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

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replaced chromatic with trientonal/fretwise
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=== Chromaticism: semitonal, trientonal and microtonal scales ===
=== Chromaticism: semitonal, trientonal and microtonal scales ===
Most 41-equal intervals suggest a specific ratio, but those only a few edosteps wide don't. Thus the remaining categories don't imply any prime subgroups. Traditional 12-equal chromaticism, which translates to runs played on every other fret, is called semitonal, a conventional term referring to the 12-equal semitone.  Playing a run of notes one fret apart is called trientonal, which means "by third-tones". In a guitar context, it can be called fretwise. Microtonal scales differ from fuzzy scales in having many sequential ^1 intervals, and no steps larger than a vm2. Thus fuzzy means partly but not fully microtonal, and a fuzzy diatonic scale could be called a diatonic/microtonal scale.  
Most 41-equal intervals suggest a specific ratio, but those only a few edosteps wide don't. Thus the remaining categories don't imply any prime subgroups. Traditional 12-equal chromaticism, which translates to runs played on every other fret, is called '''semitonal''', a conventional term referring to the 12-equal semitone.  Playing a run of notes one fret apart is called '''trientonal''', which means "by third-tones". In a guitar context, it can be called '''fretwise'''. '''Microtonal''' scales differ from fuzzy scales in having many sequential ^1 intervals, and no steps larger than a vm2. Thus fuzzy means partly but not fully microtonal, and a fuzzy diatonic scale could be called a diatonic/microtonal scale. '''Chromatic''' is an umbrella term that includes semitonal, trientonal/fretwise and microtonal.  
{| class="wikitable" style="text-align:center;"
{| class="wikitable" style="text-align:center;"
!scale type -->
!scale type -->
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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.  
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.  
Each column header is a string-hopping move. The first column heading is "-5 = 3\41 = m2", which means that you go back 5 frets when hopping, which equals 3 edosteps, which equals a plain minor 2nd. Each row header is a string-sliding move in a similar format. "+1 = 2\41 = 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.  
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.  
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|+
|+
!
!
!-5 = 3 = m2
!-5 = 3\41 = m2
!-4 = 5 = ~2
!-4 = 5\41 = ~2
!-3 = 7 = M2
!-3 = 7\41 = M2
!-2 = 9 = vm3
!-2 = 9\41 = vm3
!-1 = 11 = ^m3
!-1 = 11\41 = ^m3
!-0 = 13 = vM3
!-0 = 13\41 = vM3
!--1 = 15 = ^M3
!--1 = 15\41 = ^M3
|-
|-
!+1 = 2 = vm2
!+1 = 2\41 = vm2
Laquinyo
Laquinyo


(P8, P12/5)
(P8, P12/5)
|solid block
|solid block
3L 16s = 19
gen = vM3
'''3L 16s = 19'''


L=3, s=2
L=3, s=2
|solid block
|solid block
3L 13s = 16
gen = vM3
'''3L 13s = 16'''


L=5, s=2
L=5, s=2
|solid block
|solid block
3L 10s = 13
gen = vM3
'''3L 10s = 13'''


L=7, s=2
L=7, s=2
|solid block
|solid block
3L 7s = 10
gen = vM3
'''3L 7s = 10'''


L=9, s=2
L=9, s=2
|solid block
|solid block
3L 4s = 7
gen = vM3
'''3L 4s = 7'''


L=11, s=2
L=11, s=2
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L=15, m=7, s=2
L=15, m=7, s=2
|-
|-
!+2 = 4 = ^m2
!+2 = 4\41 = ^m2
Sasa-tritribizo
Sasa-tritribizo
|checkerboard
|checkerboard
8L 3s = 11
gen = ^M3
'''8L 3s = 11'''
 
L=4, s=3
L=4, s=3
|alternate frets
|alternate frets
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65445-4454
65445-4454
|checkerboard
|checkerboard
3L 5s = 8
gen = ^M3
'''3L 5s = 8'''


L=7, s=4
L=7, s=4
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4947-494
4947-494
|checkerboard
|checkerboard
3L 2s = 5
gen = ^M3
'''3L 2s = 5'''


L=11, s=4
L=11, s=4
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|
|
|-
|-
!+3 = 6 = vM2
!+3 = 6\41 = vM2
Triyo
Saquadyo


(P8, P4/3)
(P8, P5/4)
|1XL 4L 3s = 8
|1XL 4L 3s = 8


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|
|
|-
|-
!+4 = 8 = ^M2
!+4 = 8\41 = ^M2
Latrizo
Latrizo


(P8, P5/3)
(P8, P5/3)
|diagonal lines
|diagonal lines
4L 3s = 7
gen = ^m3
'''4L 3s = 7'''


L=8, s=3
L=8, s=3
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|
|
|-
|-
!+5 = 10 = m3
!+5 = 10\41 = m3
|every 5th fret
|every 5th fret


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L=10, s=5
L=10, s=5
|wa pentatonic
|wa pentatonic
2L 3s = 5
gen = P5
'''2L 3s = 5'''


L=10, s=7
L=10, s=7
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|[[Superkleismic]]
|[[Superkleismic]]
|(P8, ccP4/9)
|(P8, ccP4/9)
|7 = 4L 3s
|'''7 = 4L 3s'''
11 = 4L 7s
11 = 4L 7s
|8 3
|'''8 3'''
5 3
5 3
| +4, -5
| '''+4, -5'''
-5, -4
-5, -4
|-
|-
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'''8 = 3L 5s'''
'''8 = 3L 5s'''


11 = 8L 3s
'''11 = 8L 3s'''
|'''11 4'''
|'''11 4'''
'''7 4'''
'''7 4'''


4 3
'''4 3'''
|'''+2, -1'''
|'''+2, -1'''
'''+2, -3'''
'''+2, -3'''


+2, -5
'''+2, -5'''
|-
|-
|16 = v4
|16 = v4