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== Overview ==
== Overview ==


This is the practical guide to Kite Guitar scales. See also [[Kite Giedraitis's Categorizations of 41edo Scales]].  
This is the practical guide to Kite Guitar scales. See also [[Kite Giedraitis's Categorizations of 41edo Scales]], which is more theoretical.  


There are many possible 41edo scales. Those discussed here are those with at least 5 notes, and which contain a plain perfect 5th. Scales that are awkward to play on the Kite guitar are avoided. An awkward scale has a step which requires a jump of more than four frets. Thus plain minor 2nds and 3rds are avoided. A scale naturally hops from one string to the next as it goes up or down. Unlike other guitars, the Kite guitar doesn't let one hop freely. For example, the 3-limit scale fragment P1 M2 M3 P4 requires 3 hops, 2 upward and 1 downward. Any scale which doesn't have exactly three upward hops per octave is awkward, because the downward hop will always be at least 6 frets, and usually 7 or more. Almost every scale with a low prime limit and/or a low odd limit is not awkward.  
There are many possible 41edo scales. Those discussed here are those with at least 5 notes, and which contain a plain perfect 5th. Scales that are awkward to play on the Kite guitar are avoided. An awkward scale has a step which requires a jump of more than four frets. Thus plain minor 2nds and 3rds are avoided. A scale naturally hops from one string to the next as it goes up or down. Unlike other guitars, the Kite guitar doesn't let one hop freely. For example, the 3-limit scale fragment P1 M2 M3 P4 requires 3 hops, 2 upward and 1 downward. Any scale which doesn't have exactly three upward hops per octave is awkward, because the downward hop will always be at least 6 frets, and usually 7 or more. Almost every scale with a low prime limit and/or a low odd limit is not awkward.  
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|^m3
|^m3
|}
|}
The edosteps that can be swapped due to fuzziness are underlined. For example, in the first pentatonic scale, downing the 2nd makes <u>7 6</u> 11 - 6 7 become <u>6 7</u> 11 - 6 7. The two step sizes almost always differ by one, something like <u>4 6</u> becoming <u>5 5</u> never occurs. Hence the fuzziness never affects the overall step sizes. The only exceptions are the dorian and locrian scales, which are problematic anyway.  
The edosteps that are affected by fuzziness are underlined. For example, the downmajor scale is P1 (v)M2 vM3 P4 P5 vM6 vM7 P8. The edosteps are <u>76</u>47-674. Downing the 2nd makes <u>67</u>47-674. The two underlined step sizes almost always differ by one, something like <u>4 6</u> becoming <u>5 5</u> rarely occurs. Hence the fuzziness rarely affects the overall step sizes. The only exceptions are the dorian and locrian scales and the dodecatonic scales, all of which are problematic anyway.  


The step sizes column shows the sizes used. Two modes of a scale will have the same step sizes, so modes are also grouped by step sizes. The largest-to-smallest ratio L/s indicates how even the scale is. 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 even [[5-edo|equipentatonic]] feel.
The step sizes column shows the sizes used. Two modes of a scale will have the same step sizes, so modes are also grouped by step sizes. The largest-to-smallest ratio L/s indicates how even the scale is. 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 even [[5-edo|equipentatonic]] feel.
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The moves column is perhaps the most practical information in the table. It says how many frets to move up or down as you ascend the scale. Positive numbers refer to forward moves that move up the fretboard on a single string. Negative numbers refer to backwards moves that move up a string, then down the fretboard. The moves are not listed in order of size. Rather, forward moves are listed, then backward moves. In each category, they are listed by how often they occur in the scale. Assuming no excess string-hopping, there will always be 3 backward moves per octave. There are at most only two sizes of backward moves, the first one occuring twice and the second one once.  
The moves column is perhaps the most practical information in the table. It says how many frets to move up or down as you ascend the scale. Positive numbers refer to forward moves that move up the fretboard on a single string. Negative numbers refer to backwards moves that move up a string, then down the fretboard. The moves are not listed in order of size. Rather, forward moves are listed, then backward moves. In each category, they are listed by how often they occur in the scale. Assuming no excess string-hopping, there will always be 3 backward moves per octave. There are at most only two sizes of backward moves, the first one occuring twice and the second one once.  


