Fabrice Mogini
4.0 FULLY-EXTENDED TUNINGS.
4.1 FROM SYMMETRY TO POLYTUNINGS.
Another way again to consider Messiaen's mode is to imagine that it originates
from the juxtaposition of two four-note equal temperaments starting from a
different fundamental.
In twelve notes per octave this is as if we had built a four-note equal tuning
starting from C, juxtaposed to the same tuning starting from a B._Another
of Messiaen's modes has the progression:
[1tone/1 tone/1 semitone/1 semitone]
repeated three times to obtain the octave.
Diagram . Messiaen's mode number 2 of limited transposition.
This mode corresponds to the juxtaposition of a six-note equal temperament
and a four-note equal temperament, starting both from the same root. The encounter
of these two tunings gives us exactly all the notes of Messiaen's symmetric
mode.
If two tunings make a mode, it must be that this mode itself is part of a
larger tuning system. It must be possible to refer to this larger set and
use directly all of its notes.
It must also be possible to recall each tuning separately or even come back
to the mode.
When these two tunings operate together, symmetry can be perceived as a simple
and single pattern with the attributes of a tempered tuning. This feature
gives us a new type of tuning system: Polytunings.
4.2 POLYTUNINGS.
Using four and six equal notes per octave together is simple because both
tunings are contained in the twelve-note equal temperament. Twelve is indeed
divisible by six and four.
This works as well with larger numbers and even with an odd number as nine,
opposed to an even number of steps per octave as twelve.
In that particular case, a 36 equal-note system would still contain the degrees
of a twelve-tone system as well as the notes of the nine-tone temperament.
This idea of a 36 equal note system is only an explanation of how the two
tunings together can sound harmonious.
In reality, all the other notes of the 36 note scale should not be played
otherwise we would just play with this larger equal temperament.
The point is to be able to still feel the 12 and the 9 note system as well
as the larger one of thirty-six.
Tunings with few notes per octave have the advantage of simplicity over larger
ones such as the thirty-six notes system because larger intervals are easier
to distinguish.
These polytunings favour simplicity but permit the language possibilities
of larger complex systems.
With polytunings, Each temperament is a tonality within a common tuning system.
In that sense polytunings have much in common with pantonality, described
by Reti (1958):
"Pantonality is the phenomenon of 'movable tonics', that is, a structural
state in which several tonics exert their gravitational pull simultaneously,
counteractingly as it were, regardless of whether any of the various tonics
ultimately becomes a concluding one".
4.3 TOWARDS A TOTAL PITCH FIELD.
Using an equal temperament with a very large number of notes per octave supposes
the use of very small intervals that might be audible but not clearly distinguished
from one another by the ear.
There is as well a distinction to be made between what we are able perceive
and what is finally chosen by the attention. The role of attention in perception
is a subject that goes far beyond this study (and my knowledge). However,
I suspect that the assemblage into patterns (as well as personal routines,
cultural usage and personal intent) is determinant in the stimulation of the
neural system and focuses the attention.
My aim was then to organise the total pitch field into self-similar divisions
of the octave that contribute to a tuning of a higher order.
Instead of having very small divisions and a single equal temperament, different
scalar systems are used together.
This term is here different from the musical notion of a scale. The word scale
is here a synonym for dimension (e.g. scale from a road map).
This is a compromise between a free choice of pitch along the total field
and restricting the set we can choose from to a simple rule: each note can
be classified into a pattern made of equal intervals. This whole equal tempered
structure is mapped to several scalar dimensions.
4.4 CONNECTIONS BETWEEN THREE OR MORE EQUAL TUNINGS/TRANSITIVITY.
For practical applications, the notes that are common to two tunings can be
used as pivots. By using pivots, One can either use these tunings simultaneously
or change from one to the other.
When more than two tunings are used, transitivity relationships are building
bridges between two tunings that have normally no note in common apart from
fundamental and octave.
Example with three tunings a, b and c:
Observations:
_Tuning a has its fourth note aligned with the third of tuning b.
_Tuning b has its seventh note aligned with the tenth of tuning c.Conclusions:
_Tunings a and c can be played together if b is used at the same time. b creates
a link between a and c.
_Each tuning has anyway its fundamental and octaves aligned with those of
other equal t emperaments.
This transitivity guaranties a complex network with a maximum of linked tunings.
We can re-order so many tunings with these links that we start to have access
to the total pitch field.4.5 EXAMPLE OF AN EXTENDED PITCH FIELD ORGANISED
WITH POLYTUNINGS.
