Tempered lattices have some kind of approximation built in. Although the usual lattices aren't always (or even typically) used to represent JI, tempered lattices always need to be used with some kind of temperament.
Because they're usually two-dimensional, tempered lattices can also be used as generalized keyboard mappings. As such, they will favour harmony over melody, although some of these mappings are also intended to work not as badly for melody as they otherwise would.
Paul Erlich probably did tetrahedral lattices first. They're a fairly obvious idea. But there are other ways of showing the same harmony, for example this template from Dave Keenan:
7
|
|
5 |
| |
| |
1----3----9
The numbers indicate frequency ratios. It becomes a 2-dimensional lattice by assuming the septimal kleisma 225:224. That means it's only suitable for temperaments that remove the kleisma. With meantone names, it's a regular square lattice as used by the likes of Fokker.
G#---D#---A#---E#---B# | | | | | | | | | | E----B----F#---C#---G# | | | | | | | | | | C----G----D----A----E | | | | | | | | | | Ab---Eb---Bb---F----C | | | | | | | | | | Fb---Cb---Gb---Db---Ab
A twist on that idea is this template
7 \ ---5 / \ / 3---1
It also has the septimal kleisma built in, but is based on a 3:4:5 chord instead of 4:5:6. As these two-dimensional lattices can be used as generalised keyboard mappings, you may not always want to privilege the usual root position. Also, if you tune the basic chord as 6:7:8:10, the pitch increases from left to right. That makes the keyboard not as bad melodically as it would otherwise be, and also makes the lattice easier to read.
Here are more notes:
Ct--Ft--A#--D#--G# / \ / \ / \ / \ / \ G#--C#--F#--B---E---A \ / \ / \ / \ / \ / A---D---G---C---F / \ / \ / \ / \ / F---Bb--Eb--Ab--Db \ / \ / \ / \ / Gb--B+--E+--A+
It happens that the diagonals on that lattice are the generators for the miracle and magic temperaments. The horizontal is the meantone/schismic generator, as usual. Like other triangular lattices, it can be mapped to a honeycomb keyboard.
Another template is
5
/
7 /
3-----1
add a few notes to it
C#----F# / \ Eb/ \ / Ct\ / Ft\ A-----D-----G \ B+/ \ E+/ \ / C#\ / Bb----Eb
The lattice could be simplified by setting Ct=B+ and Ft=E+ in the middle. The template becomes
5
/
/
3--7--1
This means that 7:6 and 8:7 become tempered to the same interval. There isn't a common term for such an interval, or a notation for the resulting temperament. So it's difficult to describe, but this is the lattice for it. It ends up square, and so would work well on a ZTar. Here's what it would look like in 29-equal:
20--26--3---9--15 / \ / \ / \ / \ / \ / \ 17--23--0---6--12 \ / \ / \ / \ / \ / \ / 14--20--26--3---9
My 9-limit lattice could be simplified if the intervals Eb-E and E-E+ were tempered to be the same. That's would mean setting 25:24 and 36:35 to be equivalent. Which happens to be the case for my magic temperament. So here's a 22 note magic MOS on the simplified lattice
Eb----Bb----F-----C
B#/ \ Gb/ \ Db/ \ Ab/ Et
G+ / D#\ / A#\ / E#\ / Cb
B-----F#----C#----G#
G / \ D / \ A / \ E /
Eb / Bb\ / F \ / C \ /
B#----Gb----Db----Ab----Et
G+/ \ D#/ \ A#/ \ E#/ Cb
/ B \ / F#\ / C#\ / G#
G-----D-----A-----E
Eb/ \ Bb/ \ F / \ C /
B# / Gb\ / Db\ / Ab\ /
G+----D#----A#----E#----Cb
/ \ B / \ F#/ \ C#/ G#
/ G \ / D \ / A \ / E
Eb----Bb----F-----C
B#/ \ Gb/ \ Db/ \ Ab/ Et
G+ / D#\ / A#\ / E#\ / Cb
B-----F#----C#----G#
G / \ D / \ A / \ E /
Eb / Bb\ / F \ / C \ /
B#----Gb----Db----Ab----Et
G+/ \ D#/ \ A#/ \ E#/ Cb
/ B \ / F#\ / C#\ / G#
G-----D-----A-----E
That uses meantone names, although it isn't a meantone temperament so they don't work very well. Still, I don't have a better form of magic notation. The lattice would suit a honeycomb keyboard as stands. Or, it could be made to fit a rectangular keyboard by looking less like a lattice.
