Kite's thoughts on 41edo Lattices - Revision history

TallKite: TallKite moved page 41edo Lattices to Kite's thoughts on 41edo Lattices: Editor VectorGraphics has repeatedly edited pages I've written about my own research and inserted misinformation. On discord he is openly hostile to me. This move is necessary to avoid a toxic work environment for me.

2025-08-20T01:28:15Z

TallKite moved page 41edo Lattices to Kite's thoughts on 41edo Lattices: Editor VectorGraphics has repeatedly edited pages I've written about my own research and inserted misinformation. On discord he is openly hostile to me. This move is necessary to avoid a toxic work environment for me.

← Older revision	Revision as of 01:28, 20 August 2025
(No difference)

Fredg999: Text replacement - "Ups and Downs Notation" to "Ups and downs notation"

2025-08-20T01:07:31Z

Text replacement - "Ups and Downs Notation" to "Ups and downs notation"

← Older revision		Revision as of 01:07, 20 August 2025
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	== Lattices ==		== Lattices ==
	=== The 5-limit (ya) Lattice ===		=== The 5-limit (ya) Lattice ===
	This lattice uses [[Ups and ~~Downs Notation~~\|ups and downs notation]] to name the [[41-edo\|41-edo (aka 41-equal)]] notes:		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]]		[[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. 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.		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.

TallKite: changed "off zone" to "complex zone"

2024-11-18T22:38:27Z

changed "off zone" to "complex zone"

← Older revision		Revision as of 22:38, 18 November 2024
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	== Commas Part II ==		== 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! Depending on the melody, this can result in the scale temporarily containing offperfect tonics, 4ths and 5ths, which can be disconcerting.		The Kite guitar fretboard is made up of alternating rainbow zones and complex zones. Playing a Gu or Ru pump on the Kite guitar forces one to either shift a pitch or walk into the complex zone. Going into the complex 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 complex 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, 4ths and 5ths, 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.

TallKite at 10:13, 28 June 2023

2023-06-28T10:13:11Z

← Older revision		Revision as of 10:13, 28 June 2023
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	Chord progressions can be mapped out on the lattice as a series of chord shapes. Often two adjacent chords have a common tone, and the progression "walks" around the lattice. If there are no common tones, the progression "jumps" from one area to another. Often there are several different places to jump. More on this later.		Chord progressions can be mapped out on the lattice as a series of chord shapes. Often two adjacent chords have a common tone, and the progression "walks" around the lattice. If there are no common tones, the progression "jumps" from one area to another. Often there are several different places to jump. More on this later.

	This lattice shows the notes as intervals, and shows how many edosteps each note is from the tonic D.		This lattice shows the notes as intervals, and shows how many [[Arrow\|arrows]] (edosteps) each note is from the tonic D.
	[[File:The Kite Tuning lattices-1.png\|none\|thumb\|499x499px]]		[[File:The Kite Tuning lattices-1.png\|none\|thumb\|499x499px]]
	This lattice is smaller than the previous one, but it still extends far enough to include all 41 notes, except 16 and 25 (v4 and ^5). By coincidence, each number appears exactly once.		This lattice is smaller than the previous one, but it still extends far enough to include all 41 notes, except 16 and 25 (v4 and ^5). By coincidence, each number appears exactly once.
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	== Commas ==		== Commas ==
	A comma is a just intonation ratio that is small, less than (roughly) 50¢. In 41-edo aka 41-equal, a comma maps to a small number of ~~edosteps~~, usually 0 or 1, occasionally 2. The technical term for a comma that maps to 0 ~~edosteps~~ is '''vanishing''' comma. On the Kite guitar, mapping to 0, 1 or 2 ~~edosteps~~ can be called a no-fret, half-fret or one-fret comma.		A comma is a just intonation ratio that is small, less than (roughly) 50¢. In 41-edo aka 41-equal, a comma maps to a small number of arrows, usually 0 or 1, occasionally 2. The technical term for a comma that maps to 0 arrows is '''vanishing''' comma. On the Kite guitar, mapping to 0, 1 or 2 arrows can be called a no-fret, half-fret or one-fret comma.

