Dancing with snowflakes: Hexatonic scale in 11-EDO

-------------------------------------------------------------
Dancing like snowflakes
A hexatonic scale in 11-EDO: Some first impressions
-------------------------------------------------------------

As explained in my preceding post, I have been somewhat overwhelmed by
the beauty of 11-EDO and its unique kind of consonance when joined
with a customized timbre designed to maximize the consonance of the
655-cent and 545-cent intervals as a "quasi-3:2" and "quasi-4:3."

Regretting the inadequacies of language, and expressing my desire to
get some music on tape where the timbres can be recorded as well as
the notes, I will try to share some detail about the 11-EDO hexatonic
scale that I'm using, my keyboard layout, and a quick example of
note-against-note writing inspired by the recent counterpoint
discussions.

Seeking to put the most important thing at the beginning, I should
explain that my title "Dancing with snowflakes" was prompted by an
experience with a progression I'll describe as a kind of epilogue.

--------------------------------
1. The hexatonic scale in 11-EDO
--------------------------------

The hexatonic scale that I've come up with 11-EDO -- and I wonder if
others have done this before, because it seems such an attractive and
"natural" pattern -- has this structure, using the note names I apply
on my 22-EDO keyboard based on a regular chain of 22-EDO fifths:

22-EDO note name 11-EDO steps cents
F 0 0.00
G 2 218.18
A 4 436.36
B 6 654.55
C# 8 872.73
Eb 9 981.82
F 11 1200.00

Thus the pattern of 11-EDO steps is 2-2-2-2-1-2, or L-L-L-L-s-L.

A critical aspect of this scale, as I use it, is that the timbre is
"Chowningized" for maximal consonance for the 6-step interval of
~654.55 cents -- virtually identical to 54:37 -- so as to have the
effect of a "quasi-3:2."

This timbral factor may influence my melodic as well as harmonic
perceptions of this scale.

Thus the lower "tetrachord" F-B suggests to me a medieval European
Lydian (diatonic mode of F-F), and the upper tetrachord B-F a rather
Mixolydian effect (diatonic mode of G-G). The curious paradox is that
F-G-A-B seems at once like the lower "fifth" of the scale, and like a
tritone (which it literally is); the upper tetrachord B-C#-Eb-F seems
like the complementary "fourth."

At the same time, when I heard this, it struck me as quite different
from a European diatonic scale, and rather like a kind of scale I have
heard in some recordings of Cambodian music, or music from some of the
ethnic communities in the highlands of Vietnam.

Part of the reason may have been the "Celeste" metallophone timbre I
use as a favorite for 11-EDO, but the hexatonic structure itself also
seemed to fit more with these musics than with a European diatonic
scheme.

If I played a drone and stayed within the upper tetrachord, moving
between I am tempted to call the "octave" and "quasi-fifth," for
example a "quasi-fifth" drone of F3-B3 with figures of F5-Eb5-C#5-B4,
then I could hear a kind of medieval European modality, with the
"3:2-like" consonance coming to the forefront and seeming to shape
both vertical and melodic dimensions.

Fifths and fourths also play an important role as vertical elements in
the music of Southeast Asia I mentioned (a big attraction for me), and
it is fascinating to find this kind of exquisite color in 11-EDO
intervals varying by over 47 cents from the simplest ratios of 3:2 and
4:3.

Here's a Scala file for this MOS scale:

! hexat11a.scl
!
Hexatonic scale (2-2-2-2-1-2) in 11-tET
6
!
218.18182
436.36364
654.54545
872.72727
981.81818
2/1

When I tried the SHOW DATA feature, Scala reported that the scale was
"strictly proper," had "Myhill's property," and was "maximally even."
This suggests to me that the hexatonic scale may already be well known
to people involved with 11-EDO, since these are much-sought
attributes.

