41edo Lattices
Overview
This page explains lattices and commas for anyone wanting to compose for the Kite guitar. Lattices let you visualize scales, chords, and chord progressions.
Lattices
The 5limit (ya) Lattice
This lattice uses ups and downs notation to name the 41edo (aka 41equal) notes:
The middle row is a chain of 5ths. Moving one step to the right (aka fifthwards) adds a 5th, and one step to the left (fourthwards) adds a 4th. Just like the 12equal circle of 5ths, octave equivalence is assumed and each note represents an entire pitch class. Moving diagonally rightandup (the 1:00 direction) adds a downmajor 3rd, 5/4. This yoward step adds prime 5. Moving guward leftanddown subtracts prime 5. (The terms yo and gu come from color notation.) Since moving 5thwards/4thwards adds/subtracts prime 3, and the octave is prime 2, every octavereduced 5limit ratio appears exactly once in the lattice. Thus the lattice is a "map" of all possible 5limit notes.
Every interval appears in the lattice as a vector. For example a major 7th is one step fifthward and one step yoward, making a vector in the 2:00 direction that spans two triangles. This vector is called a monzo. Every 5limit monzo translates directly to a series of sideways and diagonal steps, plus octaves.
The middle row is the plain row. The row immediately above it is the down row. Then doubledown (aka dud), tripledown (aka trud), etc. (There's also dup and trup for ^^ and ^^^.) Why does one go up to the down row? For a full explanation of this and lattices in general, see chapter 1.3 of Kite's book, Alternative Tunings: Theory, Notation and Practice.
If this were just intonation, the lattice would extend infinitely in all directions. But because this is 41equal, the dud row can be rewritten as a dup row. For example, vvB = ^^Bb. And trud notes would in practice almost always be written as up notes. So the lattice wraps around on itself, like a world map in which the western tip of Alaska appears on both the far right and the far left. More on this later.
Every chord type has a certain shape. Downmajor aka 5over or yo chords such as D vF# A appear as upwardpointing triangles. Upminor aka 5under or gu chords e.g. D ^F A are downwardpointing triangles. The downmajor7 chord (D vF# A vC#) is two adjacent triangles, as is the upminor7 chord (D ^F A ^C). The downmajor6 chord (D vF# A vB) has the same shape as the upminor7 chord, which tells you that they are chord homonyms (same notes, different roots).
Chord progressions can be mapped out on the lattice as a series of chord shapes. Often two adjacent chords have a common tone, and the progression "walks" around the lattice. If there are no common tones, the progression "jumps" from one area to another. Often there are several different places to jump. More on this later.
This lattice shows the notes as intervals, and shows how many arrows (edosteps) each note is from the tonic D.
This lattice is smaller than the previous one, but it still extends far enough to include all 41 notes, except 16 and 25 (v4 and ^5). By coincidence, each number appears exactly once.
The 7limit (yaza) lattice
Moving zowards in the 2:00 direction from the D exactly in the middle to the nearby vC adds 7/4. This direction can be thought of as a third dimension, making the down7 chord D vF# A vC be a tetrahedron protruding upwards from the page. Likewise the upminor6 chord D ^F A ^B is a tetrahedron sinking down into the page. The lattice has three layers. One layer is the original 5limit lattice. All the notes that protrude upwards form a 2nd layer, and all the sinking notes make the 3rd layer. Whereas the 5limit lattice lets you move three steps in the yoward direction, this 7limit lattice only lets you move one step zoward. We will remedy this in Lattices Part II.
In the 5limit lattice, each note has a unique name. To find vvD#, go to the dud row and look among the notes with sharps. But in the 7limit lattice, multiple notes have the same name. In just intonation, they would sound different, but in 41equal they are identical.
Here is this lattice as intervals. Each color represents a separate layer of the lattice.
With so many more notes than the previous lattice of intervals, each number now appears multiple times. In particular, both 0 and 1 appear twice. More on that later.
