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→Scale Shapes (downmajor tuning): first attempt at a major expansion, needs work...
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There are many possible scales. Those listed here are select ones with a low prime limit and/or a low odd limit.  
There are many possible scales. Those listed here are select ones with a low prime limit and/or a low odd limit.  


Every scale can be thought of as a chord, e.g. the 12edo major pentatonic scale is a 6add9 pentad. Many pentads and heptads have an innate comma which 41edo does not temper out. Thus many Kite Guitar scales are "fuzzy", meaning a scale degree may vary by 1 edostep. In the tables below, a note that may be either a M2 or a vM2 is indicated by (v)M2. In general, major scales have a fuzzy 2nd and minor scales have a fuzzy 4th. Harmonic and subharmonic scales, which are segments of the harmonic and subharmonic series, are not fuzzy.
Every scale can be thought of as a chord, e.g. the 12edo major pentatonic scale is a 6add9 pentad. Many pentads and heptads have an innate comma which 41edo does not temper out. Thus many Kite Guitar scales are "fuzzy", meaning a scale degree may vary by 1 edostep. In the tables below, a note that may be either a M2 or a vM2 is indicated by (v)M2. In general, major scales have a fuzzy 2nd and minor scales have a fuzzy 4th. But the chord progression may make other degrees fuzzy. For example, Iv - IVv - Vv7 - Iv requires a fuzzy 4th.


The modes of a scale are grouped together. Not every mode is shown. Two modes of a scale will use the same prime subgroup, so modes are grouped by subgroup.  
The modes of a scale are grouped together. Not every mode is shown. Two modes of a scale will use the same prime subgroup, so modes are grouped by subgroup.  


Each scale has steps of various sizes, shown in the far right columns as both intervals and edosteps. Two modes of a scale will have the same step sizes, so modes are also grouped by step sizes. The largest-to-smallest ratio can be calculated directly from the edosteps. For example, the downminor and upmajor heptatonic scales have a very large L/s ratio of 8/2 = 4, giving them a lopsided feel. But the downminor/upmajor ''pentatonic'' scales have a very small L/s ratio of only 9/7 = 1.29, giving them a 5-edo-ish feel.
Each scale has steps of various sizes, shown in the far right columns as both intervals and edosteps. Two modes of a scale will have the same step sizes, so modes are also grouped by step sizes. The largest-to-smallest ratio can be calculated directly from the edosteps. For example, the downminor heptatonic scale has a very large L/s ratio of 8/2 = 4, giving it a lopsided feel. But the downminor ''pentatonic'' scale has a very small L/s ratio of only 9/7 = 1.29, giving it an [[5-edo|equipentatonic]] feel.


Harmonic and subharmonic may be abbreviated as har- and subhar-, e.g. harmajor pentatonic.
Harmonic and subharmonic scales are segments of the harmonic and subharmonic series. They are not fuzzy. Harmonic and subharmonic may be abbreviated as har- and subhar-, e.g. harmajor pentatonic. Pentatonic scales use (sub)harmonics 5-10, and heptatonic scales use (sub)harmonics 7-14.


=== Pentatonic Scales ===
=== Pentatonic Scales ===
Every pentatonic scale has 5 modes, but only those modes with a non-fuzzy root and 5th are listed. The harmonic and subharmonic scales are named after the triad implied by the 3rd and 5th, minus the up or down. Note that the harmonic ''major'' scale contains a ''minor'' 7th, and the harmonic ''minor'' scale contains a ''major'' 6th. Likewise with the subharmajor and subharminor scales. A harmonic diminished pentatonic scale would be P1 ^m3 d5 ^m6 ^m7 P8 = 5:6:7:8:9. But it's not very plausible, and would be heard as one of the other modes.
Every pentatonic scale has 5 modes, but only those modes with a non-fuzzy 5th are listed.  
{| class="wikitable left-9 center-all"
{| class="wikitable left-9 center-all"
|+
|+
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|^m7
|^m7
|P8
|P8
|style="text-align: left"|^m7,(^)11 chord
| style="text-align: left" |^m7,(^)11 chord
|-
|-
! rowspan="2" |za
! rowspan="2" |za
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|^M6
|^M6
|P8
|P8
|style="text-align: left"|^6,(^)9 chord
| style="text-align: left" |^6,(^)9 chord
|}
The harmonic and subharmonic scales are named after the triad implied by the 3rd and 5th, minus the up or down. Note that the harmonic ''major'' scale contains a ''minor'' 7th, and the harmonic ''minor'' scale contains a ''major'' 6th. Likewise with the subharmajor and subharminor scales. A harmonic diminished pentatonic scale would be P1 ^m3 d5 ^m6 ^m7 P8 = 5:6:7:8:9. But it's not very plausible, and would be heard as one of the other modes.
{| class="wikitable left-9 center-all"
|+
!subgroup
!name
! colspan="6" |scale
!as a chord
! colspan="2" |step sizes
|-
|-
! rowspan="2" |yaza
! rowspan="2" |yaza
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|vM6
|vM6
|P8
|P8
|style="text-align: left"|vm6,11 = 6:7:8:9:10
| style="text-align: left" |vm6,11 = 6:7:8:9:10
|-
|-
! rowspan="3" |"
! rowspan="3" |"
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|^M6
|^M6
|P8
|P8
|style="text-align: left"|^m6,11 = 12/(12:10:9:8:7)
| style="text-align: left" |^m6,11 = 12/(12:10:9:8:7)
|-
|-
!subharmonic diminished
!subharmonic diminished
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|vm7
|vm7
|P8
|P8
|style="text-align: left"|vm7(b5),vm6 = 14/(14:12:10:9:8)
| style="text-align: left" |vm7(b5),vm6 = 14/(14:12:10:9:8)
|}
|}