As you might put it, Paul, I find that the hexatonic scale has a
coherence analogous to that of a seven-note diatonic scale, and yet
quite different. Again, timbre may play a role: having the same
interval serve as both tritone and quasi-fifth is quite incredible. It
is concordant, in a very novel way, with melody and harmony taking new
shapes.

By the way, a bit of humor: I was considering, this weekend,
announcing that this scale was "maximally even" in the simple sense of
using only the "even-numbered" degrees of 22-EDO. What a pleasant
surprise it was to run Scala and find out that a more "strictly
proper" definition (pun intended) also applies.

-----------------------------------
2. Keyboard layout in 22-EDO/11-EDO
-----------------------------------

When I first became involved with the timbre question for 11-EDO last
summer, I found it rather difficult to play in 11-EDO degrees, and
only 11-EDO degrees, on my 22-note keyboard arrangement. The hexatonic
scale has largely solved this problem for me, and a bit of practice
should pretty much solve the rest.

Here I should explain that my 22-EDO layout has two 12-note manuals
each tuned to a regular chain of 22-EDO fifths (13/22 octave) in a
pattern of Eb-G#, with the manuals 2/22 octave apart. Thus, using an
F-F octave which fits the coming diagram for the 11-EDO notes, and
with the plus sign (+) showing a note raised by 1/11 octave (~109.09
cents):

273 491 600 981 1091
5 9 11 18 20
F#+/Ab G#+/Bb Bb+ C#+/Eb Eb+
F+ G+ A+ B+ C+ D+ E+ F+
2 6 10 14 15 19 23 24
109 327 545 764 818 1036 1255 1309
------------------------------------------------------------------
164 382 491 873 982
3 7 9 16 18
F# G# Bb C# Eb
F G A B C D E F
0 4 8 12 13 17 21 22
0 218 436 655 709 927 1145 1200

This arrangement makes available the regular intervals for a 12-note
tuning on each keyboard, and results in a bit of redundancy, since
there are 24 keys and only 22 notes per octave -- sometimes a
desirable feature, as has been recognized at least since the keyboard
designs of Scipione Stella and Fabio Colonna for their 31-note
meantone instruments in the early 17th century.

With the hexatonic scale, I found it easy to find the following 11-EDO
subset, this time with steps shown in 11-EDO rather than 22-EDO terms:

873 1091
9 10
C#+/Eb Eb+
F+ G+ A+ B+ F+
1 3 5 7 12
109 327 545 764 1309
---------------------------------------------------------------
873 982
8 9
C# Eb
F G A B F
0 2 4 6 11
0 218 436 655 1200

Visually as well as conceptually, this is easy to find. The lower
keyboard has the "basic" notes of the hexatonic scale, while the upper
keyboard has the "supplementary" or -- one might say in a free
metaphor -- "chromatic" steps filling out a complete 11-EDO set. The
note Eb, equivalent to C#+, gets a place on both keyboards.

----------------------------------------------------------
3. Hexatonic interval names and concord/discord categories
----------------------------------------------------------

Taking a leaf from Paul Erlich's book -- more specifically, his famed
article on the decatonic scale[1], I'll show a system for naming
intervals based on counting steps in this scale, along with the
categories of concord/discord I tend to use for counterpoint and
harmony.

Please let me emphasize that concord/discord are very much matters of
style, with the 655-cent and 545-cent intervals here treated as
quasi-3:2 and quasi-4:3 consonances serving as the primary stable
concords along with unisons and octaves. This very "equivalence," or
musical metaphor, may point to an _inharmonic_ timbre of the kind
designed to make it convincing.