Commas
A comma is a just intonation ratio that is small, less than (roughly) 50¢. In 41edo aka 41equal, a comma maps to a small number of arrows, usually 0 or 1, occasionally 2. The technical term for a comma that maps to 0 arrows is vanishing comma. On the Kite guitar, mapping to 0, 1 or 2 arrows can be called a nofret, halffret or onefret comma.
5limit (ya) commas
Certain notes are colored red, green and blue for emphasis. (The color choice is arbitrary.) The green notes are all half a fret sharper than the red D. The vector from the red D to any of these green notes is a halffret comma. The three halffret commas are named via color notation, and the name tells you which row it's on: Triyo is 3 rows up, Gu is one row down, and Sagugu is two rows down. Each blue note is the inverse of a green note, i.e. a descending comma.
The most important comma historically is the Gu comma, with ratio 81/80. Consider the progression Im  bIII  bVII  IV  Im. On the lattice, it becomes D^m  ^Fv  ^Cv  ^Gv  ^D^m, and it walks you from the red D to the green ^D. This is called a comma pump. Such pumps are a major issue in just intonation. On the Kite guitar fretboard, this comma pump walks you fifthwards, which is towards the nut. The progression I  vi  ii  V  I becomes Dv  vB^m  vE^m  vAv  vDv, which is a descending Gu pump that walks you from the red D to the blue vD. On the guitar, it walks you fourthwards, towards the bridge.
The Gu comma vanishes (i.e. becomes nofret) in meantone temperaments like 12equal, and you can play such progressions without worrying. But when translating 12equal songs to 41equal, one has to deal with Gu comma pumps quite often. One usually uses a small pitch shift, which is often not very noticeable. So if your chord progression walks from the red D to a red, green or blue note, it in effect jumps back to the tonic. This is desirable musically because the alternative is backtracking more or less the way you came, more predictable. (For example, Dv  vB^m  vE^m  vAv  vE^m  Gv  Dv backtracks.)
The other commas can be pumped too, but they rarely are. Like Gu but unlike Triyo, Sagugu is also a nofret comma in 12equal. So you can sit down with a 12equal guitar or keyboard and play a progression that pumps Sagugu, and you'll have done something quite unique! Perhaps this comma is rarely pumped because it requires two yoward or guward root movements, and fourthward/fifthward root movements are more common. An example might be I^m  ^bVIv  ^bIIv  ^bII^m  [^^bbVII=VI]v  IIv  Vv  Vv7  I^m. To play this in 12equal, just ignore the ups and downs: Dm  Bb  Eb  Ebm  [Cb=B]  E  A  A7  D.
So those are the halffret commas. The onefret commas such as D to vvD# (Yoyo) and D to ^³Ebb (Trigu) aren't colored because such commas are too big to "fudge". What about nofret commas? They are too remote to appear in the lattice. The next lattice zooms out to reveal the nearest three, colored and labeled in red. The unlabeled red notes are just the descending versions of these three commas. For nofret commas, there is no sonic difference in 41equal between an ascending comma pump and a descending one, and both versions can be treated as the same.
The three nofret commas are the Laquinyo, Saquadyo and Layo commas. Layo is another comma that 12equal tempers out, but is very rarely pumped. Saquadyo is the comma that equates the dup row with the dud row. There is also a new halffret comma, Laquadyo.
The sheer remoteness of ya nofret commas tells us that most ya music on the Kite guitar that travels around the lattice widely will tend to have pitch shifts, unless it backtracks. This descending Layo pump is an exception:
 I^m7  ^bVIv7 (4x)
 ^bIIv7  [^bV=#IV]v7  VIIv7  IIIv7  VIv7  IIv7  Vv7  Vv9  I^m7
7limit (yaza) commas
Once we add prime 7, we get nonremote, easily pumpable nofret commas.
This lattice introduces three nofret commas, Zotriyo, Ruyoyo and Saruyo. The latter two are reasonably close and fairly pumpable, especially Ruyoyo. An example of a Saruyo pump is I Will Survive. You can play such progressions without worrying. Lame joke: without fretting, that's why it's called a nofret comma!
