible to connect a Lydian scale to a Locrian scale by reversing the rotation of the spiral once the tritone is reached. The tritone thus becomes, together with the origin, the welding point between the two orientations: on one side it closes the Lydian spiral, on the other it opens the Locrian spiral in the opposite direction.
Polymodal Tonality
In the traditional sense, the leading tone is a note placed a semitone below the arrival note that tends to resolve upward: for example B tends toward C. The same principle, however, can also be thought of in reverse: a note placed a semitone above the arrival note can resolve downward, as Db tends toward C. In this way, two types of attraction can be distinguished: a lower leading tone, which rises toward its resolution, and an upper leading tone, which descends.
The Lydian mode can be read as an ascending scale on the tonic because it contains two lower leading tones:
| Scale | Notes | Lower Leading Tones | Resolution |
|---|---|---|---|
| C Lydian | C - D - E - F# - G - A - B | B, F# | B → C, F# → G |
| Function | 1 - 2 - 3 - #4 - 5 - 6 - 7 | 7, #4 | 7 → 1, #4 → 5 |
The note B is a semitone below C and tends to resolve toward the tonic; the note F# is a semitone below G and tends to resolve toward the fifth. Lydian thus contains two upward drives: B → C and F# → G. For this reason it can be interpreted as a field that rises toward the two main poles of the tonic: 1 and 5. Lydian is therefore an ascending scale.
Locrian functions in a mirror-like manner. It does not contain a perfect fifth, but a diminished fifth; for this reason its descending attractions are not oriented toward 1 and 5, but toward 1 and 4:
| Scale | Notes | Upper Leading Tones | Resolution |
|---|---|---|---|
| C Locrian | C - Db - Eb - F - Gb - Ab - Bb | Db, Gb | Db → C, Gb → F |
| Function | 1 - b2 - b3 - 4 - b5 - b6 - b7 | b2, b5 | b2 → 1, b5 → 4 |
The note Db is a semitone above C and tends to resolve toward the tonic; the note Gb is a semitone above F and tends to resolve toward the fourth. Locrian thus contains two downward drives, Db → C and Gb → F, and for this reason it is a descending scale.
At this point, keeping C and G as poles of resolution, the Lydian mode and the Locrian mode can be read as two opposite ways of approaching these same points:
Thus C Lydian and G Locrian both converge on C and G, but from opposite directions. C Lydian tends toward C and G from below; G Locrian tends toward G and C from above. The former organizes an ascending tension on the tonic root-fifth dyad, the latter a descending tension on the dominant. In this sense Lydian rises toward the tonal poles, while Locrian descends toward the same poles.
These two movements embody the movement and tension/resolution between tonic and dominant, extending the framework of tonal music to the twelve tones of the chromatic scale. For this reason this framework can be defined as “polymodal tonality.”
Harmonic Series
The root of the Lydian-Locrian system can be formulated as a generative hypothesis based on the harmonic series. In this reading, the aim is not to derive a scale directly from the harmonic series, but to observe certain significant features found between tonic and dominant, and between dominant and return to the tonic. From these features, interval cells are extracted which, when repeated, generate the Lydian field and the Locrian field respectively.