Kite's thoughts on 41edo Lattices: Difference between revisions
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== Lattices == | == Lattices == | ||
=== The 5-limit (ya) Lattice === | === The 5-limit (ya) Lattice === | ||
This lattice uses [[Ups and | 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. | ||
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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 [[Arrow|arrows]] (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 | 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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=== 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 | 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 | ||
!as a yazala comma | |||
!yazala | !as a yazalatha comma | ||
!yazalatha | |||
|- | |- | ||
|red vCx | |red vCx | ||
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== Commas Part II == | == Commas Part II == | ||
The Kite guitar fretboard is made up of alternating rainbow zones and | 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. | ||