This lattice also introduces the halffret Ru comma, a very important comma because it's so nearby, and so easy to pump. The other new halffret commas are Lazoyoyo, Sazoyo, Laruyo and Zotrigu.
Lattices Part II
The Full Yaza Lattice
As previously noted, the 7limit lattice is limited to only three layers, zo = 7over, ru = 7under and ya = no7s. This is rather limiting. For example, any comma with a 7exponent greater than 1 or less than 1 won't appear in this lattice, e.g. 50/49 or 49/48. The solution to this is to use the Bizozogu microcomma. This comma is only 0.7¢ and is nearly impossible to hear. It is a nofret comma with a 7exponent of 4. Strangely enough, it doesn't vanish in 12equal! This comma equates the layer above zo with the one below ru. Thus the lattice wraps around and there are only 4 layers total. To fit all 4 layers onto one plane, the lattice becomes rectangular. The gray lines that form triangles are still there, but the notes inside the triangles are shifted slightly to make a straight row. (Actually only nearly straight, to preserve readability.) In addition, unnamed dots have been added to the other rows. These dots form the 4th layer.
The previous lattice represents 7limit JI, with 4 primes, thus a rank4 tuning. This lattice represents a rank3 temperament of za JI. Since the comma that is tempered is so small, the two tunings sound identical to the human ear. But the structure of the lattice fundamentally changes. The previous lattice was 3D, but this one can be viewed as both 3D and 2D. To navigate this lattice, one could step as before 4thwd/5thwd, yoward/guward and zoward/ruward. But as a 2D lattice, one steps rightward/leftward and upward/downward. Each horizontal step is onehalf as long as a triangleside. Thus on the middle row, from D to A is two rightward steps. Thus one rightward step is half a 5th, i.e. a neutral 3rd of 12\41. From D up to G# is a vertical step. Thus one upward step is just over half an octave (21\41), and two upward steps octavereduces to a halffret comma.
Before, there was a onetoone correspondence between notes in the lattice and JI ratios. But now, any ratio can have the Bizozogu microcomma added to it and that new ratio will map to the same spot in the lattice. For example, the unnamed dot between D and A represents both 60/49 and 49/40. As the former, it's the 6th of a G#^m6 chord, and would be spelled ^E#. As the latter, it's the 7th of an Abv7 chord, and would be spelled vGb. It could also be spelled ^^F or vvF#. This is why the dots are unnamed!
Here's the lattice as intervals. How many zeros and ones can you find?
Here's the lattice with the commas marked:
Each red, green or blue spot on the lattice represents multiple commas that differ by the Bizozogu microcomma. The new commas are:
 nofret: Labizoyo, Zozoyo, Rurutriyo, Latrizo and Satrizoagu
 halffret: Laquinzo, Zozo, Biruyo, Latrizoagugu, Saquadru and Triruagu
Using 2D steps, the Ruyoyo comma is always 5 rightward steps and 3 upwards steps. Using 3D steps, from the middle red D to Ruyoyo has a familiar shape: two 5thwd steps, two yoward steps and a ruward step. From the red ^Ebb to the red D has a similar shape.
But notice the shape of the interval from Zozo/Biruyo to Laquinzo/Lazoyoyo. This too is the Ruyoyo comma, but it's one and a half 5thwd steps, one yoward step and one zoward step. From the Ru comma to Zozo/Biruyo is the same steps in the opposite order, and the shape is rotated 180 degrees. So the 2D shape of the comma doesn't change, but the 3D shape does. When you're tracing a chord progression on the full yaza lattice, it's very helpful to be able to spot the Ruyoyo comma in these different shapes.
Something similar happens with the Ru comma. From Ruyoyo to Zozo is a Ru comma, as is Zozoyo to Sazoyo.
The 11limit (yazala) and 13limit (yazalatha) Lattice
This lattice can also be applied to higher prime limits by assuming the Lulu and Thuthu commas vanish. These are not microcommas, and are fairly audible at 8¢ and 17¢ respectively. Thus this wouldn't be a very accurate lattice for JI. But it works well for 41equal because both commas are nofret commas. Moving 5 steps to the right represents the 11/8 interval. Moving 1 step to the left represents the 13/8 interval. A single rightward step becomes both 11/9 and 27/22, as well as 39/32 and 16/13, all neutral 3rds.