=== Heptatonic Scales ===
=== Heptatonic Scales ===
Only a few of the modes are listed. See also the octotonic scales that omit the 15th harmonic.
Five of the seven ya modes are formed from this collection of notes:
 
(to do: add lattice)
 
Five of the seven za modes are formed from this collection:
 
(to do: add lattice)
 
The two dorian scales and the two locrian scales are not from these lattices, and are not actually modes of the other scales.  
 
To be consistent, the two dorian scales should have a fuzzy tonic. To avoid this, and to provide all six triads, there are ''two'' fuzzy notes. Note that the 6th of the <u>up</u>dorian scale can be <u>down</u>ed.
 
To be consistent, the uplocrian or downlocrian scale should have an upflat or downflat 5th. To get a plain flat 5th, and thus a more consonant 5:6:7 or 7/(7:6:5) tonic triad, the 5th is fuzzy as well as the 3rd.
{| class="wikitable center-all"
{| class="wikitable center-all"
|+
|+
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! colspan="2" |step sizes
! colspan="2" |step sizes
|-
|-
! rowspan="2" |ya
! rowspan="5" |ya
(2.3.5)
(2.3.5)
!downlydian
|P1
|M2
|vM3
|vA4
|P5
|(v)M6
|vM7
|P8
| rowspan="5" |^m2, vM2, M2
| rowspan="5" |4 6 7
|-
!downmajor
!downmajor
|P1
|P1
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|vM7
|vM7
|P8
|P8
| rowspan="2" |^m2, vM2, M2
|-
| rowspan="2" |4 6 7
!downmixolydian
|P1
|vM2
|vM3
|P4
|(v)5
|vM6
|m7
|P8
|-
|-
!upminor
!upminor
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|P8
|P8
|-
|-
! rowspan="2" |za
!upphrygian
|P1
|^m2
|^m3
|P4
|P5
|^m6
|(^)m7
|P8
|-
!"
!updorian
|P1
|M2
|^m3
|(^)4
|P5
|(v)M6
|^m7
|P8
|^m2, ~2, vM2, M2
|4 5 6 7
|-
!"
!uplocrian
|P1
|^m2
|(^)m3
|P4
|(^)d5
|^m6
|m7
|P8
|m2, ^m2, vM2, M2, ^M2
|3 4 6 7 8
|-
! rowspan="5" |za
(2.3.7)
(2.3.7)
!uplydian
|P1
|M2
|^M3
|^A4
|P5
|(^)M6
|^M7
|P8
| rowspan="5" |vm2, M2, ^M2
| rowspan="5" |2 7 8
|-
!upmajor
|P1
|(^)M2
|^M3
|P4
|P5
|^M6
|^M7
|P8
|-
!upmixolydian
|P1
|^M2
|^M3
|P4
|(^)5
|^M6
|m7
|P8
|-
!downminor
!downminor
|P1
|P1
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|vm7
|vm7
|P8
|P8
| rowspan="2" |vm2, M2, ^M2
| rowspan="2" |2 7 8
|-
|-
!upmajor
!downphrygian
|P1
|P1
|(^)M2
|vm2
|^M3
|vm3
|P4
|P4
|P5
|P5
|^M6
|vm6
|^M7
|(v)m7
|P8
|-
!yaza
!downdorian
|P1
|M2
|vm3
|(v)4
|P5
|(v)M6
|vm7
|P8
|vm2, ~2, M2, ^M2
|2 5 7 8
|-
!"
!downlocrian
|P1
|vm2
|(v)m3
|P4
|(v)d5
|vm6
|m7
|P8
|P8
|}
|vm2, m2, vM2, M2, ^M2
 
