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

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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.

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 5-limit monzo translates directly to a series of sideways and diagonal steps, plus octaves.

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 5-limit 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 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, [http://www.tallkite.com/AlternativeTunings.html Alternative Tunings: Theory, Notation and Practice].

The middle row is the plain row. The row immediately above it is the down row. Then double-down (aka dud), triple-down (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, [http://www.tallkite.com/AlternativeTunings.html 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.

If this were just intonation, the lattice would extend infinitely in all directions. But because this is 41-equal, 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 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).

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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.

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.

In the 5-limit 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 7-limit lattice, multiple notes have the same name. In just intonation, they would sound different, but in 41-equal they are identical.

== Commas ==

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

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

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 half-fret comma. The three half-fret commas are named via [[color notation]], and the name tells you which row it's on: [[250/243|Triyo]] is 3 rows up, [[81/80|Gu]] is one row down, and [[2048/2025|Sagugu]] is two rows down. ~~The~~ blue ~~notes are~~ the inverse of ~~the~~ green ~~notes~~, i.e. descending ~~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 half-fret comma. The three half-fret commas are named via [[color notation]], and the name tells you which row it's on: [[250/243|Triyo]] is 3 rows up, [[81/80|Gu]] is one row down, and [[2048/2025|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.

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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.

So those are the half-fret commas. The one-fret commas such as D to vvD# ([[25/24|Yoyo]]) and D to ^³Ebb ([[128/125|Trigu]]) aren't colored because such commas are too big to "fudge". What about no-fret 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 no-fret commas, there is no sonic difference between an ascending comma pump and a descending one, and both versions can be treated as the same.

So those are the half-fret commas. The one-fret commas such as D to vvD# ([[25/24|Yoyo]]) and D to ^³Ebb ([[128/125|Trigu]]) aren't colored because such commas are too big to "fudge". What about no-fret 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 no-fret commas, there is no sonic difference in 41-equal between an ascending comma pump and a descending one, and both versions can be treated as the same.

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

The three no-fret commas are the [[Magic|Laquinyo]], [[20000/19683|Saquadyo]] and [[32805/32768|Layo]] commas. Layo is another comma that 12-equal tempers out, but is very rarely pumped. Saquadyo is the comma that equates the ~~double-up~~ row with the ~~double-down~~ row. There is also a new half-fret comma, [[16875/16384|Laquadyo]].

The three no-fret commas are the [[Magic|Laquinyo]], [[20000/19683|Saquadyo]] and [[32805/32768|Layo]] commas. Layo is another comma that 12-equal 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 half-fret comma, [[16875/16384|Laquadyo]].

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:

@@ Line 8: / Line 8: @@
 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.
-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 5-limit monzo translates directly to a series of sideways and diagonal steps, plus octaves.
+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 5-limit 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 double-down, triple-down, etc. Why does one go <u>up</u> to the <u>down</u> row? For a full explanation of this and lattices in general, see chapter 1.3 of Kite's book, [http://www.tallkite.com/AlternativeTunings.html Alternative Tunings: Theory, Notation and Practice].
+The middle row is the plain row. The row immediately above it is the down row. Then double-down (aka dud), triple-down (aka trud), etc. (There's also dup and trup for ^^ and ^^^.) Why does one go <u>up</u> to the <u>down</u> row? For a full explanation of this and lattices in general, see chapter 1.3 of Kite's book, [http://www.tallkite.com/AlternativeTunings.html 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.
+If this were just intonation, the lattice would extend infinitely in all directions. But because this is 41-equal, 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 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).
@@ Line 22: / Line 22: @@
 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.
-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.
+In the 5-limit 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 7-limit lattice, multiple notes have the same name. In just intonation, they would sound different, but in 41-equal they are identical.
 == Commas ==
@@ Line 29: / Line 29: @@
 === 5-limit (ya) commas ===
 [[File:41equal lattice 5-limit with commas.png|none|thumb|466x466px]]
-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 half-fret comma. The three half-fret commas are named via [[color notation]], and the name tells you which row it's on: [[250/243|Triyo]] is 3 rows up, [[81/80|Gu]] is one row down, and [[2048/2025|Sagugu]] is two rows down. The blue notes are the inverse of the green notes, i.e. descending 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 half-fret comma. The three half-fret commas are named via [[color notation]], and the name tells you which row it's on: [[250/243|Triyo]] is 3 rows up, [[81/80|Gu]] is one row down, and [[2048/2025|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.
@@ Line 37: / Line 37: @@
 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.
-So those are the half-fret commas. The one-fret commas such as D to vvD# ([[25/24|Yoyo]]) and D to ^³Ebb ([[128/125|Trigu]]) aren't colored because such commas are too big to "fudge". What about no-fret 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 no-fret commas, there is no sonic difference between an ascending comma pump and a descending one, and both versions can be treated as the same.
+So those are the half-fret commas. The one-fret commas such as D to vvD# ([[25/24|Yoyo]]) and D to ^³Ebb ([[128/125|Trigu]]) aren't colored because such commas are too big to "fudge". What about no-fret 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 no-fret commas, there is no sonic difference in 41-equal between an ascending comma pump and a descending one, and both versions can be treated as the same.
 [[File:41equal lattices big.png|none|thumb|477x477px]]
-The three no-fret commas are the [[Magic|Laquinyo]], [[20000/19683|Saquadyo]] and [[32805/32768|Layo]] commas. Layo is another comma that 12-equal tempers out, but is very rarely pumped. Saquadyo is the comma that equates the double-up row with the double-down row. There is also a new half-fret comma, [[16875/16384|Laquadyo]].
+The three no-fret commas are the [[Magic|Laquinyo]], [[20000/19683|Saquadyo]] and [[32805/32768|Layo]] commas. Layo is another comma that 12-equal 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 half-fret comma, [[16875/16384|Laquadyo]].
 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: