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

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(A work in progress...)

== Overview ==

This page explains commas and lattices for the [[Kite Guitar|Kite guitarist]].

== Lattices ==

~~Lattices are a great tool for exploring just intonation (aka JI) and also near-just music in the larger edos.~~

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

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

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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, and you'll have done something quite unique!

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. This~~ is another comma that 12-equal tempers out, but is very rarely pumped.

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 are too remote. This lattice zooms out to reveal the nearest three, plus their descending versions.

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

The three new 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 half-fret comma, [[16875/16384|Laquadyo]].

~~There are two more remote no~~-fret commas. ~~From Sagugu to Triyo~~ is ~~the [~~[~~Magic|Laquinyo~~]~~] comma. From Gu to Triyo is the Saquadyo comma. Neither of these vanish in 12~~-~~equal. Saquadyo is the comma that equates the double~~-~~up row with the double~~-~~down row~~.

The sheer remoteness of ya zero-fret commas tells us that most ya music on the Kite guitar that travels around the lattice widely will tend to have pitch shifts. This Layo pump that I wrote is an exception:

I^m7 - ^bVIv7 (4x)

^bIIv7 - [^bV=#IV]v7 - VIIv7 - IIIv7 - VIv7 - IIv7 - Vv7 - Vv9

I^m7 - ^bVIv7 (4x)

Once we add prime 7, we get easily pumpable zero-fret commas.

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

[[File:41equal lattice 7-limit with commas.png|none|thumb|482x482px]]

This lattice introduces two no-fret commas, [[5120/5103|Saruyo]] and [[225/224|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 in 41-equal between an ascending comma pump and a descending one, and both versions can be treated as the same.

This lattice introduces two no-fret commas, [[5120/5103|Saruyo]] and [[225/224|Ruyoyo]]. Both are reasonably close and fairly pumpable, especially Ruyoyo. The unlabeled red notes are just the descending versions of these two commas. Since the comma maps to zero frets, there is no difference in 41-equal between an ascending comma pump and a descending one, and both versions can be treated as the same.

An example of a Saruyo pump is [[Kite Guitar Translations by Kite Giedraitis#I Will Survive .28Gloria Gaynor.29|I Will Survive]].

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You can play such progressions without worrying. Lame joke: without fretting, that's why it's called a no-fret comma!

This lattice also introduces the green [[64/63|Ru]] comma, very important because it's so nearby~~. Like Gu~~, ~~this one..~~.

This lattice also introduces the green [[64/63|Ru]] comma, very important because it's so nearby, and so easy to pump.

== Lattices Part II ==

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As previously noted, the 7-limit lattice is limited to only three layers, zo = 7-over, ru = 7-under and noza = no-7s. This is rather limiting. For example, any comma with a 7-exponent 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 assume the [[2401/2400|Bizozogu]] microcomma is tempered out. This comma is only 0.7¢ and is nearly impossible to hear. Strangely enough, it doesn't vanish in 12-equal, although it does in 41-equal! This comma equates the layer above zo with the one below ru. 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.

[[File:41equal lattice 11-limit.png|none|thumb|544x544px]]

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. From D up to G# is a vertical step. Thus one upward step is just over half an octave, and two upward steps octave-reduces to a half-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. From D up to G# is a vertical step. Thus one upward step is just over half an octave, and two upward steps octave-reduces to a half-fret comma.

Before, there was a one-to-one correspondence between notes in the lattice and JI ratios. But now, any ratio can have the Bizozogu ~~comma~~ 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!

Before, there was a one-to-one 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 this new lattice with commas marked:

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

~~The commas are unnamed because one~~ spot on the lattice represents multiple commas. ~~For example~~, ~~the green dot directly above the red D in the middle represents both the~~ [[49/48|Zozo]] ~~and~~ [[50/49|Biruyo]] ~~commas.~~

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

* no-fret: Labizoyo, Zozoyo/Rurutriyo, [[1029/1024|Latrizo]] and Satrizo-agu/Saruyo

* half-fret: Laquinzo/Lazoyoyo, Sazoyo, [[49/48|Zozo]]/[[50/49|Biruyo]], Latrizo-agugu/Laruyo, Saquadru and Zotrigu/Triru-agu

(TO DO: add links to the ratios)

Using 2-D steps, the Ruyoyo comma is ~~now~~ 5 rightward steps and 3 upwards steps. Using 3-D steps, from the red D to ~~the red vCx~~ has a familiar shape: two 5thwd steps, two yoward ~~stets~~ and a ruward step. From the red ^Ebb to the red D has a similar shape.

Using 2-D steps, the Ruyoyo comma is always 5 rightward steps and 3 upwards steps. Using 3-D 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 ~~between the two green notes directly above the red D~~ to the ~~red vCx.~~ it's one and a half 5thwd steps, one yoward step and one zoward step. From ~~the green ^D (~~the Ru comma ~~in the 4th lattice)~~ to ~~the green dot~~ is the same steps. So the 2-D shape of the comma doesn't change, but the 3-D 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.

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 2-D shape of the comma doesn't change, but the 3-D 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.

