something really comforting

Joseph Pherson wrote,

> As I mentioned, there is something really comforting about 12 notes
per octave and repeating octaves at 12... particularly among the
"keyboard-centric" set, in which I, for better or worse, number...

STRANGE CREATIONS:

I remember reading in H.H. Stuckenschmidt's "Twentieth Century Music"
a description of Schoenberg's attempts to infuse traditional forms
into the new non-tonal language, and how that had given rise to such
"strange creations" as the passacaglia "Nacht," and the fugues and
mirror canons of "Der Mondfleck"...

Last week as I was digging around trying to find an old photocopied
version of Brian McLaren's "A Brief History of Microtonality In the
20th Century," I happened across some of my old attempts to try and
"infuse," or rather project, some traditional forms and shapes into 20
equal tuning. I was hoping to do this so that I could attempt to start
afresh when I first got my 20 equal guitar back in the early '90s...
the problem with this for me ended up being that I was really focusing
too much attention on an area that really didn't have a whole lot to
do with my actual music... however, because of the odd relations to
some traditional forms and or designs, this process did give rise to
some pretty "strange creations." Anyway, I decided to try and spruce
it up a little bit and roll it out here at the TD.

AN 11-TONE SYSTEM IN TWENTY EQUAL TUNING:

What I was looking for was a non-5 way to organize and operate in 20
equal. First I tried to construct a consistent circle of "fifths" on
the 11/20. To do this I added four more note letters -- H, I, J, and
K -- to the traditional C through G arrangement, and organized them
like so:

C
F 540 660 K
Ib 1080 120 H
Ab 420 780 E
Db 960 240 B
Gb 300 900 J
Jb 840 360 G
Bb 180 1020 D
Eb 720 480 A
Hb 60 1140 I
Kb 600 600 F#
Cb 1140 60 C#
Fb 480 720 K#
Ibb 1020 180 H#
Abb 360 840 E#
Dbb 900 300 B#
Gbb 240 960 J#
Jbb 780 420 G#
Bbb 120 1080 D#
Ebb 660 540 A#
Hbb 1200 0 I#

This then corresponded to a basic 11-tone scale of:

C H B G A F K E J D I

Or in 20 equal scale steps and cents - 0 2 4 6 8 9 11 13 15 17 19 20,
at 0 120 240 360 480 540 660 780 900 1020 1140 1200.

You could also look at this 11-tone, pseudo diatonic scale as a
sequence of three pentads (which would of course be a I-VI-VII, I-IV-V
analogue):
_____ _____
/ \ / \
F E D C B A K J I H G
\_____/

This in turn could be mapped 11 9 to a traditional white and black key
keyboard arrangement as say:

Hb Bb Gb Ab Kb Eb Jb Db Ib
C# H# B# G# F# K# E# J# D#
| C | H | B | G | A | F | K | E | J | D | I | C |

This 20 equal, 11-tone scale could also be embedded into 11 basic
interval classes, i.e., a 1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th
11th 12th arrangement that would create a "modal" rotation of:

I. 1 2 3 4 5 6 7 8 9 10 11 12
II. 1 2 3 4 b5 6 7 8 9 10 b11 12
III. 1 2 3 b4 b5 6 7 8 9 b10 b11 12
IV. 1 2 b3 b4 b5 6 7 8 b9 b10 b11 12
V. 1 b2 b3 b4 b5 6 7 b8 b9 b10 b11 12
VI. 1 2 3 4 5 #6 7 8 9 10 11 12
VII. 1 2 3 4 5 6 7 8 9 10 b11 12
VIII. 1 2 3 4 b5 6 7 8 9 b10 b11 12
IX. 1 2 3 b4 b5 6 7 8 b9 b10 b11 12
X. 1 2 b3 b4 b5 6 7 b8 b9 b10 b11 12
XI. 1 b2 b3 b4 b5 6 b7 b8 b9 b10 b11 12
XII. 1 2 3 4 5 6 7 8 9 10 11 12

And mapped to a modified staff as:

B A K J I H G F E:

"Believe and know Jack, it's high-government future elections!"

G F E D C B A K:

"G-fed, c-bak."

(well, no... perhaps not... but you get the idea):

-----------------------------------------E-----------------
K
---------------------------------------F-------------------
A
-------------------------------------G---------------------
B
-----------------------------------H-----------------------
C
---------------------------------I-------------------------
D
-------------------------------J---------------------------
E
-----------------------------K-----------------------------
F
---------------------------A-------------------------------
G
-------------------------B---------------------------------
H
--C--
I
-------------------D---------------------------------------
J
-----------------E-----------------------------------------
K
---------------F-------------------------------------------
A
-------------G---------------------------------------------
B
-----------H-----------------------------------------------
C
---------I-------------------------------------------------
D
-------J---------------------------------------------------
E
-----K-----------------------------------------------------
F
---A-------------------------------------------------------
G
-B-

This would be the series of 11-tone, 20 equal keys and their
corresponding accidentals:

