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

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autopatrolled, Sysops
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expanded the 11-limit and 13-limit section and the table at the end, other changes too
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=== 7-limit (yaza) commas ===
=== 7-limit (yaza) commas ===
Once we add prime 7, we get easily pumpable no-fret commas.[[File:41equal lattice 7-limit with commas.png|none|thumb|482x482px]]
Once we add prime 7, we get easily pumpable no-fret commas.[[File:41equal lattice 7-limit with commas.png|none|thumb|482x482px]]
This lattice introduces three no-fret commas, [[875/864|Zotriyo]], [[5120/5103|Saruyo]] and [[225/224|Ruyoyo]]. 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!
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!


This lattice also introduces the half-fret [[64/63|Ru]] comma, a very important comma because it's so nearby, and so easy to pump.
This lattice also introduces the half-fret [[64/63|Ru]] comma, a very important comma because it's so nearby, and so easy to pump. The other new half-fret commas are [[525/512|Lazoyoyo]], [[2240/2187|Sazoyo]], [[3645/3584|Laruyo]] and [[126/125|Zotrigu]].


== Lattices Part II ==
== Lattices Part II ==
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=== The Full Yaza Lattice ===
=== The Full Yaza Lattice ===


As previously noted, the 7-limit lattice is limited to only three layers, zo = 7-over, ru = 7-under and ya = 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 use the [[2401/2400|Bizozogu]] microcomma. This comma is only 0.7¢ and is nearly impossible to hear. It is a no-fret comma with a 7-exponent of 4. Strangely enough, it doesn't vanish in 12-equal! This comma equates the layer above zo with the one below ru. Thus the lattic ewraps around and there are only 4 layers total. 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.
As previously noted, the 7-limit lattice is limited to only three layers, zo = 7-over, ru = 7-under and ya = 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 use the [[2401/2400|Bizozogu]] microcomma. This comma is only 0.7¢ and is nearly impossible to hear. It is a no-fret comma with a 7-exponent of 4. Strangely enough, it doesn't vanish in 12-equal! This comma equates the layer above zo with the one below ru. Thus the lattice wraps around and there are only 4 layers total. 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]]
[[File:41equal lattice 11-limit.png|none|thumb|544x544px]]
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 (21\41), 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 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.


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


* no-fret: [[33075/32768|Labizoyo]], [[245/243|Zozoyo]]/[[4000/3969|Rurutriyo]], [[1029/1024|Latrizo]] and [[10976/10935|Satrizo-agu]]/[[5120/5103|Saruyo]]
* no-fret: [[33075/32768|Labizoyo]], [[245/243|Zozoyo]], [[4000/3969|Rurutriyo]], [[1029/1024|Latrizo]] and [[10976/10935|Satrizo-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]]
* half-fret: [[16807/16384|Laquinzo]], [[49/48|Zozo]], [[50/49|Biruyo]], [[83349/81920|Latrizo-agugu]], [[65536/64827|Saquadru]] and [[1728/1715|Triru-agu]]


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.
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.
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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.  
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. From Ruyoyo to Zozo is a Ru comma, as i Zozoyo to Sazoyo.
Something similar happens with the Ru comma. From Ruyoyo to Zozo is a Ru comma, as is Zozoyo to Sazoyo.


=== The 11-limit (yazala) and 13-limit (yazalatha) Lattice ===
=== 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.
This lattice can also be applied to higher prime limits by assuming the [[243/242|Lulu]] and [[512/507|Thuthu]] commas vanish. 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 it works well for 41-equal because both commas are no-fret commas. Moving 5 steps to the right represents the 11/8 interval. Moving 1 step to the left represents the 13/8 interval. A single 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, to Bizozogu add Luyoyo and/or Luzozogu and replace the yaza commas with yazala commas.)
Each red, green and blue note can be re-interpreted as a higher-prime-limit comma. Some examples:
{| class="wikitable"
|+
!note
!yaza comma
| rowspan="8" |'''is also'''
!yazala commas
!yazalatha commas
|-
|red vCx
|Ruyoyo
|[[385/384|Lozoyo]]
|[[105/104|Thuzoyo]], [[275/273|Thuloruyoyo]]
|-
|red dot
|Zozoyo
|[[100/99|Luyoyo]], [[245/242|Biluzo-ayo]]
|[[325/324|Thoyoyo]]
|-
|green dot
|Zozo
|[[45/44|Loyo]]
|[[65/64|Thoyo]]
|-
|red D
|Bizozogu
|Lulu, [[441/440|Luzozogu]]
|Thuthu, [[144/143|Thulu]], [[352/351|Thulo]], [[196/195|Thuzozogu]]
|-
|green ^D
|Ru
|[[56/55|Luzogu]]
|[[78/77|Tholuru]], [[91/90|Thozogu]], [[169/168|Thothoru]]
|-
|green ^D
|Gu
|[[99/98|Loruru]], [[121/120|Lologu]]
|[[66/65|Thulogu]]
|-
|green ^^Ebb
|Zotrigu
|[[176/175|Lorugugu]]
|[[351/350|Thorugugu]]
|}


