Yahoo Tuning Groups Ultimate Backup tuning a 9-tone "Liszt scale"?

Kyle Gann wrote,

> So much of Liszt's late music is based on augmented triads that one
can analyze passages of a late work like *Nuages gris* in Tcherepnin
scale: move through any three augmented triads, and you've got a
Tcherepnin scale. His *Sketches for a Harmony of the Future*,
however - the manuscript that was seen by friends but never surfaced
after Liszt's death - seems to have had less to do with scales and
more to do with building chords from intervals other than thirds,
especially fourths and tritones. And I can't find a passage of much
length in the scale.

Thanks for posting the "Tcherepnin scale" and Liszt info Kyle, I
really appreciate it! You might be interested to know that the reason
I had started looking in this general direction in the first place was
not the augmented triad (the 6s3L, or Tcherepnin scale having, despite
some enharmonically mangled spellings, a major third on every scale
degree) but rather the "bimodal triad"...

A while back I had stumbled onto an internet article by the Cuban
composer Enrique Ubieta (have ever heard of him?):

<http://www.ubieta.com/bio.htm>

His bimodal triad (he refers to his 1-b3-3-5 bimodal chord as a
"triad") for one reason or another caught my imagination, and I began
to experiment a bit with the idea of a "bimodal scale" to go with the
bimodal chord.

Initially I experimented with some 10-tone scales based on a simple
extrapolation of the diatonic scale as a I-IV-V composite; in other
words, if the diatonic major scale can be seen as an F A C E G B D,
I-IV-V composite, what happens if you run the bimodal chord through
the same premise?

In Ubieta's paper "Bimodalism -- A New Dimension and Ethos in
Harmony," he seems to describe two basic interpretations of his
bimodal chord:

<http://www.ubieta.com/bimodalism/BimodalHarmony.htm>

One interpretation is a harmonic series identity... "the essence of
the harmonic discipline of Bimodalism lies in the simultaneous
blending of major and minor modes in triads with the same fundamental
root."

The other as overtonal/undertonal symmetry... "consequently, both
remote tonalities and modulation concepts are senseless in Bimodalism,
as it is intrinsically a chain of 12 equivalent and harmonically
linked chords, forming, in fact, a true unitonality."

So I tried running the simplest interpretations of each of these two
possibilities through the I-IV-V premise.

The first would have the bimodal chord as a 12:14:15:18, and would
give a 10-tone C D D# E F G G# A A# B, I-IV-V bimodal scale:

5/3-------5/4------15/8
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
14/9.---/-,7/6.-\-/-,7/4. \ / \
`4/3'-----`1/1'-----`3/2-------9/8

The second would take the bimodal chord as an overtonal undertonal
composite of a 4:5:6 and a 1/(6:5:4). This would give a 10-tone C D Eb
E F G Ab A Bb B, I-IV-V bimodal scale:

5/3-----5/4----15/8
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
4/3-----1/1-----3/2-----9/8
\ / \ / \ /
\ / \ / \ /
\ / \ / \ /
8/5-----6/5-----9/5

Both of these scales are consistent with a two step size cardinality
interpretation. Twelve-tone equal temperament, which for the usual
"practical" reasons is the tuning Ubieta is concerned with, could be
seen as a consistent, if somewhat less than desirable (as in the case
of the 12:14:15:18), tempering of either of these interpretations with
both being represented by the same 8s2L 2111211111 step structure.

Of these two step size, 8s2L (LsssLsssss) temperaments:

2 10
12
14 22
16 26 34 32
18 30 40 38 46 56 54 42

(etc.)

I remember most liking the ones that sharpened up the distinctions
between the interval classes (these would be the "strictly proper
mediants," i.e., those that fall between either "2s+L" and "s+L", or
"s+L" and "s+2L" on the Stern-Brocot Tree; 22-tET being the simplest,
or first "2s+L" and "s+L" mediant mapping here) and didn't rub up too
close to "s+L" thereby causing the intervals within classes to nearly
converge.

All-in-all, the 10-tone scales (while plenty interesting) were not
quite what I was looking for; as their most interesting
characteristics had -- despite the best intentions of the scales
origin -- little to do with the bimodal chord.

While Ubieta doesn't seem to be concerned the least with a bimodal
scale that takes care of its own business (i.e., creates its own
bimodal chords as an intrinsic part, or byproduct, of the scales
design), I thought it would be worth a look... and that's where my
interest in these 9-tone scales came into the picture.

Nine-tone equal temperament would be the simplest model to frame this
in; as you'd have an 0 2 3 5, 1-b3-3-5 or "1-3-4-6" forming on every
rotation or scale degree. What I had hoped to do however, was find a
way to paraphrase his basic harmonic concept by recasting it in some
sort of abstracted "diatonic" model.

