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

Line 45:

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]].

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 descending Layo pump ~~that I wrote~~ is an exception:

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 descending Layo pump is an exception:

I^m7 - ^bVIv7 (4x)

* I^m7 - ^bVIv7 (4x)

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

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

* I^m7 - ^bVIv7 (4x)

I^m7 - ^bVIv7 (4x)

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

Line 57:

Line 55:

=== 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, 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.

This lattice introduces three no-fret commas, [[875/864|Zotriyo]], [[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]].~~

(TO DO: add Zotriyo to the lattice)

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

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!

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

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

== Lattices Part II ==

Line 79:

Line 77:

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 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.

Line 90:

Line 86:

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

This lattice can also be applied to higher prime limits by assuming the [[243/242|Lulu]] and [[512/507|Thuthu]] commas. These are not microcommas, and are fairly audible at 8¢ and 17¢ respectively. Thus this wouldn't be a very accurate lattice for JI. But this lattice works well for 41-equal because both commas are no-fret commas. The rightward step becomes both [[11/9]] and [[27/22]], as well as [[39/32]] and [[16/13]], all neutral 3rds.

(TO DO: in this lattice, replace Bizozogu with Lulu and replace the yaza commas with yazala commas. Same with Thuthu.)

== Commas Part II ==

The Kite guitar fretboard is made up of alternating rainbow zones and off zones. Playing a Gu or Ru pump on the Kite guitar forces one to either shift a pitch or walk into the off zone. Going into the off zone makes the tonic drift sharp or flat by half a fret. But playing a Saruyo pump forces one to walk up the neck clear through the off zone into the next rainbow zone. The tonic drifts flat and then sharp!

(work in progress...)

The Kite guitar fretboard is made up of alternating rainbow zones and off zones. Playing a Gu or Ru pump on the Kite guitar forces one to either shift a pitch or walk into the off zone. Going into the off zone makes the tonic drift sharp or flat by half a fret. But playing a Saruyo pump forces one to walk up the neck clear through the off zone into the next rainbow zone. The tonic drifts flat and then sharp! Depending on the melody, this can result in the scale temporarily containing offperfect tonics and fifths.

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.

5-limit JI is a rank-3 tuning, and any edo is a rank-1 tuning. Thus any two linearly independent (i.e. non-redundant) commas suffice to reduce 5-limit JI to an edo. The choice of which two commas determines which edo we end up in. There are many pairs of commas that will result in 41edo. Let's choose Laquinyo and Layo. Since they are linearly independent, any 5-limit zero-fret comma can be expressed uniquely as the sum or difference of these two commas. For example, the Wa-41 comma is one Laquinyo comma minus five Layo commas. Laquinyo is a no-walk comma and Layo is a walk-once comma. Therefore we can determine how far a comma walks by the number of Layo commas it contains. Thus Wa-41 is a walk-five-times comma. (See Kite's [[Kite Guitar Exercises and Techniques by Kite Giedraitis#The Circle of 5ths|circle of 5ths exercise]]).

~~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-fet~~ commas ~~are walk-halfway, walk-one-and-a-half, etc~~.

For 7-limit, we need three zero-fret commas. Since Ruyoyo is a no-walk comma, let's use Laquinyo, Layo and Ruyoyo. Again, the number of Layo commas determines how far the comma pump walks.

~~Rank~~ and Co-~~rank~~

Every comma's [[monzo]] has a certain power of three. The comma's walking distance corresponds roughly to one eighth of this power of three.

{| class="wikitable"

|+selected 41-edo commas

!no-walk

!walk-halfway

!walk-once

!walk-one-and-a-half

|-

|Laquinyo = -1

|Gu = 4

|Layo = 8

|Lalawa = 12

|-

|Ruyoyo = 2

|Ru = -2

|Saruyo = -6

|

|}

5-limit JI is a rank-3 tuning, and any edo is a rank-1 tuning. Thus any two linearly independent (i.e. non-redundant) commas suffice to reduce 5-limit JI to an edo. The choice of which two commas determines which edo we end up in.

[[Category: Kite Guitar]]

[[Category:41edo]]

