Re: Notation for 22-tET etc.

In TD 226.10 I wrote:

>So here's a symmetric decatonic as chains of fifths.
>
>A\ E\ B\ F#\ C#\
>Eb Bb F C G
>
>Here we extend it both ways to encompass all of 22-tET.
>
>C\ G\ D\ A\ E\ B\ F#\ C#\ G#\ D#\ A#\
>Gb Db Ab Eb Bb F C G D A E
>
>Now we rearrange them in pitch order.
>
>C Db C#\ D\ D Eb D#\ E\ E F Gb
>F#\ G\ G Ab G#\ A\ A Bb A#\ B\ B

Manuel has informed me that this is the same as the naming scheme used for
22-tET in Scala.

To see it in Scala do:
equaltemp 22
set notation E22
show

Dan Stearns has sent me a post he made back in January that has many
similarities to the "periodic table" part of my last post.

Paul E., thanks for reposting those chords. You can change the + to / and -
to \ now, as you thought you ought.

Regards,
-- Dave Keenan
http://dkeenan.com

[Dave Keenan:]
>Dan Stearns has sent me a post he made back in January that has many
similarities to the "periodic table" part of my last post.

Which FWIW was:

1f, 1F
2f, 1F + 1f, 2F
3f, 1F + 2f, 2F + 1f, 3F
4f, 1F + 3f, 2F + 2f, 3F + 1f, 4F
5f, 1F + 4f, 2F + 3f, 3F + 2f, 4F + 1f, fF
6f, 1F + 5f, 2F + 4f, 3F + 3f, 4F + 2f, 5F + 1f, 6F
Ff

THE RIGHT TRIANGLE OF "f" AND "F" (i.e., 5&7):
I use this 'right triangle of 5&7' to efficiently define the effects that
fifth size has (when mapped +2+2-5+2+2+2-5 onto a circle of clockwise
n/12*7 integer fifths, and counterclockwise d/12*5 integer fourths) on the
W and h step diatonic seconds of the +W+W+h+W+W+W+h heptad (@: 1st, 2nd,
3rd, 4th, 5th, 6th, 7th, 8th).

THE EXTERIOR SET:
1, 2, 3, 4, 6, 8, 9, 11, 13, 16, 18, and 23 (@ a F:f relationship of 1:0,
1:1, 2:1, 2:2, 4:2, 5:3, 5:4, 6:5, 8:5, 9:7, 11:7, and 13:10), would be the
twelve (and only) equidistant divisions of the octave comprised of an
entire intervallic inventory that lies outside this (1/35th of an octave)
'F/f aperture...':

If F<4/7ths of an octave; h>W, and if F>3/5ths of an octave,
h=-n.

THE PERIMETER SET:
5, 7, 10, 14, 15, 21, 20, 28, 25, 35, 30, and 35 (@ 1f, 1F, 2f, 2F, 3f, 4f,
3F, 5f, 4F, 6f, Ff and fF), would be the [twelve] equidistant divisions of
the octave that define the parameters of the 'F/f aperture...':

fh=0 and FW=Fh.

THE INTERIOR SET:
12, 17, 19, 22, 24, 26, 27, 29, 31, 33, 32, and 34 (@ 1F+1f, 1F+2f, 2F+1f,
1F+3f, 2F+2f, 3F+1f, 1F+4f, 2F+3f, 3F+2f, 4F+1f, 1F+5f, and 2F+4f) would be
the twelve equidistant divisions of the octave that lie inside the borders
of the F/f aperture where W>h and h>0 when mapped (+2+2-5+2+2+2-5) onto a
circle of clockwise n/12*F "F's," and counterclockwise n/12*f "f's."

EDO'S >f*F & F*f:
All equidistant divisions of the octave after f*F (35) have a fifth/fourth
inside of the F/f parameters... therefore all equidistant divisions of the
octave after fF&Ff (mapped onto a circle of clockwise n/12*F integer "F's"
and counterclockwise n/12*f integer "f's," +2+2-5+2+2+2-5), will also have
a diatonic heptad (1-2-3-4-5-6-7-8 @ +W+W+h+W+W+W+h) where W>h, and h>0.

Dan

Dave Keenan wrote,

>Paul E., thanks for reposting those chords. You can change the + to / and -
>to \ now, as you thought you ought.

OK, here goes:

Pentachordal decatonic scale:
F#\ G G#\ Bb B\ C C#\ Eb E\ F (F#\)

The tetrads formed by scalar template 1,4,7,9:

1. F#\ Bb C#\ E\ = F#\ A#\\ C#\ E\ (~4:5:6:7)
2. G B\ Eb F (dissonant)
3. G#\ C E\ F#\ (dissonant)
4. Bb C#\ F G = Bb Db/ F G (~1/6:1/5:1/4:1/7)
5. B\ Eb F#\ G#\ (dissonant)
6. C E\ G Bb (~4:5:6:7)
7. C#\ F G#\ B\ = C#\ E#\\ G#\ B\ (~4:5:6:7)
8. Eb F#\ Bb C = Eb Gb/ Bb C (~1/6:1/5:1/4:1/7)
9. E\ G B\ C#\ (~1/6:1/5:1/4:1/7)
T. F G#\ C Eb (dissonant)

Symmetrical decatonic scale:
B\ C C#\ Eb E\ F F#\ G A\ Bb (B\)

The tetrads formed by scalar template 1,4,7,9:

