# Conversation with a Dog



## millionrainbows (Jun 23, 2012)

Here is a conversation I had which concerns the Allen Forte book _*The Structure of Atonal Music*_ and John Rahn's _*Basic Atonal Theory,*_ with mention of Howard Hanson's *Harmonic Materials of Modern Music.*

Dawg, my next concern is with how, exactly (or if) these Hanson ideas differ (if at all) from the Forte sets.

As far as I can determine, they seem very similar, each one dealing with the 'interval content' of each set, or scale. I know that Forte arranges these sets in 'normal order' in order to eliminate redundancy; yet, this could possibly blind one to the possibilities inherent in the 'un-normalized' sets, if we approach all of them as scales, not sets.

The harmonic results might differ, as we saw in the videos on the George Russell thread; how starting on 'F' instead of 'C' changes the harmonic halo.

So, if we use the Forte sets as 'scales,' then this sets up a root, and an heirarchy; and that's what tonality does. We could further establish tonality by free repetition of the appropriate notes. This approach sees each 'set' as a scale, with a definite starting point, whether or not it has been 'normalized'.

The normalization process classifies similar sets as being equivalent in interval content; yet, the 'modes' of the normalized sets might yield quite different sonorities, just as D dorian differs from C major, yet they are members of the same 'interval set.' And when we begin constructing chords within these 'scales' or 'sets,' then the harmonic results become more apparent.

Am I on the right track here, or do I need to re-read Forte, or have I read it the wrong way?
-----------------------------------------------------------------------------------

millions,

you are absolutely correct here. you must allow each set to be transposed through its constituent modes in order to explore its diversity. the normalization process is merely for "inventory" purposes, so that some kind of logical ordering may take place. rahn's criteria is slightly different than forte's but this affects only a half-dozen of some 352 scale/arpeggio/set types. for the record, i use rahn's. the inventory problem cannot be understated: it is very difficult to keep track of 4096 pitch sets. i can barely remember the names of a half-dozen neighbors across the street, let alone 4096 of anything. here's how it breaks down:

...there are 2048 inversions (modes) of 352 pitch sets for a total of 4096 tonal combinations. the term inversion is a bit problematic in that inversions as applied to chords is different than what is used in the case of an inverted melody. for those who are more curious and don't yet have a headache or some form of chronic insomnia:

0 tones have 0 inversions / modes of 1 pitch set in 1 key for a total of 1 tonal combination

1 tone has 1 inversion / mode of 1 pitch set in 12 keys for a total of 12 tonal combinations

2 tones have 10 inversions / modes of 5 pitch sets in 12 keys with a total of 60 tonal combinations
plus 1 inversion / mode of 1 pitch set in 6 keys with a total of 6 tonal combinations
equals 11 inversions / modes of 6 pitch sets for a total of 66 tonal combinations

3 tones have 54 inversions / modes of 18 pitch sets in 12 keys with a total of 216 tonal combinations
plus 1 inversion / mode of 1 pitch set in 4 keys with a total of 4 tonal combinations
equals 55 inversions / modes of 19 pitch sets for a total of 220 tonal combinations

4 tones have 160 inversions / modes of 40 pitch sets in 12 keys with a total of 480 tonal combinations
plus 4 inversions / modes of 2 pitch sets in 6 keys with a total of 12 tonal combinations
plus 1 inversion / mode of 1 pitch set in 3 keys with a total of 3 tonal combinations
equals 165 inversions / modes of 43 pitch sets for a total of 495 tonal combinations

5 tones have 330 inversions / modes of 66 pitch sets in 12 keys for a total of 792 tonal combinations

6 tones have 450 inversions / modes of 75 pitch sets in 12 keys with a total of 900 tonal combinations
plus 9 inversions / modes of 3 pitch sets in 6 keys with a total of 18 tonal combinations
plus 2 inversions / modes of 1 pitch set in 4 keys with a total of 4 tonal combinations
plus 1 inversion / mode of 1 pitch set in 2 keys with a total of 2 tonal combinations
equals 462 inversions / modes of 80 pitch sets for a total of 924 tonal combinations

7 tones have 462 inversions / modes of 66 pitch sets in 12 keys for a total of 792 tonal combinations

8 tones have 320 inversions / modes of 40 pitch sets in 12 keys with a total of 480 tonal combinations
plus 8 inversions / modes of 2 pitch sets in 6 keys with a total of 12 tonal combinations
plus 2 inversions / modes of 1 pitch set in 3 keys with a total of 3 tonal combinations
equals 330 inversions / modes of 43 pitch sets for a total of 495 tonal combinations

9 tones have 162 inversions / modes of 18 pitch sets in 12 keys with a total of 216 tonal combinations
plus 3 inversions / modes of 1 pitch set in 4 keys with a total of 4 tonal combinations
equals 165 inversions / modes of 19 pitch sets for a total of 220 tonal combinations

10 tones have 50 inversions / modes of 5 pitch sets in 12 keys with a total of 60 tonal combinations
plus 5 inversions / modes of 1 pitch set in 6 keys with a total of 6 tonal combinations
equals 55 inversions / modes of 6 pitch sets for a total of 66 tonal combinations

11 tones have 11 inversions / modes of 1 pitch set in 12 keys for a total of 12 tonal combinations

12 tones have 1 inversion / mode of 1 pitch set in 1 key for a total of 1 tonal combination

therefore, there are 2048 inversions / modes of 352 pitch sets for a total of 4096 tonal combinations

whew!


----------

