Kite Guitar Scales: Difference between revisions

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If N goes into 41 X times with a remainder of Y, then the near-N-edo scale has steps YL and (N-Y)s, where L=X+1 and s=X. This near-N-edo scale is altered slightly so that there are only 3 odd numbers, and the rest are even. This avoids an awkward scale and also tends to make the intervals well tuned. For example, the unaltered whole-tone scale would have thirds of mostly 14/11 with some 5/4, but the the altered one has thirds of mostly 5/4 with some 9/7.   
If N goes into 41 X times with a remainder of Y, then the near-N-edo scale has steps YL and (N-Y)s, where L=X+1 and s=X. This near-N-edo scale is altered slightly so that there are only 3 odd numbers, and the rest are even. This avoids an awkward scale and also tends to make the intervals well tuned. For example, the unaltered whole-tone scale would have thirds of mostly 14/11 with some 5/4, but the the altered one has thirds of mostly 5/4 with some 9/7.   


The alteration is done so that it produces only 1 additional step size which is either 1 edostep larger than L or else 1 edostep smaller than s. If possible (and it often is), the alteration is done so that this new step size occurs only once. This is ideal because almost all steps are within the original L-to-s range, and the original (small) L/s ratio still describes the overall sound of the scale. If the new step size occurs more than once, the 3 step sizes are named L, m and s. If it only occurs once, the new step size is named either XL or xs, for extra large/small. The new step occurs more than once for near-edos 8 and 12-17, and not at all for near-edos 11 and 19.   
The alteration is done so that it produces only 1 additional step size which is either 1 edostep larger than L or else 1 edostep smaller than s. If possible (and it often is), the alteration is done so that this new step size occurs only once. This is ideal because almost all steps are within the original L-to-s range, and the original (small) L/s ratio still describes the overall sound of the scale. If the new step size occurs more than once, the 3 step sizes are named L, m and s. If it only occurs once, the new step size is named either XL or xs, for extra large/small. The new step is bolded in the table below. It occurs more than once for near-edos 8 and 12-17, and not at all for near-edos 11 and 19.   


{| class="wikitable"
{| class="wikitable"
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!4
!4
|11 10 9
|11 10 9
|2L 1s 1xs
|2L 1s '''1xs'''
|1.1
|1.1
|0
|0
| +5, -1, -2
| +5, -1, -2
|a dim7 tetrad
|a dim6 or dim7 tetrad
|-
|-
!5
!5
|9 8 7
|9 8 7
|2L 2s 1xs
|2L 2s '''1xs'''
|1.125
|1.125
|3-4
|3-4
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!6
!6
|8 7 6
|8 7 6
|1XL 3L 2s
|'''1XL''' 3L 2s
|1.17
|1.17
|0
|0
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!7
!7
|7 6 5
|7 6 5
|1XL 4L 2s
|'''1XL''' 4L 2s
|1.2
|1.2
|4-5
|4-5
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!8
!8
|6 5 4
|6 5 4
|3L 3m 2s
|3L 3m '''2s'''
|1.5
|1.5
|0
|0
| +3, +2, -4
| +3, +2, -4
|2 dim7 tetrads
|2 dim6/dim7 tetrads
|-
|-
!9
!9
|6 5 4
|6 5 4
|1XL 3L 5s
|'''1XL''' 3L 5s
|1.25
|1.25
|
|
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!10
!10
|5 4 3
|5 4 3
|2L 7s 1xs
|2L 7s '''1xs'''
|1.25
|1.25
|5-6
|5-6
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!12
!12
|4 3 2
|4 3 2
|7L 3m 2s
|7L 3m '''2s'''
|2.0
|2.0
|
|
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=== Tritonic and Tetratonic ===
=== Tritonic and Tetratonic ===
Tritonic scales are augmented triads, discussed here. Tetratonic scales are dim7 tetrads, discussed here.
Tritonic scales are augmented triads. The moves are -0 and --1, meaning same fret up 1 string, and up 1 fret up 1 string. Tetratonic scales are dim6/dim7 tetrads. Both augmented and dim6/dim7 chords are discussed on the [[Kite Guitar Chord Shapes (downmajor tuning)|chords page]].


