Durations, dynamics and other aspects of music other than the pitch can be freely chosen by the composer, and there are also no general rules about which tone rows should be used at which time (beyond their all being derived from the prime series, as already explained). However, individual composers have constructed more detailed systems in which matters such as these are also governed by systematic rules (see serialism).
How To Make A Serialism Matrix
On the other hand, consider the matrix that rotates the coordinate system through a counterclockwise angle . The coordinates of the fixed vector in the rotated coordinate system are now given by a rotation matrix which is the transpose of the fixed-axis matrix and, as can be seen in the above diagram, is equivalent to rotating the vector by a counterclockwise angle of relative to a fixed set of axes, giving
where is the transpose and is the identity matrix. Equation (15) is the identity which gives the orthogonal matrix its name. Orthogonal matrices have special properties which allow them to be manipulated and identified with particular ease.
Any twelve-tone composition is based on several forms of a row.One can easily create and refer to a table of row forms when analyzingor composing a serial composition. This table is called variously atwelve-tone matrix, twelve-by-twelve array, or magicsquare.
To complete the matrix in columns instead or rows, add the number inthe first column to each pitch class number in the first column. Thenumber at the top of each column serves as the reference pitch for eachtransposition of the inversion.
When the matrix is complete, one can find any retrograde form byreading a row backwards (right to left). One can find any retrogradeinversion form by reading an inversion backwards (bottom to top). Thereference pitch of the prime and retrograde forms is always in thefirst column. The reference pitch of the inversion and theretrograde inversion is always in the top row.
As one final piece of technical, terminological preamble, we introduce the matrix (plural: matrices). This is a neat, compact way of setting out all of the 48 rows in a row class on one 12-by-12 grid. By convention:
Thus my question: under what stipulations will a twelve-tone matrix have both of these diagonals using only their own pitch? (By "twelve tone" I mean a row that uses all twelve pitch classes once and only once.) Does this occur only when the row form is completely symmetrical, or are there outside possibilities that would also create such a matrix?
It's starts out ugly, but the general strategy is to figure out relations between these intervals to try to reduce them down. From this first row we can construct a full matrix. Here it is (with only 1st row, 1st column, and diagonals completed).
Where we are now is very close to the end. These are the minimum required conditions to find a 12-TET matrix with a both diagonals constant. The goal from here is to construct a row such that each tone only appears once, and the intervalic relationships provided are observed. So long as the first interval is, for example, a=7, then the second last row element must be 6+7=1, and so on.
This grid is a called a matrix. We could then fill in each row of our matrix with the transpositions of the original pc set and each column with inversions. For example, where would the other two pc sets that we figured out in our original example, T1 and T1I, fit in this grid?
The great thing about this system is that once each of the transpositions (rows) are filled in correctly, each column will be a correctly transposed version of the inverted tone rows. Once filled in, a matrix such as this shows every transposition and inversion of any aggregate pc set.
Using our terminology of P, R, I, and RI, complete the tone row matrix above by filling in each row and then labeling. Is it necessary to go in a particular order? What is the easiest way to fill it out for you? Knowing that we use P, I, R, and RI to label each direction of the matrix, how would you differentiate each tone row? Would retrograde tone rows be labeled by their starting pitch or their corresponding prime label?
As with most things, some enterprising people have created a shortcut to all of this work. There are now free-to-use matrix calculators such as the 12-tone assistant at this excellent website. These are great to speed up your analysis, but as a student, make sure that you understand the principles of why and how these work before becoming entirely reliant on them. Even when trying to use this website, you must understand exactly what output you are trying to achieve before you can choose the correct option.
Admittedly, it is unlikely that you will ever be required to create a tone row matrix without access to a calculator (unless you are taking a music theory exam.) But the tedium of transposing each of the rows by hand will help you notice patterns in the tone row and help you to remember the nuances of the tone row as you analyze. This often provides insight into the analysis that would be missed by those relying on a calculator to create the matrix.
Note: Desktop/lap sized screen recomended for best user experience (the matrix cannot be effectively resized for optimal mobile viewing due to the 12x12 nature). If you must using a small screen device you will need to scroll horizontally on the matrix and staff to see them in their entirety.