G+
|
G+--B#--Eb--G
| | | |
G---B---D#--Gb
| | | |
Gb--Bb--D---F#
| | | |
F#--A#--Db--F
| | | |
F---A---C#--E#
| | | |
E#--Ab--C---E
| | | |
E---G#--Cb--Et
|
Et
You can loop that to get more octaves on something like a ZBoard. Or get two repetitions on the 6 "strings" of a ZTar, with two more octave transpositions down the neck. Or, you could fret a real guitar to 19- or 22-equal, and tune the strings in major thirds to get this lattice.
For a really crazy lattice, try this!
A#----------E#----------B#---------F##---------C##---------G##
/ \ / \ / \ / \ / \ / \
/ \ C#/ / \ G#/ / \ D#/ / \ A#/ / \ E#/ / \
/ \ / \ / \ / \ / \ / \
/ A#\ \ / E#\ \ / B#\ \ / F#// \ / C#// \ / G#// \
/ \ / \ / \ / \ / \ / \
F#----------C#----------G#----------D#----------A#----------E#----------B#
\ / \ / \ / \ / \ / \ /
\ A/ / \ E/ / \ B/ / \ F#/ / \ C#/ / \ G#/ /
\ / \ / \ / \ / \ / \ /
\ / C#\ \ / G#\ \ / D#\ \ / A#\ \ / E#\ \ /
\ / \ / \ / \ / \ / \ /
A-----------E-----------B-----------F#----------C#----------G#
/ \ / \ / \ / \ / \ / \
/ \ C/ / \ G/ / \ D/ / \ A/ / \ E/ / \
/ \ / \ / \ / \ / \ / \
/ A\ \ / E\ \ / B\ \ / F#\ \ / C#\ \ / G#\ \
/ \ / \ / \ / \ / \ / \
F-----------C-----------G-----------D-----------A-----------E-----------B
\ / \ / \ / \ / \ / \ /
\ Ab/ / \ Eb/ / \ Bb/ / \ F/ / \ C/ / \ G/ /
\ / \ / \ / \ / \ / \ /
\ / C\ \ / G\ \ / D\ \ / A\ \ / E\ \ /
\ / \ / \ / \ / \ / \ /
Ab----------Eb----------Bb----------F-----------C-----------G
/ \ / \ / \ / \ / \ / \
/ \ Cb/ / \ Gb/ / \ Db/ / \ Ab/ / \ Eb/ / \
/ \ / \ / \ / \ / \ / \
/ Ab\ \ / Eb\ \ / Bb\ \ / F\ \ / C\ \ / G\ \
/ \ / \ / \ / \ / \ / \
Fb----------Cb----------Gb----------Db----------Ab----------Eb----------Bb
\ / \ / \ / \ / \ / \ /
\ Ab\\ / \ Eb\\ / \ Bb\\ / \ Fb/ / \ Cb/ / \ Gb/ /
\ / \ / \ / \ / \ / \ /
\ / Cb\ \ / Gb\ \ / Db\ \ / Ab\ \ / Eb\ \ /
\ / \ / \ / \ / \ / \ /
Abb---------Ebb---------Bbb----------Fb----------Cb----------Gb
The chromatic semitone 25:24 is divided into 3 equal parts. Hence we don't have "half sharps" so I used / and \ for raising and lowering instead. A septimal diesis 36:35 is exactly 2/3 of a 25:24, so this is already a two-dimensional lattice. But the 9-limit template is so big, it might not be worth the trouble
5
/ \
/ \
/ \
/ \
/ \
1-----------3-----------9
\ /
\ /
\ /
\ /
\ /
/ \
/ \
/ \
/ \
/ \
-----------
\ /
\ 7 /
\ /
\ /
\ /
It works with 46, 50 and 53-equal. The temperament covering 46 and 53 needs 30 notes (or a 32 note MOS) for a complete 9-limit chord, accurate to 1.7 cents. With microtempering, you can get the lattice to work without deviating more than half a cent from the 9-limit. Whatever, here's what it looks like with 46-equal
13----------41----------22----------3-----------30----------11----------38
\ / \ / \ / \ / \ / \ /
\ 27 / \ 8 / \ 35 / \ 16 / \ 43 / \ 24 /
\ / \ / \ / \ / \ / \ /
\ / 40 \ / 21 \ / 2 \ / 29 \ / 10 \ /
\ / \ / \ / \ / \ / \ /
26----------7-----------34----------15----------42----------23
