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

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== A review of 12-equal scales ==
== A review of 12-equal scales ==
There are three broad categories of 12-equal scales: pentatonic, diatonic and chromatic:
There are three broad categories of 12-equal scales: pentatonic, diatonic and chromatic:
{| class="wikitable" style="text-align:center;"
{| class="wikitable" style="text-align:center;"
!scale type -->
! colspan="2" |pentatonic
! colspan="3" |diatonic
! colspan="2" |chromatic
|-
|-
!scale steps
! Scale type →
|M2
! colspan="2" | Pentatonic
|m3
! colspan="3" | Diatonic
|m2
! colspan="2" | Chromatic
|M2
|(A2)
|A1 or m2
|(M2)
|-
|-
!semitones per scale step
! scale steps
|2
| M2
|3
| m3
|1
| m2
|2
| M2
|(3)
| (A2)
|1
| A1 or m2
|(2)
| (M2)
|-
|-
!example scale
! semitones per scale step
| colspan="2" |C D E G A C
| 2
| colspan="3" |C D E F G A B C
| 3
| colspan="2" |C Db D Eb E F F# G Ab A Bb B C
| 1
| 2
| (3)
| 1
| (2)
|-
|-
!scale steps in semitones
! example scale
| colspan="2" |2 2 3 2 3
| colspan="2" | C D E G A C
| colspan="3" |2 2 1 2 2 2 1
| colspan="3" | C D E F G A B C
| colspan="2" |1 1 1 1 1 1 1 1 1 1 1 1
| colspan="2" | C Db D Eb E F F# G Ab A Bb B C
|-
! scale steps in semitones
| colspan="2" | 2 2 3 2 3
| colspan="3" | 2 2 1 2 2 2 1
| colspan="2" | 1 1 1 1 1 1 1 1 1 1 1 1
|}
|}
Strictly speaking, "diatonic" means a maximally even 5L2s scale, but here it's used more loosely. The maximally even requirement is relaxed. Also, occasional augmented 2nds are allowed, and the harmonic minor scale is considered to be diatonic. But in fact, common diatonic scales tend to avoid adjacent minor 2nds. Likewise, pentatonic scales tend to avoid adjacent minor 3rds. Hemitonic pentatonic scales such as C D Eb G Ab C and C E F G B C are for our purposes considered to be diatonic scales with missing notes, as are most hexatonic scales.
Strictly speaking, "diatonic" means a maximally even 5L2s scale, but here it's used more loosely. The maximally even requirement is relaxed. Also, occasional augmented 2nds are allowed, and the harmonic minor scale is considered to be diatonic. But in fact, common diatonic scales tend to avoid adjacent minor 2nds. Likewise, pentatonic scales tend to avoid adjacent minor 3rds. Hemitonic pentatonic scales such as C D Eb G Ab C and C E F G B C are for our purposes considered to be diatonic scales with missing notes, as are most hexatonic scales.


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== Prime subgroups ==
== Prime subgroups ==
Imperfect degrees in 12-equal have two qualities, major and minor, and each one implies two [[Color notation|colors]].  
Imperfect degrees in 12-equal have two qualities, major and minor, and each one implies two [[Color notation|colors]].
 
{| class="wikitable" style="text-align:center;"
{| class="wikitable" style="text-align:center;"
!quality
| colspan="2" |minor
| colspan="2" |major
|-
|-
!color
! Quality
|4thwd wa
| colspan="2" | Minor
|gu
| colspan="2" | Major
|yo
|5thwd wa
|-
|-
!prime
! color
|3-under
| 4thwd wa
|5-under
| gu
|5-over
| yo
|3-over
| 5thwd wa
|-
! prime
| 3-under
| 5-under
| 5-over
| 3-over
|}
|}
12-equal accurately represents only primes 2, 3 and 5 (as well as 17 and 19, and various other higher primes). 41-equal accurately represents primes 7, 11 and 13 as well. There are 7 qualities:
12-equal accurately represents only primes 2, 3 and 5 (as well as 17 and 19, and various other higher primes). 41-equal accurately represents primes 7, 11 and 13 as well. There are 7 qualities:
{| class="wikitable" style="text-align:center;"
{| class="wikitable" style="text-align:center;"
!quality
|downminor
|minor
|upminor
|mid
|downmajor
|major
|upmajor
|-
|-
!color
! quality
|zo
| downminor
|4thwd wa
| minor
|gu
| upminor
|lo/lu/tho/thu
| mid
|yo
| downmajor
|5thwd wa
| major
|ru
| upmajor
|-
|-
!prime
! color
|7-over
| zo
|3-under
| 4thwd wa
|5-under
| gu
|11-over/under, 13-over/under
| lo/lu/tho/thu
|5-over
| yo
|3-over
| 5thwd wa
|7-under
| ru
|-
! prime
| 7-over
| 3-under
| 5-under
| 11-over/under, 13-over/under
| 5-over
| 3-over
| 7-under
|}
|}
In [[color notation]], these subgroups are named wa = 2.3 = 3-limit, ya = 2.3.5 = 5-limit, za = 2.3.7, and ila = 2.3.11. The subgroups can combine, e.g. yaza = 2.3.5.7. Note that 7-limit includes both za and yaza. 41-equal doesn't distinguish between the ila subgroup and the tha subgroup 2.3.13, so tha is lumped in with ila.  
In [[color notation]], these subgroups are named wa = 2.3 = 3-limit, ya = 2.3.5 = 5-limit, za = 2.3.7, and ila = 2.3.11. The subgroups can combine, e.g. yaza = 2.3.5.7. Note that 7-limit includes both za and yaza. 41-equal doesn't distinguish between the ila subgroup and the tha subgroup 2.3.13, so tha is lumped in with ila.  


