Kite's thoughts on 41edo Lattices: Difference between revisions

== Lattices ==

=== The 5-limit (ya) Lattice ===

[[File:41equal lattice 5-limit.png|none|thumb|456x456px]]

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 12-equal circle of 5ths, octave equivalence is assumed and each note represents an entire pitch class. 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. (The terms yo and gu come from [[color notation]].) Since moving 5thwards/4thwards adds/subtracts prime 3, and the octave is prime 2,  every octave-reduced 5-limit ratio appears exactly once in the lattice. Thus the lattice is a "map" of all possible 5-limit notes.  

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 remedy this in Lattices Part II.

== Commas ==

=== 5-limit (ya) commas ===

[[File:41equal lattice 5-limit with commas.png|none|thumb|466x466px]]

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 other commas can be pumped too, but they rarely are. Like Gu but unlike Triyo, 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 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 12-equal, just ignore the ups and downs: Dm - Bb - Eb - Ebm - [Cb=B] - E - A - A7 - D.  

[[File:41equal lattices big.png|none|thumb|477x477px]]

The sheer remoteness of ya no-fret 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:

=== 7-limit (yaza) commas ===

This lattice introduces three no-fret commas, [[875/864|Zotriyo]], [[225/224|Ruyoyo]] and [[5120/5103|Saruyo]]. The latter two are reasonably close and fairly pumpable, especially Ruyoyo. An example of a Saruyo pump is [[Kite Guitar Translations by Kite Giedraitis#I Will Survive .28Gloria Gaynor.29|I Will Survive]]. You can play such progressions without worrying. Lame joke: without fretting, that's why it's called a no-fret comma!

The previous lattice represents 7-limit JI, with 4 primes, thus a rank-4 tuning. This lattice represents a rank-3 [[Regular temperament|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 3-D, but this one can be viewed as both 3-D and 2-D. To navigate this lattice, one could step as before 4thwd/5thwd, yoward/guward and zoward/ruward. But as a 2-D lattice, one steps rightward/leftward and upward/downward. Each horizontal step is one-half as long as a triangle-side. 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 octave-reduces to a half-fret comma.

[[File:41equal lattice 11-limit with commas.png|none|thumb|558x558px]]

Each red, green or blue spot on the lattice represents multiple commas that differ by the Bizozogu microcomma. The new commas are:

{| class="wikitable"

|+

!note

!yaza comma

|-

|red vCx

== Commas Part II ==

We can call Saruyo a walk-once comma, and Ru and Gu walk-halfway commas. All no-fret commas are no-walk, walk-once, walk-twice, etc. All half-fret commas are walk-halfway, walk-one-and-a-half, etc.