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Kite Giedraitis's Categorizations of 41edo Scales: Difference between revisions

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==41-edo MOS scales==
==41-edo MOS scales==
Most MOS scales either lack a perfect 5th or are awkward to play on the Kite Guitar. Awkward scales require more than 3 string hops per octave, or moves by more than 4 frets. Moves are explained in [[Kite Guitar Scales]]. 
We can find all non-awkward MOS scales by requiring that one step size be an odd number of edosteps and the other be even, and further requiring that there are exactly 3 of the first step size. Then we simply make a table with odd step sizes on the top and even ones on the side. Not all odd/even combinations make a MOS scale, because 41 minus the 3 odd steps isn't always a multiple of the even step size. In those cases a 3rd step size is used once. It's named either XL or xs or m. Often there is more than one 3rd step possible. Alternatively, we can avoid the 3rd step size by allowing non-octave scales, as in the Bohlen-Pierce scale in the bottom row.
Each column header is a string-hopping move. The first column heading is "-5 = 3 = m2", which means that you go back 5 frets when hopping, which equals 3\41, which equals a plain minor 2nd. Each row header is a string-sliding move in a similar format. "+1 = 2 = vm2" means go up 1 fret = 2\41 = a downmiinor 2nd.
Geometrically, each column header defines a diagonal line, except the vM3 column which defines a line parallel to the frets. Each row header says how many frets apart these lines are. These geometrical patterns make (possibly non-octave) MOS scales. Often the first line of each table entry describes this geometry. Once one masters these geometrical patterns, one can flit about the fretboard and use MOS scales to quickly span large intervals.
TO DO: make fretboard diagrams of these geometrical patterns.
The Laquinyo scales use +1 moves and make a solid block on the fretboard. The Laquinyo generator is a vM3.
The Sasa-tritribizo scales use +2 moves and -1 / -3 / -5 moves and make a "checkerboard" pattern on the fretboard. The Sasa-tritribizo generator is an ^M3.
{| class="wikitable"
|+
!
!-5 = 3 = m2
!-4 = 5 = ~2
!-3 = 7 = M2
!-2 = 9 = vm3
!-1 = 11 = ^m3
!-0 = 13 = vM3
!--1 = 15 = ^M3
|-
!+1 = 2 = vm2
Laquinyo
(P8, P12/5)
|solid block
3L 16s = 19
L=3, s=2
|solid block
3L 13s = 16
L=5, s=2
|solid block
3L 10s = 13
L=7, s=2
|solid block
3L 7s = 10
L=9, s=2
|solid block
3L 4s = 7
L=11, s=2
|double harmonic vminor
2L 1m 4s = 7
L=13, m=7, s=2
|vminor Sakura
2L 1m 2s = 5
L=15, m=7, s=2
|-
!+2 = 4 = ^m2
Sasa-tritribizo
|checkerboard
8L 3s = 11
L=4, s=3
|alternate frets
2L 7s 1xs = 10
L=5, s=4, xs=3
twin vminor pentatonic
1XL 3L 5s = 9
XL=6, L=5, s=4
65445-4454
|checkerboard
3L 5s = 8
L=7, s=4
|alternate frets
2L 1m 4s = 7
L=9, m=7, s=4
double harmonic:
4947-494
|checkerboard
3L 2s = 5
L=11, s=4
|90-degree zigzag
2L 1m 2s = 5
L=13, m=7, s=4
^minor Sakura scale:
7,4,13,4,13
Indochinese scale:
13,4,7,13,4
|
|-
!+3 = 6 = vM2
Triyo
(P8, P4/3)
|1XL 4L 3s = 8
XL=8, L=6, s=3
|ya equi-hepta
1XL 4L 2s = 7
XL=7, L=6, s=5
|every 3rd fret
whole-tone
1XL 3L 2s = 6
XL=8, L=7, s=6
|diagonal lines
3x9 + 8 + 6 = 5
3L 1m 1s = 4L 1s
|ya pentatonic
2L 1m 2s = 5
L=11, m=7, s=6
|
|
|-
!+4 = 8 = ^M2
Latrizo
(P8, P5/3)
|diagonal lines
4L 3s = 7
L=8, s=3
|every 4th fret
dots only
1XL 2L 3s = 6
XL=10, L=8, s=5
2L 1m 3s 1xs = 7
L=8, m=6, s=5, xs=4
6585-854
|2L 3s 1xs = 6
L=8, s=7, xs=4
787-874
1XL 2L 2s = 5
XL=11, L=8, s=7
787-8,11
|za pentatonic
2L 2s 1xs = 5
L=9, s=8, xs=7
|
|
|
|-
!+5 = 10 = m3
|every 5th fret
3L 1m 2s = 6
L=10, m=5, s=3
|Bohlen-Pierce
P12 = 4L 5s = 9
L=10, s=5
|wa pentatonic
2L 3s = 5
L=10, s=7
|
|
|
|
|}
=== (Notes to myself) ===
Changing awkward very-near-equal scales to non-awkward less-equal scales:
pentatonic: 8 8 8 8 9 --> 9 9 7 8 8,  9 9 9 8 6, 7 7 7 10 10
hexatonic: 7 7 7 7 7 6 --> 7 7 7 6 6 8, 7 7 9 6 6 6, 9 9 7 6 6 4, ...
heptatonic: 6 6 6 6 6 6 5 --> 5 5 7 6 6 6 6, 7 7 7 6 6 4 4
octatonic: 5 5 5 5 5 5 5 6 --> 5 5 5 6 6 6 4 4
nonatonic: 5 5 5 5 5 4 4 4 4 --> 5 5 5 4 4 4 4 4 6
decatonic: 4 4 4 4 4 4 4 4 4 5 --> 5 5 3 4 4 4 4 4 4 4
eleven: 4 4 4 4 4 4 4 4 3 3 3 -- not awkward!!!
twelve: 4 4 4 4 4 3 3 3 3 3 3 3 --> 3 3 3 4 4 4 4 4 4 4 2 2
The double harmonic scale C ^Db vE F G ^Ab vB C has these chords:
Cv  CvM7
^Dbv, ^Dbv7, ^DbvM7, ^Dv7(v5), ^Dbvm, ^Dbvm7, ^Dbvm7(b5)
vE^m, vE^m6,
F^m, F^m,vM7
*[[List of edo-distinct 41et rank two temperaments]]
*[[List of edo-distinct 41et rank two temperaments]]
Mathematically, 41edo has 20 edo-distinct temperaments, and each one has infinite MOS scales. This table only lists musically useful ones. MOS scales listed are those with:
Mathematically, 41edo has 20 edo-distinct temperaments, and each one has infinite MOS scales. This table only lists musically useful ones. MOS scales listed are those with:
Line 426: Line 647:
* s >= 2
* s >= 2
* L/s <= 4.5
* L/s <= 4.5
* L/s <= 2 if 1L
* L/s <= 2 if there's only one L
{| class="wikitable right-1 right-2"
{| class="wikitable right-1 right-2"
|+Table of Temperaments by generator
|+Table of 41edo Temperaments by generator
|-
|-
!edosteps
!edosteps
Line 665: Line 886:
|
|
|
|
|}
|}  
The moves column is explained  Most MOS scales either lack a perfect 5th or are awkward to play on the Kite Guitar. Awkward scales require more than 3 string hops per octave, or moves by more than 4 frets.
 
