The root of the Lydian-Locrian System can be formulated as a generative hypothesis starting from the harmonic series. In this reading, the goal is not to derive a scale directly from the harmonic series, but to observe certain significant traits found between tonic and dominant, and between dominant and the return to the tonic. From these traits, interval cells are extracted which, when repeated, generate the Lydian field and the Locrian field respectively.
In the tables below, the “harmonics” column indicates the number of the harmonics of the C series from which the cell is derived. The “intervals” column indicates the consecutive shifts in semitones.
From the first to the fourth harmonic
| Notes | Harmonics | Intervals | Notes obtained by repeating the intervals | Scale |
|---|---|---|---|---|
| C - G | 2 - 3 | 5 | C - G - D - A - E - B - F# | C Lydian |
| G - C | 3 - 4 | 4 | G - C - F - Bb - Eb - Ab - Db | G Locrian |
In the first case, the ascending fifth is repeated: starting from C, one obtains C - G - D - A - E - B - F#, which, reordered, produces C Lydian. In the second case, the opposite movement is repeated, G - C: starting from G, one obtains G - C - F - Bb - Eb - Ab - Db, which, reordered, produces G Locrian.
| Notes | Harmonics | Intervals | Notes obtained by repeating the intervals | Scale |
|---|---|---|---|---|
| C - E - G | 4 - 5 - 6 | +4, +3 | C - E - G - B - D - F# - A | C Lydian |
| G - Bb - C | 6 - 7 - 8 | +3, +2 | G - Bb - C - Eb - F - Ab - Db | G Locrian |
From the fourth to the eighth harmonic
| Notes | Harmonics | Intervals | Notes obtained by repeating the intervals | Scale |
|---|---|---|---|---|
| C - D - E - F# - G | 8 - 9 - 10 - 11 - 12 | +2, +2, +2, +1 | C - D - E - F# - G - A - B | C Lydian |
| G - Ab - Bb - C | 12 - 13 - 14 - 16 | +1, +2, +2 | G - Ab - Bb - C - Db - Eb - F | G Locrian |
| In the C - E - G group, obtained from harmonics 4 - 5 - 6, a major third and a minor third appear. Repeating this cell produces C - E - G - B - D - F# - A, which, reordered, gives C Lydian. In the G - Bb - C group, obtained from harmonics 6 - 7 - 8, a minor third and a major second appear; repeating this cell produces G - Bb - C - Eb - F - Ab - Bb - Db, which, reordered, gives G Locrian. |
From the eighth to the fourteenth harmonic
In the segment C - D - E - F# - G, obtained from harmonics 8 - 9 - 10 - 11 - 12, the cell 2-2-2-2b appears. Repeating it until seven notes are completed produces C - D - E - F# - G - A - B, that is, C Lydian. In the segment G - Ab - Bb - (C), obtained from harmonics 12 - 13 - 14 - (16), the cell 2b, 2, 2 appears instead. Repeating it until seven notes are completed produces G - Ab - Bb - C - Db - Eb - F, that is, G Locrian.
Beyond the fourteenth harmonic
Harmonic 15, approximable as B, is excluded from this cell because it marks a different threshold in the behavior of the harmonic series. Up to the fourteenth harmonic, the intervals between the notes considered still retain a sufficient content of information: they are not simple chromatic fillings, but recognizable distances, capable of orienting the construction of a scale. A fifth, a third, a tone or a semitone placed in a certain mutual relationship can still shape a specific, non-chromatic structure.
From the fifteenth harmonic onward, however, the series increasingly tends toward chromatic saturation. The segment:
G - Ab - Bb - B - C
already shows a compression of the sonic space: the notes begin to occupy increasingly close positions, to the point of suggesting a chromatic continuity. From that moment on, the only scale that the series could naturally indicate is the chromatic scale; but for that very reason, its content of information becomes unusable.
By “information” here is meant the capacity of an interval to select, orient, and structure. An interval is informative when it says something about the shape of the scale: for example, it favors a fifth over a fourth, a major third over a minor one, a succession of tones and semitones over another. In this sense, information does not depend solely on the presence of a note, but on that note’s capacity to create a structure.
However, when the intervals all become semitones, or even distances smaller than a semitone, this capacity diminishes drastically. If everything is close to everything, no interval emerges with enough strength to shape a scale anymore. It is therefore not simply a matter of saying that the series no longer informs the Lydian-Locrian system: it no longer informs, with sufficient clarity, any kind of scale within the equally tempered system of twelve notes per octave.
In this hypothesis, the fifteenth harmonic thus represents the point at which the generative model loses definition. Before this threshold, the intervals still possess a content of information capable of producing form; after this threshold, the harmonic series tends toward chromaticism and then toward a progressive compression toward the unison.
This limit depends on the reference to the twelve-tone tempered system per octave. The idea that, beyond the fifteenth harmonic, the harmonic series loses informative capacity holds only in this context, that is, if one assumes as a horizon a scale organized in twelve chromatic pitches. The question should therefore be verified in systems with a greater number of octave subdivisions, where intervals that in 12-ET turn out to be too compressed or ambiguous might instead retain a structural function. The limit does not necessarily concern the harmonic series in itself, but the way in which a given tuning system is able to receive and organize its information.
integrations
- analysis on other ETs
- connection with moments of symmetry
- cycles of individual intervals (major 2nd = whole-tone = half Lydian/half Locrian: holds for every single interval); verification with pairs of intervals, etc.