=== Pentatonic ===
=== Pentatonic ===
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=== Octotonic ===
=== Octotonic ===
The 3rd step size occurs twice in these scales, making them less near-equal and less MOS-like than the other near-equal scales so far. Every scale contains a section of 2L 1m, making that section identical to an equi-heptatonic tetrachord. For example, the first scale in the table below has P5 ~6 ^m7 P8, as does the equi-minor scale. Every scale also contains a section of 2L 1s, making a down-4th, and thus an up-5th upon octave inversion. But this does not motivate fuzziness, because octotonic chords are naturally constructed from stacking "octa-thirds", i.e. using every other note of the scale. Chords avoid both perfect and off-perfect 5ths in favor of the dim 5th. The down-4th is likewise avoided.
The [[Bohlen-Pierce]] or B-P 13-edt scale is a non-octave scale that is contained in 41-edo. (Technically, the 41edo scale is 13-edt stretched by half a cent.) B-P has only one step size, 5 edosteps = ~2. These steps are very near to one-eighth of an octave, so it can be thought of as a near-8edo scale. Unfortunately B-P is very awkward to play on the Kite guitar. It also has the wolfy ^5 and v8 intervals.


Because of the prominence of the "octa-5th" (i.e. tritone) in octatonic chords, this interval plays a role analogous to the perfect 5th in other scales. Every octotonic scale contains eight tritones. The most consonant tritone is the dim 5th. Of course all eight tritones can't be dim 5ths without fuzziness, but half of them can be. In particular, the tonic chord can be a dim7 chord that contains two dim 5ths. The only two such chords that are playable are the ^dim7 and vdim7 chords. If we require that the remaining four notes of the scale make another such chord, there are only two near-equal octotonic scales. Each has two main modes, depending on which of the dim7 chords is considered to be the tonic chord.   
The 3rd step size (4 edosteps) occurs twice in these scales, making them less near-equal and less MOS-like than the other near-equal scales so far. Every scale contains a section of 2L 1m, making that section identical to an equi-heptatonic tetrachord. For example, the first scale in the table below has P5 ~6 ^m7 P8, as does the equi-minor scale. Every scale also contains a section of 2L 1s, making a down-4th, and thus an up-5th upon octave inversion. Every scale also contains 1L 2m, which makes another down-4th. Fortunately, octotonic chords are naturally constructed from stacking "octa-thirds", i.e. using every other note of the scale. Chords avoid both perfect and off-perfect 5ths in favor of the dim 5th. The down-4th is likewise avoided.
 
Because of the prominence of the "octa-5th" (i.e. tritone) in octatonic chords, this interval plays a role analogous to the perfect 5th in other scales. Every octotonic scale contains eight tritones. The most consonant tritone is the dim 5th = 7/5. Of course all eight tritones can't be dim 5ths without fuzziness, but half of them can be. In particular, the tonic chord can be a dim7 chord that contains two dim 5ths. The only two such chords that are playable are the ^dim7 and vdim7 chords. If we require that the remaining four notes of the scale make another such chord, there are only two near-equal octotonic scales. Each has two main modes, depending on which of the dim7 chords is considered to be the tonic chord.   
 
The scales are named after the root of the non-tonic dim7 chord. This chord is always upped or downed (^dim7 vs. vdim7) to match the root. If the tonic chord is upped or downed the opposite way, the two dim7 chords, and hence the entire scale, can easily be deduced from the name: the <u>up</u>flat-2 octotonic scale has an <u>up</u>dim7 chord on the ^bII and a <u>down</u>dim7 chord on the I. The octave inverse of ^b2 is vM7, thus the other main mode of the upflat-2 scale is the down-7 scale. If the tonic chord is upped or downed the same way, we must add that direction to the name: the up-3 up scale has an updim7 chord on ^III and an updim7 chord on I.   