The Laquinyo scales use +1 and make a solid block.
 
The Sasa-tritribizo scales use +2 and -1 / -3 / -5 and make a "checkerboard" pattern on the fretboard.
 
{| class="wikitable"
|+
!
!--1 = 15 = ^M3
!-0 = 13 = vM3
!-1 = 11 = ^m3
!-2 = 9 = vm3
!-3 = 7 = M2
!-4 = 5 = ~2
!-5 = 3 = m2
|-
!+1 = 2 = vm2
|vminor Sakura
5 = 2x15 + 7 + 2x2
|double harmonic vminor
7 = 2x13 + 7 + 4x2
|solid block
7 = 3L 4s
|solid block
10 = 3L 7s
|solid block
13 = 3L 10s
|solid block
16 = 3L 13s
|solid block
19 = 3L 16s
|-
!+2 = 4 = ^m2
|
|90-degree zigzag
Sakura scale: 7,4,13-4,13
 
Indochinese scale
 
5 = 2L 1m 2s
|checkerboard
5 = 3L 2s
|alternate frets
double harmonic
 
7 = 2L 1m 4s
 
m = 7 = -3
|checkerboard
8 = 3L 5s
|alternate frets
twin vminor
 
10 = 2x5 + 7x4 + 3
 
9 = 6 + 3x5 + 5x4
 
65445-4454
|checkerboard
11 = 8L 3s
|-
!+3 = 6 = vM2
|
|
|ya pentatonic
5 = 2L 1m 2s
 
m = 7 = -3
|diagonal lines
5 = 3x9 + 8 + 6
 
3L 1m 1s = 4L 1s
|every 3rd fret
whole-tone
 
6 = 1XL 3L 2s
 
XL = 8 = +4
|ya equi-hepta
7 = 1XL 4L 2s
 
XL = 7 = -3
|
|-
!+4 = 8 = ^M2
|
|
|
|za pentatonic
5 = 2L 2s 1xs
 
xs = 7 = -3
|??????
6 = 2x8 + 3x7 + 4
 
787-874
 
5 = 787-8,11
|every 4th fret
dots only
 
6 = 10 + 2x8 + 3x5
 
7 =  2x8 + 3x5 + 6 + 4
 
6585-854
|7 = 4L 3s
|-
!+5 = 10 = m3
|
|
|
|
|5 = 2L 3s
|Bohlen-Pierce
9 = 4L 5s = P12
|
|}
 
penta: 8 8 8 8 9 --> 9 9 7 8 8,  9 9 9 8 6, 7 7 7 10 10
 
hexa: 7 7 7 7 7 6 --> 7 7 7 6 6 8, 7 7 9 6 6 6, 9 9 7 6 6 4, ...
 
hepta: 6 6 6 6 6 6 5 --> 5 5 7 6 6 6 6, 7 7 7 6 6 4 4
 
octa: 5 5 5 5 5 5 5 6 --> 5 5 5 6 6 6 4 4
 
nona: 5 5 5 5 5 4 4 4 4 --> 5 5 5 4 4 4 4 4 6
 
deca: 4 4 4 4 4 4 4 4 4 5 --> 5 5 3 4 4 4 4 4 4 4
 
eleven: 4 4 4 4 4 4 4 4 3 3 3
 
twelve: 4 4 4 4 4 3 3 3 3 3 3 3 --> 3 3 3 4 4 4 4 4 4 4 2 2
 
Double harmonic: 4947-494
 
C ^Db vE F G ^Ab vB C
 
Cv  CvM7
 
^Dbv, ^Dbv7, ^DbvM7, ^Dv7(v5), ^Dbvm, ^Dbvm7, ^Dbvm7(b5)
 
vE^m, vE^m6,
 
F^m, F^m,vM7


[[Category:Kite Guitar]]
[[Category:Kite Guitar]]
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