	=== 5-limit (ya) commas ===		=== 5-limit (ya) commas ===
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	!yaza comma		!yaza comma
	!as a yazala comma		!as a yazala comma
	!as a yazalatha ~~commas~~		!as a yazalatha comma
	\|-		\|-
	\|red vCx		\|red vCx

TallKite: added lattices of intervals, some rewriting too

2022-10-01T07:43:46Z

added lattices of intervals, some rewriting too

← Older revision		Revision as of 07:43, 1 October 2022
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	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.		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 (aka dud), triple-down (aka trud), etc. (There's also dup and trup for ^^ and ^^^.) 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].		The middle row is the plain row. The row immediately above it is the down row. Then double-down (aka '''dud'''), triple-down (aka '''trud'''), etc. (There's also '''dup''' and '''trup''' for ^^ and ^^^.) 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 dud row can be rewritten as a dup row. For example, vvB = ^^Bb. And trud notes would in practice almost always be 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.		If this were just intonation, the lattice would extend infinitely in all directions. But because this is 41-equal, the dud row can be rewritten as a dup row. For example, vvB = ^^Bb. And trud notes would in practice almost always be 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.

	Every chord type has a certain shape. Downmajor aka 5-over or yo chords such as D vF# A appear as upward-pointing triangles. Upminor aka 5-under or gu chords are downward-pointing triangles. The downmajor7 chord is two adjacent triangles, as is the upminor7 chord. The downmajor6 chord has the same shape as the upminor7 chord, which tells you that they are chord homonyms (same notes, different roots).		Every chord type has a certain shape. Downmajor aka 5-over or yo chords such as D vF# A appear as upward-pointing triangles. Upminor aka 5-under or gu chords e.g. D ^F A are downward-pointing triangles. The downmajor7 chord (D vF# A vC#) is two adjacent triangles, as is the upminor7 chord (D ^F A ^C). The downmajor6 chord (D vF# A vB) has the same shape as the upminor7 chord, which tells you that they are chord homonyms (same notes, different roots).

	Chord progressions can be mapped out on the lattice as a series of chord shapes. Often two adjacent chords have a common tone, and the progression "walks" around the lattice. If there are no common tones, the progression "jumps" from one area to another. Often there are several different places to jump. More on this later.		Chord progressions can be mapped out on the lattice as a series of chord shapes. Often two adjacent chords have a common tone, and the progression "walks" around the lattice. If there are no common tones, the progression "jumps" from one area to another. Often there are several different places to jump. More on this later.

			This lattice shows the notes as intervals, and shows how many edosteps each note is from the tonic D.
			[[File:The Kite Tuning lattices-1.png\|none\|thumb\|499x499px]]
			This lattice is smaller than the previous one, but it still extends far enough to include all 41 notes, except 16 and 25 (v4 and ^5). By coincidence, each number appears exactly once.

	=== The 7-limit (yaza) lattice ===		=== The 7-limit (yaza) lattice ===
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	In the 5-limit lattice, each note has a unique name. To find vvD#, go to the dud 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 dud 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.

			Here is this lattice as intervals. Each color represents a separate layer of the lattice.[[File:The Kite Tuning lattices-2.png\|thumb\|477x477px\|link=https://en.xen.wiki/w/File:The_Kite_Tuning_lattices-2.png\|alt=\|none]]With so many more notes than the previous lattice of intervals, each number now appears multiple times. In particular, both 0 and 1 appear twice. More on that later.

