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

Line 1:

== Overview ==

This page explains lattices and commas for anyone wanting to compose for the [[Kite Guitar|Kite guitar]]. Lattices ~~are the primary tool Kite Giedraitis uses to compose~~.

This page explains lattices and commas for anyone wanting to compose for the [[Kite Guitar|Kite guitar]]. Lattices let you visualize scales, chords, and chord progressions.

== Lattices ==

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

This lattice uses [[Ups and Downs Notation|ups and downs notation]]:

This lattice uses [[Ups and Downs Notation|ups and downs notation]] to name the [[41-edo|41-edo (aka 41-equal)]] notes:

[[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. 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.

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.

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, 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, 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].

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.

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". ~~There are~~ no-fret commas~~, but 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 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]].

@@ Line 1: / Line 1: @@
 == Overview ==
-This page explains lattices and commas for anyone wanting to compose for the [[Kite Guitar|Kite guitar]]. Lattices are the primary tool Kite Giedraitis uses to compose.
+This page explains lattices and commas for anyone wanting to compose for the [[Kite Guitar|Kite guitar]]. Lattices let you visualize scales, chords, and chord progressions.
 == Lattices ==
 === The 5-limit (ya) Lattice ===
-This lattice uses [[Ups and Downs Notation|ups and downs notation]]:
+This lattice uses [[Ups and Downs Notation|ups and downs notation]] to name the [[41-edo|41-edo (aka 41-equal)]] notes:
 [[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. 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.
+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.
+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, 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, 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].
 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.
@@ 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". There are no-fret commas, but 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 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]].