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

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Kite's personal thoughts on 41-edo scales as they relate to [[the Kite Guitar]]. See also [[Kite Guitar Scales]].
Kite's personal thoughts on 41-equal scales as they relate to [[the Kite Guitar]]. See also [[Kite Guitar Scales]].


__TOC__
__TOC__


== A review of 12-edo scales ==
== A review of 12-equal scales ==
There are three broad categories of 12-edo 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-edo 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-edo accurately represents only primes 2, 3 and 5 (as well as 17 and 19, and various other higher primes). 41-edo 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-edo doesn't distinguish between the ila subgroup and the tha subgroup 2.3.13, so tha is lumped in with ila.


== 41-edo scales ==
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.  
41-edo has an enormous variety of scales. There are many thousands of unconventional scales, but we will focus on the ones that map compactly to the JI lattice. These are scales that contain numerous perfect 5ths. Two notes a perfect fifth apart generally have the same quality. So compact scales use only a few qualities, and thus a small prime subgroup.


In practice, 41-edo scales tend to be "fuzzy", meaning that one or two scale notes may sometimes shift by an edostep. For example, a major scale may contain both a M2 and a vM2, and use whichever one is required by the harmony at the moment.
== 41-equal scales ==
41-equal has an enormous variety of scales. There are many thousands of unconventional scales, but we will focus on the ones that map compactly to the JI lattice. These are scales that contain numerous perfect 5ths. Two notes a perfect fifth apart generally have the same quality. So compact scales use only a few qualities, and thus a small prime subgroup.


There are five broad categories of 41-edo scales: pentatonic, diatonic, semitonal, chromatic and microtonal.
In practice, 41-equal scales tend to be "fuzzy", meaning that one or two scale notes may sometimes shift by an edostep. For example, a major scale may contain both a M2 and a vM2, and use whichever one is required by the harmony at the moment.
 
There are five broad categories of 41-equal scales: pentatonic, diatonic, semitonal, fretwise and microtonal. The three latter ones fall under the general category of chromatic.


=== 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.  


A scale needn't have every single step size on the list in order to be in the category, just most of them. In practice, a non-wa pentatonic scale will often lack a m3 step, as in the examples. But a fuzzy pentatonic scale often will have a m3, e.g. C D vE G vA/A C. Ya and za scales generally contain a wolf 5th (either an ^5 or a v5), and would often become fuzzy to avoid the wolf.  
A scale needn't have every single step size on the list in order to be in the category, just most of them. In practice, a non-wa pentatonic scale will often lack a m3 step, as in the examples. But a fuzzy pentatonic scale often will have a m3, e.g. C D vE G vA/A C. Ya and za scales generally contain an off-5th (either an ^5 or a v5), and would often become fuzzy to avoid the wolf.  


In addition to these broad categories, every 41-edo scale has a unique name that uses ups and downs. The 4 pentatonic examples above are major pentatonic, downmajor pentatonic, downminor pentatonic and double-upminor pentatonic.  
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 wolf 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 wolf 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)
|}
|}


=== Semitonal, chromatic and microtonal scales ===
=== Chromaticism: semitonal, fretwise and microtonal scales ===
Most 41-edo intervals suggest a specific ratio, but those only a few edosteps wide don't. Thus the remaining categories don't imply any prime subgroups. On the Kite guitar, playing a run of notes one fret apart inherits the term "chromatic" from 12-edo. This term is especially appropriate if using color notation to describe 41-edo intervals, since chromatic means "colorful". Traditional 12-edo chromaticism, which translates to runs played on every other fret, is called semitonal, a conventional term referring to the 12-edo semitone. 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 '''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" |chromatic
! 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.  


From this we can deduce that chromatic scales are often 19 tones, and microtonal ones are often 22. We can also deduce that a semitonal scale of 12 notes usually has two vm2's. If there are more vm2's, the scale is semitonal/chromatic. Scales of 1, 2 and 3 edosteps are chromatic/microtonal.   
From this we can deduce that fretwise scales are often 19 tones, and microtonal ones are often 22. We can also deduce that a semitonal scale of 12 notes usually has two vm2's. If there are more vm2's, the scale is semitonal/fretwise. Scales of 1, 2 and 3 edosteps are fretwise/microtonal.   