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=== Pentatonic scales ===
=== Pentatonic scales ===
There are four basic categories of pentatonic scales, one for each of the prime subgroups:
There are four basic categories of pentatonic scales, one for each of the prime subgroups:
{| class="wikitable" style="text-align:center;"
{| class="wikitable" style="text-align:center;"
!scale type -->
! colspan="2" |wa pentatonic
! colspan="4" |ya pentatonic
! colspan="4" |za pentatonic
! colspan="4" |ila pentatonic
|-
|-
!scale steps
! Scale type →
|M2
! colspan="2" | Wa pentatonic
|m3
! colspan="4" | Ya pentatonic
|vM2
! colspan="4" | Za pentatonic
|M2
! colspan="4" | Ila pentatonic
|m3
|^m3
|M2
|^M2
|vm3
|m3
|~2
|M2
|m3
|~3
|-
|-
!edosteps per scale step
! scale steps
|7
| M2
|10
| m3
|6
| vM2
|7
| M2
|10
| m3
|11
| ^m3
|7
| M2
|8
| ^M2
|9
| vm3
|10
| m3
|5
| ~2
|7
| M2
|10
| m3
|12
| ~3
|-
|-
!example scale
! edosteps per scale step
| colspan="2" |C D E G A C
| 7
| colspan="4" |C D vE G vA C
| 10
| colspan="4" |C vEb F G vBb C
| 6
| colspan="4" |C vvE F G vvB C
| 7
| 10
| 11
| 7
| 8
| 9
| 10
| 5
| 7
| 10
| 12
|-
|-
!scale steps in edosteps
! example scale
| colspan="2" |7 7 10 7 10
| colspan="2" | C D E G A C
| colspan="4" |7 6 11 6 11
| colspan="4" | C D vE G vA C
| colspan="4" |9 8 7 9 8
| colspan="4" | C vEb F G vBb C
| colspan="4" |12 5 7 12 5
| colspan="4" | C vvE F G vvB C
|-
! scale steps in edosteps
| colspan="2" | 7 7 10 7 10
| colspan="4" | 7 6 11 6 11
| colspan="4" | 9 8 7 9 8
| colspan="4" | 12 5 7 12 5
|}
|}
The za scale is the most equally distributed, thus arguably the most pentatonic-friendly of the subgroups. The za pentatonic example has an L/s ratio of only 1.29, whereas the ya example has a 1.83 ratio, and the ila example has a 2.4 ratio.  
The za scale is the most equally distributed, thus arguably the most pentatonic-friendly of the subgroups. The za pentatonic example has an L/s ratio of only 1.29, whereas the ya example has a 1.83 ratio, and the ila example has a 2.4 ratio.  
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In addition to these broad categories, every 41-equal scale has a unique name that uses ups and downs. The 4 pentatonic examples above are major pentatonic, downmajor pentatonic, downminor pentatonic and dupminor pentatonic. [[Kite Guitar Exercises and Techniques by Kite Giedraitis|Rotating]] these scales makes the minor (wa), upminor (ya), upmajor (za) and dudmajor (ila) pentatonic scales.  
In addition to these broad categories, every 41-equal scale has a unique name that uses ups and downs. The 4 pentatonic examples above are major pentatonic, downmajor pentatonic, downminor pentatonic and dupminor pentatonic. [[Kite Guitar Exercises and Techniques by Kite Giedraitis|Rotating]] these scales makes the minor (wa), upminor (ya), upmajor (za) and dudmajor (ila) pentatonic scales.  


