B ASIC ELEMENTS OF MY MUSICAL LANGUAGE

A. Tcherepnin

1. Nine-step scale
(The terms tetrachord, hexachord, etc. are used in two senses-- as a succession of notes when written as an unbroken word; as a chord when written tetra/chord, penta/chord, hexa/chord.)

2. Interpoint

3. Pentatonic scales

4. Chromatic tetrachords and the eight-step scale

5. Georgian harmony

6. Hard and soft intervals and harmony

Some observations

(In Basic Elements of My Musical Language, a manuscript written on twenty-four pages of music paper in January 1962, Alexander Tcherepnin discussed the distinctive technical innovations in his scores and discussed the philosophical basis of his music. Five sections of this treatise [slightly abridged] appear on this website.)

I. Nine-step scale

Major-minor tetrachords are constructed within the interval of a major third using two half-steps and one whole-step:

Major-minor hexachords are constructed within the interval of a major seventh using alternations of half-step and one-and-a-half-step intervals:

with one octave added to complete the row:

As shown in the example, mode II is the inversion of mode I.

Note that C, E and Ab are common to both hexachords and that the major third interval between them is the same distance covered by the tetrachord of Mode I.

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The nine-step scale, which results from the addition of two major-minor hexachords is therefore based on three interlocking major-minor tetrachords and can have three modes:

Because the interlocked tetrachords have identical intervals, the nine step major-minor scale can have three points of departure (three tonics) in the same row, which are indicated by changes in notation (not by transposition).

As each nine step row can have three modes and three tonics for each of the modes (differing by notation), this results in a total of 9 nine-step scales for each row: 3 fundamental and 6 derivative (through change of notation).

Each nine step row can be transposed [only] three times in "fixed Doh" terms, [i. e., by a half-step, a whole step, or a step-and-a-half], since any further transposition merely yields a one of the four fundamental rows (or scales), with the tonic (fundamental tone of the start) placed a major third above or below.

There can therefore be 4 fundamental nine-step scales, each with three modes (12 scales altogether) and 8 derivative nine-step scales (obtained by change of notation), each with 3 modes each (24 altogether).

This brings the number of nine-step-scale "tonalities" or modes to 36.

The four fundamental nine-step scales are as follows:
(notation in ascending fifths)

The notation will indicate the position of the point of departure (the tonic) and can be said to indicate the position of the tonal center.

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If used in major thirds or minor sixths the scale will keep those intervals audibly intact (although some of the thirds will be written as diminished fourths, and some of the sixths will be written as augmented fifths):

If used in minor thirds or in major sixths, the intervals will audibly vary between minor thirds and major seconds (when in thirds) or between major sixths and minor sevenths (when in sixths):

If used in fourths or fifths the intervals will audibly vary between perfect and augmented fourths (when in fourths) or between perfect and diminished fifths (when in fifths).

If used in minor seconds or major sevenths the intervals will vary audibly between major and minor seconds (when in seconds) or major and minor sevenths (when in sevenths).

In arpeggiated form, the nine-step scale can be presented in the following ways:

The three-voice nine-step triads and their inversions

The fundamental perfect chord of nine-step scale harmony is the major-minor tetrachord.

The major-minor tetra/chord and its inversions:

in five-part settings in which the fundamental tetra/chord is considered as stable (and final) any note of the fundamental tetra/chord can be doubled; the penta/chord introduces the element of instability.

The minor penta/chord is tonal in its "resolution":

The major penta/chord can effect modulation by nine-step resolutions that arrive at a new tonal center:

In six-part settings the penta/chord (major or minor) is considered as stable, while the hexa/chord is unstable, finding resolution in a penta/ chord, and so on, up to the point at which--in ten-part harmony--the entire nine-step scale becomes a stable chord, with instability provided by an extra tonal appoggiatura (in ten-part settings), two extra appoggiaturas (eleven-part) or three appoggiaturas (twelve-part).

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