=== Harmonic scales ===
=== Harmonic scales ===
In Western music, harmonies often require notes that the melody doesn't. For example, "Auld Lang Syne" has a pentatonic melody but diatonic harmonies. Often the melody is diatonic but the harmonies are at least somewhat chromatic. The score will have accidentals in the piano part but not the vocal part. The scale used by the melody is the melodic scale, and the one used to construct chords is the harmonic scale. 41-edo yaza harmonic scales are usually semitonal or chromatic.   
In Western music, harmonies often require notes that the melody doesn't. For example, "Auld Lang Syne" has a pentatonic melody but diatonic harmonies. Often the melody is diatonic but the harmonies are at least somewhat chromatic. The score will have accidentals in the piano part but not the vocal part. The scale used by the melody is the melodic scale, and the one used to construct chords is the harmonic scale. 41-equal yaza harmonic scales are usually semitonal or fretwise.   


In 12-edo, a song is generally in a major or minor key, and uses a major or minor scale. A ya piece in 41-edo often is as well. But unlike 12-edo, 41-edo allows the use of yaza chords such as 4:5:6:7. If this is one's tonic chord, both major and minor are used simultaneously. A simple Iv7 - IVv7 progression has both a downmajor 3rd and a downminor 3rd. Clearly the major/minor duality no longer applies. Instead, there is an up/down duality.     
In 12-equal, a song is generally in a major or minor key, and uses a major or minor scale. A ya piece in 41-equal often is as well. But unlike 12-equal, 41-equal allows the use of yaza chords such as 4:5:6:7. If this is one's tonic chord, both major and minor are used simultaneously. A simple Iv7 - IVv7 progression has both a downmajor 3rd and a downminor 3rd. Clearly the major/minor duality no longer applies. Instead, there is an up/down duality.     


For ya or za scales, one chooses a 7-note subset of the 12 notes, and lets the imperfect degrees be either major or minor, or some combination. For yaza scales, choose a 12-note subset, and let all but the tonic, 4th and 5th be either upped or downed. (The M2 and m7 may also be plain.) Up is utonal and down is otonal. Combining upped and downed intervals in a 41-edo scale creates double-up and double-down intervals, i.e. mid intervals. This increases the odd limit and/or the prime limit, so scales tend not to mix up and down.
For 12-equal ya scales, one chooses a 7-note subset of the 12 notes, and lets the imperfect degrees be either major or minor, or some combination. For 41-equal yaza scales, choose a 12-note subset, and let all but the tonic, 4th and 5th be either upped or downed. (The M2 and m7 may also be plain.) Up is utonal and down is otonal. Combining upped and downed intervals in a 41-equal scale creates dup and dud intervals, i.e. mid intervals. This increases the odd limit and/or the prime limit, so yaza scales tend not to mix up and down.


Harmonic scales aren't played sequentially to create melodies, and having more than 3 odd intervals isn't awkward. Often a harmonic scale is fuzzy, and uses pitch shifts of one edostep. Such a scale could be classified as diatonic/microtonal or semitonal/microtonal.
Harmonic scales aren't played sequentially to create melodies, and having more than 3 odd intervals isn't awkward. Often a harmonic scale is fuzzy, and uses pitch shifts of one edostep. Such a scale could be classified as diatonic/microtonal or semitonal/microtonal.
Line 420: Line 436:
yaza: D is the obvious tonic for either scale
yaza: D is the obvious tonic for either scale


==41-edo 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]]. 
 
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.
 
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\41 = m2
! -4 = 5\41 = ~2
! -3 = 7\41 = M2
! -2 = 9\41 = vm3
! -1 = 11\41 = ^m3
! -0 = 13\41 = vM3
! --1 = 15\41 = ^M3
|-
!+1 = 2\41 = vm2
Laquinyo
 
(P8, P12/5)
| solid block
gen = vM3
'''3L 16s = 19'''
 
L=3, s=2
| solid block
gen = vM3
'''3L 13s = 16'''
 
L=5, s=2
| solid block
gen = vM3
'''3L 10s = 13'''
 
L=7, s=2
| solid block
gen = vM3
'''3L 7s = 10'''
 
L=9, s=2
| solid block
gen = vM3
'''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\41 = ^m2
Sasa-tritribizo
| checkerboard
gen = ^M3
'''8L 3s = 11'''
 
L=4, s=3
 
2.9.15.7.11.13
 
(P8, cm7/5)
 
(1 5 5 5 2 7)
 
(0 -5 -3 -6 4 -9)
 
or (P8, c<sup>6</sup>P5/18)
| 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
gen = ^M3
'''3L 5s = 8'''
 
L=7, s=4
| alternate frets
2L 1m 4s = 7
 
L=9, m=7, s=4
 
double harmonic:
 
4947-494
| checkerboard
gen = ^M3
'''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\41 = vM2
Saquadyo
 
(P8, P5/4)
| 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\41 = ^M2
Latrizo
 
(P8, P5/3)
| diagonal lines
gen = ^m3
'''4L 3s = 7'''
 
L=8, s=3
 
(P8, ccP4/9)
 
Latrizo & Zotriyo & Luyoyo
 
(1 4 5 2 4)
 
(0 -9 -10 3 -2)
| 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\41 = 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
gen = P5
'''2L 3s = 5'''
 
L=10, s=7
|
|
|
|
|}
 
=== (TO DO / Notes to myself) ===
Expand each entry in the table above to include JI interpretation, comma list and RTT mapping, as with +4  and -5
 