These subgroups can be combined to make another four subgroups. Yala pentatonic scales tend to have off 5ths, and thus may be fuzzy. A yazala pentatonic scale must be fuzzy, in order to contain so many different step sizes.
These subgroups can be combined to make another four subgroups. Yala pentatonic scales tend to have off 5ths, and thus may be fuzzy. A yazala pentatonic scale must be fuzzy, in order to contain so many different step sizes.
 
{| class="wikitable" style="text-align:center;"
{| class="wikitable" style="text-align:center;"
!scale type -->
! colspan="6" |yaza pentatonic
! colspan="6" |yala pentatonic
! colspan="6" |zala pentatonic
! colspan="8" |yazala pentatonic
|-
|-
!scale steps
! Scale type →
|vM2
! colspan="6" | Yaza pentatonic
|M2
! colspan="6" | Yala pentatonic
|^M2
! colspan="6" | Zala pentatonic
|vm3
! colspan="8" | Yazala pentatonic
|m3
|-
|^m3
! scale steps
|~2
| vM2
|vM2
| M2
|M2
| ^M2
|m3
| vm3
|^m3
| m3
|~3
| ^m3
|~2
| ~2
|M2
| vM2
|^M2
| M2
|vm3
| m3
|m3
| ^m3
|~3
| ~3
|~2
| ~2
|vM2
| M2
|M2
| ^M2
|^M2
| vm3
|vm3
| m3
|m3
| ~3
|^m3
| ~2
|~3
| vM2
| M2
| ^M2
| vm3
| m3
| ^m3
| ~3
|-
|-
!edosteps
! edosteps
|6
| 6
|7
| 7
|8
| 8
|9
| 9
|10
| 10
|11
| 11
|5
| 5
|6
| 6
|7
| 7
|10
| 10
|11
| 11
|12
| 12
|5
| 5
|7
| 7
|8
| 8
|9
| 9
|10
| 10
|12
| 12
|5
| 5
|6
| 6
|7
| 7
|8
| 8
|9
| 9
|10
| 10
|11
| 11
|12
| 12
|-
|-
!example
! example
| colspan="6" |C D vE G vBb C
| colspan="6" | C D vE G vBb C
| colspan="6" |C D vE G vvB C
| colspan="6" | C D vE G vvB C
| colspan="6" |C vEb F G vvB C
| colspan="6" | C vEb F G vvB C
| colspan="8" |C ^Eb/vvE F G vBb C
| colspan="8" | C ^Eb/vvE F G vBb C
|-
|-
!edosteps
! edosteps
| colspan="6" |7 6 11 9 8  (harmonics 6-10)
| colspan="6" | 7 6 11 9 8  (harmonics 6-10)
| colspan="6" |7 6 11 12 5
| colspan="6" | 7 6 11 12 5
| colspan="6" |9 8 7 12 5
| colspan="6" | 9 8 7 12 5
| colspan="8" |11/12 6/5 7 9 8
| colspan="8" | 11/12 6/5 7 9 8
|}
|}


