Tcherepnin
The Tcherepnin scale, or 3L 6s, is a common structural scale which repeats at one third of the octave; as such, it is often associated with the Augmented temperament which splits the octave into three major thirds. The scale is named after the music theorist Alexander Tcherepnin, who utilized its 12edo tuning to create a double-tonal sound with two different types of thirds; the pattern was also noted by Olivier Messiaen, who listed it as the third scale in his list of modes of limited transposition.
Because the Tcherepnin scale was popularized by progressive 20th century musicians such as Allan Holdsworth and Toru Takemitsu, a great deal of composition theory can be attained through analysis of these pieces, which can be applied to composition in xenharmonic systems which include it.
General scale theory
The Tcherepnin scale is a mode of limited transposition, which means that certain rotations of this scale are identical to one another; while the scale has nine notes per octave, there are only three unique rotations: a bright mode, a dark mode, and a symmetrical mode. The symmetrical mode was the one used by Tcherepnin, and the bright mode was the one documented by Messiaen.
| Brightness | Pattern | Degree Qualities |
|---|---|---|
| + | LssLssLss | MMPMMPMMP |
| 0 | sLssLssLs | mMPmMPmMP |
| - | ssLssLssL | mmPmmPmmP |
Degrees, intervals, and notation
The degrees and intervals of the Tcherepnin scale can be classed in two main ways: diamond MOS notation, where ordinal numbers correspond to the precise number of steps, or quasi-diatonic notation, where interval names are based on diatonic categories with the unchanging 400c and 800c intervals being dubbed the naiadic and cocytic degrees. The diamond MOS system yields more convenient methods of interval arithmetic in a vacuum, while the quasi-diatonic system yields familiar categories and preserves octave complements and chord constructions from diatonic music.
Degrees can be said to tend bright or dark based on whether their quality in the symmetrical mode is equivalent to that in the bright mode or the dark mode; in quasi-diatonic notation, odd numbers tend bright while even numbers tend dark, and the degrees labeled with Greek symbols do not have more than one quality.
The interval just below the cocytic (the "major sixth" in Tcherepnin ordinals, or "major fifth" in quasidiatonic ordinals) clusters around the size of a pentatonic generator, or an antipentic generator in hard tunings; 15edo, with a step size ratio of 3:1, represents the midpoint where the fifth generates 5edo. With this in mind, the intervals and degrees of the Tcherepnin scale can be compared to those of the scales that this fifth yields.
| Degree
(Quasi) |
Degree
(Tcher) |
Quality | Cent Range | Diatonic |
|---|---|---|---|---|
| 2 | 1-step | minor | 0 - 133 | m2 |
| major | 133 - 400 | M2 | ||
| 3 | 2-step | minor | 0 - 266 | d3 |
| major | 266 - 400 | m3 | ||
| ς | 3-step | perfect | 400 | M3~d4 |
| 4 | 4-step | minor | 400 - 533 | P4 |
| major | 533 - 800 | A4 | ||
| 5 | 5-step | minor | 400 - 666 | d5 |
| major | 666 - 800 | P5 | ||
| ι | 6-step | perfect | 800 | A5~m6 |
| 6 | 7-step | minor | 800 - 933 | M6 |
| major | 933 - 1200 | A6 | ||
| 7 | 8-step | minor | 800 - 1066 | m7 |
| major | 1066 - 1200 | M7 |
Where one of the two notations must be chosen to avoid ambiguity, this page will prefer Quasi-diatonic notation. When quality must be specified for a degree, this page will use ♯/♭ to represent raising or lowering by a MOS chroma, and the symmetrical mode will be taken as the default qualities. Intervals between two pitches will use M or m with the Arabic numerals, or P with the Greek symbols, to specify in-scale qualities.
Chords and tonality
Alexander Tcherepnin considered the most characteristic part of the scale to be its separation of the diatonic minor and major third onto different degrees, allowing both to be reached within the same mode. If we consider the 3 and ς degrees to be the main building blocks of chords, there are four types of triad constructions: chthonics (quasi-diatonic degrees 1 - 3 - 4), minor (degrees 1 - 3 - 5), major (1 - ς - 5), and augmented (1 - ς - ι). This leads to eight total chords, with three qualities of chthonic, two qualities of minor, two qualities of major, and one quality of augmented.
Each degree of the scale has one of each type of triad, with the augmented triad being the same in every mode:
| Mode | Chthonic | Minor | Major |
|---|---|---|---|
| Bright | M3, M4 | M3, M5 | Pς, M5 |
| Symmetrical | M3, m4 | ||
| Dark | m3, m4 | m3, m5 | Pς, m5 |
Where the qualities of these triads need be distinguished, they can be described with their characteristic mode and construction type, such as "bright major" for {Pς, M5} or "dark chthonic" for m3, m4.
It is notable that in 12edo tuning, the bright chthonic triad is enharmonically equivalent to the diminished chord from the diatonic scale. Many contemporary musicians who use the Tcherepnin scale take advantage of this fact, and curious xen composers might treat their usage of it as a model for how to fit chthonic chords and diatonic-like tertian chords into the same harmonic framework.
Harmonic systems
There are several possibilities for establishing a tonal framework in the Tcherepnin scale, based both on analysis of usage of the scale in music both xen and otherwise, and on applying fundamental principles to the scale as a self-standing system.
A common thread among contemporary treatment of the scale is how to work around the limited transposition; the two major types of approaches to this are to establish an axis system where there are multiple tonics, or to establish a modal system where there is no tonic at all.
Tcherepnin's axis system
An axis system is a system whereby functions repeat at some "axis" that subdivides the octave. In the case of the Tcherepnin scale, the axis occurs between degrees 1, ς, and ι, making them all tonics under this analysis; this is the primary framework seen in the works of Alexander Tcherepnin himself.
Because the m4 and M5 degrees resemble a diatonic generator, a significant amount of diatonic harmony can be transferred to this scale; Alexander Tcherepnin himself took advantage of this similarity, using the major triads on the 4 and 5 degrees for the functions of diatonic IV and V. Thus degrees 1, ς, and ι have tonic functions; 2, 4, and 6 have subdominant functions; and degrees 3, 5, and 7 have dominant functions. Note also that these degree functions correspond to brightness tendencies, with tonic degrees being the same in all modes, subdominant degrees tending dark, and dominant degrees tending bright.
Tcherepnin primarily employed major and minor chords in his axis-based harmony, with augmented chords being occasionally used, and chthonics only consisting of the bright quality; however, all eight chord types can be applied to this system. Note that chthonic chords, the main type of chord that Tcherepnin did not use, necessarily contain one degree for each of the functions; for instance, the chthonic chord on the 1 degree contains 1 (a tonic), 3 (a dominant), and 4 (a subdominant). This tonal ambiguity makes it a very useful tension with a plethora of resolution options, acting as a nexus that is limited only by voice leading.
Holdsworth's harmonic system
Jazz guitarist Allan Holdsworth made usage of the Tcherepnin scale in several pieces, and used unconventional constructions of chords to create a flowing modal sound. Unlike Tcherepnin's axis system, however, Holdsworth prioritized the qualities and relative intervals within the chords rather than the absolute degrees which they contained. Because the limited transposition provides a very unclear sense of tonality, Holdsworth's approach appears to favor tension and release paradigms that explore the scale modally, rather than establishing any clear center of gravity.