=== The 11-limit (yazala) and 13-limit (yazalatha) Lattice ===

@@ Line 1: / Line 1: @@
+(A work in progress...)
+== Overview ==
+This page explains commas and lattices for the [[Kite Guitar|Kite guitarist]].
 == Lattices ==
-Lattices are a great tool for exploring just intonation (aka JI) and also near-just music in the larger edos.
 === The 5-limit (ya) Lattice ===
 This lattice uses [[Ups and Downs Notation|ups and downs notation]]:
@@ Line 38: / Line 41: @@
 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, and you'll have done something quite unique!
-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. This is another comma that 12-equal tempers out, but is very rarely pumped.
+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 are too remote. This lattice zooms out to reveal the nearest three, plus their descending versions.
+[[File:41equal lattices big.png|none|thumb|477x477px]]
+The three new 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 half-fret comma, [[16875/16384|Laquadyo]].
-There are two more remote no-fret commas. From Sagugu to Triyo is the [[Magic|Laquinyo]] comma. From Gu to Triyo is the Saquadyo comma. Neither of these vanish in 12-equal. Saquadyo is the comma that equates the double-up row with the double-down row.
+The sheer remoteness of ya zero-fret commas tells us that most ya music on the Kite guitar that travels around the lattice widely will tend to have pitch shifts. This Layo pump that I wrote is an exception:
+I^m7 - ^bVIv7  (4x)
+^bIIv7 - [^bV=#IV]v7 - VIIv7 - IIIv7 - VIv7 - IIv7 - Vv7 - Vv9
+I^m7 - ^bVIv7  (4x)
+Once we add prime 7, we get easily pumpable zero-fret commas.
 === 7-limit (yaza) commas ===
 [[File:41equal lattice 7-limit with commas.png|none|thumb|482x482px]]
-This lattice introduces two no-fret commas, [[5120/5103|Saruyo]] and [[225/224|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 in 41-equal between an ascending comma pump and a descending one, and both versions can be treated as the same.
+This lattice introduces two no-fret commas, [[5120/5103|Saruyo]] and [[225/224|Ruyoyo]]. Both are reasonably close and fairly pumpable, especially Ruyoyo. The unlabeled red notes are just the descending versions of these two commas. Since the comma maps to zero frets, there is no difference in 41-equal between an ascending comma pump and a descending one, and both versions can be treated as the same.
 An example of a Saruyo pump is [[Kite Guitar Translations by Kite Giedraitis#I Will Survive .28Gloria Gaynor.29|I Will Survive]].
@@ Line 50: / Line 63: @@
 You can play such progressions without worrying. Lame joke: without fretting, that's why it's called a no-fret comma!
-This lattice also introduces the green [[64/63|Ru]] comma, very important because it's so nearby. Like Gu, this one...
+This lattice also introduces the green [[64/63|Ru]] comma, very important because it's so nearby, and so easy to pump.
 == Lattices Part II ==
@@ Line 58: / Line 71: @@
 As previously noted, the 7-limit lattice is limited to only three layers, zo = 7-over, ru = 7-under and noza = no-7s. This is rather limiting. For example, any comma with a 7-exponent 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 assume the [[2401/2400|Bizozogu]] microcomma is tempered out. This comma is only 0.7¢ and is nearly impossible to hear. Strangely enough, it doesn't vanish in 12-equal, although it does in 41-equal! This comma equates the layer above zo with the one below ru. 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.
 [[File:41equal lattice 11-limit.png|none|thumb|544x544px]]
-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. From D up to G# is a vertical step. Thus one upward step is just over half an octave, and two upward steps octave-reduces to a half-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. From D up to G# is a vertical step. Thus one upward step is just over half an octave, and two upward steps octave-reduces to a half-fret comma.
-Before, there was a one-to-one correspondence between notes in the lattice and JI ratios. But now, any ratio can have the Bizozogu comma 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!
+Before, there was a one-to-one 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 this new lattice with commas marked:
 [[File:41equal lattice 11-limit with commas.png|none|thumb|558x558px]]
-The commas are unnamed because one spot on the lattice represents multiple commas. For example, the green dot directly above the red D in the middle represents both the [[49/48|Zozo]] and [[50/49|Biruyo]] commas.
+Each green or red spot on the lattice represents multiple commas that differ by the Bizozogu microcomma. The new commas are:
+* no-fret: Labizoyo, Zozoyo/Rurutriyo, [[1029/1024|Latrizo]] and Satrizo-agu/Saruyo
+* half-fret: Laquinzo/Lazoyoyo, Sazoyo, [[49/48|Zozo]]/[[50/49|Biruyo]], Latrizo-agugu/Laruyo, Saquadru and Zotrigu/Triru-agu
+(TO DO: add links to the ratios)
-Using 2-D steps, the Ruyoyo comma is now 5 rightward steps and 3 upwards steps. Using 3-D steps, from the red D to the red vCx has a familiar shape: two 5thwd steps, two yoward stets and a ruward step. From the red ^Ebb to the red D has a similar shape.
+Using 2-D steps, the Ruyoyo comma is always 5 rightward steps and 3 upwards steps. Using 3-D 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 between the two green notes directly above the red D to the red vCx. it's one and a half 5thwd steps, one yoward step and one zoward step. From the green ^D (the Ru comma in the 4th lattice) to the green dot is the same steps. So the 2-D shape of the comma doesn't change, but the 3-D 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.
+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 2-D shape of the comma doesn't change, but the 3-D 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.
 === The 11-limit (yazala) and 13-limit (yazalatha) Lattice ===