Kite's thoughts on 41edo Lattices: Difference between revisions
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== Overview == | == Overview == | ||
This page explains commas | This page explains lattices and commas for the [[Kite Guitar|Kite guitarist]]. | ||
== Lattices == | == Lattices == | ||
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=== The 7-limit (yaza) lattice === | === The 7-limit (yaza) lattice === | ||
[[File:41equal lattice 7-limit.png|none|thumb|463x463px]] | [[File:41equal lattice 7-limit.png|none|thumb|463x463px]] | ||
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 | 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 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. | ||
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(work in progress...) | (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. | 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, which can be disconcerting. | ||
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. | ||
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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. | 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. | ||
Every comma's [[monzo]] has a certain power of three. The comma's walking distance corresponds roughly to | Every comma's [[monzo|monzolattice 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 yaza commas, before dividing by 8, add 2 for every "zo" in the name and subtract 2 for every "ru". | |||
{| class="wikitable" | {| class="wikitable" | ||
|+selected 41-edo commas | |+selected 41-edo commas | ||
! | |||
!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 | |||
|Laquinyo = -1 | |Laquinyo = -1 | ||
|Gu = 4 | |Gu = 4 | ||
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|Lalawa = 12 | |Lalawa = 12 | ||
|- | |- | ||
|Ruyoyo = 2 | | | ||
|Ru = -2 | |Laquadyo = 3 | ||
|Saruyo = -6 | |Saquadyo = -9 | ||
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|- | |||
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|Sagugu = -4 | |||
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|- | |||
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|Triyo = -5 | |||
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|- | |||
! rowspan="2" |yaza | |||
|Zotriyo = -3 + 2 | |||
|Zotrigu = 2 + 2 | |||
|Latrizo = 1 + 6 | |||
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|- | |||
|Ruyoyo = 2 - 2 | |||
|Ru = -2 - 2 | |||
|Saruyo = -6 - 2 | |||
| | | | ||
|} | |} | ||