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James

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Posts: 292
Reply with quote  #1 
This month we posted my chapter "The Mathematics of Four-Note Chords" and Ted's "Systematically Invertible 4 Note Chord Types."  Here are links to these two items:

http://www.tedgreene.com/images/lessons/v_system/15_The_Mathematics_of_Four-Note_Chords-and-Beyond.pdf

and

http://www.tedgreene.com/images/lessons/v_system/SysInvertible_4-NoteChordTypes-43_1985-05-18_withText.pdf

In my Mathematics chapter, I talk about the 165 four-note chords if you don't exclude inversions.  On Ted's sheet, he writes out all 165, crossing out those that are inversions or transposition/inversions of previous chords.  The ones that are not crossed out are the 43 four-note chords if you do exclude inversions.  If you look at Ted's sheet from the end to the beginning, you see he has 1 chord under the heading CA, 3 under CAb, 6 under CG, 10 under CGb, 15 under CF, 21 under CE, 28 under CEb, 36 under CD, and 45 under CDb.  These are the triangular numbers I talk about in the Mathematics chapter.

I should have pointed out these relationships between Ted's page and the Mathematics of Four-Note Chords but I just noticed them.  It's a nice confirmation.
James

Registered:
Posts: 292
Reply with quote  #2 
Here's another interesting item related to the math of the 43 four-note chord types and the 165 inversions of those four-note chord types.  On a page entitled "V-1 by Intervals" dated 11-24-92 in Ted's personal notes, he wrote this:

43 x 4 = 172 PRESUMED total.
165 = ACTUAL total.  Why?!???

Well, the answer to Ted's question is in my Mathematics of Four-Note Chords and Beyond.  Internal symmetry in three special cases out of the 43 produces fewer than four systematic inversions.  For details, see:

http://www.tedgreene.com/images/lessons/v_system/15_The_Mathematics_of_Four-Note_Chords-and-Beyond.pdf
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