To see how this works, consider the two ya pentatonic scales. Their two main moves are +3 and -1. Any short sequence of moves that alternates between +3 and -1 will be some fragment of these scales. Likewise +4 and -2 moves evoke the two za pentatonic scales. And +3 and -3 moves, with some +2 moves, evoke the ya heptatonic modes.  
To see how this works, consider the two ya pentatonic scales. Their two main moves are +3 and -1. Any short sequence of moves that alternates between +3 and -1 (or -3 and +1 if descending) will be some fragment of these scales. Likewise +4 and -2 moves evoke the two za pentatonic scales. And +3 and -3 moves, with some +2 moves, evoke the ya heptatonic modes. For longer sequences, one's natural inclination to stay in the same region of the fretboard, and to repeat at the octave, will guide one when to include the other moves.  


== Pentatonic Scales ==
== Pentatonic Scales ==
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!as a chord
!as a chord
!as chains of 5ths
!as chains of 5ths
!as edosteps
!edosteps
!step sizes
!step sizes
!steps
!steps
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! colspan="6" |scale
! colspan="6" |scale
!as a chord
!as a chord
!as edosteps
!edosteps
!step sizes
!step sizes
!steps
!steps
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! colspan="8" |scale
! colspan="8" |scale
!as chains of 5ths
!as chains of 5ths
!as edosteps
!edosteps
!step sizes
!step sizes
!steps
!steps
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! colspan="8" |scale
! colspan="8" |scale
!as chains of 5ths
!as chains of 5ths
!as edosteps
!edosteps
!step sizes
!step sizes
!steps
!steps
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! colspan="9" |scale
! colspan="9" |scale
!as a chord
!as a chord
!as edosteps
!edosteps
!step sizes
!step sizes
|-
|-
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<br>
</tt>
</tt>
Five of the seven za modes are formed from this collection:
Five of the seven za modes are formed from this collection:
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<br>
</tt>
</tt>
In both cases, the D is fuzzy. But the two dorian scales and the two locrian scales are not from these lattices, and are not actually modes of the other scales.
In both cases, the D is fuzzy. But the two dorian scales and the two locrian scales are not from these lattices, and are not actually modes of the other scales.
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! colspan="8" |scale
! colspan="8" |scale
!as chains of 5ths
!as chains of 5ths
!as edosteps
!edosteps
!step sizes
!step sizes
!steps
!steps
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|7 6 (5) 4
|7 6 (5) 4
|varies
|varies
|too many
|varies
|-
|-
!"
!"
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|(8) 7 6 4 (3)
|(8) 7 6 4 (3)
|varies
|varies
|too many
|varies
|-
|-
! rowspan="5" |za
! rowspan="5" |za
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|8 7 (6) (3) 2
|8 7 (6) (3) 2
|varies
|varies
|too many
|varies
|-
|-
!"
!"
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|"
|"
|varies
|varies
|too many
|varies
|}
|}
It would also be possible to define the modes based on the harmonic and subharmonic scales. For example, the downmixolydian scale could be P1 M2 vM3 P4 P5 vM6 vm7 P8, which contains a 4:5:6:7:9 chord. But this scale has two wolf 5ths.
It would also be possible to define the modes based on the harmonic and subharmonic scales. For example, the downmixolydian scale could be P1 M2 vM3 P4 P5 vM6 vm7 P8, which contains a 4:5:6:7:9 chord. But this scale has two wolf 5ths.


== Near-equidistant Scales ==
== Near-equidistant Scales ==
Certain Asian music uses very "lopsided" scales such as P1 M3 P4 P5 M7 P8 (SE Asia) and P1 M2 m3 P5 m6 P8 (Japan). While there is a certain charm to these, scales with equal or roughly equal sizes are also attractive. The only such 12edo scales are the whole tone scale and the full 12-note gamut. Since 41 is a prime number, it has no strictly equal scales. But there are many nearly-equal scales. We've seen how the upmajor and downminor pentatonic scales are nearly equi-pentatonic.
Certain Asian music uses very "lopsided" scales such as P1 M3 P4 P5 M7 P8 (SE Asia) and P1 M2 m3 P5 m6 P8 (Japan). While there is a certain charm to these, scales with equal or roughly equal sizes are also attractive. The only such 12edo scales are the whole tone scale and the full 12-note gamut. Since 41 is a prime number, it has no strictly equal scales. But there are many nearly-equal scales.  
 
=== Pentatonic ===
We've already seen how the upmajor and downminor pentatonic scales are nearly equi-pentatonic, with L/s = 1.29.


=== Heptatonic ===
=== Heptatonic ===
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! colspan="8" |scale
! colspan="8" |scale
!as chains of 5ths
!as chains of 5ths
!as edosteps
!edosteps
!step sizes
!step sizes
!steps
!steps
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|}
|}


=== Dodecatonic ===
=== Dodecatonic (twelve-tone) ===
"The Flight of the Bumblebee" has simple 5-limit triads, but a scale that is clearly dodecatonic. The evenly-spaced 12edo scale is quite fitting for this piece, nicely evoking the random movements of flying insects. How would this piece translate to the Kite Guitar? Poorly, because the scale would be either very awkward to play (all plain notes, lots of hopping between strings), or very uneven (steps of 2, 3, 4 and 5, L/s ratio of 2.5). The obvious uneven scale is the [[Duodene|harmonic duodene]], with 3 fuzzy notes to avoid wolf 5ths.
"The Flight of the Bumblebee" has simple 5-limit triads, but a scale that is clearly dodecatonic. The evenly-spaced 12edo scale is quite fitting for this piece, nicely evoking the random movements of flying insects. How would this piece translate to the Kite Guitar? Poorly, because the scale would be either very awkward to play (all plain notes, lots of hopping between strings), or very uneven, with an L/s ratio of at least 2.
 
Here is one mode of the 3-limit pythagorean dodecatonic scale, which is the closest one can get to 12-edo on the Kite Guitar. There are 7 small steps of 3 edosteps, each of which requires a string-hop. The moves column is deceptive, because repeated -5 moves will drive one near the nut and run out of room. Thus some of the -5 moves will have to be +8 frets but -1 string.
 
{| class="wikitable center-all"
!subgroup
!name
! colspan="13" |scale
!as a chain of 5ths
!edosteps
!step sizes
!steps
!moves
|-
!wa
(2.3)
!3-limit
dodecatonic
|P1
|m2
|M2
|m3
|M3
|P4
|A4
|P5
|m6
|M6
|m7
|M7
|P8
|m2637P415M2637A4
|3434-343-
34343
|4 3
L/s = 1.5
|5L 7s
or 12L
| +2, -5
|}
The obvious uneven scale is the [[Duodene|harmonic duodene]], with 3 fuzzy notes to avoid wolf 5ths. Note that the fuzzy major 2nd and minor 7th affect the step sizes. Both vM2-^m3 and vM6-^m7 are 5 edosteps.


{| class="wikitable center-all"
{| class="wikitable center-all"
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! colspan="13" |scale
! colspan="13" |scale
!as chains of 5ths
!as chains of 5ths
!edosteps
!step sizes
!step sizes
|-
|-
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|P8
|P8
|A4^m2637  m7P415M2  vM2637vA4
|A4^m2637  m7P415M2  vM2637vA4
|5 4 3 2
|4<u>34</u>2-4<u>43</u>-
42<u>43</u>4
|(5) 4 3 2
L/s = 2.5
L/s = 2.5
|}
|}
Is there an easily playable chromatic-sounding scale with nearly equal steps? See below.
For an even scale with small steps that's not awkward, see the next section.


=== Decatonic - "Ten is the new twelve" ===
=== Decatonic - the semitonal scale ===
Is there an easily playable chromatic-sounding scale with nearly equal steps? Imagine such a scale expressed in edosteps. To avoid awkward string-hopping, we need three odd numbers and the rest even. If the even number is 8, we get the near-equipentatonic scales, because one-eighth of 41 is about 5. If the even number is 6, we get the near-equiheptatonic scales, because one-sixth of 41 is about 7. The next even number is 4, which makes a decatonic scale.
Is there an easily playable chromatic-sounding scale with nearly equal steps? Imagine such a scale expressed in edosteps. To avoid awkward string-hopping, we need three odd numbers and the rest even. The even numbers should all be the same. The odd numbers should be 1 greater or 1 less. If the even number is 8, we get the near-equipentatonic scales, because one-eighth of 41 is about 5. If the even number is 6, we get the near-equiheptatonic scales, because one-sixth of 41 is about 7. The next even number is 4, which makes a decatonic scale. Thus the saying that on the Kite Guitar, "ten is the new twelve".
 
However, the term for these scales is not chromatic but '''semitonal''', because the steps are roughly the size of a 12edo semitone. Chromatic refers to movement by a single fret, see the next section.