Eight equal temperaments with a small number
of notes per octave. Results given in cents:10 notes per octave:
0,120,240,360,480,600,720,840,960,1080,1200;
12 notes per octave:
0,100,200,300,400,500,600,700,800,900,1000,1100,1200;
14 notes per octave:
0,85,71,171,257,342,428,514,600,685,942,1028,1114,1200;
15 notes per octave:
0,80,160,240,320,400,480,560,640,720,800,880,960,1040,1120,1200;
16 notes per octave:
0,75,150,225,300,375,450,525,600,675,750,825,900,975,1050,1125,1200;
17 notes per octave:
0,70,141,211,282,352,423,494,564,635,705,776,847,917,988,1058,1129,1200;
18 notes per octave:
0,66,133,200,266,333,400,466,533,600,666,733,800,866,933,1000,1066,1133,
1200;19 notes per octave:
0,63,126,189,252,315,378,442,505,568,631,694,757,821,884,947,1010,1073,
1136,1200
Extended Pitch field (Polytuning) of 99 notes per octave, organised in eight
simple equal temperaments. Results given in cents:
[0, 63, 66, 70, 71, 75, 80, 85, 100, 120, 126, 133, 141, 150, 160, 171, 189,
200, 211, 225, 240, 252, 257, 266, 282, 300, 315, 320, 333, 342, 352, 360,
375, 378, 400, 423, 428, 442, 450, 466, 480, 494, 500, 505, 514, 525, 533,
560, 564, 568, 600, 631, 635, 640, 666, 675, 685, 694, 700, 705, 720, 720,
733,750, 757, 776, 800, 800, 821, 825, 840, 847, 866, 880, 884, 900, 917,
933, 942, 947, 960, 975, 988, 1000, 1010, 1028, 1040, 1050, 1058, 1066, 1073,
1080, 1100, 1114,1040,1125,1129,1133,1136,1200];
Note:
Among 10 and 19 notes per octave, two tunings (11 and 13) were omitted. This
was a subjective choice: they are too similar to the famous twelve equal-note
temperament and could be taken as an 'out of tune' version of it.
4.6 TECHNIQUES FOR COMPARING EQUAL TEMPERAMENTS.
Frequencies are calculated with a logarithmic scale. It is necessary to use
a different system to compare easily intervals or degrees from different equal
temperaments. The system in cents seems adapted to this task because it is
linear.
There are 1200 cents per octave, which corresponds to an interval of 100 cents
for the equal-tempered semitone. For a ten-note equal tuning the smallest
interval has 120 cents. We observe that the sixth degree in 10 e.t. has the
same value than the seventh degree in 12 e.t. (600 cents).
This technique of observation (going through lists) can be fastidious for
the comparison of tunings with a larger number of elements.
The difference between the smallest interval of each of these temperaments
is 20 cents: this is a valuable clue for the arithmetic deduction of common
frequencies.
A list of equal temperaments such as Moreno's (discussed earlier on) is the
starting point of an extensive comparison.
4.7 COMMON FREQUENCIES / TOLERANCE.
Polytunings is a system that uses common frequencies as pivots that connect
different temperaments.
With certain pairs of equal temperaments, we will not find exactly matching
frequencies so it is indispensable to choose on which criteria two frequencies
are near enough from each other to be considered as equivalent (pivot)in the
context of our larger tuning.
Let us come back to the comparison of 10 and 12 e.t.
The smallest interval found in both tunings is 100 cents (12 e.t.).
This will be the smallest value we can understand as part of the chromatic
patterns of these tunings, while twenty cents is the smallest value for the
actual interaction of the two systems.
Half of this value, 50 cents, would be equally out of tune between two adjacent
notes in this temperament.
A much smaller value would then be near enough from one of the two adjacent
frequencies to become its possible substitute.
This is a matter of resolution: when a note is very near from one present
in the tuning, it is assimilated to its chromatic pattern. If the frequency
is too far, it is detected as alien to the pattern's progression.
I personally accept twenty per cent of the smallest interval found in any
two tunings as an interval acceptable to represent the next adjacent note.
To compare many different tunings, the smallest interval can be taken as a
basic value from which a tolerance threshold will be chosen. This choice being
subjective, it is a good idea to make its value dependant on the context instead
of fixing it for good.