/ \ / \ / \ / \ / \ / \
/ \ 39 / \ 20 / \ 1 / \ 28 / \ 9 / \
/ \ / \ / \ / \ / \ / \
/ 25 \ / 6 \ / 33 \ / 14 \ / 41 \ / 22 \
/ \ / \ / \ / \ / \ / \
11----------38----------19----------0-----------27----------8-----------35
\ / \ / \ / \ / \ / \ /
\ 24 / \ 5 / \ 32 / \ 13 / \ 40 / \ 21 /
\ / \ / \ / \ / \ / \ /
\ / 37 \ / 18 \ / 45 \ / 26 \ / 7 \ /
\ / \ / \ / \ / \ / \ /
23----------4-----------31----------12----------39----------20
/ \ / \ / \ / \ / \ / \
/ \ 36 / \ 17 / \ 44 / \ 25 / \ 6 / \
/ \ / \ / \ / \ / \ / \
/ 22 \ / 3 \ / 30 \ / 11 \ / 38 \ / 19 \
/ \ / \ / \ / \ / \ / \
9-----------35----------16----------43----------24----------5-----------32
11-limit lattices would need 4 dimensions (5 if you include the 9-direction) to be exact for JI. So, I always use simplified lattices.
There are two important 11-limit commas that can be eliminated to simplify graphing. One gives the comma (-3 -1 -1 0 2)H or 121/120. This makes two intervals of 11/8 equivalent to one of 15/8. Here's a lattice showing a 15/8:
10------15 / \ / / \ / 8------12
So, we can stick the 11 midway between the 8 and the 15. That gives an 11-limit complete chord looking like this:
5
/ \
/ \
/ 11
/ 7 \
/ \
1-----------3-----------9
B-----------F#----------C#----------G#----------D#----------A#
\ / \ / \ / \ / \ /
\ Ab / \ Eb / \ Bb / \ F / \ C /
C+ / G+ / D+ / A+ / E+ /
\ / B# \ / Fx \ / Cx \ / Gx \ /
\ / \ / \ / \ / \ /
D-----------A-----------E-----------B-----------F#
/ \ / \ / \ / \ / \
/ \ Cb / \ Gb / \ Db / \ Ab / \
/ Et / Bt / F+ / C+ / G+
/ G# \ / D# \ / A# \ / E# \ / B# \
/ \ / \ / \ / \ / \
Bb----------F-----------C-----------G-----------D-----------A
\ / \ / \ / \ / \ /
\ Abb / \ Ebb / \ Bbb / \ Fb / \ Cb /
Ct / Gt / Dt / At / Et /
\ / B \ / F# \ / C# \ / G# \ /
\ / \ / \ / \ / \ /
Db----------Ab----------Eb----------Bb----------F
The other approximation is to set the neutral third of 11/9 to be exactly half a fifth of 3/2. This gives the comma (-1 5 0 0 -2)H or 243/242. The 11-limit complete chord isn't so compact as above.
5
/ \
/ \
/ \
/ 7 \
/ \
1-----------3-----------9-----11
B-----D+----F#----A+----C#----E+----G#----B+----D#----F#+---A#
\ / \ / \ / \ / \ /
\ Ab / \ Eb / \ Bb / \ F / \ C /
\ / \ / \ / \ / \ /
\ / B# \ / Fx \ / Cx \ / Gx \ /
\ / \ / \ / \ / \ /
D-----F+----A-----C+----E-----G+----B-----D+----F#
/ \ / \ / \ / \ / \
/ \ Cb / \ Gb / \ Db / \ Ab / \
/ \ / \ / \ / \ / \
/ G# \ / D# \ / A# \ / E# \ / B# \
/ \ / \ / \ / \ / \
Bb----Dt----F-----At----C-----Et----G-----Bt----D-----F+----A
\ / \ / \ / \ / \ /
\ Abb / \ Ebb / \ Bbb / \ Fb / \ Cb /
\ / \ / \ / \ / \ /
\ / B \ / F# \ / C# \ / G# \ /
\ / \ / \ / \ / \ /
Db----Ft----Ab----Ct----Eb----Gt----Bb----Dt----F
Well, now you know the two lattices. Which should you use? Of course, you should use both, in different contexts. Otherwise, why would I bother telling you both? Think of the trouble I could have saved!
