Non permutational serialism (2): Static and dynamic intervals

Though it is permuting notes, the function Train() is not ‘permutational’ because it does not compose permutations but only iterate through the notes of some series in order to modify the corresponding values in another one. This could certainly be done ‘permutationally’ but with much more computation and it’d be useless in this context, since the goal of this set of functions is precisely to obtain sequences different from typical permutational ones. With Trainsposition() we had an already interesting example since each modifying note was given two parameters (in addition to its index): an integer value and a sign. The amount of strictly permutational manipulations needed to perform such a computation would be fairly large.

Static intervals

Now instead of applying one modification to each note in a series and take a picture of the result after each step in the process, we want to have a sequence of intervals (they can be represented by a series too) that will run through our original series and stop at some stage. Each interval will be applied to a gradually decreasing number of notes in the original row. Let’s name ‘origin’ the series to be modified. The number of concerned notes in the origin is decreasing because each time an interval has reached the last available note, is remains as a permanent modification. So for a twelve-tone origin, as soon as the first interval has reached the last note, it remains there and this note index is not available anymore for further modification. Then the second interval only touches eleven notes in the origin, and so on till the last change which only concerns one note. This is a train of static intervals.

These schoenbergian examples are taken from a 2008 Etude for clarinet.

Series S:   05814B62793A
Reverse R:  A39726B41850
Inverse I:  074B816A5392
0pposite O: 2935A618B470

Ex. 1:

*** DESCR: seq_obj: Seq_2 after Build_tree: object tree: 
25814B62793A 
65814162793A 
B5814562993A 
27814A62193A 
1B816162693A
5481A0629930 
278134828934
A6A161020939
AA2159549930
677199885B3B
73A365715333
A39726B41850 
funt: Static(A=05814B62793A,oct,B=A39726B41850,oct,ord=05814B62793A) ready: 1 ***
*** seq_proc : Tkrow_2 Sequence object: recorded

In this sequence we reach at step 12 the schoenbergian Reverse of Series S. The notes are modified following the order given by the same Series S. Now we could be interessed by the series of intervals that is applied in this case. To acquire this knowledge, Csgrouper provides – on the ‘Series’ tab – an analytic function called Statana(). Putting our origin (05814B62793A) into fields A and ord and our target (A39726B41850) into field B and applying function Statana() we obtain the resulting series (261215A2265A) and a sequence of signs (++-+++—–+) that have to be applied to S in order to get R under the static intervals sequential method. You can verify that the first value (2) has to be added (+) to note index (A=10, i.e. the eleventh note in S = 3) in order to get (5) which is the eleventh note in R. To get there, (2) had first to change the value of each other note in S. This knowledge isn’t required to run Static() though, but the same sequence of values can then be devoted to another role somewhere else in a work for instance.

Dynamic intervals

Now we can also have our origin modified permanently by each interval. Instead of falling back to the original value when a note has been modified, we keep the modified value. Then next time the interval changes this note, it is no more the value that was shown in the origin. This is a train of dynamic intervals, and it almost makes melodies sound like random sequences. However, the ultimate state of the series is completely determined too. Thus sequences of dynamic intervals are useful in music because from an original pattern they create complications leading to other defined musical patterns. The complication itself shows properties that the ear may perhaps sometimes apprehend or recognize, most of all when this form is used in several places in a piece. Again the internal parameters of Dynamic() can be known by feeding the same fields and appying Dynana() (checking the box ‘exp’ will further expand the results).

Ex. 2:

Now we start from where we left off in Static(), i.e. R (A39726B41850) and we want to reach O (2935A618B470) according to order (A39726B41850):

*** DESCR: seq_obj: Seq_2 after Build_tree: object tree: 
A39726B41890
A39B26B41850
A39726B41030
A39526B818B0
A31126B41610
A39326321250
A37726BA1474
A339669018B0
A75126541A9A
A39B06765216
A1B38ABA1038
2935A618B470 
funt: Dynamic(A=A39726B41850,oct,B=2935A618B470,oct,ord=A39726B41850) ready: 1 ***
*** seq_proc : Tkrow_2 Sequence object: recorded
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