Author Topic: Interval Projection  (Read 13143 times)

millions

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Re: Interval Projection
« Reply #45 on: August 13, 2011, 04:24:33 AM »
I thought this chart was interesting; from WIK, it shows the various 'interval projections' I have been discussing. These are the same ideas as Hanson's.

Pitch class multiplication modulo 12

When dealing with pitch class sets, multiplication modulo 12 is a common operation. Dealing with all twelve tones, or a tone row, there are only a few numbers which one may multiply a row by and still end up with a set of twelve distinct tones. Taking the prime or unaltered form as P0, multiplication is indicated by Mx, x being the multiplicator:


The following table lists all possible multiplications of a chromatic twelve-tone row:

 M   M × (0,1,2,3,4,5,6,7,8,9,10,11) mod 12
 0   0   0   0   0   0   0   0   0   0   0   0   0
 1   0   1   2   3   4   5   6   7   8   9   10   11
 2   0   2   4   6   8   10   0   2   4   6   8   10
 3   0   3   6   9   0   3   6   9   0   3   6   9
 4   0   4   8   0   4   8   0   4   8   0   4   8
 5   0   5   10   3   8   1   6   11   4   9   2   7
 6   0   6   0   6   0   6   0   6   0   6   0   6
 7   0   7   2   9   4   11   6   1   8   3   10   5
 8   0   8   4   0   8   4   0   8   4   0   8   4
 9   0   9   6   3   0   9   6   3   0   9   6   3
10   0   10   8   6   4   2   0   10   8   6   4   2
11   0   11   10   9   8   7   6   5   4   3   2   1

Note that only M1, M5, M7, and M11 give a one to one mapping (a complete set of 12 unique tones). This is because each of these numbers is relatively prime to 12. Also interesting is that the chromatic scale is mapped to the circle of fourths with M5, or fifths with M7, and more generally under M7 all even numbers stay the same while odd numbers are transposed by a tritone. This kind of multiplication is frequently combined with a transposition operation. It was first described in print by Herbert Eimert, under the terms "Quartverwandlung" (fourth transformation) and "Quintverwandlung" (fifth transformation) (Eimert 1950, 29–33), and has been used by the composers Milton Babbitt (Morris 1997, 238 & 242–43; Winham 1970, 65–66), Robert Morris (Morris 1997, 238–39 & 243), and Charles Wuorinen (Hibbard 1969, 157–58). This operation also accounts for certain harmonic transformations in jazz (Morris 1982, 153–54).

Look at this chart as being "stacks" of a certain interval, going left to right.
The top row "0" means nothing.
1X row (beginning with "1") is an ascending chromatic scale.
2X row is a whole-tone scale (stacking seconds, or "2").
3X row is a diminished scale
4X row is major thirds, or an augmented chord;
5X row is stacks of fourths (eventually giving all 12 notes)
6X row is the tritone, invertible with itself;
7X row is stacks of fifths (eventually giving all 12 notes)
8X is the inversion of row 4, also augmented
9X is the inversion of row 3, diminished
10X is the inversion of row 2, whole tone
11X is the inversion of row 1X, a descending chromatic scale
« Last Edit: July 22, 2012, 09:32:38 AM by millions »
"In Spring! In the creation of art, it must be as it is in Spring!" -Arnold Schoenberg
"The trouble with New Age music is that there's no evil in it."-Brian Eno

millions

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Re: Interval Projection
« Reply #46 on: September 06, 2011, 06:03:21 PM »
The difference in Hanson's approach is that he wants to derive usable scales out of all the projections, not just the 1, 5, 7, and 11; so in the case of the 'recursive' intervals which tend to stay within the octave, such as 3:   0   3   6   9   0   3   6   9   0   3   6   9 (stacks of minor thirds), Hanson does a chromatic alteration at the point where the recursive pattern begins again.
"In Spring! In the creation of art, it must be as it is in Spring!" -Arnold Schoenberg
"The trouble with New Age music is that there's no evil in it."-Brian Eno

millions

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Re: Interval Projection
« Reply #47 on: November 09, 2012, 12:08:43 PM »
Dawg, or anybody who is listening: I'm finally figuring out how Hanson, Elliott Carter, and other "pseudo-serialists" are using the sets that Forte indexed. Elliott Carter came to these sets independently.
The key lies in which sets they are interested in. For Carter, these were the "all-interval tetrachords" and the "all-triad hexachords." This allows them to break-down the sets into triads, and "stack" them to control (somewhat) the vertical or harmonic aspects of a row, to create music easier on the ear, while remaining true to serial sets.
The idea started with Babbitt, but he was not interested in reconciling tonality with serialism.
George Perle's book "Twelve-Tone Tonality" deals with this subject. His music still sounds quite modern, but more ear-friendly than, say, Boulez.

And Dawg, your response to "why are the Forte sets reduced to normal order" left me wondering. You said it was for convenience, which I can see, but this "normalizing" of the interval-relations is biased, I think, towards atonal or serial considerations of ordered rows and interval relations, not as "tonal entities" or scales with pitch-based starting points.

What do you think?

Here's what dawg said:

 "...biased, I think, towards atonal or serial considerations of ordered rows and interval relations, not as "tonal entities" or scales with pitch-based starting points."
 
i agree. all of the "normalized" lists (and/or "descriptions" if you will) that i've seen are ordered according to only numerical (as in, say, a list of zip codes) criteria - ordered by numerical order rather than actual location; therefore, the complete lack of a tonal context is manifest...
 
one way to fix this is by (try this, really) applying the set theory description (such as 01234) to the circle of fifths rather than the circle of half-steps, thus 01234 = C G D A E rather than C C# D D# E. perhaps not perfectly unbiased in the tonal realm but most certainly "better."
 
do it though: apply 01234 and 01235, etc... to the circle of fifths and see if this does not fit better in terms of tonal consonance. it is very "russellian," as LCC is based initially on fifths.
 
 
dawg :o)
« Last Edit: November 12, 2012, 08:58:12 AM by millions »
"In Spring! In the creation of art, it must be as it is in Spring!" -Arnold Schoenberg
"The trouble with New Age music is that there's no evil in it."-Brian Eno