4.8 CHANGEABLE TOLERANCE THRESHOLDS.
In a musical context it is possible to imagine fast passing notes that are
dissonant but in the slow and regular parts of the music, the dissonance would
be experienced as increased.
The notion of 'in tune' can be alternated with 'out of tune' if this done
at the right time and place.
It could be then ideal to be able to change the tolerance threshold (limits
of the accepted dissonance) accordingly to the compositional context.
When I compose, I can choose the threshold value (or its changes) beforehand.
With two musicians improvising it is more difficult to predict what setting
is best because it depends on a context that is not known yet.
This is what happened when I had a first improvisation with Nick Collins,
researcher for Middlesex University, London.
He was very familiar with alternative equal tunings but had not tried to mix
them together as a unique tuning yet.
After our first live cession with mixed equal temperaments we realised how
difficult it is to find in real time which notes should be used as pivots
between these tunings (especially when using tunings with a large number of
notes per octave).
From this first attempt Nick Collins came up with some lines of programming
in SuperCollider1, that would allow performers to change in real time this
tolerance limit, using a the mouse of the computer. With this changeable tolerance
threshold, one can pass from having all the notes of both tunings available
at once, to a situation where only the few notes with exactly matching frequency
are selected, with of course all the intermediary stages possible between
these two extremes.
4.9 POLES OF ATTRACTION / NODES.
The recognition of simple intervals does not only occur when two tones are
exactly related to each other by a simple interval.
As soon as they are near enough from each other (simple ratio area), the recognition
is possible. In the same way, each simple ratio is next to an adjacent one.
This is how it is possible to use the twelve-note equal temperament instead
of just ratios.
Apart from the ear limitation (interval too small to be distinguished), culture
dictates when a frequency ratio is perceived as a node in the music tuning
system.
The nodes are defining areas of simple ratios, any other note being attracted
by another nodal area. This is how two or more temperaments can be played
together without having to use any tolerance threshold: each note out of the
area will be resolved.
4.10 SIMULATING NODES FOR NOISE-BANDS.
Noise-bands are made out of a cloud of frequencies. In noise-bands, no nodal
structure can be perceived but the sense of linear distance (adjacency) is
still present.
If we give more importance to one of these frequencies by adding its octave,
there is a chance to create a pole of attraction. This means that we can use
noise in a harmonic context, once this node is tuned to a chosen temperament.
When frequencies are subtracted from noise, the same thing happens because
some relationships will be more obvious than others in the frequency cloud.
These nodes created are mapped in the same way to the temperament's nodes.
Nodal structures reproduce themselves at the octave and, extending this idea
to the use of many temperaments, we can deduce that a single common note will
be enough to distinguish a nodal area and justify the use of the whole noise-band.
The doubling at the octave to reinforce a node is a way of combining different
adjacent systems.
4.11 SAMPLE ANALYSIS / FINDING AN ADEQUATE TUNING.
We have seen how spectralist composers construct the best tuning possible
to match the partial distribution of a sound.
A group of sounds in a time sequence could also correspond to a particular
tuning.
Instead of trying to create this new tuning, we can start to find which one
in the list of equal tempered tunings is the most suitable. We will then rely
on a resolution to the smallest adjacent note in the tuning to quantize the
values with a small deviation.
The most common or important frequencies found in a sound sample will be the
nodes for this whole sequence. If these nodes are similar in their structure
to those of an equal tempered tuning, we can surely map the sample to the
whole tuning.
An example with a bird sample will illustrate this technique.
Sounds with a musical quality, in the classic definition, must have a definite
and clear pitch. Some sounds like those of a bird have the clearness required
to be musical.
Unfortunately, if we were to use a whole sample of it, the sequence of frequencies
(a tuning of its own) will have little chance to be found among those of the
twelve- note equal temperament: a bird song might be musical but not in the
context of our usual tuning.
Technique:
_The software SuperCollider (by J.Mc.Cartney) is implemented with a pitch
follower that I use to detect and print the frequencies of a bird song (Trush).
Trush frequencies:
[1743,1802,
1690,
1620,
1431,1178,1174,1270,1252,
1405,1309,1334,1189,1550,1501];
Note: the underlined numbers represent frequencies that will be found in one
of our equal temperaments.
_The numbers are then used with a program that compares them with the frequencies
from a dozen different equal temperaments.
Each match and neighbour frequency are stored into a new list.
Finally the program iterates over each number from that new list to find which
temperament got the larger number of matching or neighbour frequencies.