A good rule of thumb is, if you're using a temperament, the lattice should be consistent with it. Some temperaments work with both. The commas are (-3 -1 -1 0 2)H and (-1 5 0 0 -2)H. Add them, and you get (-4 4 -1 0 0)H, or a syntonic comma. So, for both 11-limit commas to be removed, the syntonic comma must be removed. That leaves a meantone-like scale. Specifically, it's a meantone-like 7+3 scale. Examples are 38, 31 and 24-equal. 17-equal also works, although it isn't very meantone-ish. The only equal temperament that is 11-limit consistent, and approximates both commas to a unison, is 31.
Other scales where the former (neutral third-based) lattice is consistent are 34, 41 and 72-equal. The latter (neutral second-based) lattice is consistent with 15, 22, 29, 46 and 53-equal.
There must be scales that don't work with either approximation. For equal temperaments, the only ones I can find with fewer than 100 notes and consistent in the 11-limit are 26, 80, 87 and 94.
More generally, the neutral second-based lattice is better for strict 11-limit harmony, because the complete chord is more compact. The neutral third-based lattice is the thing where you bring in neutral thirds but don't care about this overtone rubbish. Also, that lattice is less ambiguous, because 12/11 and 11/10 are both 11-limit intervals, and so should be distinguished. The residue between 11/9 and 3/2 is 27/22. Complex enough to sweep under the carpet. For JI, you may want to alter the second-based lattice so that the all notes are distinguished. I reckon the simplest thing's to use a temperament, but that's just (or tempered) me.
Paul Erlich has supplied the 11-limit tonality diamond drawn on both types of lattice. (This was in Tuning digest 378.) Here are the diagrams:
10/9---------5/3---------5/4
/ \ / \ / \
/ \ / \ 10/7 / \
/ 11/9----+-----&-----+---11/8
/ 14/9---------7/6---------7/4/ \
/ \ / \ / \ /***/ / \
16/9---------4/3------+--1/1--/-+----3/2---------9/8
\ / * / \ / / / \ /
\ 8/7--\---/-/12/7-/---\--9/7 /
16/11---+-\-/-@-----+---18/11 /
\ / 7/5 \ / \ /
\ / \ / \ /
8/5---------6/5---------9/5
Some symbols are used in place of ratios for reasons of space: * for 14/11, *** for 11/7 @ for both 12/11 and 11/10 (they're equivalent) and & for both 11/6 and 20/11.
That other diagram:
10/11-10/9---------5/3---------5/4
/ \ / \ / \ / \
/ \ / \ / \ 10/7 / \
/ X \ / \ / \ / \
/ 14/11-\14/9--------7/6---------7/4 \
/ / \ \ / \ / \ / \ / \
16/11-16/9--12/11--4/3-18/11\--1/1--/11/9--3/2--11/6---9/8--11/8
\ / \ / \ / \ / \ \ / /
\ 8/7--\---/-12/7--/---\-9/7-\-/11/7 /
\ / \ / \ / \ X /
\ / 7/5 \ / \ / \ /
\ / \ / \ / \ /
8/5---------6/5---------9/5--20/11
B-----D+----F#----A+----C#----E+----G#----B+----D#----F#+---A#
\ / \ / \ / \ / \ /
\ Ab / Ct\ Eb / Gt\ Bb / Dt\ F / At\ C /Et
\ / \ / \ / \ / \ /
\ / B# \ / Fx \ / Cx \ / Gx \ F#/
\ / Gbb \ / Dbb \ / Abb \ / Ebb \ /
D-----F+----A-----C+----E-----G+----B-----D+----F#
That's looking mighty confusing. Ct and B# are close in pitch, as are F+ and Gbb. How about pretending they really are equal? That gives us the following lattice