C H B G A F K E J D I C
K E J D I C H B G A F# K
H B G A F# K E J D I C# H
E J D I C# H B G A F# K# E
B G A F# K# E J D I C# H# B
J D I C# H# B G A F# K# E# J
G A F# K# E# J D I C# H# B# G
D I C# H# B# G A F# K# E# J# D
A F# K# E# J# D I C# H# B# G# A
I C# H# B# G# A F# K# E# J# D# I
F# K# E# J# D# I C# H# B# G# A# F#
C# H# B# G# A# F# K# E# J# D# I# C#
Cb Hb Bb Gb Ab Fb Kb Eb Jb Db Ib Cb
Kb Eb Jb Db Ib Cb Hb Bb Gb Ab F Kb
Hb Bb Gb Ab F Kb Eb Jb Db Ib C Hb
Eb Jb Db Ib C Hb Bb Gb Ab F K Eb
Bb Gb Ab F K Eb Jb Db Ib C H Bb
Jb Db Ib C H Bb Gb Ab F K E Jb
Gb Ab F K E Jb Db Ib C H B Gb
Db Ib C H B Gb Ab F K E J Db
Ab F K E J Db Ib C H B G Ab
Ib C H B G Ab F K E J D Ib
F K E J D Ib C H B G A F

Minor keys are an 8th higher than their corresponding major key, so C
Major is also E Minor, Cb (or I) Major is also Eb (or K#) minor, and
so on. So as I'm calling the basic chord in this system a five note
1-3-5-7-9 (or 0 4 8 11 15 and 0 240 480 660 900 in 20ths of an octave
and cents), i.e., a major pentad, I'm also calling a 1-3-b5-7-9 (i.e.,
0 4 7 11 15 at 0 240 420 660 900) the minor pentad. So with that in
mind, here are some of the basic 11-tone 20 equal chords that would be
analogous to the traditional major, minor, dominant 7th, diminished
7th, and augmented (and it should be fairly easy for anyone to
extrapolate other common chords from these, i.e., the min7th analogy
in the 11-tone 20 equal system is a min11th at 1-3-b5-7-9-b11, etc.):

maj 1 3 5 7 9
min 1 3 b5 7 9
dom 11th 1 3 5 7 9 b11
dim 11th 1 b3 b5 b7 b9 bb11
aug 1 3 5 #7 #9

Though this scale fits very nicely into the numerically and
intervalically consistent conventions of the triadic, triangular
lattice as:

B-----J-----G-----D-----A-----I
\ / \ / \ / \ / \ / \
\ / \ / \ / \ / \ / \
F-----C-----K-----H-----E-----B

The basic chord here is a 1-3-5-7-9, C B A K J pentad, and not a
1-3-5, 4:5:6, C E G (or 1-4-7, 13:16:19, C G K) type triad. So it
would be preferable to attempt to plot the pentads into a two
dimensional compromise, say something along the lines:

C
/|\
/ | \
/ | \
/ | \
K,---+----B
\ , | / \
\ |, / \
\ | /' \
\|/ ' \
J--------'A

By rotating this to a more conventional orientation, a two-dimensional
compromise of sorts is struck where the major and minor pentads a bit
easier to catch at a glance.

major:

,A
, |
, / |
, |
,B. |
, | `. / |
, | `. |
, | `. |
C.-------+--/----,J
`. | ,
`. | ,
`. |/,
`K

minor:

,B.
, | `.
, / | `.
, | `.
C.-------+-------,J
| `. / | ,
| `. | ,
| `. | ,
| / ,K
| ,
| ,
|/,
Ab

The full 11-tone scale using both of these lattice orientations would
be:

,D.
, | `.
, / | `.
, | `.
,E.-------+-------,A.
, | `. / | , | `.
, | `. | , / | `.
, | `. | , | `.
F.-------+--/----,B.-------+-------,I
`. | , | `. / | , |
`. | , | `. | , / |
`. |/, | `. | , |
`C.-------+--/----,J. |
`. | , | `. / |
`. | , | `. |
`. |/, | `. |
`K.-------+--/----,G
`. | ,
`. | ,
`. |/,
`H
Or, using the first:

F
/|\
/ | \
/ | \
/ | \
C,---+----E
/|\ , | /|\
/ | \ |, / | \
/ | \ | /' | \
/ | \|/ |' \
K,---+----B----+---'D
/|\ , | /|\ | /
/ | \ |, / | \ | /
/ | \ | /' | \ | /
/ | \|/ |' \|/
H,---+----J----+---'A
\ , | / \ | /
\ |, / \ | /
\ | /' \ | /
\|/ ' \|/
G--------'I

WELL, WHAT OF IT THEN?:

In the annotated version of Charles Ives' essay "Some Quarter-tone
Impressions" that I posted earlier in the week, Ives says, "Chords of
four or more notes, as I hear it, seem to be a more natural basis than
triads." And Rick Tagawa commented that "Ives' feeling that the basic
chords should have four or more notes finds support in the theories of
Joseph Yasser, an advocate of a 19-tone tuning (in A Theory of
Evolving Tonality, New York, 1932). The basic chords of Yaser's system
are hexads." So with that in mind, I'd be interested to know what
sorts of subjective listening reactions people might have to the basic
equilibriums of these 11-tone, 20 equal "major" and "minor" type
analogous... Do these distorted -- or perhaps saturated -- proportions
line up favorably with the expectations that would be caused by their
traditional diatonic structural similarities (abstract though they
certainly are)?

Let me know...

Dan Stearns
<http://stations.mp3s.com/stations/55/117_west_great_western.html>