The twin downminor scale consists of two downminor pentatonic scales, offset from each other by two frets. Mode #1 is (12:13:14:15:16:17:18)/12 plus (12:13:14:15:16)/8, except that prime 17 isn't well tuned.
The twin downminor scale consists of two downminor pentatonic scales, offset from each other by two frets. Mode #1 is (12:13:14:15:16:17:18)/12 plus (12:13:14:15:16)/8, except that prime 17 isn't well tuned.
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!name
!name
! colspan="11" |scale
! colspan="11" |scale
!as edosteps
!edosteps
!step sizes
!step sizes
!steps
!steps
!moves
!moves
|-
|-
! rowspan="2" |yalaza
! rowspan="2" |yazala
(2.3.5.7.11)
(2.3.5.7.11)
!twin downminor #1
!twin downminor #1
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|
|
|}
|}
The twin downminor scale works well for the blues. It lacks a M2, so over the V chord, shift the scale so that it's rooted on the 5th. Likewise shift the root to the 4th over the IV chord.
{| class="wikitable center-all"
!subgroup
!name
! colspan="11" |scale
!edosteps
!step sizes
!steps
!moves
|-
! rowspan="3" |yazala
(2.3.5.7.11)
!twin downminor on I
|P1
|~2
|vm3
|vM3
|(v)4
|d5
|P5
|~6
|vm7
|vM7
|P8
|544-<u>43</u>4-5444
| rowspan="3" |5 4 3
L/s = 1.67
| rowspan="3" |2L 7m 1s
or 2L 8s
or 10L
| rowspan="3" | +2, -4, -5
|-
!twin downminor on IV
|P1
|~2
|vm3
|vM3
|P4
|~5
|vm6
|vM6
|(v)m7
|vM7
|P8
|544-45-44<u>43</u>4
|-
!twin downminor on V
|P1
|m2
|M2
|~3
|v4
|d5
|P5
|~6
|vm7
|vM7
|P8
|345-444-5444
|}
=== Nineteen-tone - The chromatic scale ===
Let's continue the analysis started in the previous section. The next even number below 4 is 2. This implies many steps of a single fret, and three string-hopping steps of 3 edosteps. (1 edostep is just too small!) This makes a 19-note scale. There's not much to say about these scales. All the modes sound fairly similar. Using the full 19 note scale is somewhat overkill, unless your song is about bumblebees.
The steps are so small that they really can't be said to imply any prime subgroup. The L/s ratio is 1.5 and the steps are 3L 16s. The moves are +1 and -5. If there are 6 or 7 notes per string, it's a MOS scale of the [[Magic|Laquinyo]] temperament, which has a (P8, P12/5) [[pergen]]. If not, it's a MODMOS scale of Laquinyo. For example, the 2nd scale in the table is MODMOS because the large steps are not evenly distributed throughout the scale.
{| class="wikitable center-all"
!name
! colspan="20" |scale
!edosteps
|-
!chromatic
|P1
|vm2
|^m2
|(v)M2
|vm3
|^m3
|vM3
|^M3
|P4
|d5
|~5
|P5
|vm6
|^m6
|vM6
|vm7
|^m7
|vM7
|^M7
|P8
|22<u>32</u>22-22322-2223-2222
|-
!"
|P1
|vm2
|^m2
|vM2
|^M2
|^m3
|vM3
|^M3
|P4
|~4
|~5
|P5
|vm6
|^m6
|vM6
|^M6
|^m7
|vM7
|^M7
|P8
|222232-22232-2222-3222
|}
Although the full scale is a bit much, pieces of it are nice. Chromatic melodies with movements of a single fret are quite novel and pleasant. 59¢ is just barely large enough to sound like a 2nd and not a quartertone. Any of the above scales can be spiced up this way.
One can make a scale that's not very even, but still quite interesting, by using a smaller MOS, for example 13. This scale is 3 chromatic runs separated by 3 major 2nds.
{| class="wikitable center-all"
!name
! colspan="14" |scale
!edosteps
!step sizes
!steps
!moves
|-
!?
|P1
|vm2
|^m2
|^m3
|vM3
|^M3
|P4
|P5
|vm6
|^m6
|vM6
|^M6
|^M7
|P8
|2272-227-222272
|7 2, L/s = 3.5
|3L 10s
| +2, -3
|}
=== Microtonal scales ===
These use step sizes of 1 and 2 edosteps only. If there are only 3 small steps, it is a 22-note MOS or MODMOS of Laquinyo. With more than 3 small steps, they are quite awkward to play, with much string-hopping and fret-leaping.
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