Note that every unit (the space between two blocks) is meant to describe the temporal extent of some event (e.g., a sound, or a silence). Thus the very last three blocks of such a multiple alignment are actually the starting points of the first three units of a new alignment. Moreover, the number of units between two positions of the alignment depends on the metric choice. For instance, considering a quarter note as unit, the distance between the sixth block of the first sequence of distances (position 160 in the alignment) and the third block of the second sequence of distances (position 76 in the alignment) is a half note and 5 thirty-second notes (2 + 5/8) according to the metric of the first sequence of distances (i.e., 5 units between position 1 a 160). The distance between the same two blocks is approximately 2 + 1/5 according to the metric of the second sequence of distances, and approximately 1 + 2/5 + 1/7 according to the metric of the third sequence of distances.

Of course it is possible to measure the linear distances between all points of the alignment but it is also necessary to introduce some kind of approximation to avoid complex musical notation. We can approximate those distances using the closest option available on a preordered table of note values and derive three different patterns starting from the original multiple alignment.

After converting all the linear distances into note values, we can decide which kind of event (sound, silence, dynamics, etc.) must be bound to the fundamental elements of each pattern. In the following example, I used the third pattern as template and assigned dynamics, articulations and rests respectively to the elements of the distance sequences derived from the third (blue blocks), the second (red blocks), and the first (black blocks) hexachord. By way of information, the serialization of these three parameters were obtained from the diagonals of the matrix of permutations of the generating row.

As you might know, there are three categories of Scotch: malt, grain and blended whisky. Blended whisky (Johnnie Walker, Ballantine’s, J&B, etc.) is a mixture of 1/3 single malt and 2/3 grain whiskies from several distilleries. The wide success of those international brands is probably due to the common compliance of their conformed tastes. Sadly, the less distinctive, the more successful. Grain whisky is made from a mash of cereal grains (usually malted and unmated barley, wheat and maize). There are not so many different grain distilleries. Indeed bottled grain whisky is still a bit of an odd choice! Finally malt whisky is made from 100% malted barley. Malts are divided in single (no blends) and vetted (blends of malts). Lagavulin and Oban are two out of six so-called ‘Classic Malts’ marketed since 1988 by Diageo. The other four are Dalwhinnie, Cragganmore, Talisker e Glenkinchie.

16 years old Lagavulin is made on Islay, “the Queen of the Hebrides”. Its rubbish color is warm like its taste. When you nose it, you get scent of leather and wood on fire. The taste is way smokey but gentle and sweet, like an old cask cellar in winter or a mild cigar’s aroma. Smooth peaty tones return in the end. Unfortunately the design of the bottle suggests that it was realized after 1999 (the royal seal has been replaced with a sailing ship drawing on the label). I was told that the late 1980′s batches were the best. Anyway, that dram of Lagavulin has been a revelation for me being used to bourbon. Now I am curious to try out Oban. I will report my impressions here later.

A   x . . x
B   x x .
C   . . . x . x

First, I assign a specific combination of them to the voices I want to overlap

B     C           B
x x . . . . x . x x x .

A       A       B     A
x . . x x . . x x x . x . . x

At this point, there are several ways to align the two strings. I could fix the beat and scroll one string upon the other, like this

|x x . |. . . |x . x |x x . |<============
======>|x . . |x x . |. x x |x . x |. . x |

This kind of approach doesn’t rescue from the risk of rhythmic monotony. As an alternative, I could stretch or shrink one string proportionately to the length of the other string

or partially in its length

|x x . . . |.         x         .         x         x         x         .         |
|x . . x x |.      .      x      x      x      .      x      .      .      x      |

Here we go! It now appears that the method gained a broadened applicability. It would be interesting to apply this kind of musical pattern transformations to simulate a mosaic tessellation like the following “Maya-ish” example, where I used just two patterns (. x x and . x x x).

Everything’s composed—but not written yet.
W. A. Mozart

At this point it’s possible to arrange the four non-standard rows according to the preferred scheme. I followed an outline that I had projected in consideration of the requirement of an overall irregular rhythmic structure.

This row is its own transposed retrograde inversion just like the row from the fifth movement of Schoenberg’s Serenade, Op. 24. Thus the second hexachord is the retrograde inversion of the first one, which summarizes the feature of the entire row (see the green square in the matrix). The matrix is symmetric along the red diagonal in the figure below, being P0 = RI9 and I0 = R3, P9 = RI6 and I3 = R3, and so on.

Moreover, the middle eight notes of the row compose an octatonic scale.

Not so bad for just three notes!

J. S. Bach – Christmas oratorio, BWV 248 – Chorale ‘Ach mein herzliebes Jesulein’ (1734)
J.S. Bach – Canonic Variations on ‘Von Himmel hoch’, BWV 769 (1748)
J.S. Bach/I. Stravinsky – Chorale Variations on ‘Von Himmel hoch’, BWV 769 (arr. 1956)
I. Stravinsky – Threni (1958)

The arrangement of Bach’s variations (deserved round of applause for the organist David Hyde!) was originally made as a companion piece for Canticum sacrum (1955), Stravinsky’s first piece (premiered in Venice) to contain a movement entirely based on a tone row, “Surge, aquilo”. Even though I might have considered myself lucky enough, Threni (commissioned from the director of Venice Biennale) was actually my personal good reason for not missing out this outstanding concert. As in Berg’s Concerto, when the dodecaphonic magma surprisingly merges into occasional tonal flashes (such as in “Perceiving Hope” from De Elegia Tertia or in De Elegia Quinta) it’s like gasping relief after surfacing from the abyss. Really uplifting. Although the confused acoustics of the Albert Hall (in spire of the recent optimization) didn’t justice to the meek performance by BBC Singers, soloists and the London Sinfonietta with Atherton conducting, I greatly enjoyed the concert and the serious cultural commitment by British people and institutions. Italy should learn a thing or two…

I threaded the original row into two complex rhythmic sets based on the combination of five simple rhythmic patters (described previously) and I aligned them:

A D E A B C
C A D E A B

• • • – – – – – - • – • – • • • • • – • – – • • - – – – – • • • • • – • – • – • - • • – – • • •
- • • – – • • • • • • – – – – – - • – • – • • • • • – • – – • • - – – – – • • • • • – • – • – •

Then I systematically transposed one voice letting the other lying still and vice-versa (I wrote a Perl script to generate the LilyPond output shown above). I obtained the twenty-two pitch variants of the two voices alignment based on the original Copland’s row. Note that each variant/staff could be transposed further on with a resulting amount of 276 pitch variants of the same pattern alignment!

At the end, I arbitrarily chose two variants (this is the not-serial part!). I assigned them respectively to the left hand and to the right hand and I composed the following piano piece: