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Posted by Caleb Morgan Arbitrarily, we choose a limited-interval series: #8.123 in the table provided in my first post on Self-sim series. The #8 series is: C D D# F G G# A# B C# E F# A Notice that the series has the following interval pattern in semitones: 2,1,2,2,1,2,1,2,3,2,3,3 (last 3 is wrap-around) Next, we can choose a transformation of this series: p/9/4, or primary series down a minor 3rd and rotated 4 positions. E F G Ab Bb C# Eb F# A B C D We start with a simple chromatic scale, and align our p/9/4 of limited-interval series #8 underneath it: C C# D D# E F F# G Ab A Bb B If we were thinking of these two series as if they were a cyclical permutation, we’d get: (C E Bb) (C# F) (D G F# Eb Ab A B) next we align this with a simple chromatic scale underneath, and do our “X-swap” C E Bb C# F D G F# Eb Ab A B Result: A rather Berg-ian series that strongly implies C minor! Maybe too tonal for school! Remember that all self-similarity is partial and incomplete, with the exception of Mallalieu series. In this case, we’re looking to see whether the FIRST FIVE NOTES occur within SIX notes of each other in transformations of the series. They do, in many transformations. So this is good enough to be a useful MOF series. Look at the standard transformations of the series. I capitalize the occurrences of the first five notes ( C,Eb,F,Ab,Db) p/8/5: p/5/6: p/10/10: r/9/10: i/6/7: Ri/10/2: Ri/9/3: Ri/8/8: Ri/0/11: Next post: further explanation of 2 ^ n mod 13 and the vast number of other series that can be derived from the indices of discrete logarithms.
on 9/27/2006, 6:07 am, in reply to "mallalieu & self-sim series"
68.166.234.51
note: Because it's difficult to control fonts and formatting, in the following post the series don't align vertically. This makes it hard to see the very simple process. If you're interested, simply copy the series by hand and align them yourself, simply 1 to 1.
Here’s some more examples of a fairly simple technique to make self-similar 12-tone rows, or MOFS, as they are called. (Multiple Order Function Series).
The numbers after the period indicate the intervals in the series.
E F G Ab Bb C# Eb F# A B C D
C C# D D# E F F# G Ab A Bb B
C Eb F Ab Db E G F# A Bb D B
How does this series have the quasi-fractal, self-similar, multiple-order function?
It’s ok if we have to “wrap around” or repeat one cycle of the series.
p/11/4:
C Eb f# F Ab a C# a# b d e g
C Eb d F f# a# g Ab b C# e a
C b d Eb g e F Ab a# C# f# a
C a a# c# Eb f# b d F e g Ab c a a# C#
C a g# b g f# Eb e c# a# F d c a Ab b g f# eb e
C#
C a ab e g f# Eb c# a# F d b c a Ab e g f# eb C#
C c# e Eb f# a d F g a# b Ab c C#
C Eb d F Ab c# e f# a a# g b
C Eb F Ab a f# a# b d C# e g
C c# a# d Eb f# F Ab b e g a c C#
There are also multiple occurrences of the first-five-note motif in the 5m transformations and the 5p (every 5th note) transformations, and also the 5m/5p transformations.
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