In that case the 19 equal temperament was selected; it is not the best one
we can conceive (as the spectralists would do) but the most adapted if we
want to use equal temperaments and include perhaps the sample with some polytunings.
Here is the selected tuning and its frequencies starting from fundamental
of 880 up to one octave).
19 notes equal temperament frequencies:
[880,912,946,981,1018,1056,1095,
1136,1178,1222,1267,1314,1363,1414,
1466,1521,1577,1636,1697,1760];
Note: the underlined numbers represent frequencies that will be found in the
original sample or that are good substitutes for them.
An electronic instrument in this tuning can be used simultaneously with the
sample. An example of this musical application is given in the accompanying
C.D.
Future developments will include a proper formula to calculate which tuning
is the most appropriate without having to compare lists which is fastidious
and c.p.u. intensive for the computer.
4.12 PATTERN-BASED TUNINGS.
For Burns and Ward (1982) a tuning system does not operate independently from
other compositional factors. Melodies are perceived in gestalts or patterns
rather than a succession of intervals.
Sharing this idea with them, I worked on a tuning system based on melodic
permutations. The intervals are not fitting in the frame of an octave but
the way they go beyond the octave limit is dictated by the melodic permutations.
I use a limited set of intervals with different sizes that are not multiples
of each other. I would not classify these intervals among ratios since ratios
are still deduced from the octave.
These intervals have different sizes but each frequency available in the tuning
is not fixed because it is not based on a frame but yields from the last note.
To decide on these intervals I have divided a unit of 1 in four non equal
parts: 1 = 0.14 + 0.22 + 0.26 + 0.36
These four portions have sizes that correspond to the prime numbers: 7 + 11
+ 13 + 19
This unit of 1 was only chosen to measure the intervals and make sure they
have no multiple in common.
Each new frequency in the pattern yields a node from which the melodic development
takes place.
_If the melody starts from a frequency located at the position zero, we have
the melodic choice to go up or down in frequency by 0.14, 0.22, 0.26 or 0.36.
I will choose here 0 + 0.26.
_This note yields again and the same choice process takes us to a new frequency
(for instance: 0 + 0.26 - 0.22).
It is soon that we will realise that it is unlikely that we get back to one
of the precedent frequencies or its octave:
( 0 + 0.26 - 0.22 + 0.36 - 0.14 + 0.22).
Because there is a limited amount of intervals available in the system, we
tend to perceive them as patterns. We do not have anymore this split between
melody and tuning. Here, melodic patterns dictate the tuning.
In most tuning systems there are many intervals we can choose from this is
why we tend to follow the melodic line more than intervals themselves.
There are only four different patterns (four unequal sizes) in this system.
These proportions are constantly heard and become more important than the
frequency on which we end up. At last, this fractal system can take us anywhere
along the total pitch field while preserving a sense of unity: the chaotic
permutations of four unequal size intervals create a progression that is easy
to follow since it is based on patterns.
I call this type of organisation a fractal tuning since it is made of self-similar
patterns which reproduce themselves in different dimensions (constant change
of scalar system).
4.13 PERCEPTION OF PATTERNS.
REGULARITY / IRREGULARITY.
Our senses can detect quantitative differences in the received stimuli but
are responding more to geometric or proportional differences.
Schwaller de Lubiz and Lamy (1982) wrote on symmetry in nature and its role
in perception:
"The content of our experience results from an abstract architecture
composed of harmonic waves of energy, nodes of relationality, melodic forms
springing forth from the eternal realm of geometric proportion".
Regularity and irregularity work together in a dynamic balance and are two
sides of a same coin.
A tuning can be interpreted as a higher degree of regularity arranged in scales
and privileged intervals by the composer. Such sub-divisions fragment the
tuning in an irregular way for expression to be possible.
If everything was regular it would be difficult to notice anything so we need
a certain contrast for perception to be active by attention.
The 'irregular' takes place among the 'regular' and can be noticed in the
context of such a differential system.
Most composers or even improvisers take into account the predominance of a
tuning system (culture) and regard this aspect as the regular element in the
balance 'regularity/irregularity'.
Permutations, variations and the deconstruction into scales are perceptible
patterns that create contrast within this tuning.
Some elements can be compared to others in the system and are exposed to the
attention's focus: it is a constant interplay between patterns from different
levels.
Eventually, this balance will be resolved when reconsidering the tuning first.
This will alter the way contrasting sub-sets relate to this larger tuning,
thus changing the music language.
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