source file: m1400.txt Date: Wed, 29 Apr 1998 15:27:17 -0400 Subject: Reply to Joe Monzo From: "Paul H. Erlich" >In traditional 12-eq harmonic theory, with extended >chords built upwards in 3rds, if the near-quartertone >identities of 11 and 13 are assumed to be implied >by the "flat 5th / sharp 11th" and "flat 13th", >respectively I think any good jazz musician would assume otherwise. >, the rational implications of the chord- >members of a major chord line up pretty much as >follows: >27/16 13th >13/8 flat 13th >11/8 sharp 11th >4/3 11th (21/16 is another possibility) >19/16 sharp 9th (7/6 is another possibility)* >9/8 9th >17/16 flat 9th >15/8 major 7th >7/4 minor 7th >25/16 augmented 5th >3/2 5th >11/8 diminished 5th (23/16 ???) >5/4 3rd >1/1 root >* See the discussion about these two particular >intervals between Paul Erlich and and myself >re: Hendrix Chord [Tuning Digest #1376], as to >whether 19 is admissible as a rational implication >of the "sharp 9th". I think 19 is much more admissible as a rational implication of the sharp 9th than 11 or 13 are admissible as rational implications of anything in 12tET (despite what Schoenberg may have had to say). Let's analyze a few common jazz chords (which are normally played, of course, in 12tET): 1. 13flat9 (meaning: root, maj .3rd, min. 7th, min. 9th, maj. 13th; the perfect 5th is often omitted in jazz) I think this chord derives its flavor from the major triad formed on the maj.13th with the min. 9th and maj. 3rd. This triad resonates clearly and forms a polyharmonic structure against the root. Tuning the triad 27:34:40 would destroy this effect and render the chord pretty ugly (IMO). Hence in this case the maj. 13th should be 5/3 and the min. 9th should be 25/12. 2. sharp11flat9 (meaning: root, maj. 3rd, min. 7th, min. 9th, aug. 11th) Here, there is a strongly resonating dominant seventh chord formed on the aug. 11th with the min. 7th, min. 9th, and maj. 3rd. If we fix the maj. 3rd at 5/4, then the aug. 11th should be 10/7, the min. 7th should be 25/14, and the min. 9th should be 15/7. Of course, the fact that the min. 7th is also close to 7/4 helps to reinforce the root along with the maj. 3rd. It is very common in jazz to have a maj. 3rd - min. 7th tritone imply two different roots, one with their traditional roles and one with their roles reversed. This requires a tuning, such as 12tET, where the 50:49 "septimal sixth-tone" vanishes. 3. 6/9 (meaning root, maj. 3rd, perf. 5th. maj. 6th, maj. 9th) Here, we have a chain of 5ths, but also a major 3rd and two minor thirds. If all these intervals are to be consonant, no just interpretation will really do. This chord requires a tuning where the 80:81 syntonic comma vanishes. Other examples can be pointed out, but I think that's enough to make my point for now. By the way, the simplest ET consistent in the 19-limit is 80tET.