Kite's thoughts on 41edo Lattices

New title: Crash course in tuning theory for Kite Guitar players

This is a continuation of the non-technical explanation.

Frequency Ratios

Every well-tuned interval is a frequency ratio. The easiest way to learn these ratios is to learn the intervals between notes in the harmonic series.

Lattices

Lattices are a great tool for exploring just intonation and also the larger edos.

The 5-limit (ya) Lattice

This lattice uses ups and downs notation:

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. Moving diagonally right-and-up (the 1:00 direction) adds a downmajor 3rd, 5/4. This yoward step adds prime 5. Moving guward left-and-down subtracts prime 5. Since moving 5thwards/4thwards adds/subtracts prime 3, every 5-limit monzo translates directly to a series of sideways and diagonal steps, and every octave-reduced 5-limit ratio appears exactly once in the lattice. Thus the lattice is a "map" of all possible 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.

The middle row is the plain row. The row immediately above it is the down row. Then double-down, triple-down, etc. 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 41-equal, the double-down row could be rewritten as a double-up row. For example, vvB = ^^Bb. And triple-down notes would in practice almost always 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 5-over or yo chords such as D vF# A appear as upward-pointing triangles. Upminor aka 5-under or gu chords are downward-pointing triangles. The downmajor7 chord is two adjacent triangles, as is the upminor7 chord. The downmajor6 chord 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.

The 7-limit (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 5-limit lattice. All the notes that protrude upwards form a 2nd layer, and all the sinking notes make the 3rd layer. Whereas the 5-limit lattice lets you move three steps in the yoward direction, this 7-limit lattice only lets you move one step zoward. We will fix this later.

In the 5-limit lattice, each note has a unique name. To find vvD#, go to the double-down row and look among the notes with sharps. But in the 7-limit lattice, multiple notes have the same name. In just intonation, they would sound different, but in 41-equal they are identical.

Commas

A comma is a just intonation ratio that is a narrow interval of less than (roughly) 50¢. In 41edo (or any edo for that matter), a comma is mapped to a small number of edosteps, usually 0 or 1, occasionally 2. The technical term for a comma that maps to 0 edosteps is vanishing comma. On the Kite guitar, mapping to 0, 1 or 2 edosteps can be called a no-fret, half-fret or one-fret comma.

5-limit commas

Certain notes are colored red and green for emphasis. (The color choice is somewhat 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 half-fret comma. The three half-fret 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.

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 D to ^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 pump that walks you towards the bridge.

A comma is an interval that's large enough to be audible but small enough to be fudge-able.

Rather than letting the tonic drift, ... pitch shift

In 12-equal, the Gu comma is a no-fret comma, and you can play such progressions without worrying. Lame joke: without fretting, that's why it's called a no-fret comma!

But when translating songs to 41-equal, one has to deal with Gu over and over...

The other commas can be pumped too, but they rarely are. Sagugu is also a no-fret comma in 12-equal. So you can sit down with a 12-equal guitar or keyboard and play a progression that walks to (or from) Sagugu,

The one-fret commas such as D to vvD# (Yoyo) aren't included because such commas are too big to fudge. The zero-fret commas would be included, but they are too remote to fit on this lattice. However they can be deduced from the green notes, because from one green note to another is a no-fret comma. For example, from Sagugu to Gu is the Layo comma. If we imagine a vector running from Sagugu to Gu, and then slide that vector over to start at the red D, we can see that Layo is on the down row, a few steps past the righthand edge. If the lattice were bigger, there would be a red vC# there.

There are a few more remote no-fret commas. From Sagugu to Triyo is the Laquinyo comma. From Gu to Triyo is the Saquadyo comma. Saquadyo is the one that equates the double-up row with the double-down row.

7-limit commas

This lattice introduces two no-fret commas, Saruyo and Ruyoyo. Both are reasonably close and fairly pumpable. The unlabeled red notes are just the descending versions of these two commas. Since the comma maps to zero frets, there is no difference between an ascending comma pump and a descending one, and both versions can be treated as the same.

This lattice also introduces the green Ru comma, very important because it's so nearby. Like Gu, this one...

Kite's thoughts on 41edo Lattices

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