=== Diatonic scales ===
=== Diatonic scales ===
There are four basic categories of diatonic scales. In practice, a non-wa scale will often lack a m2 step, unless it's fuzzy. The ya and za diatonic scales have off 5ths, and thus tend to be fuzzy. The ila scale is the most equally distributed, thus arguably the most diatonic-friendly. Ya is also fairly equal. Za scales tend to have a very lopsided L/s ratio.  
There are four basic categories of diatonic scales. In practice, a non-wa scale will often lack a m2 step, unless it's fuzzy. The ya and za diatonic scales have off 5ths, and thus tend to be fuzzy. The ila scale is the most equally distributed, thus arguably the most diatonic-friendly. Ya is also fairly equal. Za scales tend to have a very lopsided L/s ratio.  
{| class="wikitable" style="text-align:center;"
{| class="wikitable" style="text-align:center;"
!scale type -->
! colspan="2" |wa diatonic
! colspan="4" |ya diatonic
! colspan="4" |za diatonic
! colspan="3" |ila diatonic
|-
|-
!scale steps
! Scale type →
|m2
! colspan="2" | Wa diatonic
|M2
! colspan="4" | Ya diatonic
|m2
! colspan="4" | Za diatonic
|^m2
! colspan="3" | ILa diatonic
|vM2
|M2
|vm2
|m2
|M2
|^M2
|m2
|~2
|M2
|-
|-
!edosteps
! scale steps
|3
| m2
|7
| M2
|3
| m2
|4
| ^m2
|6
| vM2
|7
| M2
|2
| vm2
|3
| m2
|7
| M2
|8
| ^M2
|3
| m2
|5
| ~2
|7
| M2
|-
|-
!example
! edosteps
| colspan="2" |C D E F G A B C
| 3
| colspan="4" |C D vE F G vA vB C
| 7
| colspan="4" |C D vEb F G vAb vBb C
| 3
| colspan="3" |C vvD Eb F G vvA Bb C
| 4
| 6
| 7
| 2
| 3
| 7
| 8
| 3
| 5
| 7
|-
|-
!edosteps
! example
| colspan="2" |7 7 3 7 7 7 3
| colspan="2" | C D E F G A B C
| colspan="4" |7 6 4 7 6 7 4
| colspan="4" | C D vE F G vA vB C
| colspan="4" |7 2 8 7 2 7 8
| colspan="4" | C D vEb F G vAb vBb C
| colspan="3" |5 5 7 7 5 5 7
| colspan="3" | C vvD Eb F G vvA Bb C
|-
! edosteps
| colspan="2" | 7 7 3 7 7 7 3
| colspan="4" | 7 6 4 7 6 7 4
| colspan="4" | 7 2 8 7 2 7 8
| colspan="3" | 5 5 7 7 5 5 7
|}
|}
There are four additional subgroups for diatonic scales:  
There are four additional subgroups for diatonic scales:  
{| class="wikitable" style="text-align:center;"
{| class="wikitable" style="text-align:center;"
!scale type -->
! colspan="6" |yaza diatonic
! colspan="5" |yala diatonic
! colspan="5" |zala diatonic
! colspan="7" |yazala diatonic
|-
|-
!scale steps
! Scale type →
|vm2
! colspan="6" | Yaza diatonic
|m2
! colspan="5" | Yala diatonic
|^m2
! colspan="5" | Zala diatonic
|vM2
! colspan="7" | Yazala diatonic
|M2
|^M2
|m2
|^m2
|~2
|vM2
|M2
|vm2
|m2
|~2
|M2
|^M2
|vm2
|m2
|^m2
|~2
|vM2
|M2
|^M2
|-
|-
!edosteps
! scale steps
|2
| vm2
|3
| m2
|4
| ^m2
|6
| vM2
|7
| M2
|8
| ^M2
|3
| m2
|4
| ^m2
|5
| ~2
|6
| vM2
|7
| M2
|2
| vm2
|3
| m2
|5
| ~2
|7
| M2
|8
| ^M2
|2
| vm2
|3
| m2
|4
| ^m2
|5
| ~2
|6
| vM2
|7
| M2
|8
| ^M2
|-
|-
!example
! edosteps
| colspan="6" |C vD vEb F G vAb vBb C
| 2
| colspan="5" |C vvD ^Eb F G ^Ab Bb C
| 3
| colspan="5" |C vvD Eb F G vAb vBb C
| 4
| colspan="7" |C D vE ^^F G vvA vBb C
| 6
| 7
| 8
| 3
| 4
| 5
| 6
| 7
| 2
| 3
| 5
| 7
| 8
| 2
| 3
| 4
| 5
| 6
| 7
| 8
|-
|-
!edosteps
! example
| colspan="6" |6 3 8 7 2 7 8
| colspan="6" | C vD vEb F G vAb vBb C
| colspan="5" |5 6 6 7 4 6 7
| colspan="5" | C vvD ^Eb F G ^Ab Bb C
| colspan="5" |5 5 7 7 2 7 8
| colspan="5" | C vvD Eb F G vAb vBb C
| colspan="7" |7 6 6 5 5 4 8  (harmonics 8-14)
| colspan="7" | C D vE ^^F G vvA vBb C
|-
! edosteps
| colspan="6" | 6 3 8 7 2 7 8
| colspan="5" | 5 6 6 7 4 6 7
| colspan="5" | 5 5 7 7 2 7 8
| colspan="7" | 7 6 6 5 5 4 8  (harmonics 8-14)
|}
|}


=== Chromaticism: semitonal, fretwise and microtonal scales ===
=== Chromaticism: semitonal, fretwise 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 '''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, fretwise and microtonal.  
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 '''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, fretwise and microtonal.
 
