Yahoo Tuning Groups Ultimate Backup tuning Miracle-31

Dave,

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> --- In tuning@y..., <manuel.op.de.coul@e...> wrote:
> >
> > Miracle-31 also contains interesting amounts of Fokker
> > circular chord mirrorings (kringspiegelingen), see my post of
> > 27-jan-1995 ...
>
> That's not in the Yahoo archive. Can anyone tell me where I can
> find it? Can you email it to me Manuel?

Never say I didn't do a nice thing for ya! <g> Besides, I'm a sucker
for such great terms as "kringspiegelingen". (msg found on a chunk of
files still residing up at Mills, and assuming since Manuel
referenced it he doesn't mind having it here...)

-------------------------------------
Date: Fri, 27 Jan 1995 22:30 +0100
From: ****@***.** (Manuel Op de Coul)
To: tuning
Subject: Circular chord mirrorings

I have been creating a number of Fokker's circular mirrorings
(kringspiegelingen) lately which I will explain. They are created by
taking a chord and repeatingly taking an inverse until the original
chord is reached again. There are two kinds, single-tie and double-
tie. With single-tie, an inverse is taken with one note as pivot.
Double-tie has two notes as pivot or you could say the interval
between the two is the pivot. For example:

Take 4:5:6, C-E-G single-tie mirrored in E gives 25/6:25/5:25/4, C#-E-
G# and double-tie mirrored in C and G gives 24/6:24/5:24/4, C-Eb-G
With the appropriate choice of pivots you end up with the original
chord. An octave is added to each scale shown below.

Double-tie circular mirroring of 4:5:6
1: 6/5 315.6414 minor third
2: 5/4 386.3139 major third
3: 4/3 498.0452 perfect fourth
4: 3/2 701.9553 perfect fifth
5: 8/5 813.6866 minor sixth
6: 5/3 884.3591 major sixth
7: 2/1 1200.000 octave

As a lattice it looks like this:

* - *
| \ | \
* - 0 - *
\ | \ |
* - *

The 6 triangles represent the chords. Start with the upper right one
and
circle clockwise.

Single-tie circular mirroring of 3:4:5
|
1: 9/8 203.9100 major whole tone
2: 6/5 315.6414 minor third
3: 5/4 386.3139 major third
4: 27/20 519.5515 wide fourth
5: 45/32 590.2239 tritone
6: 36/25 631.2828 classic diminished fifth
7: 25/16 772.6278 classic augmented fifth
8: 8/5 813.6866 minor sixth
9: 5/3 884.3591 major sixth
10: 9/5 1017.596 just minor seventh
11: 15/8 1088.269 classic major seventh
12: 2/1 1200.000 octave

The diagram:

*
| \
* - * - * - *
\ | \ |
* [0] *
| \ | \
* - * - * - *
\ |
*

The centre-tone of a single-tie mirroring is not included in the
scale. In the scale above, degree 0 is the node left to the middle.

A double-tie circular mirroring can also be made with a four-tone
chord. With 4:5:6:7 this becomes:

1: 21/20 84.46723 minor semitone
2: 7/6 266.8710 septimal minor third
3: 6/5 315.6414 minor third
4: 49/40 351.3382 larger approximation to neutral
third
5: 5/4 386.3139 major third
6: 7/5 582.5125 septimal diminished fifth
7: 3/2 701.9553 perfect fifth
8: 42/25 898.1539
9: 12/7 933.1295 septimal major sixth
10: 7/4 968.8264 harmonic seventh
11: 9/5 1017.596 just minor seventh
12: 2/1 1200.000 octave

Alternatively this can be done with one pivot tone being common in all
mirrorings, giving:

1: 8/7 231.1741 septimal whole tone
2: 7/6 266.8710 septimal minor third
3: 6/5 315.6414 minor third
4: 5/4 386.3139 major third
5: 4/3 498.0452 perfect fourth
6: 7/5 582.5125 septimal diminished fifth
7: 10/7 617.4880 septimal augmented fourth
8: 3/2 701.9553 perfect fifth
9: 8/5 813.6866 minor sixth
10: 5/3 884.3591 major sixth
11: 12/7 933.1295 septimal major sixth
12: 7/4 968.8264 harmonic seventh
13: 2/1 1200.000 octave

