5-Limit Triangular (Hexagonal) Lattices
The 5-limit lattice, like 5-limit harmony, is defined around the triads:
E
/ \
/ \
/ \
C-------G
|
A-------E
\ /
\ /
\ /
C |
A step to the right is a fifth. Up-right is a major third, and down-right is a minor third. This lattice is octave-equivalent, and so useful for defining scales that repeat every octave.
A bit more of the lattice looks like this:
B---------F#--------C#--------G#--------D#--------A#
/ \ / \ / \ / \ / \ /
/ \ / \ / \ / \ / \ /
/ \ / \ / \ / \ / \ /
/ \ / \ / \ / \ / \ /
G---------D---------A---------E---------B---------F#
\ / \ / \ / \ / \ / \
\ / \ / \ / \ / \ / \
\ / \ / \ / \ / \ / \
\ / \ / \ / \ / \ / \
Bb--------F---------C---------G---------D---------A
/ \ / \ / \ / \ / \ /
/ \ / \ / \ / \ / \ /
/ \ / \ / \ / \ / \ /
/ \ / \ / \ / \ / \ /
Gb--------Db--------Ab--------Eb--------Bb--------F
If a scale is defined on this lattice, you can instantly see what major and minor triads it contains. For example, a diatonic key looks like this:
D---------A---------E---------B
\ / \ / \ / \
\ / \ / \ / \
\ / \ / \ / \
\ / \ / \ / \
F---------C---------G---------D
Each point on the lattice is connected to 6 others, so I would call it a hexagonal lattice. However, in music theory it is usually called a triangular lattice. Triangles are used rather than squares so that a minor third appears as a more consonant interval than a major seventh. Consonance is generally related to distance on a harmonic lattice. When you bring octaves into it, a three dimensional cubic lattice is more appropriate.
Here are some more chords on a triangular lattice:
| CMaj7 | G7 | G6 |
E---B / \ / C---G |
B
/ \
G---D
\
F |
E---B \ / \ G---D |
There is some debate as to whether G7 should be like that, as Mark Nowitzky demonstrates. This is the third tuning he considers. The first one is the 7-limit tetrachord below.
I've also prepared a long list of 5-limit scales, defined on the triangular lattice.
For 7-limit harmony, the triangle becomes a tetrahedron:
E
/ \
/ \
/ A# \
/ \
C---------G
|
A---------E
\ /
\ Gb /
\ /
\ /
C |
These two chords, and subsets thereof, are 7-limit consonances.
It's been pointed out to me (ahem) that the term "tetrahedral lattice" might be misleading. The lattice does not consist entirely of tetrahedra, because they don't tesselate. However, I persist in using the term because the tetrahedra are the foundation of the harmony.
C-E-G-A# is similar to C7, C-E-G-Bb. The interval C-A# is the ratio 7/4, and so the A# is flatter than the Bb in C7. The 7-limit chord is often known as a subminor seventh, which can be written C-E-G-Bbt. The t is for a half-flat. I also use + for a half-sharp on this page.
7-limit chords are those that are based on whole number ratios that involve factors of 7. They can be written on the lattice as
5
/ \
/ \
/ 7 \
/ \
1---------3There are intervals close to 7-limit used in musics from all over the world. So why aren't we familiar with them in Europe today? I'm not an expert on the history. But . . . it looks like, in the early days of the Christian church, 7-limit scales were associated with coarse, even obscene, songs. So obviously these songs had no part in the church, which had a strong influence on classical and even folk music. Once instruments were built around 12-equal, it became very difficult to play 7-limit scales even for people who wanted to. But, with digital technology, we can appreciate them once more. Well, that's how I see it, anyway.
There is a good reason why I wrote A# instead of Bb. In meantone temperaments, A# is the flatter and so closer to just. Observing this distinction, or specifying "quartertone" flats, means two different chords won't be confused.
The 7-limit lattice is three dimensional. The A# is above the C major chord, and the Gb below the a minor. Here is a larger subset of the 7-limit lattice:
B---------F#--------C#--------G#--------D#--------A#
/ \ / \ / \ / \ / \ /
/ \ G#+ / \ D#+ / \ A#+ / \ E#+ / \ B#+ /
/ Ft \ / Ct \ / Gt \ / Dt \ / At \ /
/ \ / \ / \ / \ / \ /
G---------D---------A---------E---------B---------F#
\ / \ / \ / \ / \ / \
\ E+ / \ B+ / \ F#+ / \ C#+ / \ G#+ / \
\ / Abt \ / Ebt \ / Bbt \ / Ft \ / Ct \
\ / \ / \ / \ / \ / \
Bb--------F---------C---------G---------D---------A
/ \ / \ / \ / \ / \ /
/ \ G+ / \ D+ / \ A+ / \ E+ / \ B+ /
/ Fbt \ / Cbt \ / Gbt \ / Dbt \ / Abt \ /
/ \ / \ / \ / \ / \ /
Gb--------Db--------Ab--------Eb--------Bb--------F
Think of this as three different 5-limit planes.
Here it is with more points connected up:
B-----------------F#----------------C#----------------G#----------------D#----------------A#
/ \ / \ / \ / \ / \ /
/ \ / \ / \ / \ / \ /
/ \ G#+----/-----\----D#+----/-----\----A#+----/-----\----E#+----/-----\----B#+ /
/ \ / \ / \ / \ / \ / \ / \ / \ / \ / /
/ \ / \ / \ / \ / \ / \ / \ / \ / \ / /
/ Ft----------------Ct----------------Gt----------------Dt----------------At \ /
/ \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
/ \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
G-----------\-----D-----/-----\-----A-----/-----\-----E-----/-----\-----B-----/-----\-----F#
\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \
\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \
\ E+----\-----/-----B+----\-----/----F#+----\-----/----C#+----\-----\----G#+ \ \
\ \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ \
\ \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ \
\ / Abt---------------Ebt---------------Bbt----------------Ft----------------Ct \
\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \
\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \
Bb----/-----\-----F-----/-----\-----C-----/-----\-----G-----/-----\-----D-----/-----------A
/ \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
/ \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
/ / G+----\-----/-----D+----\-----/-----A+--- \-----/-----E+----\-----/-----B+ /
/ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
/ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
/ Fbt---------------Cbt---------------Gbt---------------Dbt---------------Abt \ /
/ \ / \ / \ / \ / \ /
/ \ / \ / \ / \ / \ /
Gb----------------Db----------------Ab----------------Eb----------------Bb----------------F
Now you know your 7-limit lattice, try some examples of 7-limit scales.
Erv Wilson was the first with tetrahedral lattices, although I don't think he ever published them. Still, 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 like 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
The 9-limit is like the 7-limit but with a major 9th added from the root. The basic chords can happily be written on a standard 7-limit lattice, like so:
E
/ \
/ \
/ A# \
/ \
C---------G---------D
|
D---------A---------E
\ /
\ Gb /
\ /
\ /
C |
But this template makes the 4:5:6:7:9 chord increase in pitch from left to right
5
1-----3-----9
\ /
\ /
7
and the different kinds of thirds all come together
E+ / \ / E \ C-----G \ Eb/ \ / Ebt
The 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.
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.
Along with some corrections for this page, 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
If more notes are added to the lattice, we get lots of them inside each 5-limit triangle:
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. So, as Ct and B# are close in pitch, as are F+ and Gbb, and so on, 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