== Commas Part II ==
== Commas Part II ==
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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.
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.
Is there a way to determine how far a comma will make us walk? Yes!


5-limit JI is a rank-3 tuning, and 41-equal is a rank-1 tuning. [[Regular temperament theory]] tells us that any two linearly independent (i.e. non-redundant) no-fret ya commas suffice to reduce 5-limit JI to 41-equal. There are many pairs of commas that will work; let's use Laquinyo and Layo. Since they are linearly independent, any ya no-fret comma can be expressed uniquely as the sum or difference of these two commas. For example, the [[41-comma|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]]).
5-limit JI is a rank-3 tuning, and 41-equal is a rank-1 tuning. [[Regular temperament theory]] tells us that any two linearly independent (i.e. non-redundant) no-fret ya commas suffice to reduce 5-limit JI to 41-equal. There are many pairs of commas that will work; let's use Laquinyo and Layo. Since they are linearly independent, any ya no-fret comma can be expressed uniquely as the sum or difference of these two commas. For example, the [[41-comma|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]]).
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Every comma's [[monzo|monzo or lattice vector]] has a certain power of three, which we can call P. The comma's walking distance corresponds roughly to P/8. These powers of three correspond to yoward/guward diagonal lines on the lattice. Thus the walking distance of a comma is roughly proportional to its horizontal distance from the line running through ^Bb, D and vF#. Note that there is only one no-walk ya comma, Laquinyo.  
Every comma's [[monzo|monzo or lattice vector]] has a certain power of three, which we can call P. The comma's walking distance corresponds roughly to P/8. These powers of three correspond to yoward/guward diagonal lines on the lattice. Thus the walking distance of a comma is roughly proportional to its horizontal distance from the line running through ^Bb, D and vF#. Note that there is only one no-walk ya comma, Laquinyo.  


For za and yaza commas, before dividing by 8, add 2 for every "zo" in the name and subtract 2 for every "ru".
For za and yaza commas, before dividing by 8, add 2 for every "zo" in the name and subtract 2 for every "ru". In the table below, the resulting number indicates which column the comma is in.
{| class="wikitable"
{| class="wikitable"
|+selected 41-equal commas
|+various 41-equal commas with their 3-exponents and (doubled) 7-exponents
!
! rowspan="2" |
!no-walk
!no-walk
!walk-halfway
!walk-halfway
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!walk-one-and-a-half
!walk-one-and-a-half
|-
|-
! rowspan="4" |ya
!(no-fret)
!(half-fret)
!(no-fret)
!(half-fret)
|-
! rowspan="4" |wa
and
 
ya
 
|Laquinyo = -1
|Laquinyo = -1
|Gu = 4
|Gu = 4
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! colspan="5" |
! colspan="5" |
|-
|-
! rowspan="5" |yaza
! rowspan="5" |za
|Zotriyo = -3 + 2
and
|Zotrigu = 2 + 2
 
|Latrizo = 1 + 6
yaza
 
 
|Zotriyo = -3 + 2 = -1
|Zotrigu = 2 + 2 = 4
|Latrizo = 1 + 6 = 7
|
|
|-
|-
|Ruyoyo = 2 - 2
|Ruyoyo = 2 - 2 = 0
|Ru = -2 - 2
|Ru = -2 - 2 = -4
|Saruyo = -6 - 2
|Saruyo = -6 - 2 = -8
|
|
|-
|-
|Zozoyo = -5 + 4
|Zozoyo = -5 + 4 = -1
|Zozo = -1 + 4
|Zozo = -1 + 4 = -3
|
|
|
|
|-
|-
|
|
|Biruyo = 0 - 4
|Biruyo = 0 - 4 = -4
|
|
|
|
|-
|-
|
|
|Laruyo = 6 - 1
|Laruyo = 6 - 1 = 5
|
|
|
|
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