Fabrice Mogini
3.0 TUNINGS NOT BASED ON THE OCTAVE OR ON A SINGLE STEP SIZE.
3.1. BEYOND THE OCTAVE.
Some researchers and composers who have experimented with alternative divisions
of a frame have questioned the choice of the octave for such a frame.
Wendy1 Carlos (), has published information about certain of these tunings.
The number of intervals per octave is not an exact positive integer. This
is equivalent to saying that the notes available are found within a frame
smaller or larger than the octave.
W. Carlos Tunings:
name number of steps per octave cents per interval
Alpha 15.385 78
Beta 18.809 63.8
Gamma 34.188 35.1
Of course there are many other possibilities, the steps are equal-tempered
but the octave is avoided.
The predominance of the octave, as we have seen earlier, is not only justified
by cultural factors but finds its root in the way we hear. The chroma analysis
shows how a frequency and its octave are equivalent.
The octave is a natural phenomenon that still subsists whenever the tuning
is not based on an exact octave division.
In a non-octavian tuning system, a pitch is still aurally equivalent to its
non-existent octave.
This octave can then create a pulling effect on the other notes and especially
those which are directly next to it. These adjacent notes can seem out of
tune because the ear recognises easily that we are in the octave region, so
the octave should be substituted to them.
This point is not relevant if the composer who uses such tunings gives the
same importance to each note. This could be imagined as a type of dodecaphonic
composition with that sort of tuning. In that case, we would use each note
and only once, in a special order, before changing the series and having the
chance to use the same note again.
Rare are the composers who are so systematic in the way they use rules. In
reality, composers tend to use, consciously or not, different weights for
each note.
If a pitch appears more often, it is more likely to be remembered. It also
has more interactions with the other notes that are then experienced through
the way they relate to this important pitch.
A new definition of tonality could then rely on the observation of repetition
in music (notes, fragment, chordal or modal structure). It is possible to
base an analysis on the probability of appearance in a composition of such
subsets (stochastic processes).
Coming back to non-octavian tunings, we can assume that a repeated note that
has the status of tonal centre will suggests its octave as an important node
in the tuning system. This is why notes in the octave region will be still
sensed as a poor approximation of the octave.
3.2. EQUAL-TEMPERED TUNINGS WITH EXPONENTIAL BASE DIFFERENT THAN
TWO.
Moreno (1992) has listed in his expanded tunings those beyond the
octave that are still based on logarithmic organisation. This kind of temperament
contains a certain number of equal steps for a frame as large as two or more
octaves.
The chroma axis still has its qualities and remains a determinant parameter
to consider when composing.
In that tuning context, Moreno supposes that there is an expended perception
of chroma.
The listener would then try to perceive exponentially related subsets as being
similar: they are perceptually equivalent cycles of the whole scale.
Moreno gives a table showing the step number, its frequency and its equivalent
in cents for each equal-tempered tuning possible (up to 24).The first note
has always the same frequency of 27.50 Hz, and the octave has a frequency
of 55 Hz. When the exponential base is larger than two, there is no octave
and 55 Hz cannot be found in the list but an interval of two or three octaves
replaces it in its role frame.
3.3. EQUAL DIVISION OF THE OCTAVE USING TWO TYPES OF STEPS OF DIFFERENT
SIZE.
Charles Lucy (1990) has created a tuning (and a scale) based on a sequence
of large and a small steps. This contrast in the size of steps is balanced
by the regularity at which it occurs.
This principle can be adopted for an extension of any equal temperament. An
adaptation for the twelve-note system for instance would suppose first that
we have two kinds of steps in permutation: a tone for the larger step and
a semitone as the smallest step.
I would personally call this a modal structure with symmetric qualities within
a normal tuning. This is because I was aware of Messiaen's modes of limited
transposition (his second mode has exactly the same step's progression1) before
I got to know the existence of Lucy's tunings.
With this particular Lucy tuning, we have eight steps per octave, made of
a permutation of small and large intervals. In reality it is nothing less
than twelve equal sized notes and only eight we can choose from.
Nevertheless modes such as messiaen's can be more difficult to conceive when
increasing the number of steps per octave. For instance playing exclusively
in that mode with forty-eight equal notes per octave supposes that sixteen
notes must be omitted.
A Lucy tuning is already done without these notes and is easier to handle:
all the notes will always be in tune for that mode.
Of course there seems to be a limitation: since notes are missing some other
types of modes or certain transpositions of this one are impossible to obtain.
However, it is possible to choose beforehand a larger tuning with more notes.
If eight notes made with different size steps do not seem enough to modulate
as much as twelve equal ones, we can just move on to twenty-four equal steps
and obtain a sixteen-note Lucy tuning that will offer more modulation possibilities.
3.4. MESSIAEN'S MODES OF LIMITED TRANSPOSITION.
As we have seen in the precedent chapter, certain modes show symmetry in their
interval progression. This quality makes them equivalent to a proper tuning
system. They relate to the octave as equal portions that are then divided
into two smaller parts.
Messiaen's mode number two is given in his book 'my method of composition'.
It corresponds to the sequence of a tone and a semitone, repeated until the
octave is reached.
Diagram . Messiaen's mode number 2 of limited transposition.
If the usual major and minor scale are so complicated to transpose, it is
because of the poor symmetry in their interval structure.
A symmetric mode can be compared with a structure made of multiples. This
is why by transposing this kind of mode we generally end up with the same
notes again, found at a new degree position in that relative transposition.
One way of seeing this mode number two, is as a four-note equal tuning where
each interval is itself divided in one third and two thirds. These different
intervals are multiples of each other and for this reason have in their symmetric
distribution the characteristics of the equal tempered system.
In other words, equal temperament is finally a very simple pattern when it
comes to perception. Its structure, seen chromatically, is made out of semitones
in a line. With Messiaen's mode, the symmetry emerges as pattern and relates
automatically to the equal tempered tuning.
This mode combines subsets from different tonalities. We can call each of
these tonalities and their relative major or minor separately if we want to
emphasise one of them. We can as well mix them together again in such a balanced
way that a pattern (the Lucy tuning) emerges and transforms the mode: from
being a scale it becomes a tuning system in itself.
4.0 FULLY-EXTENDED TUNINGS.
4.1 FROM SYMMETRY TO POLYTUNINGS.
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