-----------------------------------------------------------------------
11-EDO steps cents hexatonic name example category
-----------------------------------------------------------------------
0 0.00 unison F4-F4 perfect or
stable concord
.......................................................................
1 109.09 minor 2nd C#4-Eb4 discord
quite tense
.......................................................................
2 218.18 Major 2nd F4-G4 imperfect concord
relatively tense
.......................................................................
3 327.27 minor 3rd B3-Eb4 medial concord
relatively blending
.......................................................................
4 436.36 Major 3rd F4-A4 medial concord
relatively blending
.......................................................................
5 545.45 minor 4th B4-F5 perfect or
stable concord
.......................................................................
6 654.55 Major 4th F4-B4 perfect or
stable concord
.......................................................................
7 763.64 minor 5th G3-Eb4 medial concord
relatively blending
.......................................................................
8 872.73 Major 5th F3-C#4 medial concord
relatively blending
.......................................................................
9 981.82 minor 6th F3-Eb4 imperfect concord
relatively tense
.......................................................................
10 1090.91 Major 6th Eb4-C#5 discord
quite tense
.......................................................................
11 1200.00 octave C#4-C#5 perfect or
stable concord
-----------------------------------------------------------------------

Here's a quick impression of note-against-note writing in two parts,
with lots of 11th-13th century inspiration and a bit of influence from
later centuries also. The (') marks show phrasing, a bit like a
fermata or pause, and the style could be almost a bit like a plainsong
or chant in two parts, as in the free organum of the later 11th
century:
, ,
F5 Eb5 C#5 Eb5 C#5 F5 Eb5 B4 C#5 B4 Eb5 C#5 B4 A4 G4 F4
F4 G4 A4 B4 C#5 B4 A4 F4 Eb4 F4 F4 G4 F4 F4 Eb4 F4

--------------------------
4. Dancing with snowflakes
--------------------------

Now for the title of this article. Today I was playing and listening
to a progression of

A+3 B3
F+3 F3

This is an "altered" progression requiring inflected degrees outside
a single hexatonic set: it involves a 436-cent interval (for me a
"major third" in either 11-EDO or 22-EDO terms) expanding to a
655-cent "quasi-3:2."

While I regard this as the 11-EDO "equivalent" for the usual medieval
European progression of major third to fifth, this is quite different:
both voices move by a single 11-EDO step, or "demitone" as I call it,
whereas the medieval diatonic progression has one voice move by a
whole-tone and the other by a semitone.

That 436-655 cent progression sounded at once familiar and different,
and it had "the purity of a snowflake," as I wrote in my notebook.
Here is a four-voice variation from the same page of my notebook:

C#*5 F5
B*4 B4
A*4 B4
F*4 F4

This also had a "snowflake-like" quality that let me I know that I was
in the untrammeled realms of xenharmonic space.

----
Note
----

1. Paul Erlich, "Tuning, Tonality, and Twenty-Two-Tone Temperament,"
_Xenharmonikon_ 17:12-40 (Spring 1998), pp. 24-25.

Most appreciatively,

Margo Schulter
mschulter@value.net

Hi Margo,

So good to see you posting again! I've actually posted about the
(222212) 6-tone scale in 11-tet that you mention here a couple of
times before. In fact I wrote a piece I'm very happy with in it and
have bits and pieces of several others.

I found that scale by ear while improvising in 11-tet, and it's one of
those that you really can't go wrong with if your music is flexible
(or tailored) enough to allow incidental harmonies free reign. I used
no special timbres whatsoever with it and it worked like a dream.

To me this scale sort of sits on a fence perched between a pentatonic
and a diatonic scale (though it's always ready to keel over into the
pentatonic yard). Back when I was posting about xenharmonic 'white key
scales', this is exactly the sort of scale I had in mind. It's just
'complex' enough to be more harmonically engaging than something like
5-tet but still 'primitive' enough to work melodically any way you
have at it.

