Re: 9-limit triangular lattices

Glad you like em.

5
,' ^
,' |\
,' | \
1 -------|- 3 7-limit o-tetrad
`. | /
`. |/
7

1/7
/| `.
/ | `.
1/3 ------- 1/1 7-limit u-tetrad
^ | ,'
\ | ,'
\| ,'
1/5

5
,' ^ `
,' |\ `
,' | \ ` .
1 =======|= 3 ========= 9 9-limit o-pentad
`. | / , '
`. |/, '
7

1/7
, '/| `.
, ' / | `.
1/9 ======= 1/3 ======= 1/1 9-limit u-pentad
` ^ | ,'
` \ | ,'
` .\| ,'
1/5

I missed a few connections on the decatonics so here they are again. Note
that F#/ = Gb\, C#/ = Db\, G#/ = Ab\, and Eb = D#\\, Bb = A#\\, F = E#\\.

Erlich's symmetrical decatonic scale

B/ ------- F#/ ------- C#/
| `. /| `. /| `.
| `. / | `. / | `.
| A/ -------- E/ -------- B/
| ,' ^ | ,' ^ | ,' ^
| ,' |\ | ,' |\ | ,' |
| ,' | \| ,' | \| ,' |
F -------|- C -------|- G |
| `. | /| `. | /| `. |
| `. |/ | `. |/ | `. |
| Eb -------- Bb -------- F
| ,' ^ | ,' ^ | ,' ^
| ,' |\ | ,' |\ | ,' |
| ,' | \| ,' | \| ,' |
B/ ------| F#/ ------| C#/ |
`. | / `. | / `. |
`. |/ `. |/ `. |
A/ -------- E/ -------- B/

Erlich's pentachordal decatonic scale

B/ ------- F#/ ------- C#/ ------- G#/
| | `. /| `. / `.
| | `. / | `. / `.
| | E/ -------- B/ ------- F#/
| | ,' ^ | ,' ^ |
| | ,' |\ | ,' | |
| | ,' | \| ,' | |
F --------- C -------|- G | |
| `. /| `. | /| `. | |
| `. / | `. |/ | `. | |
| Eb -------- Bb -------- F --------- C
| ,' ^ | ,' ^ | ,' ^ ,' |
| ,' \ | ,' |\ | ,' |\ ,' |
| ,' \| ,' | \| ,' | \ ,' |
B/ ------- F#/ ------| C#/ ------| G#/ |
`. | / `. | / `. |
`. |/ `. |/ `. |
E/ -------- B/ ------- F#/

Fokker-Lumma scale
D#--------A#
,'/:\`. ,'/
F--/-:-\--C /
/|\/ : \/| /
/ |/\ : /\|/
A---------E---------B---------F#
/|\ /|\`. /,'/ \`.\:/,'/
/ | \ / | \ Db-/---\--Ab /
/ D#--------A# \ | / \ | /
/,'/:\`.\ /,'/ `.\|/ \|/
F--/---\--C--/------G---------D
/|\/ : \/| /
/ |/\ : /\|/
/ B---------F#
/,' `.\:/,'
Db--------Ab

becomes
F ========= C
,' ^ `. ,'/| `.
,' |\ `.,' / | `.
,' | \ ,' D# -------- A#
Db ------|- Ab ,' ^ | ,'
, '/| `. ,|'/|,'. \ | ,'
, ' /,| ' `. |/,' `. \| ,'
A ========= E ========= B ========= F#
,' ^ ` ,' ^ `| ,' ^ | ,'
,' |\ `,' |\ | `,' \ | ,'
,' | \ ,' ` |.\| ,' ` .\| ,'
F =======|= C =======|= G ========= D
,' ^ `. |'/| `. , |'/ , '
,' |\ `.,'|/,| ' `. |/, '
,' | \ ,' D# -------- A#
Db ------|- Ab ,' ^ | ,'
`. | / ,'. \ | ,'
`. |/,' `. \| ,'
B ========= F#

I don't think this is as enlightening as it is for the scales with a
half-octave. Note that I've repeated a lot more notes here than in the
Erlichs and the Keenan. I did it to show the whole scale in a single
5-limit plane

D# -------- A#
,' ^ ,'
,' \ ,'
,' \ ,'
A ========= E ========= B ========= F#
,' ^ ` ,' ^ ` ,' ^ ,'
,' \ `,' \ `,' \ ,'
,' \ ,' ` .\ ,' ` .\ ,'
F ========= C ========= G ========= D
,' ^ ,'
,' \ ,'
,' \ ,'
Db -------- Ab

and to show the hexany.

