PROTOMODALISM 3

by ArcoX

The idea which sprang unexpectedly into my head,
and kickstarted my protomodal project was this:

The tonic is a fixed datum, and the mode can be represented
by a six digit quaternary (Base 4) number, where each digit represents
one of the intervals between the seven tones. The intervals can be a
minor 2^{nd} (e.g. B to C), a major 2^{nd} (C to D), an
aumented 2^{nd} (C to D♯), or a doubly augmented 2^{nd} (G♭ to A♯). These four possible intervals are
represented by the numbers 0, 1, 2 and 3 respectively. My specific
definition of a heptatonic mode is that it contains all of the tones A
through G, one of which is the tonic, and each of which can occur in one of
the three variants sharp, flat or natural.

So the Major (or Ionian) mode, in which the
intervals are maj. 2^{nd}, maj. 2^{nd}, min. 2^{nd},
maj. 2^{nd}, maj. 2^{nd}, maj. 2^{nd} can be
uniquely represented by 110111 which, converted to decimal, is 1,301.
This system gives us a series of numbers from 000000 (decimal 0) to
333333 (4,095), contained within which are all the possible heptatonic
modes. Not all of these numbers will represent valid modes because the 6
digits, plus the unused seventh digit representing the interval between
the leading tone and the tonic, have to add up to 5 to complete the
octave from tonic to tonic. Think about it; the 5 equals the semitones
unused in the mode out of the octave’s twelve. So the series includes a
number of combinations which either fail to extend to a complete octave
or stretch beyond it. 333333, for instance, extends over 2 octaves. I
programmed an Excel spreadsheet to do the calculations which revealed
that the series 0 to 4,095 includes 399 possible heptatonic modes, and
precisely which tones each is composed of.

Some of the modes uniquely occur in only one
key, and some pretty odd keys at that, like B♯ and F♭. The lowest number which gives a valid mode is
000011 (5) which gives us B♯ C♯ D E♭ F♭ G♭ A; while the highest 311000 (3392) gives F♭ G♯ A♯ B♯ C♯ D E♭. Transposing some these unusual modes into
other keys may require the use of double, treble and even quadruple
accidentals so is probably better avoided.
At this stage of my research it was beginning to
feel a bit abstruse, like understanding subatomic particles or the square
root of 1, but it was getting very exciting.

The Excel spreadsheet was substantially expanded
into an essential tool for the protomodal composer:


The
modes are rationalized in an indexed list from 1 to 399. Entering any of
these numbers will display the associated mode, graphically on the
keyboard and as a list. The display also shows which keys are available
without the use of multiple accidentals. Conversely, entering a mode by
clicking seven notes on the keyboard will show the index number. Most
important of all for my composing method, a random mode can be generated
by chance and the selection filtered, if required, to omit the existing
named modes or those which I have already used in compositions. A
separate page shows which nameable chords are available in the given
mode:


The
spreadsheet has other clever functions which I shall describe if and when
their relevance becomes apparent in my continuing discourse. It’s a neat
little tool, and the protomodal method has given me a great creative
boost.

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