Enter a tone row, either in numbers");cd.write(" (0-11) or in musical notation (Ab, C#, E, etc.). Do not use");cd.write(" double flats, double sharps, etc., but enharmonic spellings are");cd.write(" ok (i.e. D# or Eb).Then click "convert it"");cd.write(" and the row will be magically converted to all of its");cd.write(" permutations in matrix form. To clear the matrix while keeping");cd.write(" the original row intact, click "clear the matrix."
");cd.write("
If you don't have a row, and would like to generate a random 12");cd.write(" tone row, simply click "generate a row." ");cd.write("The row can contain either pitches or numbers, determined by which button is selected.");cd.write(" This row can then be converted into the matrix as well (by clicking on");cd.write(" "convert it").
");cd.write("To quit the matrix converter, click "quit the matrix");cd.write(" converter."
KEYWORDS: integral serialism, array, matrix, twelve-tone, compositional process, filter technique, magic square, Latin square, Maderna, Nono, Improvvisazione N. 1, Musica su due dimensioni, Serenata N. 2, Darmstadt
[14] While each dot of the original series is shifted the same number of times (twelve times), the identity of the original series is completely obliterated after its first appearance. The original series will eventually resurface in its original form, however: since the numbers in each row, column, and diagonal of the number square in Example 5 add up to the same sum (132), the thirteenth dot in each row of the matrices will occur exactly 144 positions after its first appearance in the series of the first matrix (132 positions skipped plus the 12 positions occupied by the dots). Thus the thirteenth dots in all rows taken together will reproduce the original series. This takes place twelve full matrices after the first statement of the series (after 144 positions = 12 matrices), that is, in the thirteenth matrix, which Maderna elides with the first. The original series in the first matrix is thus doubly accounted for, starting and closing the permutation cycle.(17)
[26] Example 12 transcribes the matrices that tabulate the permutations of this series. The series itself appears in the first matrix, represented by the dots connected by straight lines. No sketch survives that documents how Maderna determined the permutations in the following matrices, but the principle is easily reconstructed, as shown in Example 13. Each column of the table lists the values used for the shifts of the pitch class shown at the top. For instance, the first dot in the first matrix of Example 12 (pitch class C), is first moved to the right by skipping over 11 empty positions, then 15, then 9, etc., following the values in the first column of Example 13 (read from top to bottom). And so forth for the remaining pitch classes.
[27] The array of Example 13 is not a magic or Latin square. Each row contains a different set of numbers (adding up to various sums). Each column, on the other hand, contains a different arrangement of the same set of 16 values: values 9 and 11 occur four times, value 10 three times, value 12 twice, and values 13, 14, and 15 once each. The sum of all values, for each column, is 176. This means that in Example 12 the permutational procedure will ultimately map the original series in the first matrix onto itself via a wrap-around: each dot of the original series in the first matrix is moved a total of 192 positions to the right (176 positions skipped plus the 16 positions occupied by the dots = 192), that is, a total of 16 matrices (192 = 16*12). The first matrix serves simultaneously as the seventeenth matrix, closing the permutation cycle.(31)
However, it is Arnold Schoenberg (also from Austria) who is best known for pioneering the form of twelve-tone serialism that caught on and influenced a number of prominent composers of the mid-20th Century.
Multidimensional scaling (MDS) is a technique that generates a map displaying the relative positions of a number of objects based on a given set of pairwise distances between these objects. The following example may help to understand the essence of MDS. Given a geographical map of North America and a scale, one can compute the aerial distances between cities. If instead the initial data is a set of pairwise distances between North American cities, one can attempt to recover the geographical map of North America (within about a symmetry and/or rotation). MDS is a methodology that uses algorithms to implement this idea. Although MDS can generate a two-dimensional map that could perhaps, or hopefully, be interpreted as latitude and longitude in the geographical example, the technique per se can be used to generate more than two dimensions from a given distance matrix. A third dimension could perhaps be interpreted as the average altitude of cities with respect to sea level (i.e., topography).
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