F ========= C
^ `. ,'/|
|\ `.,' / |
| \ ,' D# |
| Ab ,' ^ |
| / ,'. \ |
|/,' `. \|
B ========= F#

Interesting that it brings the D#:Ab 8:11 closer together.

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

[Dave Keenan, TD 222.10]

Dave Keenan is the ASCII-lattice King!

Joseph L. Monzo monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
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Dave Keenan wrote,

> 5
> ,' ^
> ,' |\
> ,' | \
>1 -------|- 3 7-limit o-tetrad
> `. | /
> `. |/
> 7
>
> 1/7
> /| `.
> / | `.
>1/3 ------- 1/1 7-limit u-tetrad
> ^ | ,'
> \ | ,'
> \| ,'
> 1/5
>
> 5
> ,' ^ `
> ,' |\ `
> ,' | \ ` .
>1 =======|= 3 ========= 9 9-limit o-pentad
> `. | / , '
> `. |/, '
> 7
>
> 1/7
> , '/| `.
> , ' / | `.
>1/9 ======= 1/3 ======= 1/1 9-limit u-pentad
> ` ^ | ,'
> ` \ | ,'
> ` .\| ,'
> 1/5
>
>I missed a few connections on the decatonics so here they are again. Note
>that F#/ = Gb\, C#/ = Db\, G#/ = Ab\, and Eb = D#\\, Bb = A#\\, F = E#\\.
>
>Erlich's symmetrical decatonic scale
>
>B/ ------- F#/ ------- C#/
>| `. /| `. /| `.
>| `. / | `. / | `.
>| A/ -------- E/ -------- B/
>| ,' ^ | ,' ^ | ,' ^
>| ,' |\ | ,' |\ | ,' |
>| ,' | \| ,' | \| ,' |
>F -------|- C -------|- G |
>| `. | /| `. | /| `. |
>| `. |/ | `. |/ | `. |
>| Eb -------- Bb -------- F
>| ,' ^ | ,' ^ | ,' ^
>| ,' |\ | ,' |\ | ,' |
>| ,' | \| ,' | \| ,' |
>B/ ------| F#/ ------| C#/ |
> `. | / `. | / `. |
> `. |/ `. |/ `. |
> A/ -------- E/ -------- B/
>
>
>Erlich's pentachordal decatonic scale
>
>B/ ------- F#/ ------- C#/ ------- G#/
>| | `. /| `. / `.
>| | `. / | `. / `.
>| | E/ -------- B/ ------- F#/
>| | ,' ^ | ,' ^ |
>| | ,' |\ | ,' | |
>| | ,' | \| ,' | |
>F --------- C -------|- G | |
>| `. /| `. | /| `. | |
>| `. / | `. |/ | `. | |
>| Eb -------- Bb -------- F --------- C
>| ,' ^ | ,' ^ | ,' ^ ,' |
>| ,' \ | ,' |\ | ,' |\ ,' |
>| ,' \| ,' | \| ,' | \ ,' |
>B/ ------- F#/ ------| C#/ ------| G#/ |
> `. | / `. | / `. |
> `. |/ `. |/ `. |
> E/ -------- B/ ------- F#/
>
>Fokker-Lumma scale
> D#--------A#
> ,'/:\`. ,'/
> F--/-:-\--C /
> /|\/ : \/| /
> / |/\ : /\|/
> A---------E---------B---------F#
> /|\ /|\`. /,'/ \`.\:/,'/
> / | \ / | \ Db-/---\--Ab /
> / D#--------A# \ | / \ | /
> /,'/:\`.\ /,'/ `.\|/ \|/
> F--/---\--C--/------G---------D
> /|\/ : \/| /
> / |/\ : /\|/
> / B---------F#
> /,' `.\:/,'
> Db--------Ab
>
>becomes
> F ========= C
> ,' ^ `. ,'/| `.
> ,' |\ `.,' / | `.
> ,' | \ ,' D# -------- A#
> Db ------|- Ab ,' ^ | ,'
> , '/| `. ,|'/|,'. \ | ,'
> , ' /,| ' `. |/,' `. \| ,'
> A ========= E ========= B ========= F#
> ,' ^ ` ,' ^ `| ,' ^ | ,'
> ,' |\ `,' |\ | `,' \ | ,'
> ,' | \ ,' ` |.\| ,' ` .\| ,'
> F =======|= C =======|= G ========= D
> ,' ^ `. |'/| `. , |'/ , '
> ,' |\ `.,'|/,| ' `. |/, '
> ,' | \ ,' D# -------- A#
>Db ------|- Ab ,' ^ | ,'
> `. | / ,'. \ | ,'
> `. |/,' `. \| ,'
> B ========= F#
>
>I don't think this is as enlightening as it is for the scales with a
>half-octave. Note that I've repeated a lot more notes here than in the
>Erlichs and the Keenan. I did it to show the whole scale in a single
>5-limit plane
>
> D# -------- A#
> ,' ^ ,'
> ,' \ ,'
> ,' \ ,'
> A ========= E ========= B ========= F#
> ,' ^ ` ,' ^ ` ,' ^ ,'
> ,' \ `,' \ `,' \ ,'
> ,' \ ,' ` .\ ,' ` .\ ,'
> F ========= C ========= G ========= D
> ,' ^ ,'
> ,' \ ,'
> ,' \ ,'
>Db -------- Ab
>
>and to show the hexany.
>
>F ========= C
>^ `. ,'/|
>|\ `.,' / |
>| \ ,' D# |
>| Ab ,' ^ |
>| / ,'. \ |
>|/,' `. \|
>B ========= F#