Each red, green and blue note can be reinterpreted as a higherprimelimit comma. Some examples:
note  yaza comma  as a yazala comma  as a yazalatha comma 

red vCx  Ruyoyo  Lozoyo  Thuzoyo, Thuloruyoyo 
red dot  Zozoyo  Luyoyo, Biluzoayo  Thoyoyo 
green dot  Zozo  Loyo  Thoyo 
red D  Bizozogu  Lulu, Luzozogu  Thuthu, Thulu, Thulo, Thuzozogu 
green ^D  Ru  Luzogu  Tholuru, Thozogu, Thothoru 
green ^D  Gu  Loruru, Lologu  Thulogu 
green ^^Ebb  Zotrigu  Lorugugu  Thorugugu 
Commas Part II
The Kite guitar fretboard is made up of alternating rainbow zones and off zones. Playing a Gu or Ru pump on the Kite guitar forces one to either shift a pitch or walk into the off zone. Going into the off zone makes the tonic drift sharp or flat by half a fret. But playing a Saruyo pump forces one to walk up the neck clear through the off zone into the next rainbow zone. The tonic drifts flat and then sharp! Depending on the melody, this can result in the scale temporarily containing offperfect tonics, 4ths and 5ths, which can be disconcerting.
We can call Saruyo a walkonce comma, and Ru and Gu walkhalfway commas. All nofret commas are nowalk, walkonce, walktwice, etc. All halffret commas are walkhalfway, walkoneandahalf, etc.
Is there a way to determine how far a comma will make us walk? Yes!
5limit JI is a rank3 tuning, and 41equal is a rank1 tuning. Regular temperament theory tells us that any two linearly independent (i.e. nonredundant) nofret ya commas suffice to reduce 5limit JI to 41equal. There are many pairs of commas that will work; let's use Laquinyo and Layo. Since they are linearly independent, any ya nofret comma can be expressed uniquely as the sum or difference of these two commas. For example, the Wa41 comma is one Laquinyo comma minus five Layo commas. Laquinyo is a nowalk comma and Layo is a walkonce comma. Therefore we can determine how far a comma walks by the number of Layo commas it contains. Thus Wa41 is a walkfivetimes comma. (See Kite's circle of 5ths exercise).
For 7limit, we need three nofret commas. Since Ruyoyo is a nowalk comma, let's use Laquinyo, Layo and Ruyoyo. Again, the number of Layo commas determines how far the comma pump walks.
Every comma's monzo or lattice vector has a certain power of three, which we can call P. The comma's walking distance corresponds roughly to P/8. These powers of three correspond to yoward/guward diagonal lines on the lattice. Thus the walking distance of a comma is roughly proportional to its horizontal distance from the line running through ^Bb, D and vF#. Note that there is only one nowalk ya comma, Laquinyo.
For za and yaza commas, before dividing by 8, add 2 for every "zo" in the name and subtract 2 for every "ru". In the table below, the resulting number indicates which column the comma is in.
nowalk  walkhalfway  walkonce  walkoneandahalf  

(nofret)  (halffret)  (nofret)  (halffret)  
wa
and ya 
Laquinyo = 1  Gu = 4  Layo = 8  Lalawa = 12 
Laquadyo = 3  Saquadyo = 9  
Sagugu = 4  
Triyo = 5  
za
and yaza

Zotriyo = 3 + 2 = 1  Zotrigu = 2 + 2 = 4  Latrizo = 1 + 6 = 7  
Ruyoyo = 2  2 = 0  Ru = 2  2 = 4  Saruyo = 6  2 = 8  
Zozoyo = 5 + 4 = 1  Zozo = 1 + 4 = 3  
Biruyo = 0  4 = 4  
Laruyo = 6  1 = 5 