# Non permutational serialism (1): Trainsposition()

Aside from what is explained in the ‘code’ section, Csgrouper offers non permutational functions, that were the origin of the whole program back in 2008. Their purpose is to transform gradually one series to another by arithmetic means.

#### Transposition

While mere ‘Transposition’ is easily replaced by setting the correct ‘tone’ value for a ‘Suite’ in the corresponding row (and could therefore be removed from the function menu if it wasn’t for the possibility of giving a negative interval value to the field ‘n’),  ‘Trainsposition’ represents the first in a set of non permutational functions I wish to describe here. However there is a last semi-permutational function to describe before, because it makes the link between permutational and non permutational functions by giving to the latter ones a general action scheme.

#### Train

The Train function is quite simple, provided you give it two Series, no matter whether they are made of distinct elements, and one Order Series, all three in the same base, it will yield a gradual replacement of the values of the first one by the values of the second one following the indexical order given as third series.

The Order Series (ord) must be made of distinct elements, and that feature is shared by all Train-like functions explained from now on.

Ex:

```*** DESCR: seq_obj: Seq_1 after Build_tree:
object tree: 102145 112145 111145 111145 111115 111111
*** seq_proc : Tkrow_1 Sequence object: recorded```

This last example taken from the output in Csgrouper’s terminal after row validation, shows well the process that will be shared by all non permutational functions. The original Series sees its first element replaced by the corresponding element in B, and so on till the whole Series is replaced (here the repetitions come from the fact that we have chosen a replacing Series made of one sole element) . Now if we had put 3 into field ‘n’ we’d have obtained:

`object tree: 111145 111110`

And this is a (small) bug since last sign should be one! But we see that the number of intermediary series made by the train is different. This feature will be further described below.

#### Trainsposition

This function will take a Series (A) as first argument (a twelve-tone row in our example) and transpose each of its notes according to the values in a second Series (B) in the same base. This transposition will however not be done at once but note by note, so if the Series has twelve notes, it will then be repeated twelve times, each time showing a new differing note till reaching a totally different Series, possibly with non-distinct signs. This behaviour makes the ear gradually used to a different note pattern.

To make things a little less predictable, we add three factors to our transformation: first, the order in which the notes are taken for transposition is not necessarily ascending but represented by any Series in the same base put into field ‘ord’. Second, unlike normal transposition, the  value combined with the original note is not necessarily unique, i.e. you can have a different transposition value for each note of your base Series. This is the content of field (B). Of course you can also choose to have the transpositions done with a sole value, then your field (B) should be filled with 12 times that value, for example, ‘111111111111’ will transpose all your original row one semitone higher. The third factor is the sense of transposition, and it will define whether you’d like to have your transposition value added or subtracted from your original note. This is the content of field ‘signs’ and it be filled with signs corresponding to the values in field (B). For example ‘-+++++++++++’ will make the function subtract from your initial note, and then add to the other ones.

Another way of modifying the behaviour of this function is to change the value of the field ‘n’ to one of the divisors of your Series base. For a twelve-tone row we see that this value can be set to 1, 2, 3, 4 or 6. Each one of these choices will have the effect of grouping together as many changes or steps in the gradual process. Having 3 in ‘n’ for instance, will limit the repetitions of a twelve-tone row to 4, each one containing 3 notes transposed. This value is also set as a global fallback value for each row by meta-field ‘Steps’, but the utility of this field raises a question: to be efficient in each possible case, it should always be set to one (because a series base can be a prime number) and thus it isn’t requiring an independant field.

Unfortunately – unlike what exists for the Static and Dynamic trains of intervals we’ll examine later –  Csgrouper doesn’t provide any analytic tool to tell quickly which Series of transpositions (B) you have to use in order to obtain a certain Series as last output of this function, but this is not very difficult to calculate.

Finally as with any Series manipulation in Csgrouper, one has to be aware of the octavial range attributed jointly (correlative fields ‘Aoct’ and ‘Boct’).