These are wonderful, Dave! BTW, is there a logic to the way you're notating
my scales, or is it more or less arbitrary?

[Paul Erlich TD 223.18]
>These are wonderful, Dave!

I must give credit where due. I wouldn't have found this if not for the
9-limit tiling that Carl Luma produced some months ago.

>BTW, is there a logic to the way you're notating
>my scales, or is it more or less arbitrary?

Quite logical (I think). I hope I'm being consistent with existing
convention for the / and \.

Your scales (and my strange 9-limit one) are viewed as parallel chains of
approximate fifths an approximate 5:7 apart, so they are.

Erlich Sym dec
A/ E/ B/ F#/ C#/
Eb Bb F C G

Erlich Pent dec
E/ B/ F#/ C#/ G#/
Eb Bb F C G

Keenan strange
A/ E/ B/ F#/ C#/ G#/
Eb Bb F C G D

otonal 7-lim tetrad
7 - 1 3
5

utonal 7-lim tetrad
5
3 1 - 7

otonal 9-lim pentad
7 - 1 3 9
5

utonal 9-lim pentad
5
9 3 1 - 7

Of course this is just the key with an equal minimum number of sharps and
flats, and in the latter two cases following the meantone convention of
favouring the G# over the Ab.

Because the half-octave is an approximate 7:10 as much as it is a 5:7 we
have the following "enharmonics".
F#/ = Gb\
C#/ = Db\
G#/ = Ab\
Eb = D#\\
Bb = A#\\
F = E#\\
etc.

So

A/ E/ B/ Gb\ Db\ Ab\
Eb Bb F C G D

and

A/ E/ B/ F#/ C#/ G#/
D#\\A#\\E#\\C G D

are equally as valid as the one above. Some names make more sense in some
chords and others in others.

I'd appreciate it if you would redraft your recent treatise on the various
tetrads available in your scales, in this notation instead of using numbers
for 22-tET scale degrees. Then I'd have some hope of following it.

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

Woops! I had \\ where it should have been // (in my last two posts on this
thread). So that should have been:

A/ E/ B/ F#/ C#/ G#/
Eb Bb F C G D
...
Because the half-octave is an approximate 7:10 as much as it is a 5:7 we
have the following "enharmonics".
F#/ = Gb\
C#/ = Db\
G#/ = Ab\
Eb = D#//
Bb = A#//
F = E#//
etc.

So

A/ E/ B/ Gb\ Db\ Ab\
Eb Bb F C G D

and

A/ E/ B/ F#/ C#/ G#/
D#//A#//E#//C G D

are equally as valid as the one above. Some names make more sense in some
chords and others in others.

In any of them it is clear that D:A/ and G#/:Eb (Ab\:Eb, G#/:D#//) are both
wolves.

For those who just tuned in, this was a notation for scales with two chains
of fifths a half-octave apart where a 4:7 is 2 fourths, and a 4:5 is two
fourths and a half-octave. e.g. Paul Erlich's decatonics or my strange
9-limit temperament. I think Graham Breed calls these diaschismic.

So does it work for you? Is there a better one? Is it new?

-- Dave Keenan
http://dkeenan.com

Dave Keenan wrote,

>>These are wonderful, Dave!

>I must give credit where due. I wouldn't have >found this if not for the
>9-limit tiling that Carl Luma produced some >months ago.

Cool! I considered an "on edge" projection of the tetrahedral lattice at one point, but only considered the case where the tetrahedra would come out looking like squares in which case many lines would be indistinct. I like your assymmetrical quadrilaterals because all 6 lines are at different angles.