James
Registered:1268171845 Posts: 332
Posted 1349478221
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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 andhttp://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:1268171845 Posts: 332
Posted 1350400445
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#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