No matter what pile (vertical sonority) it falls in, this is a great
xenharmonic scale that can do no wrong!

take care,

--Dan Stearns

----- Original Message -----
From: "M. Schulter" <MSCHULTER@VALUE.NET>
To: <tuning@yahoogroups.com>
Sent: Monday, April 29, 2002 8:01 PM
Subject: [tuning] Dancing with snowflakes: Hexatonic scale in 11-EDO

> -------------------------------------------------------------
> Dancing like snowflakes
> A hexatonic scale in 11-EDO: Some first impressions
> -------------------------------------------------------------
>
> As explained in my preceding post, I have been somewhat overwhelmed
by
> the beauty of 11-EDO and its unique kind of consonance when joined
> with a customized timbre designed to maximize the consonance of the
> 655-cent and 545-cent intervals as a "quasi-3:2" and "quasi-4:3."
>
> Regretting the inadequacies of language, and expressing my desire to
> get some music on tape where the timbres can be recorded as well as
> the notes, I will try to share some detail about the 11-EDO
hexatonic
> scale that I'm using, my keyboard layout, and a quick example of
> note-against-note writing inspired by the recent counterpoint
> discussions.
>
> Seeking to put the most important thing at the beginning, I should
> explain that my title "Dancing with snowflakes" was prompted by an
> experience with a progression I'll describe as a kind of epilogue.
>
>
> --------------------------------
> 1. The hexatonic scale in 11-EDO
> --------------------------------
>
> The hexatonic scale that I've come up with 11-EDO -- and I wonder if
> others have done this before, because it seems such an attractive
and
> "natural" pattern -- has this structure, using the note names I
apply
> on my 22-EDO keyboard based on a regular chain of 22-EDO fifths:
>
> 22-EDO note name 11-EDO steps cents
> F 0 0.00
> G 2 218.18
> A 4 436.36
> B 6 654.55
> C# 8 872.73
> Eb 9 981.82
> F 11 1200.00
>
> Thus the pattern of 11-EDO steps is 2-2-2-2-1-2, or L-L-L-L-s-L.
>
> A critical aspect of this scale, as I use it, is that the timbre is
> "Chowningized" for maximal consonance for the 6-step interval of
> ~654.55 cents -- virtually identical to 54:37 -- so as to have the
> effect of a "quasi-3:2."
>
> This timbral factor may influence my melodic as well as harmonic
> perceptions of this scale.
>
> Thus the lower "tetrachord" F-B suggests to me a medieval European
> Lydian (diatonic mode of F-F), and the upper tetrachord B-F a rather
> Mixolydian effect (diatonic mode of G-G). The curious paradox is
that
> F-G-A-B seems at once like the lower "fifth" of the scale, and like
a
> tritone (which it literally is); the upper tetrachord B-C#-Eb-F
seems
> like the complementary "fourth."
>
> At the same time, when I heard this, it struck me as quite different
> from a European diatonic scale, and rather like a kind of scale I
have
> heard in some recordings of Cambodian music, or music from some of
the
> ethnic communities in the highlands of Vietnam.
>
> Part of the reason may have been the "Celeste" metallophone timbre I
> use as a favorite for 11-EDO, but the hexatonic structure itself
also
> seemed to fit more with these musics than with a European diatonic
> scheme.
>
> If I played a drone and stayed within the upper tetrachord, moving
> between I am tempted to call the "octave" and "quasi-fifth," for
> example a "quasi-fifth" drone of F3-B3 with figures of
F5-Eb5-C#5-B4,
> then I could hear a kind of medieval European modality, with the
> "3:2-like" consonance coming to the forefront and seeming to shape
> both vertical and melodic dimensions.
>
> Fifths and fourths also play an important role as vertical elements
in
> the music of Southeast Asia I mentioned (a big attraction for me),
and
> it is fascinating to find this kind of exquisite color in 11-EDO
> intervals varying by over 47 cents from the simplest ratios of 3:2
and
> 4:3.
>
> Here's a Scala file for this MOS scale:
>
> ! hexat11a.scl