	== Commas ==		== Commas ==
	A comma is a just intonation ratio that is less than (roughly) 50¢. In 41-edo aka 41-equal, a comma maps to a small number of edosteps, usually 0 or 1, occasionally 2. The technical term for a comma that maps to 0 edosteps is '''vanishing''' comma. On the Kite guitar, mapping to 0, 1 or 2 edosteps can be called a no-fret, half-fret or one-fret comma.		A comma is a just intonation ratio that is small, less than (roughly) 50¢. In 41-edo aka 41-equal, a comma maps to a small number of edosteps, usually 0 or 1, occasionally 2. The technical term for a comma that maps to 0 edosteps is '''vanishing''' comma. On the Kite guitar, mapping to 0, 1 or 2 edosteps can be called a no-fret, half-fret or one-fret comma.

	=== 5-limit (ya) commas ===		=== 5-limit (ya) commas ===
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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 non-remote, 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]], [[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 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!

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	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.		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!

	Here's ~~this new~~ lattice with commas marked:		Here's the lattice as intervals. How many zeros and ones can you find?[[File:The Kite Tuning lattices-3.png\|thumb\|571x571px\|link=https://en.xen.wiki/w/File:The_Kite_Tuning_lattices-3.png\|alt=\|none]]Here's the lattice with the commas marked:
	[[File:41equal lattice 11-limit with commas.png\|none\|thumb\|558x558px]]		[[File:41equal lattice 11-limit with commas.png\|none\|thumb\|558x558px]]
	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:
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	{\| class="wikitable"		{\| class="wikitable"
	\|+		\|+
			yaza commas re-interpreted as a higher-prime-limit comma
	!note		!note
	!yaza comma		!yaza comma
	~~\| rowspan="8" \|'''is also'''~~		!as a yazala comma
	!yazala ~~commas~~		!as a yazalatha commas
	!yazalatha commas
	\|-		\|-
	\|red vCx		\|red vCx

TallKite at 00:45, 23 September 2022

2022-09-23T00:45:02Z

TallKite: expanded the 11-limit and 13-limit section and the table at the end, other changes too

2022-01-06T08:14:08Z

expanded the 11-limit and 13-limit section and the table at the end, other changes too

Show changes

TallKite: fixed a few things

2021-09-22T20:37:12Z

fixed a few things

TallKite: done

2021-09-22T07:02:05Z

done

Show changes

TallKite: /* Commas Part II */ still WIP

2021-09-21T10:29:13Z

Commas Part II: still WIP

← Older revision		Revision as of 10:29, 21 September 2021
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	== Overview ==		== Overview ==
	This page explains commas ~~and lattices~~ for the [[Kite Guitar\|Kite guitarist]].		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 ~~fix~~ this in Lattices Part II.		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 ~~one eighth~~ of ~~this power~~ of ~~three~~.		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
			\|
			\|-
			\|
			\|Sagugu = -4
			\|
			\|
			\|-
			\|
			\|Triyo = -5
			\|
			\|
			\|-
			! rowspan="2" \|yaza
			\|Zotriyo = -3 + 2
			\|Zotrigu = 2 + 2
			\|Latrizo = 1 + 6
			\|
			\|-
			\|Ruyoyo = 2 - 2
			\|Ru = -2 - 2
			\|Saruyo = -6 - 2
	\|		\|
	\|}		\|}

← Older revision		Revision as of 00:45, 23 September 2022
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	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.		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. This vector is called a monzo. Every 5-limit monzo translates directly to a series of sideways and diagonal steps, plus octaves.		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, [http://www.tallkite.com/AlternativeTunings.html 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 (aka dud), triple-down (aka trud), etc. (There's also dup and trup for ^^ and ^^^.) 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.		If this were just intonation, the lattice would extend infinitely in all directions. But because this is 41-equal, the dud row can be rewritten as a dup row. For example, vvB = ^^Bb. And trud notes would in practice almost always be 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.