{| class="wikitable" style="text-align:center;"
{| class="wikitable" style="text-align:center;"
!scale type -->
! colspan="4" |semitonal
! colspan="2" |fretwise
! colspan="2" |microtonal
|-
|-
!scale steps
! Scale type →
|(vm2)
! colspan="4" | Semitonal
|m2
! colspan="2" | Fretwise
|A1
! colspan="2" | Microtonal
|(~2)
|vm2
|m2
|^1
|vm2
|-
|-
!edosteps
! scale steps
|(2)
| (vm2)
|3
| m2
|4
| A1
|(5)
| (~2)
|2
| vm2
|3
| m2
|1
| ^1
|2
| vm2
|-
|-
!example
! edosteps
| colspan="4" |C vDb vD vEb vE F Gb G...
| (2)
| colspan="2" |C vDb ^Db vD vEb ^Eb vE ^E...
| 3
| colspan="2" |C vDb ^Db vD D ^D vEb ^Eb...
| 4
| (5)
| 2
| 3
| 1
| 2
|-
|-
!edosteps
! example
| colspan="4" |2 4 3 4 4 3 4...
| colspan="4" | C vDb vD vEb vE F Gb G...
| colspan="2" |2 2 2 3 2 2 2...
| colspan="2" | C vDb ^Db vD vEb ^Eb vE ^E...
| colspan="2" |2 2 2 1 1 1 2...
| colspan="2" | C vDb ^Db vD D ^D vEb ^Eb...
|-
! edosteps
| colspan="4" | 2 4 3 4 4 3 4...
| colspan="2" | 2 2 2 3 2 2 2...
| colspan="2" | 2 2 2 1 1 1 2...
|}
|}
On the Kite guitar, going up an "even" interval (one that has an even number of edosteps) keeps one on the same string, and an "odd" one takes you to the next string. An octave spans 3 strings, thus a scale often has only 3 odd intervals. The exceptions are generally either fuzzy or awkward to play. The latter include wa, ila and zala diatonic, and microtonal scales with many ^1 steps.  
On the Kite guitar, going up an "even" interval (one that has an even number of edosteps) keeps one on the same string, and an "odd" one takes you to the next string. An octave spans 3 strings, thus a scale often has only 3 odd intervals. The exceptions are generally either fuzzy or awkward to play. The latter include wa, ila and zala diatonic, and microtonal scales with many ^1 steps.  
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yaza: D is the obvious tonic for either scale
yaza: D is the obvious tonic for either scale


==41-equal MOS scales==
== 41-equal 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]].   
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.  
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\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.  
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.  
Line 436: Line 452:


{| class="wikitable"
{| class="wikitable"
|+
|-
!
!  
!-5 = 3\41 = m2
! -5 = 3\41 = m2
!-4 = 5\41 = ~2
! -4 = 5\41 = ~2
!-3 = 7\41 = M2
! -3 = 7\41 = M2
!-2 = 9\41 = vm3
! -2 = 9\41 = vm3
!-1 = 11\41 = ^m3
! -1 = 11\41 = ^m3
!-0 = 13\41 = vM3
! -0 = 13\41 = vM3
!--1 = 15\41 = ^M3
! --1 = 15\41 = ^M3
|-
|-
!+1 = 2\41 = vm2
!+1 = 2\41 = vm2
Line 450: Line 466:


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


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


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


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


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


L=11, s=2
L=11, s=2
|double harmonic vminor
| double harmonic vminor
2L 1m 4s = 7
2L 1m 4s = 7


L=13, m=7, s=2
L=13, m=7, s=2
|vminor Sakura
| vminor Sakura
2L 1m 2s = 5
2L 1m 2s = 5


Line 486: Line 502:
!+2 = 4\41 = ^m2
!+2 = 4\41 = ^m2
Sasa-tritribizo
Sasa-tritribizo
|checkerboard
| checkerboard
gen = ^M3
gen = ^M3
'''8L 3s = 11'''
'''8L 3s = 11'''
Line 501: Line 517:


or (P8, c<sup>6</sup>P5/18)
or (P8, c<sup>6</sup>P5/18)
|alternate frets
| alternate frets
2L 7s 1xs = 10
2L 7s 1xs = 10


Line 513: Line 529:


65445-4454
65445-4454
|checkerboard
| checkerboard
gen = ^M3
gen = ^M3
'''3L 5s = 8'''
'''3L 5s = 8'''


L=7, s=4
L=7, s=4
|alternate frets
| alternate frets
2L 1m 4s = 7
2L 1m 4s = 7


Line 526: Line 542:


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


L=11, s=4
L=11, s=4
|90-degree zigzag
| 90-degree zigzag
2L 1m 2s = 5  
2L 1m 2s = 5  