> !
> Hexatonic scale (2-2-2-2-1-2) in 11-tET
> 6
> !
> 218.18182
> 436.36364
> 654.54545
> 872.72727
> 981.81818
> 2/1
>
> When I tried the SHOW DATA feature, Scala reported that the scale
was
> "strictly proper," had "Myhill's property," and was "maximally
even."
> This suggests to me that the hexatonic scale may already be well
known
> to people involved with 11-EDO, since these are much-sought
> attributes.
>
> As you might put it, Paul, I find that the hexatonic scale has a
> coherence analogous to that of a seven-note diatonic scale, and yet
> quite different. Again, timbre may play a role: having the same
> interval serve as both tritone and quasi-fifth is quite incredible.
It
> is concordant, in a very novel way, with melody and harmony taking
new
> shapes.
>
> By the way, a bit of humor: I was considering, this weekend,
> announcing that this scale was "maximally even" in the simple sense
of
> using only the "even-numbered" degrees of 22-EDO. What a pleasant
> surprise it was to run Scala and find out that a more "strictly
> proper" definition (pun intended) also applies.
>
>
> -----------------------------------
> 2. Keyboard layout in 22-EDO/11-EDO
> -----------------------------------
>
> When I first became involved with the timbre question for 11-EDO
last
> summer, I found it rather difficult to play in 11-EDO degrees, and
> only 11-EDO degrees, on my 22-note keyboard arrangement. The
hexatonic
> scale has largely solved this problem for me, and a bit of practice
> should pretty much solve the rest.
>
> Here I should explain that my 22-EDO layout has two 12-note manuals
> each tuned to a regular chain of 22-EDO fifths (13/22 octave) in a
> pattern of Eb-G#, with the manuals 2/22 octave apart. Thus, using an
> F-F octave which fits the coming diagram for the 11-EDO notes, and
> with the plus sign (+) showing a note raised by 1/11 octave (~109.09
> cents):
>
> 273 491 600 981 1091
> 5 9 11 18 20
> F#+/Ab G#+/Bb Bb+ C#+/Eb Eb+
> F+ G+ A+ B+ C+ D+ E+ F+
> 2 6 10 14 15 19 23 24
> 109 327 545 764 818 1036 1255 1309
> ------------------------------------------------------------------
> 164 382 491 873 982
> 3 7 9 16 18
> F# G# Bb C# Eb
> F G A B C D E F
> 0 4 8 12 13 17 21 22
> 0 218 436 655 709 927 1145 1200
>
> This arrangement makes available the regular intervals for a 12-note
> tuning on each keyboard, and results in a bit of redundancy, since
> there are 24 keys and only 22 notes per octave -- sometimes a
> desirable feature, as has been recognized at least since the
keyboard
> designs of Scipione Stella and Fabio Colonna for their 31-note
> meantone instruments in the early 17th century.
>
> With the hexatonic scale, I found it easy to find the following
11-EDO
> subset, this time with steps shown in 11-EDO rather than 22-EDO
terms:
>
>
> 873 1091
> 9 10
> C#+/Eb Eb+
> F+ G+ A+ B+ F+
> 1 3 5 7 12
> 109 327 545 764 1309
> ---------------------------------------------------------------
> 873 982
> 8 9
> C# Eb
> F G A B F
> 0 2 4 6 11
> 0 218 436 655 1200
>
> Visually as well as conceptually, this is easy to find. The lower
> keyboard has the "basic" notes of the hexatonic scale, while the
upper
> keyboard has the "supplementary" or -- one might say in a free
> metaphor -- "chromatic" steps filling out a complete 11-EDO set. The
> note Eb, equivalent to C#+, gets a place on both keyboards.
>
>
> ----------------------------------------------------------
> 3. Hexatonic interval names and concord/discord categories
> ----------------------------------------------------------
>
> Taking a leaf from Paul Erlich's book -- more specifically, his
famed
> article on the decatonic scale[1], I'll show a system for naming
> intervals based on counting steps in this scale, along with the
> categories of concord/discord I tend to use for counterpoint and
> harmony.
>
> Please let me emphasize that concord/discord are very much matters
of
> style, with the 655-cent and 545-cent intervals here treated as
