Expanded Serialism

[RJB54]

I have updated the Non-Contiguous Segmentation topic to clarify the manner in which the pitches which result from the segmentation can be used.

[RJB54]

Multi-Directional Segmentation

The next topic to be discussed is Multi-Directional Segmentation.

Simply stated, Multi-Directional Segmentation means the row is divided into two contiguous segments with one segment starting at the first pitch of the row and moving forwards while the other segment starts at the last pitch of the row and moves backwards with the two meeting somewhere in the middle of the row.

For example, one segment contains the pitches at ordinal positions 0 through 7 while the other segment contains pitches 8 through 11. These two segments would be presented more or less simultaneously with one presentation moving from ordinal 0 to 7 and the other moving from ordinal 11 to 8.

The opposite can also occur with the two segments starting in the middle with one segment going backwards to ordinal position 0, while the other goes forwards to ordinal position 11.

Either of these processes can also be being applied to a rotated row.

For example, rotate a row by 4 which results in a row of 4,5,6,7,8,9,10,11,0,1,2,3.

In this case, the first segment could present ordinal positions 4 through 9 while the second segment would present ordinal position 3 through 0 then 11 through 10.

[RJB54]

Pitch-Set Transformation

The next topic to discuss is Pitch-Set Transformation.

This topic starts as a follow-on from the Contiguous and Non-Contiguous Segmentation topics. Once a segment has been extracted from the row using either of these segmentation techniques, the resultant segment can be manipulated using pitch-set transformations.

The general idea is that (a) you extract a segment (pitch-set) from a row, then (b) apply a pitch-set transformation to that segment (pitch-set) transforming it into a different pitch-set, then, optionally, (c) apply another transformation to the second pitch-set to create a third pitch-set, etc. This allows for the creation of more content.

In my usage, Pitch-Set Transformations will typically be used in a manner analogous to Sequence (described above) in that it provides a method of generating content based on only some of the pitches of a given row independently of the remaining pitches of that row. The main difference between a Sequence and a Pitch-Set Transformation is that a Sequence will repeat the same relative pitches in the same order on differing transpositional levels, while a Pitch-Set Transformation will modify the pitches using the various transformational processes.

For my purposes I will not make use of all of the possibilities of pitch-set transformations. This is primarily due to the fact that making use of the full possibilities of this technique will take one into the realm of free atonality. For my purposes, I do not wish to go that far.
 
The pitch-set transformations which I will make use of are:

Complementation
The complementation transformation involves transforming a pitch-set into its literal complement.

Inversion
The inversion transformation involves transforming a pitch-set into its literal inversion.

Pitch
The pitch transformation involves modifying one of more pitches of pitch-set-1 resulting in pitch-set-2.

Subset
The subset transformation involves extracting some of the pitches from pitch-set-1 resulting in pitch-set-2 which has a smaller cardinality than pitch-set-1.

Superset
The superset transformation involves adding one or more pitchs to pitch-set-1 resulting in pitch-set-2 which has a larger cardinality than pitch-set-1.

Vector
The vector transformation involves transforming pitch-set-1 into pitch-set-2 which will be a pitch-set whose interval vector is similar to that of pitch-set-1. In my usage there are three levels of vector relationships. From strongest to weakest they are: (1) when the two vectors are exactly the same (Forte called this the Z relationship), for example, the vectors for both sets are [013202], (2) when there is only one vector different between the two vectors, for example, [124310] and [114310], or (3) when the difference is that two of the vector values have been switched, for example, [231041] and [131042].

[RJB54]

Doubling

This topic is a fairly simple one dealing with the idea of two or more row pitches being doubled by non-row pitches. This technique should be applied to two or more row pitches because applying it to a single pitch is inherently ambiguous.

Conceptually, the doubling should relate to the row pitches at either a single interval or at different intervals following a given intervallic pattern.

Doubling at a single interval is simple. The row pitches are all doubled by pitches the same interval away (third, sixth, second, etc.). For example, doubling at interval 3 means that the row pitches being doubled are all doubled at the minor third.

Doubling by intervallic pattern means each row pitch is doubled by pitches whose difference changes from row pitch to row pitch by a given interval pattern similar to that of Linear Interval Cycles. For example, for a single interval cycle of 2, the first row pitch (C) is doubled at interval 2 (Bb), the second row pitch (Eb) is doubled at an interval of 4 (B), the third row pitch (F) would be doubled at an interval of 6 (B), etc. For an interval cycle using a mixed interval pattern (say a cycle of 1-2-3)  the first row pitch (C) is doubled at an interval of 1 (B), the second row pitch (Eb) is doubled at an interval of 2 (Db), the third row pitch (F) is doubled at an interval of 3 (D), etc.