|2 3 6 7 8
=== Octatonic Scales ===
|}The harmonic and subharmonic scales all have the same prime subgroup, yazalatha (2.3.5.7.11.13). Adding the 15th harmonic (the '''bolded''' note) makes an octotonic scale that uses harmonics 8-16. Again, the scales are named after the triad implied by the 3rd and 5th, minus the up or down. If there are two 3rds, the unbolded one is used. Each scale contains the similarly-named pentatonic scale, e.g. the harmajor scale contains the harmajor pentatonic scale.  
The scales listed here are all harmonic or subharmonic. The prime subgroup for all these scales is yazalatha (2.3.5.7.11.13). Omitting the 15th harmonic (the '''bolded''' note) makes a heptatonic scale that uses harmonics 7-14.  
 
The scales are named after the triad implied by the 3rd and 5th, minus the up or down. If there are two 3rds, the unbolded one is used. Each scale contains the similarly-named pentatonic scale, e.g. the harmajor octotonic scale contains the harmajor pentatonic scale.
 
One of the hallmarks of harmonic and subharmonic scales is that each step has a unique size. Unfortunately, while 41edo approximates the individual harmonics quite well, these scales do not have unique step sizes. The step sizes run 7 6 6 5 5 4 4 4. The heptatonic scales are better, 8 7 6 6 5 5 4. Only the pentatonic scales have unique step sizes.
{| class="wikitable left-11 center-all"
{| class="wikitable left-11 center-all"
|+
|+
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|P8
|P8
|8:9:10:11:12:13:14:'''15'''
|8:9:10:11:12:13:14:'''15'''
| rowspan="2" |A1/^m2, ~2, vM2, M2
| rowspan="2" |^m2, ~2, vM2, M2, ^M2
| rowspan="2" |4 5 6 7
| rowspan="2" |4 5 6 7 8
|-
|-
!harmonic minor
!harmonic minor
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|~7
|~7
|P8
|P8
|style="text-align: left"|12:13:14:'''15''':16:18:20:22
| style="text-align: left" |12:13:14:'''15''':16:18:20:22
|-
|-
!subharmonic major
!subharmonic major
Line 437: Line 563:
|P8
|P8
|18/(18:16:'''15''':14:13:12:11:10)
|18/(18:16:'''15''':14:13:12:11:10)
| rowspan="2" |A1/^m2, ~2, vM2, M2
| rowspan="2" |^m2, ~2, vM2, M2, ^M2
| rowspan="2" |4 5 6 7
| rowspan="2" |4 5 6 7 8
|-
|-
!subharmonic minor
!subharmonic minor
Line 450: Line 576:
|~7
|~7
|P8
|P8
|style="text-align: left"|24/(24:22:20:18:16:'''15''':14:13)
| style="text-align: left" |24/(24:22:20:18:16:'''15''':14:13)
|}
|}
One of the hallmarks of harmonic and subharmonic scales is that each step has a unique size. Unfortunately, in 41edo, these scales do not have unique step sizes. The heptatonic scales run 8 7 6 6 5 5 4. The octotonic step sizes are worse, 7 6 6 5 5 4 4 4. Only the pentatonic scales have unique step sizes.
== Relative and Absolute Tab ==
== Relative and Absolute Tab ==
Since the fretboard is isomorphic, any interval can be expressed in '''relative tab''' as a vector. This is particularly useful for in-person oral instruction of chord shapes. For example, in the downmajor tuning, going up 2 strings and down 1 fret always takes you up a perfect 5th. In relative tab, that move is spoken as "plus-two, minus-one", and written as (+2,-1). The downmajor 2nd is at "oh, plus-three", (0,+3). The downmajor 3rd is at "plus-one, oh", (+1,0).  
Since the fretboard is isomorphic, any interval can be expressed in '''relative tab''' as a vector. This is particularly useful for in-person oral instruction of chord shapes. For example, in the downmajor tuning, going up 2 strings and down 1 fret always takes you up a perfect 5th. In relative tab, that move is spoken as "plus-two, minus-one", and written as (+2,-1). The downmajor 2nd is at "oh, plus-three", (0,+3). The downmajor 3rd is at "plus-one, oh", (+1,0).  
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