> quasi-3:2 and quasi-4:3 consonances serving as the primary stable
> concords along with unisons and octaves. This very "equivalence," or
> musical metaphor, may point to an _inharmonic_ timbre of the kind
> designed to make it convincing.
>
>
> --------------------------------------------------------------------
---
> 11-EDO steps cents hexatonic name example category
> --------------------------------------------------------------------
---
> 0 0.00 unison F4-F4 perfect or
> stable
concord
>
......................................................................
.
> 1 109.09 minor 2nd C#4-Eb4 discord
> quite tense
>
......................................................................
.
> 2 218.18 Major 2nd F4-G4 imperfect
concord
> relatively
tense
>
......................................................................
.
> 3 327.27 minor 3rd B3-Eb4 medial
concord
> relatively
blending
>
......................................................................
.
> 4 436.36 Major 3rd F4-A4 medial
concord
> relatively
blending
>
......................................................................
.
> 5 545.45 minor 4th B4-F5 perfect or
> stable
concord
>
......................................................................
.
> 6 654.55 Major 4th F4-B4 perfect or
> stable
concord
>
......................................................................
.
> 7 763.64 minor 5th G3-Eb4 medial
concord
> relatively
blending
>
......................................................................
.
> 8 872.73 Major 5th F3-C#4 medial
concord
> relatively
blending
>
......................................................................
.
> 9 981.82 minor 6th F3-Eb4 imperfect
concord
> relatively
tense
>
......................................................................
.
> 10 1090.91 Major 6th Eb4-C#5 discord
> quite tense
>
......................................................................
.
> 11 1200.00 octave C#4-C#5 perfect or
> stable
concord
> --------------------------------------------------------------------
---
>
> Here's a quick impression of note-against-note writing in two parts,
> with lots of 11th-13th century inspiration and a bit of influence
from
> later centuries also. The (') marks show phrasing, a bit like a
> fermata or pause, and the style could be almost a bit like a
plainsong
> or chant in two parts, as in the free organum of the later 11th
> century:
> , ,
> F5 Eb5 C#5 Eb5 C#5 F5 Eb5 B4 C#5 B4 Eb5 C#5 B4 A4 G4
F4
> F4 G4 A4 B4 C#5 B4 A4 F4 Eb4 F4 F4 G4 F4 F4 Eb4
F4
>
>
> --------------------------
> 4. Dancing with snowflakes
> --------------------------
>
> Now for the title of this article. Today I was playing and listening
> to a progression of
>
> A+3 B3
> F+3 F3
>
> This is an "altered" progression requiring inflected degrees outside
> a single hexatonic set: it involves a 436-cent interval (for me a
> "major third" in either 11-EDO or 22-EDO terms) expanding to a
> 655-cent "quasi-3:2."
>
> While I regard this as the 11-EDO "equivalent" for the usual
medieval
> European progression of major third to fifth, this is quite
different:
> both voices move by a single 11-EDO step, or "demitone" as I call
it,
> whereas the medieval diatonic progression has one voice move by a
> whole-tone and the other by a semitone.
>
> That 436-655 cent progression sounded at once familiar and
different,
> and it had "the purity of a snowflake," as I wrote in my notebook.
> Here is a four-voice variation from the same page of my notebook:
>
> C#*5 F5
> B*4 B4
> A*4 B4
> F*4 F4
>
> This also had a "snowflake-like" quality that let me I know that I
was
> in the untrammeled realms of xenharmonic space.
>
>
> ----
> Note
> ----
>
> 1. Paul Erlich, "Tuning, Tonality, and Twenty-Two-Tone
Temperament,"
> _Xenharmonikon_ 17:12-40 (Spring 1998), pp. 24-25.
>
>
> Most appreciatively,
>
> Margo Schulter
> mschulter@value.net
>
>
>
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