Line 543: Line 559:


13,4,7,13,4
13,4,7,13,4
|
|  
|-
|-
!+3 = 6\41 = vM2
!+3 = 6\41 = vM2
Line 549: Line 565:


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


XL=8, L=6, s=3
XL=8, L=6, s=3
|ya equi-hepta
| ya equi-hepta
1XL 4L 2s = 7
1XL 4L 2s = 7


XL=7, L=6, s=5
XL=7, L=6, s=5
|every 3rd fret
| every 3rd fret
whole-tone
whole-tone


Line 562: Line 578:


XL=8, L=7, s=6
XL=8, L=7, s=6
|diagonal lines
| diagonal lines
3x9 + 8 + 6 = 5
3x9 + 8 + 6 = 5


3L 1m 1s = 4L 1s
3L 1m 1s = 4L 1s
|ya pentatonic
| ya pentatonic
2L 1m 2s = 5
2L 1m 2s = 5


L=11, m=7, s=6
L=11, m=7, s=6
|
|  
|
|  
|-
|-
!+4 = 8\41 = ^M2
!+4 = 8\41 = ^M2
Line 577: Line 593:


(P8, P5/3)
(P8, P5/3)
|diagonal lines
| diagonal lines
gen = ^m3
gen = ^m3
'''4L 3s = 7'''
'''4L 3s = 7'''
Line 590: Line 606:


(0 -9 -10 3 -2)
(0 -9 -10 3 -2)
|every 4th fret
| every 4th fret
dots only
dots only


Line 602: Line 618:


6585-854
6585-854
|2L 3s 1xs = 6
| 2L 3s 1xs = 6
L=8, s=7, xs=4
L=8, s=7, xs=4


Line 612: Line 628:


787-8,11
787-8,11
|za pentatonic
| za pentatonic
2L 2s 1xs = 5
2L 2s 1xs = 5


L=9, s=8, xs=7
L=9, s=8, xs=7
|
|  
|
|  
|
|  
|-
|-
!+5 = 10\41 = m3
!+5 = 10\41 = m3
|every 5th fret
| every 5th fret


3L 1m 2s = 6
3L 1m 2s = 6


L=10, m=5, s=3
L=10, m=5, s=3
|Bohlen-Pierce
| Bohlen–Pierce
P12 = 4L 5s = 9
P12 = 4L 5s = 9