B-----D+----F#----A+----C#----E+----G#----B+----D#----F#+---A#
\ / \ / \ / \ / \ /
\ / \ / \ / \ / \ /
E# \ Ab / B# \ Eb / Fx \ Bb / Cx \ F / Gx \ C / Et
\ / \ / \ / \ / \ /
\ / \ / \ / \ / \ /
Bt----D-----F+----A-----C+----E-----G+----B-----D+----F#----A+
/ \ / \ / \ / \ / \
/ \ / \ / \ / \ / \
Fb / G# \ Cb / D# \ Gb / A# \ Db / E# \ Ab / B# \ Eb
/ \ / \ / \ / \ / \
/ \ / \ / \ / \ / \
Bb----Dt----F-----At----C-----Et----G-----Bt----D-----F+----A
\ / \ / \ / \ / \ /
\ / \ / \ / \ / \ /
E \ Abb / B \ Ebb / F# \ Bbb / C# \ Fb / G# \ Cb / D#
\ / \ / \ / \ / \ /
\ / \ / \ / \ / \ /
Bbt---Db----Ft----Ab----Ct----Eb----Gt----Bb----Dt----F-----At
This approximation is good for 31- 41- and 72-equal, and therefore miracle temperament. It makes these lattices the best way of visualising scales in my decimal notation. It includes the approximation 2401:2400, so it's a simplified 7-limit lattice before we even get to the 11-limit. As a square lattice comes out, it could be fitted to a Ztar or ZBoard. Here's what it looks like in 41-equal
36----7-----19----31----2-----14----26----38----9-----21----33
\ / \ / \ / \ / \ /
\ / \ / \ / \ / \ /
15 \ 27 / 39 \ 10 / 22 \ 34 / 5 \ 17 / 29 \ 41 / 12
\ / \ / \ / \ / \ /
\ / \ / \ / \ / \ /
35----6-----18----30----1-----13----25----37----8-----20----32
/ \ / \ / \ / \ / \
/ \ / \ / \ / \ / \
14 / 26 \ 38 / 9 \ 21 / 33 \ 4 / 16 \ 28 / 40 \ 11
/ \ / \ / \ / \ / \
/ \ / \ / \ / \ / \
34----5-----17----29----0-----12----24----36----7-----19----31
\ / \ / \ / \ / \ /
\ / \ / \ / \ / \ /
13 \ 25 / 37 \ 8 / 20 \ 32 / 3 \ 15 / 27 \ 39 / 10
\ / \ / \ / \ / \ /
\ / \ / \ / \ / \ /
33----4-----16----28----40----11----23----35----6-----18----30
Another simplification of the neutral-third lattice is
5
1-------3-------9---11
\ /
\ /
\ /
7
G#--B+--D#--F#+-A#
/ \ / \
G+/ B \ D+/ F#\ A+
/ \ / \
G---Bt--D---F+--A
\ / \ /
Gt\ Bb/ Dt\ Ft/ At
\ / \ /
Gb--Bbt-Db--Fbt-Ab
It sort of works with the neutral-third family 7, 24, 31, 38, ... but really it's a mapping for 31-equal as that's 11-limit consistent. So here it is in 31-equal
15--24--2---11--20 / \ / \ 14/ 23\ 1 / 10\ 19 / \ / \ 13--22--0---9---18 \ / \ / 12\ 21/ 30\ 8 / 17 \ / \ / 11--20--29--7---16
Another neat mapping for a ZTar. Unison vectors are 176:175 or (4 0 -2 -1 1)H and 243:242 or (-1 5 0 0 -2)H. These combine to give 31104:30625 or (7 5 -4 -2)H.
Lastly, here's a mapping that contains both neutral thirds and neutral seconds! It's like Dave Keenan's kleismic lattice, but with an 11 added in two different places:
7
|
|
|
|
|
5 |
| |
| |
| 11 |
| |
| |
1-----3-----9--11
So the whole lattice can be written with the 5-limit triads connected in a somewhat unusual fashion.
B---D+--F#--A+--C# |\ |\ | | \ | \ | | \ | \ | A C+ E G B | \ | \ | | \ | \ | | \| \| G---Bt--D---F+--A |\ |\ | | \ | \ | | \ | \ | F At C Et G | \ | \ | | \ | \ | | \| \| Eb--Gt--Bb--Dt--F
Again, works with 31-equal, not much else.
23--1---10--19--28 |\ |\ | | \ | \ | | \ | \ | 18 27 5 14 23 | \ | \ | | \ | \ | | \| \| 13--22--0---9---18 |\ |\ | | \ | \ | | \ | \ | 8 17 26 4 13 | \ | \ | | \ | \ | | \| \| 3---12--21--30--8