1. B\ Eb F#\ A\ = B\ D#\\ F#\ A\ (~4:5:6:7)
2. C E\ G Bb (~4:5:6:7)
3. C#\ F A\ B\ (dissonant)
4. Eb F#\ Bb C = Eb Gb/ Bb C (~1/6:1/5:1/4:1/7)
5. E\ G B\ C#\ (~1/6:1/5:1/4:1/7)
6. F A\ C Eb (~4:5:6:7)
7. F#\ Bb C#\ E\ = F#\ A#\\ C#\ E\ (~4:5:6:7)
8. G B\ Eb F (dissonant)
9. A\ C E\ F#\ (~1/6:1/5:1/4:1/7)
T. Bb C#\ F G = Bb Db/ F G (~1/6:1/5:1/4:1/7)

Next, the "major sevenths" and "minor sevenths" constructed
from 5\limit intervals:

Pentachordal decatonic scale:
F#\ G G#\ Bb B\ C C#\ Eb E\ F (F#\)

The tetrads formed by scalar template 1,4,7,10:

1. F#\ Bb C#\ F = F#\ A#\\ C#\ E#\\(~8:10:12:15)
2. G B\ Eb F#\ (dissonant)
3. G#\ C E\ G (dissonant)
4. Bb C#\ F G#\ = Bb Db/ F Ab/ (~10:12:15:9)
5. B\ Eb F#\ Bb = B\ D#\\ F#\ A#\\ (~8:10:12:15)
6. C E\ G B\ (~8:10:12:15)
7. C#\ F G#\ C = C#\ E#\\ G#\ B#\\ (~8:10:12:15)
8. Eb F#\ Bb C#\ = Eb Gb/ Bb Db/ (~10:12:15:9)
9. E\ G B\ Eb (dissonant)
T. F G#\ C E\ (dissonant)

Symmetrical decatonic scale:
B\ C C#\ Eb E\ F F#\ G A\ Bb (B\)

The tetrads formed by scalar template 1,4,7,10:

1. B\ Eb F#\ Bb = B\ D#\\ F#\ A#\\ (~8:10:12:15)
2. C E\ G B\ (~8:10:12:15)
3. C#\ F A\ C (dissonant)
4. Eb F#\ Bb C#\ = Eb Gb/ Bb Db/ (~10:12:15:9)
5. E\ G B\ Eb (dissonant)
6. F A\ C E\ (~8:10:12:15)
7. F#\ Bb C#\ F = F#\ A#\\ C#\ E#\\(~8:10:12:15)
8. G B\ Eb F#\ (dissonant)
9. A\ C E\ G (~10:12:15:9)
T. Bb C#\ F A\ (dissonant)

Pentachordal decatonic scale:
F#\ G G#\ Bb B\ C C#\ Eb E\ F (F#\)

The tetrads formed by scalar template 1,5,7,9:

1. F#\ B\ C#\ E\ (7sus4)
2. G C Eb F (~9:6:7:8)
3. G#\ C#\ E\ F#\ (~9:6:7:8)
4. Bb Eb F G (~1/6:1/9:1/8:1/7)
5. B\ E\ F#\ G#\ (~1/6:1/9:1/8:1/7)
6. C F G Bb (7sus4)
7. C#\ F#\ G#\ B\ (7sus4)
8. Eb G Bb C (~14:18:21:12)
9. E\ G#\ B\ C#\ (~14:18:21:12)
T. F Bb C Eb (7sus4)

Symmetrical decatonic scale:
B\ C C#\ Eb E\ F F#\ G A\ Bb (B\)

The tetrads formed by scalar template 1,5,7,9:

1. B\ E\ F#\ A\ (7sus4)
2. C F G Bb (7sus4)
3. C#\ F#\ A\ B\ (~9:6:7:8)
4. Eb G Bb C (~14:18:21:12)
5. E\ A\ B\ C#\ (~1/6:1/9:1/8:1/7)
6. F Bb C Eb (7sus4)
7. F#\ B\ C#\ E\ (7sus4)
8. G C Eb F (~9:6:7:8)
9. A\ C#\ E\ F#\ (~14:18:21:12)
T. Bb Eb F G (~1/6:1/9:1/8:1/7)

And finally,

Pentachordal decatonic scale:
F#\ G G#\ Bb B\ C C#\ Eb E\ F (F#\)

The tetrads formed by scalar template 1,4,6,9:

1. F#\ Bb C E\ = Gb/ Bb C E\ (French 6th)
2. G B\ C#\ F = G B\ C#\ E#\\ (French 6th)
3. G#\ C Eb F#\ (dissonant)
4. Bb C#\ E\ G (diminished 7th)
5. B\ Eb F G#\ (dissonant)

Since the scalar template is its own second inversion,
the rest of the chords are just the second inversions
of the above.

Symmetrical decatonic scale:
B\ C C#\ Eb E\ F F#\ G A\ Bb (B\)

The tetrads formed by scalar template 1,4,6,9:

1. B\ Eb F A\ = Cb/ Eb F A\ (French 6th)
2. C E\ F#\ Bb = C E\ F#\ A#\\ (French 6th)
3. C#\ F G B\ = C#\ E\\ G B\\ (French 6th)
4. Eb F#\ A\ C (diminished 7th)
5. E\ G Bb C#\ (diminished 7th)

same deal with 2nd inversions.