@@ Line 45: / Line 45: @@
 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]].
-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 descending Layo pump that I wrote is an exception:
+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 descending Layo pump is an exception:
-I^m7 - ^bVIv7  (4x)
+* I^m7 - ^bVIv7  (4x)
+* ^bIIv7 - [^bV=#IV]v7 - VIIv7 - IIIv7 - VIv7 - IIv7 - Vv7 - Vv9
-^bIIv7 - [^bV=#IV]v7 - VIIv7 - IIIv7 - VIv7 - IIv7 - Vv7 - Vv9
+* I^m7 - ^bVIv7  (4x)
-I^m7 - ^bVIv7  (4x)
 Once we add prime 7, we get easily pumpable zero-fret commas.
@@ Line 57: / Line 55: @@
 === 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, 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.
+This lattice introduces three no-fret commas, [[875/864|Zotriyo]], [[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]].
+(TO DO: add Zotriyo to the lattice)
-You can play such progressions without worrying. Lame joke: without fretting, that's why it's called a no-fret comma!
+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!
-This lattice also introduces the green [[64/63|Ru]] comma, very important because it's so nearby, and so easy to pump.
+This lattice also introduces the half-fret [[64/63|Ru]] comma, very important because it's so nearby, and so easy to pump.
 == Lattices Part II ==
@@ Line 79: / Line 77: @@
 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
+* no-fret: [[33075/32768|Labizoyo]], [[245/243|Zozoyo]]/[[4000/3969|Rurutriyo]], [[1029/1024|Latrizo]] and [[10976/10935|Satrizo-agu]]/[[5120/5103|Saruyo]]
-* half-fret: Laquinzo/Lazoyoyo, Sazoyo, [[49/48|Zozo]]/[[50/49|Biruyo]], Latrizo-agugu/Laruyo, Saquadru and Zotrigu/Triru-agu
+* half-fret: [[16807/16384|Laquinzo]]/[[525/512|Lazoyoyo]], [[2240/2187|Sazoyo]], [[49/48|Zozo]]/[[50/49|Biruyo]], [[83349/81920|Latrizo-agugu]]/[[3645/3584|Laruyo]], [[65536/64827|Saquadru]] and [[126/125|Zotrigu]]/[[1728/1715|Triru-agu]]
-(TO DO: add links to the ratios)
 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.
@@ Line 90: / Line 86: @@
 === The 11-limit (yazala) and 13-limit (yazalatha) Lattice ===
 This lattice can also be applied to higher prime limits by assuming the [[243/242|Lulu]] and [[512/507|Thuthu]] commas. These are not microcommas, and are fairly audible at 8¢ and 17¢ respectively. Thus this wouldn't be a very accurate lattice for JI. But this lattice works well for 41-equal because both commas are no-fret commas. The rightward step becomes both [[11/9]] and [[27/22]], as well as [[39/32]] and [[16/13]], all neutral 3rds.
+(TO DO: in this lattice, replace Bizozogu with Lulu and replace the yaza commas with yazala commas. Same with Thuthu.)
 == Commas Part II ==
-The Kite guitar fretboard is made up of alternating rainbow zones and off zones. Playing a Gu or Ru pump on the Kite guitar forces one to either shift a pitch or walk into the off zone. Going into the off zone makes the tonic drift sharp or flat by half a fret. But playing a Saruyo pump forces one to walk up the neck clear through the off zone into the next rainbow zone. The tonic drifts flat and then sharp!
+(work in progress...)
+The Kite guitar fretboard is made up of alternating rainbow zones and off zones. Playing a Gu or Ru pump on the Kite guitar forces one to either shift a pitch or walk into the off zone. Going into the off zone makes the tonic drift sharp or flat by half a fret. But playing a Saruyo pump forces one to walk up the neck clear through the off zone into the next rainbow zone. The tonic drifts flat and then sharp! Depending on the melody, this can result in the scale temporarily containing offperfect tonics and fifths.
+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.
+-limit JI is a rank-3 tuning, and any edo is a rank-1 tuning. Thus any two linearly independent (i.e. non-redundant) commas suffice to reduce 5-limit JI to an edo. The choice of which two commas determines which edo we end up in. There are many pairs of commas that will result in 41edo. Let's choose Laquinyo and Layo. Since they are linearly independent, any 5-limit zero-fret comma can be expressed uniquely as the sum or difference of these two commas. For example, the Wa-41 comma is one Laquinyo comma minus five Layo commas. Laquinyo is a no-walk comma and Layo is a walk-once comma. Therefore we can determine how far a comma walks by the number of Layo commas it contains. Thus Wa-41 is a walk-five-times comma. (See Kite's [[Kite Guitar Exercises and Techniques by Kite Giedraitis#The Circle of 5ths|circle of 5ths exercise]]).
-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-fet commas are walk-halfway, walk-one-and-a-half, etc.
+For 7-limit, we need three zero-fret commas. Since Ruyoyo is a no-walk comma, let's use Laquinyo, Layo and Ruyoyo. Again, the number of Layo commas determines how far the comma pump walks.
-Rank and Co-rank
+Every comma's [[monzo]] has a certain power of three. The comma's walking distance corresponds roughly to one eighth of this power of three.
+{| class="wikitable"
+|+selected 41-edo commas
+!no-walk
+!walk-halfway
+!walk-once
+!walk-one-and-a-half
+|-
+|Laquinyo = -1
+|Gu = 4
+|Layo = 8
+|Lalawa = 12
+|-
+|Ruyoyo = 2
+|Ru = -2
+|Saruyo = -6
+|
+|}
--limit JI is a rank-3 tuning, and any edo is a rank-1 tuning. Thus any two linearly independent (i.e. non-redundant) commas suffice to reduce 5-limit JI to an edo. The choice of which two commas determines which edo we end up in.
+[[Category: Kite Guitar]]
+[[Category:41edo]]