janzen@sunfun.DEC (Thomas E. Janzen CSS GNG CWO 714 850-7849 SUNFUN::JANZEN) (08/17/84)
There has been a question about all-intervals rows, i.e., twelve-tone sets.
Here are some examples.
A,Bb,Ab,B,G,C,Gb,C#,F,D,E,D#. This is the most famous (and its own retrograde)
Eb,Gb,Db,G,C,D,B,Bb,Ab,E,F,A. This is in "Studies in Counterpoint" by
Ernst Krenek, G. Shirmer, N.Y., 1940, page 37.
A,E,F,Bb,Ab,B,G,Db,C,D,Gb,Eb. I wrote this for my piano piece, "Animations",
1977, unpublished, and used it in parts of "Siddhartha", too.
Such sets need not be their own retrogrades ("symmetrical?"), but may be.
Here is an example of an A.I.R. that is its own retrograde:
C#,D,E,B,D#,F#,C,A,F,Bb,Ab,G. Krenek, ibid.
Note that to verify that these are A.I.R.s, the intervals in each set must
be measured in the same direction, either up or down. Otherwise, in the
last example, the C#-D in the beginning is a minor second just as is the
Ab-G at the end. One can say that the C#-D is a minor second (1 semi)
(going up), and the Ab-G (going up) is a major seventh (11 semis).
Note the intervals' names:
Unison - 0 semitones
Minor Second - 1 semitone
Major Second - 2 semitones
Minor Third - 3 semitones
Major Third - 4 semitones
Perfect fourth - 5 semitones
Tritone - 6 semitones
Perfect Fifth - 7 semitones
Minor Sixth - 8 semitones
Major Sixth - 9 semitones
Minor Seventh - 10 semitones
Major Seventh - 11 semitones.
I now have begun a Pascal program to find AIRs. Years ago I wrote a trial and
error program in TRS80 Basic. I think it worked but I can't find it now.
Thomas E. Janzen Digital Equipment Corp.
3390 Harbor Blvd. Costa Mesa, CA 92626
Thu 16-Aug-1984 09:00 PDT