In this scale there are 8 chords with one common tone that takes all
8 possible functions in the supraharmonic or subharmonic seventh chord
(4, 5, 6, 7, 1/4, 1/5, 1/6, 1/7) just once where a function is never
immediately preceded of followed by the inverse function. This
question is the mathematical problem known as "Le proble`me des
me'nages", or how to seat couples around a table, men and women
alternating, while no wife sits next to her husband.

Some more scales:

Double-tie circular mirroring of 6:7:8
1: 8/7 231.1741 septimal whole tone
2: 7/6 266.8710 septimal minor third
3: 4/3 498.0452 perfect fourth
4: 3/2 701.9553 perfect fifth
5: 12/7 933.1295 septimal major sixth
6: 7/4 968.8264 harmonic seventh
7: 2/1 1200.000 octave

Double-tie circular mirroring of 3:5:7
1: 7/6 266.8710 septimal minor third
2: 6/5 315.6414 minor third
3: 7/5 582.5125 septimal diminished fifth
4: 10/7 617.4880 septimal augmented fourth
5: 5/3 884.3591 major sixth
6: 12/7 933.1295 septimal major sixth
7: 2/1 1200.000 octave

Double-tie circular mirroring of 4:5:7
1: 8/7 231.1741 septimal whole tone
2: 5/4 386.3139 major third
3: 7/5 582.5125 septimal diminished fifth
4: 10/7 617.4880 septimal augmented fourth
5: 8/5 813.6866 minor sixth
6: 7/4 968.8264 harmonic seventh
7: 2/1 1200.000 octave

Single-tie circular mirroring of 6:7:8
1: 9/8 203.9100 major whole tone
2: 8/7 231.1741 septimal whole tone
3: 7/6 266.8710 septimal minor third
4: 9/7 435.0843 septimal major third
5: 21/16 470.7811 narrow fourth
6: 72/49 666.2584
7: 49/32 737.6521
8: 12/7 933.1295 septimal major sixth
9: 7/4 968.8264 harmonic seventh
10: 27/14 1137.039 septimal major seventh
11: 63/32 1172.736 octave - septimal comma
12: 2/1 1200.000 octave

Single-tie circular mirroring of 4:5:7
1: 50/49 34.97563
2: 8/7 231.1741 septimal whole tone
3: 400/343 266.1498
4: 5/4 386.3139 major third
5: 125/98 421.2895
6: 64/49 462.3483
7: 25/16 772.6278 classic augmented fifth
8: 8/5 813.6866 minor sixth
9: 80/49 848.6623
10: 7/4 968.8264 harmonic seventh
11: 25/14 1003.802 middle minor seventh
12: 2/1 1200.000 octave

Double-tie circular mirroring of 3:5:7:9
1: 21/20 84.46723 minor semitone
2: 7/6 266.8710 septimal minor third
3: 63/50 400.1086 equal major third
4: 9/7 435.0843 septimal major third
5: 27/20 519.5515 wide fourth
6: 7/5 582.5125 septimal diminished fifth
7: 3/2 701.9553 perfect fifth
8: 14/9 764.9162 septimal minor sixth
9: 49/30 849.3835 17/4-tone
10: 5/3 884.3591 major sixth
11: 9/5 1017.596 just minor seventh
12: 2/1 1200.000 octave

Fokker wrote a number of compositions in these like Harmonische
transformaties, Kabbelende die"zegolven, Kalenderblaadjes, Preludia
geometrica and others. Whether it is possible to circularly mirror
more-than-four-tone chords and using all tones as pivot, I wouldn't
know.
-------------------

Cheers,
Jon

Miracle-31

🔗manuel.op.de.coul@eon-benelux.com

🔗Dave Keenan <D.KEENAN@UQ.NET.AU>

🔗Jon Szanto <JSZANTO@ADNC.COM>