Yahoo Tuning Groups Ultimate Backup mills-tuning-list Response to Paul Erlich

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

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Response to Paul Erlich

🔗monz@juno.com (Joseph L Monzo)

🔗mr88cet@texas.net (Gary Morrison)