	Every chord type has a certain shape. Downmajor aka 5-over or yo chords such as D vF# A appear as upward-pointing triangles. Upminor aka 5-under or gu chords are downward-pointing triangles. The downmajor7 chord is two adjacent triangles, as is the upminor7 chord. The downmajor6 chord has the same shape as the upminor7 chord, which tells you that they are chord homonyms (same notes, different roots).		Every chord type has a certain shape. Downmajor aka 5-over or yo chords such as D vF# A appear as upward-pointing triangles. Upminor aka 5-under or gu chords are downward-pointing triangles. The downmajor7 chord is two adjacent triangles, as is the upminor7 chord. The downmajor6 chord has the same shape as the upminor7 chord, which tells you that they are chord homonyms (same notes, different roots).
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	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.		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 dud 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.

	== Commas ==		== Commas ==
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	=== 5-limit (ya) commas ===		=== 5-limit (ya) commas ===
	[[File:41equal lattice 5-limit with commas.png\|none\|thumb\|466x466px]]		[[File:41equal lattice 5-limit with commas.png\|none\|thumb\|466x466px]]
	Certain notes are colored red, green and blue for emphasis. (The color choice is arbitrary.) The green notes are all half a fret sharper than the red D. The vector from the red D to any of these green notes is a half-fret comma. The three half-fret commas are named via [[color notation]], and the name tells you which row it's on: [[250/243\|Triyo]] is 3 rows up, [[81/80\|Gu]] is one row down, and [[2048/2025\|Sagugu]] is two rows down. ~~The~~ blue ~~notes are~~ the inverse of ~~the~~ green ~~notes~~, i.e. descending ~~commas~~.		Certain notes are colored red, green and blue for emphasis. (The color choice is arbitrary.) The green notes are all half a fret sharper than the red D. The vector from the red D to any of these green notes is a half-fret comma. The three half-fret commas are named via [[color notation]], and the name tells you which row it's on: [[250/243\|Triyo]] is 3 rows up, [[81/80\|Gu]] is one row down, and [[2048/2025\|Sagugu]] is two rows down. Each blue note is the inverse of a green note, i.e. a descending comma.

	The most important comma historically is the Gu comma, with ratio 81/80. Consider the progression Im - bIII - bVII - IV - Im. On the lattice, it becomes D^m - ^Fv - ^Cv - ^Gv - ^D^m, and it walks you from the red D to the green ^D. This is called a comma pump. Such pumps are a major issue in just intonation. On the Kite guitar fretboard, this comma pump walks you fifthwards, which is towards the nut. The progression I - vi - ii - V - I becomes Dv - vB^m - vE^m - vAv - vDv, which is a descending Gu pump that walks you from the red D to the blue vD. On the guitar, it walks you fourthwards, towards the bridge.		The most important comma historically is the Gu comma, with ratio 81/80. Consider the progression Im - bIII - bVII - IV - Im. On the lattice, it becomes D^m - ^Fv - ^Cv - ^Gv - ^D^m, and it walks you from the red D to the green ^D. This is called a comma pump. Such pumps are a major issue in just intonation. On the Kite guitar fretboard, this comma pump walks you fifthwards, which is towards the nut. The progression I - vi - ii - V - I becomes Dv - vB^m - vE^m - vAv - vDv, which is a descending Gu pump that walks you from the red D to the blue vD. On the guitar, it walks you fourthwards, towards the bridge.
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	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.		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". 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.		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 in 41-equal 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]]		[[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]].		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 dup row with the dud row. There is also a new half-fret comma, [[16875/16384\|Laquadyo]].

	The sheer remoteness of ya no-fret commas tells us that most ya music on the Kite guitar that travels around the lattice widely will tend to have pitch shifts, unless it backtracks. This descending Layo pump is an exception:		The sheer remoteness of ya no-fret commas tells us that most ya music on the Kite guitar that travels around the lattice widely will tend to have pitch shifts, unless it backtracks. This descending Layo pump is an exception:

← Older revision		Revision as of 20:37, 22 September 2021
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	== Overview ==		== 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 ==		== Lattices ==
	=== The 5-limit (ya) Lattice ===		=== 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]]		[[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.		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.
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	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.		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]]		[[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]].		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]].