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, 41-equal 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:
* 5-13 notes
* 5-13 notes
* 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 41-equal Temperaments by generator
|-
|-
!edosteps
!  
!Cents
!  
!Temperament(s)
! colspan="2" | Temperament(s)
![[Pergen]]
!  
!MOS Scales
!  
!L s
!  
!moves
!  
|-
|-
|1 = ^1
! edosteps
|29.27
! Cents
|Sepla-sezo = {{monzo| -100 33  0 17 }}
! Color name
|(P8, P4/17)
! Other names
|lopsided, s=1
! [[Pergen]]
|
! MOS Scales
|
! L s
! moves
|-
|-
|2 = vm2
| 1 = ^1
|58.54
| 29.27
|[[Hemimiracle]]
| Sepla-sezo = {{monzo| -100 33  0 17 }}
|(P8, P5/12)
|  
|20 notes
| (P8, P4/17)
|
| lopsided, s=1
|
|  
|  
|-
|-
|3 = m2
| 2 = vm2
|87.80
| 58.54
|88cET (approx), [[Octacot]]
| Latrizo&Ruyoyo&Luluzozoyo
|(P8, P5/8)
| [[Hemimiracle]]
|13 = 1L 12s
| (P8, P5/12)
|5 3
| 20 notes
| -5, -4
|  
|  
|-
|-
|4 = ^m2
| 3 = m2
|117.07
| 87.80
|[[Miracle]]
| Zozoyo&Bizozogu
|(P8, P5/6)
| 88cET (approx), [[Octacot]]
|10 = 1L 9s
| (P8, P5/8)
|5 4
| 13 = 1L 12s
| +2, -4
| 5 3
| -5, -4
|-
|-
|5 = ~2
| 4 = ^m2
|146.34
| 117.07
|[[Bohlen-Pierce]], [[Bohpier]]
| Latrizo&Bizozogu
|(P8, P12/13)
| [[Miracle]]
|8 = 1L 7s
| (P8, P5/6)
| 10 = 1L 9s
| 5 4
|  +2, -4
|-
| 5 = ~2
| 146.34
| Zozoyo Noca
Zozoyo&Rutribiyo
| [[Bohlen–Pierce]], [[Bohpier]]
| (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
|[[Tetracot]], [[Bunya]], [[Monkey]]
| Saquadyo
|(P8, P5/4)
| [[Tetracot]], [[Bunya]], [[Monkey]]
|7 = 6L 1s
| (P8, P5/4)
| 7 = 6L 1s
13 lopsided
13 lopsided
|6 5
| 6 5
| +3, -4
| +3, -4
|-
|-
|7 = M2
| 7 = M2
|204.88
| 204.88
|[[Baldy]]
| Wawa Layo
|(P8, c<sup>3</sup>P4/20)
Wawa Layo&Ruyoyo
|6 = 5L 1s
| [[Baldy]]
| (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
|[[Rodan]], [[Guiron]]
| Latrizo&Zozoyo
|(P8, P5/3)
| [[Rodan]], [[Guiron]]
|5 = 1L 4s
| (P8, P5/3)
| 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
|[[Septimin]]
| Ruyoyo&Quinzo-ayo
|(P8, ccP4/11)
| [[Septimin]]
|5 = 4L 1s
| (P8, ccP4/11)
| 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
|[[Quasitemp]]
| Zotriyo&Bizozogu
|(P8, c<sup>3</sup>P4/14)
| [[Quasitemp]]
|lopsided, s=1
| (P8, c<sup>3</sup>P4/14)
|
| lopsided, s=1
|
|  
|  
|-
|-
|11 = ^m3
| 11 = ^m3
|321.95
| 321.95
|[[Superkleismic]]
| Tritriyo
|(P8, ccP4/9)
Latrizo&Zotriyo
|7 = 4L 3s
| [[Superkleismic]]
| (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
|[[Hemififths]], [[Karadeniz]]
| Saruyo&Bizozogu
|(P8, P5/2)
| [[Hemififths]], [[Karadeniz]]
|7 = 3L 4s
| (P8, P5/2)
| 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
|[[Magic|Magic (Latrizo)]], [[Witchcraft]]
| Laquinyo&Ruyoyo
|(P8, P12/5)
Ruyoyo&Zozoyo
|'''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
|[[Hocus]]
| Laquinyo&Ruyoyo&Lulu
|(P8, c<sup>3</sup>P4/10)
| [[Hocus]]
|lopsided, s=1
| (P8, c<sup>3</sup>P4/10)
|
| lopsided, s=1
|
|  
|  
|-
|-
|15 = ^M3
| 15 = ^M3
|439.02
| 439.02
|Sasa-tritribizo = {{monzo| 5 -35 0 18 }}
| Sasa-tritribizo = {{monzo| 5 -35 0 18 }};
|(P8, c<sup>6</sup>P5/18)
Wawa Laquinyo
|'''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
|[[Barbad]]
| Zotriyo&Quinru-aquadyo
|(P8, c<sup>7</sup>P4/19)
| [[Barbad]]
|5 = 3L 2s
| (P8, c<sup>7</sup>P4/19)
| 5 = 3L 2s
8 = 5L 3s
8 = 5L 3s


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


5 2
5 2
| -2, -3
| -2, -3
+1, -3
+1, -3


+1, -4
+1, -4
|-
|-
|17 = P4
| 17 = P4
|497.56
| 497.56
|[[Schismatic]] ([[Helmholtz]],
| Layo
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


4 3
4 3
| +5, -3
| +5, -3
-3, -5
-3, -5


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


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


5 3
5 3
| -4, -0
| -4, -0
+4, -4
+4, -4


-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)
|
|7 = 2L 5s
| (P8, c<sup>7</sup>P4/16)
| 7 = 2L 5s
9 = 2L 7s
9 = 2L 7s


Line 645: Line 951:


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


Line 651: Line 957:


4 3
4 3
| -5, -0
| -5, -0
+5, -5
+5, -5


Line 658: Line 964:
+2, -3
+2, -3
|-
|-
|20 = d5
| 20 = d5
|585.37
| 585.37
|[[Pluto]]
| Rurutriyo&Satrizo-agu
|(P8, c<sup>3</sup>P4/7)
| [[Pluto]]
|lopsided, s=1
| (P8, c<sup>3</sup>P4/7)
|
| lopsided, s=1
|
|  
|}
|  
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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