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


L=10, s=7
L=10, s=7
|
|  
|
|  
|
|  
|
|  
|}
|}


Line 680: Line 696:
* L/s <= 2 if there's only one L
* L/s <= 2 if there's only one L
{| class="wikitable right-1 right-2"
{| class="wikitable right-1 right-2"
|+Table of 41-equal Temperaments by generator
|+ Table of 41-equal Temperaments by generator
!
!
! colspan="2" |Temperament(s)
!
!
!
!
|-
|-
!edosteps
!  
!Cents
!  
!Color name
! colspan="2" | Temperament(s)
!Other names
!  
![[Pergen]]
!  
!MOS Scales
!  
!L s
!  
!moves
|-
|-
|1 = ^1
! edosteps
|29.27
! Cents
|Sepla-sezo = {{monzo| -100 33  0 17 }}
! Color name
|
! Other names
|(P8, P4/17)
! [[Pergen]]
|lopsided, s=1
! MOS Scales
|
! L s
|
! moves
|-
|-
|2 = vm2
| 1 = ^1
|58.54
| 29.27
|Latrizo&Ruyoyo&Luluzozoyo
| Sepla-sezo = {{monzo| -100 33  0 17 }}
|[[Hemimiracle]]
|  
|(P8, P5/12)
| (P8, P4/17)
|20 notes
| lopsided, s=1
|
|  
|
|  
|-
|-
|3 = m2
| 2 = vm2
|87.80
| 58.54
|Zozoyo&Bizozogu
| Latrizo&Ruyoyo&Luluzozoyo
|88cET (approx), [[Octacot]]
| [[Hemimiracle]]
|(P8, P5/8)
| (P8, P5/12)
|13 = 1L 12s
| 20 notes
|5 3
|
|
|-
| 3 = m2
| 87.80
| Zozoyo&Bizozogu
| 88cET (approx), [[Octacot]]
| (P8, P5/8)
| 13 = 1L 12s
| 5 3
|  -5, -4
|  -5, -4
|-
|-
|4 = ^m2
| 4 = ^m2
|117.07
| 117.07
|Latrizo&Bizozogu
| Latrizo&Bizozogu
|[[Miracle]]
| [[Miracle]]
|(P8, P5/6)
| (P8, P5/6)
|10 = 1L 9s
| 10 = 1L 9s
|5 4
| 5 4
|  +2, -4
|  +2, -4
|-
|-
|5 = ~2
| 5 = ~2
|146.34
| 146.34
|Zozoyo Noca; Zozoyo&Rutribiyo
| Zozoyo Noca  
|[[Bohlen-Pierce]], [[Bohpier]]
Zozoyo&Rutribiyo
|(P8, P12/13)
| [[Bohlen–Pierce]], [[Bohpier]]
|8 = 1L 7s
| (P8, P12/13)
| 8 = 1L 7s
9 lopsided
9 lopsided
|6 5
| 6 5
|  +3, -4
|  +3, -4
|-
|-
|6 = vM2
| 6 = vM2
|175.61
| 175.61
|Saquadyo
| Saquadyo
|[[Tetracot]], [[Bunya]], [[Monkey]]
| [[Tetracot]], [[Bunya]], [[Monkey]]
|(P8, P5/4)
| (P8, P5/4)
|7 = 6L 1s
| 7 = 6L 1s
13 lopsided
13 lopsided
|6 5
| 6 5
|  +3, -4
|  +3, -4
|-
|-
|7 = M2
| 7 = M2
|204.88
| 204.88
|Wawa Layo / Wawa Layo&Ruyoyo
| Wawa Layo  
|[[Baldy]]
Wawa Layo&Ruyoyo
|(P8, c<sup>3</sup>P4/20)
| [[Baldy]]
|6 = 5L 1s
| (P8, c<sup>3</sup>P4/20)
| 6 = 5L 1s
11 lopsided
11 lopsided
|7 6
| 7 6
|  +3, -3
|  +3, -3
|-
|-
|8 = ^M2
| 8 = ^M2
|234.15
| 234.15
|Latrizo&Zozoyo
| Latrizo&Zozoyo
|[[Rodan]], [[Guiron]]
| [[Rodan]], [[Guiron]]
|(P8, P5/3)
| (P8, P5/3)
|5 = 1L 4s
| 5 = 1L 4s
6, 11 lopsided
6, 11 lopsided
|9 8
| 9 8
|  +4, -2
|  +4, -2
|-
|-
|9 = vm3
| 9 = vm3
|263.41
| 263.41
|Ruyoyo&Quinzo-ayo
| Ruyoyo&Quinzo-ayo
|[[Septimin]]
| [[Septimin]]
|(P8, ccP4/11)
| (P8, ccP4/11)
|5 = 4L 1s
| 5 = 4L 1s
9 = 5L 4s
9 = 5L 4s
|9 5
| 9 5
5 4
5 4
|  -2, -4
|  -2, -4
+2, -4
+2, -4
|-
|-
|10 = m3
| 10 = m3
|292.68
| 292.68
|Zotriyo&Bizozogu
| Zotriyo&Bizozogu
|[[Quasitemp]]
| [[Quasitemp]]
|(P8, c<sup>3</sup>P4/14)
| (P8, c<sup>3</sup>P4/14)
|lopsided, s=1
| lopsided, s=1
|
|  
|
|  
|-
|-
|11 = ^m3
| 11 = ^m3
|321.95
| 321.95
|Tritriyo / Latrizo&Zotriyo
| Tritriyo  
|[[Superkleismic]]
Latrizo&Zotriyo
|(P8, ccP4/9)
| [[Superkleismic]]
|'''7 = 4L 3s'''
| (P8, ccP4/9)
| '''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
|-
|-
|12 = ~3
| 12 = ~3
|351.22
| 351.22
|Saruyo&Bizozogu
| Saruyo&Bizozogu
|[[Hemififths]], [[Karadeniz]]
| [[Hemififths]], [[Karadeniz]]
|(P8, P5/2)
| (P8, P5/2)
|7 = 3L 4s
| 7 = 3L 4s
10 = 7L 3s
10 = 7L 3s
|7 5
| 7 5
5 2
5 2
|  -4, -3
|  -4, -3
+1, -3
+1, -3
|-
|-
|13 = vM3
| 13 = vM3
|380.49
| 380.49
|Laquinyo&Ruyoyo / Ruyoyo&Zozoyo
| Laquinyo&Ruyoyo  
|[[Magic]]
Ruyoyo&Zozoyo
|(P8, P12/5)
 
