source file: m1403.txt Date: Sun, 3 May 1998 11:45:06 -0400 Subject: Response to Paul Erlich From: monz@juno.com (Joseph L Monzo) Paul Erlich provided some alternate rational interpretations of 12-eq chord tones in Tuning Digest # 1400, in response to a list I had posted the day before. I did not mean to imply that the ratios in my table are the only ones, or even the best ones, which could be implied by chord-tones used in the 12-eq scale. I was merely using that tabulation as an illustration of the similarities between Ken Wauchope's observations about perceptual limitations [Tuning Digest #1398], and mine about musical usage and implications, of prime-limits in music. I specifically intended *not* to find which ratios were best implied by the 12-eq notes, but rather exactly the opposite, namely, how primes above our usual limit (that is, above 5 or even 7) could be mapped to 12-eq notes. I agree with Erlich that certain 12-eq chords in certain contexts can imply ratios which fall under lower prime limits and which exhibit both smaller-integer and lower- prime internal relationships than the higher-prime ratios I indicated. I have always said that many different rational interpretations of 12-eq notes are possible, and Erlich provides good reasons for his choices. >> 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).>Perhaps, as Erlich suggests, 11 and 13 are far enoughaway from their closest 12-eq representations that itis not valid to assume that they can be implied by *any*12-eq notes, and so they deserve to be considered entirelyunique chord identites that can only be represented innon-12-eq tunings. > Let's analyze a few common jazz chords (which are > normally played, of course, in 12tET): Erlich correctly points out, with his first two examples, that in the higher tones of complex chords there are implications of other, simpler chords. Let's take a closer look: > [Example] 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. Here is my matrix graph illustrating all the notes described by myself and Erlich. Powers of 3 go left-to-right, powers of 5 go front-to-back, and higher primes transpose this 3x5 matrix to different levels from top-to-bottom. For each of the notes used, I give the 12-eq description (qualified by our names if necessary), the JustMusic prime-factor notation, the ratio, and the Semitones (= cents/100). 3^1 and 3^2 are only placeholders. (For the best view use 8-pt. Courier New font): Monzo -9 17^1 17/16 1.05 | | m7 7^1 7/4 9.69 | Erlich -9 | 3^-1 * 5^2 | 25/12 | 0.70 | \ | \ 3rd | Erlich 13 _ - 5^1 | 3^-1 * 5^1 - 5/4 | Monzo 13 5/3 3.86 | _ - 3^3 8.84 \ | _ - (3^2)- 27/16 \ \ | _ - (3^1)- 9.06 \ root - \ _ - n^0 (3^-1) - 1/1 0.00 Analysis:********* (The ratios and semitones are given within 1 octave. The proportions are given consecutively. The analysis of the individual chords gives the smallest integer which represents the identity.) 12-eq ratio s-t. prop. Monzo Erlich polytonal ===== ===== ==== ===== ===== ================ 13 (M) 27/16 9.06 81 27 13 (E) 5/3 8.84 80 1 -9 (M) 17/16 1.05 51 17 -9 (E) 25/24 0.70 50 5 m7 7/4 9.69 42 7 7 maj 3 5/4 3.86 30 5 5 ----- 3 root 1/1 0.00 24 1 1 __________ ______ ___ ___ 3^-1 * 5^1 (3^-1) n^0 n^0 In the "proportions" column, the absolute identites level all ratios to one common nexus, in this case, the "missing fundamental" of 4/3 (= 3^-1). This makes it easy to see the rational intervals which make up the discrepencies between our two interpretations: 81/80 (22 cents) for the 13th, and 51/50 (34 cents) for the -9. The column with my name shows how this chord fits into my rational interpretation, giving an otonal chord on a "root" of 1/1 (= n^0), with the identities 1, 5, 7, 17 and 27. The last two columns show Erlich's 7-limit polytonal interpretation, giving an otonal chord on 1/1 with identities 1, 5, and 7, and an otonal chord on 5/3 (= 3^-1 * 5^1) with identities 1, 3, and 5. The 5/4 is thus both the 5-identity in the 3-note dominant-7th-chord on 1/1, and the 3-identity in the major triad on 5/3, which "resonates clearly". > [Example] 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. In my matrix graph: Monzo -9 17^1 17/16 1.05 | | Monzo +11 11^1 11/8 5.51 | | Monzo m7 7^1 7/4 9.69 | maj. 3rd | 5^1 | 5/4 | 3.86 | \ | \ | root n^0 1/1 0.00 Erlich m7 | 5^2 * 7^-1 | 25/14 | 10.04 | \ | \ | Erlich -9 Erlich +11 | __ -- 3^1 * 5^1 * 7^-1 5^1 * 7^-1 |-- 15/14 10/7 | 1.19 6.17 | \ | \ | \ | (7^-1) Analysis: ********* 12-eq ratio s-t. prop. Monzo Erlich polytonal ===== ===== ==== ==== ====== ========================== -9 (E) 15/14 1.19 120 15 ----- 3 -9 (M) 17/16 1.05 119 17 +11 (E) 10/7 6.17 80 5 ----- 1 +11 (M) 11/8 5.51 77 11 m7 (E) 25/14 10.04 50 25 ----- 5 m7 (M) 7/4 9.69 49 7 3rd 5/4 3.86 35 5 5 ----- 35 ----- 7 root 1/1 0.00 28 1 1 ----- 7 __________ ______ ___ ___ ______ 5^1 * 7^-1 (7^-1) n^0 n^0 (7^-1) The fourth column displays the discrepancies: 120/119 (14 cents) for the -9, 80/77 (66 cents) for the +11, 50/49 (35 cents) for the m7. The fifth column shows an otonal chord on 1/1 with identities 1,5,7,11, and 17. The last three columns give Erlich's chord, which has 5/4 functioning as both the 5-identity in an otonal chord on 1/1 and the 7-identity in an otonal chord on 10/7. This is the "strongly resonating dominant seventh chord formed on the aug. 11th with the min. 7th, min. 9th, and maj. 3rd". The interpretation as an otonal chord on 8/7 puts all of Erlich's ratios over a nexus, while those on either side of it show the polytonal significance. In jazz harmony, a commonplace progression is a dominant-7th flat-5 (or sharp-11) chord which resolves a half-step lower, with a pair of tritones (root--flat 5 and maj 3--m7) which each function doubly as themselves and as the other in the "tritone substitution" of flat-II for V. This has also been analyzed by me in connection with Schoenberg in exactly the same septimal terms which Erlich uses here -- more about that below. > > [Example] 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. Matrix Graph: maj 3rd 5^1 maj 3rd 5/4 6th _ - 3^4 3.86 9th _ - 3^3 - 81/64 \ _ - 3^2- 27/16 4.08 \ 5th - 9/8 9.06 root _ - 3^1 2.04 n^0 - 3/2 1/1 7.02 0.00 Analysis: ********* 12-eq ratio s-t. prop. IDs ===== ===== ==== ==== ====== 9th 9/8 2.04 144 9 6th 27/16 9.06 108 27 5th 3/2 7.02 96 3 3rd (P) 81/64 4.08 81 5 3rd (j) 5/4 3.86 80 81 root 1/1 0.00 64 1 ______ ___ n^0 n^0 (I had enough room here to show only one syntonic comma, that between the "just" and Pythagorean major 3rds. The 13th and possibly even the 9th could also be interpreted as 5-limit.) No argument from me here. In fact, jazz harmony as it developed thru be-bop and beyond could not have happened without accepting *a priori* the 12-eq scale. Charlie Parker specifically stated that he realized how to perform what he had been hearing in his head when he found that he could improvise melodies that were made up of the higher partials of the tune's chord changes. (That the just harmonies could still be implied to some degree in the complex and fast-changing 12-eq harmonic motion should probably be ascribed to the fact that this music was generally played very fast. That the need for precison of intonation is in inverse proportion to the tempo is a point that has been noted often before.) This double implication (poly-tonality vs. higher partials) was exactly the kind of ambiguity cited by Schoenberg: see "Theory of Harmony", p 418, regarding an 11-note chord from measures 382-383 of his "Erwartung". He shows how part of this complex chord can be explained as two different dominant-7th, flat-9th chords a tritone apart, with the four upper tones forming a diminished-7th quadrad which the two chords have in common in 12-eq. Analyzing them by assuming that the 12-eq 7ths represent the 7/4 of the chords gives exactly the result observed by Erlich regarding the 50/49 "6th-tone". The 12-eq scale's ability to represent all these different implications *simultaneously* is exactly what Schoenberg desired to exploit, and it presents the best case for his retention of 12-eq, as well as for its general acceptance throughout most of this century. Of course, the debates in this forum are precisely about which ratios can be represented, and how well. Joseph L. Monzo monz@juno.com _____________________________________________________________________ You don't need to buy Internet access to use free Internet e-mail. Get completely free e-mail from Juno at http://www.juno.com Or call Juno at (800) 654-JUNO [654-5866]