I posted some possible 14 and 23-tET interpretations of one such
quasi-diatonic model a while back... These particular quasi-diatonic
interpretations came about by taking the first two terms of the
bimodal chord -- i.e., the 1-b3-3 -- as an identity (or "terms") to
generate 9-tone scales where each scale degree is connected to a
centralized tonic (in the example, "a" and "b" simply represent a
minor and major third respectively; which again are generally culled
from an identity in the form of t:a:b, "i" indicates an inversion; it
would read as a given condition "inverted", and "t" is any given
pitch; or "tonic"):

b-a
|
|
|
|
|
ai | b
\ | /
\ | /
\ | /
\ | /
\|/
(a+b)i----------t---------a+b
/|\
/ | \
/ | \
/ | \
/ | \
bi | a
|
|
|
|
|
(b-a)i

The 14 and 23-tET scales I posted represented a tempering of t:a:b =
18:21:23, where if you lattice out the t:a:b = 18:21:23 scale in 14
and 23-tET you have a situation that is analogous to the diatonic
scale where the comma -- a 7889/7776 in the following 14-tET
example -- is absorbed by the temperament and an additional consonant
chord (an additional bimodal chord here) results.

Here's the t:a:b = 18:21:23 scale:

23/21
/|\
/ | \
/ | \
/ | \
/ | \
12/7----+---23/18
/|\ | /|\
/ | \ | / | \
/ | \ | / | \
/ | \ | / | \
/ | \|/ | \
216/161--+----1/1----+--161/108
\ | /|\ | /
\ | / | \ | /
\ | / | \ | /
\ | / | \ | /
\|/ | \|/
36/23---+----7/6
\ | /
\ | /
\ | /
\ | /
\|/
42/23

Here's the 14-tET example with the additional "consonant" chord:

8-----------2
/|\ /|\
/ | \ / | \
/ | \ / | \
/ | \ / | \
/ | \ / | \
3-----+----11-----+-----5
\ | /|\ | /|\
\ | / | \ | / | \
\ | / | \ | / | \
\ | / | \ | / | \
\|/ | \|/ | \
6-----+-----0-----+-----8
\ | /|\ | /|\
\ | / | \ | / | \
\ | / | \ | / | \
\ | / | \ | / | \
\|/ | \|/ | \
9-----+-----3-----+----11
\ | / \ | /
\ | / \ | /
\ | / \ | /
\ | / \ | /
\|/ \|/
12-----------6

As 14-tET is the simplest interpretation of a sL mapping where s < L,
and both s and L are > 0 (i.e., a "s+2L" or a "2s+L"), it is also a
proper scale: a proper scale has instances of shared intervals amongst
interval classes, and in this case the shared (or "ambiguous")
intervals are the augmented and diminished 3rds and 4ths, the 5ths and
6ths, and the 7ths and 8ths:

0 171 257 429 514 686 771 943 1029 1200
0 86 257 343 514 600 771 857 1029 1200
0 171 257 429 514 686 771 943 1114 1200
0 86 257 343 514 600 771 943 1029 1200
0 171 257 429 514 686 857 943 1114 1200
0 86 257 343 514 686 771 943 1029 1200
0 171 257 429 600 686 857 943 1114 1200
0 86 257 429 514 686 771 943 1029 1200
0 171 343 429 600 686 857 943 1114 1200
0 171 257 429 514 686 771 943 1029 1200

23-tET, like 14, is "consistent" (if you allow that the
"18:21:23:26&5/6ths" bimodal chord can be rightfully seen in such a
light), thereby allowing all instances of the identity's cross-set or
interval matrix to always be the best rounded
[(LOG(N)-LOG(D))*(T/LOG(2))] representations (were "N" and "D" are the
numerator and denominator of the relevant consonant ratios, and "T" is
the temperament). Unlike 14-tET however, 23-tET (as an sL mapping that
falls between "s+L" and "s+2L" on the Stern-Brocot Tree) is strictly
proper. This allows for uniquely articulated representations of all
interval classes (23-tET recognizes the diaschisma-like 285768/279841
while still hiding the syntonic comma-like 7889/7776):

I. 0 157 261 417 522 678 783 939 1043 1200
II. 0 104 261 365 522 626 783 887 1043 1200
III. 0 157 261 417 522 678 783 939 1096 1200
IV. 0 104 261 365 522 626 783 939 1043 1200
V. 0 157 261 417 522 678 835 939 1096 1200
VI. 0 104 261 365 522 678 783 939 1043 1200
VII. 0 157 261 417 574 678 835 939 1096 1200
VIII. 0 104 261 417 522 678 783 939 1043 1200
IX. 0 157 313 417 574 678 835 939 1096 1200
X. 0 157 261 417 522 678 783 939 1043 1200

Lately I've been focusing on 9-tone interpretations that represent
just a slight compromise (rather than a quasi-diatonic reworking) of
Ubieta's basic 'bimodal chord for every scale degree' idea.

Anyway, that's where my interest in these 9-tone scales came from, and
how the 6s3L (Liszt/Tcherepnin) scale quandaries came about.

thanks again for the info,

a 9-tone "Liszt scale"?

🔗D.Stearns <STEARNS@CAPECOD.NET>

🔗Monz <MONZ@JUNO.COM>

🔗Mats �ljare <oljare@hotmail.com>

🔗D.Stearns <STEARNS@CAPECOD.NET>