|'''10 = 3L 7s'''
Saquinzo&Zozoyo
| [[Magic]]
| (P8, P12/5)
| '''10 = 3L 7s'''
'''13 = 3L 10s'''
'''13 = 3L 10s'''
|'''9 2'''
| '''9 2'''
'''7 2'''
'''7 2'''
|'''+1, -2'''
| '''+1, -2'''
'''+1, -3'''
'''+1, -3'''
|-
|-
|14 = M3
| 14 = M3
|409.76
| 409.76
|Laquinyo&Ruyoyo&Lulu
| Laquinyo&Ruyoyo&Lulu
|[[Hocus]]
| [[Hocus]]
|(P8, c<sup>3</sup>P4/10)
| (P8, c<sup>3</sup>P4/10)
|lopsided, s=1
| lopsided, s=1
|
|  
|
|  
|-
|-
|15 = ^M3
| 15 = ^M3
|439.02
| 439.02
|Sasa-tritribizo = {{monzo| 5 -35 0 18 }}; Wawa Laquinyo / Wawa Ruyoyo&Zozoyo
| Sasa-tritribizo = {{monzo| 5 -35 0 18 }};  
|
Wawa Laquinyo  
|(P8, c<sup>6</sup>P5/18)
 
|'''5 = 3L 2s'''
Wawa Ruyoyo&Zozoyo
|  
| (P8, c<sup>6</sup>P5/18)
| '''5 = 3L 2s'''
'''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
|468.29
| 468.29
|Zotriyo&Quinru-aquadyo
| Zotriyo&Quinru-aquadyo
|[[Barbad]]
| [[Barbad]]
|(P8, c<sup>7</sup>P4/19)
| (P8, c<sup>7</sup>P4/19)
|5 = 3L 2s
| 5 = 3L 2s
8 = 5L 3s
8 = 5L 3s


13 = 5L 8s
13 = 5L 8s
|9 7
| 9 7
7 2
7 2


Line 876: Line 902:
+1, -4
+1, -4
|-
|-
|17 = P4
| 17 = P4
|497.56
| 497.56
|Layo / Saruyo&Ruyoyo
| Layo  
|[[Schismatic]] ([[Helmholtz]],
Saruyo&Ruyoyo
| [[Schismatic]] ([[Helmholtz (temperament)|Helmholtz]],
[[Garibaldi]], [[Cassandra]])
[[Garibaldi]], [[Cassandra]])
|(P8, P5)
| (P8, P5)
|5 = 2L 3s
| 5 = 2L 3s
7 = 5L 2s
7 = 5L 2s


12 = 5L 7s
12 = 5L 7s
|10 7
| 10 7
7 3
7 3


Line 895: Line 922:
+2, -5
+2, -5
|-
|-
|18 = ^4
| 18 = ^4
|526.83
| 526.83
|Laquinyo&Latrizo
| Laquinyo&Latrizo
|[[Trismegistus]]
| [[Trismegistus]]
|(P8, c<sup>6</sup>P5/15)
| (P8, c<sup>6</sup>P5/15)
|5 = 2L 3s
| 5 = 2L 3s
7 = 2L 5s
7 = 2L 5s


9 = 7L 2s
9 = 7L 2s
|13 5
| 13 5
8 5
8 5


Line 913: Line 940:
-4, -5
-4, -5
|-
|-
|19 = ~4
| 19 = ~4
|556.10
| 556.10
|Sasa-quadquadlu = {{monzo| 57 -1 0 0 -16 }}
| Sasa-quadquadlu = {{monzo| 57 -1 0 0 -16 }}
|
|  
|(P8, c<sup>7</sup>P4/16)
| (P8, c<sup>7</sup>P4/16)
|7 = 2L 5s
| 7 = 2L 5s
9 = 2L 7s
9 = 2L 7s


Line 924: Line 951:


13 = 2L 11s
13 = 2L 11s
|13 3
| 13 3
10 3
10 3


Line 937: Line 964:
+2, -3
+2, -3
|-
|-
|20 = d5
| 20 = d5
|585.37
| 585.37
|Rurutriyo&Satrizo-agu
| Rurutriyo&Satrizo-agu
|[[Pluto]]
| [[Pluto]]
|(P8, c<sup>3</sup>P4/7)
| (P8, c<sup>3</sup>P4/7)
|lopsided, s=1
| lopsided, s=1
|
|  
|
|  
|}  
|}  


[[Category:Kite Guitar]]
[[Category:Kite Guitar]]
Retrieved from "https://en.xen.wiki/w/Kite_Giedraitis%27s_Categorizations_of_41edo_Scales"