{| class="wikitable center-all"
{| class="wikitable center-all"
Line 1,030: Line 1,034:
!moves
!moves
|-
|-
! rowspan="2" |yaza
! rowspan="4" |yaza
(2.3.5.7)
(2.3.5.7)
!???
!upflat-2
|P1
|P1
|^m2
|^m2
Line 1,044: Line 1,048:
|Ivdim7 + ^bII^dim7
|Ivdim7 + ^bII^dim7
|4565-4566
|4565-4566
| rowspan="2" |6 5 4
| rowspan="4" |6 5 4
L/s = 1.5
L/s = 1.5
| rowspan="2" |3L 3m 2s
| rowspan="4" |3L 3m 2s
or 8L
or 8L
| rowspan="2" | +3, +2, -4
| rowspan="4" | +3, +2, -4
|-
|-
!???
!down-7
|P1
|P1
|~2
|~2
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|P8
|P8
|I^dim7 + vVIIvdim7
|I^dim7 + vVIIvdim7
|5654-5664
|-
!
|P1
|^m2
|vm3
|^M3
|d5
|P5
|~6
|^m7
|P8
|I^dim6 + ^bII^dim7
|4565-4566
|-
!
|P1
|~2
|^m3
|v4
|d5
|^5
|M6
|vM7
|P8
|Ivdim6 + vVIIvdim7
|5654-5664
|5654-5664
|-
|-
! rowspan="2" |"
! rowspan="2" |"
!???
!down-2
|P1
|P1
|vM2
|vM2
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| rowspan="2" |"
| rowspan="2" |"
|-
|-
!???
!upflat-7
|P1
|P1
|~2
|~2
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|Ivdim7 + ^bVII^dim7
|Ivdim7 + ^bVII^dim7
|5456-5466
|5456-5466
|-
! rowspan="2" |"
!up-3 up
|P1
|~2
|^m3
|^M3
|d5
|vm6
|M6
|^m7
|P8
|I^dim7 + ^III^dim7
|5645-6546
| rowspan="2" |"
| rowspan="2" |"
| rowspan="2" |"
|-
!downflat-6 up
|P1
|~2
|^m3
|v4
|d5
|vm6
|M6
|vM7
|P8
|I^dim7 + vbVI^dim7
|5654-6564
|}
|}
=== Dodecatonic (twelve-tone) ===
=== 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, with an L/s ratio of at least 2.  
"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.  
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=== Decatonic - the semitonal scale ===
=== 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. 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".
Is there an easily playable chromatic-sounding scale with nearly equal steps? One such is the decatonic scale. 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 section on 19-tone scales.


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.
If the steps are nearly equal, it follows that every other note will make a nearly-equal pentatonic scale. Thus these scales consist of two intertwined za pentatonic scales which are (usually?) either both upmajor or both downminor. The scales are named after the "one" of the non-tonic scale, similar to how octotonic scales are named.


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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! rowspan="2" |yazala
! rowspan="2" |yazala
(2.3.5.7.11)
(2.3.5.7.11)
!twin downminor #1
!down-7 downminor
|P1
|P1
|~2
|~2
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| rowspan="2" | +2, -4, -5
| rowspan="2" | +2, -4, -5
|-
|-
!twin downminor #2
!upflat-2 downminor
|P1
|P1
|^m2
|^m2
Line 1,222: Line 1,283:
|-
|-
! rowspan="2" |"
! rowspan="2" |"
!twin upmajor
!down-7 upmajor
|P1
|P1
|m2
|m2
Line 1,285: Line 1,346:
|
|
|}
|}
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.
The down-7 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"
{| class="wikitable center-all"
!subgroup
!subgroup
Line 1,297: Line 1,358:
! rowspan="3" |yazala
! rowspan="3" |yazala
(2.3.5.7.11)
(2.3.5.7.11)
!twin downminor on I
!down-7 downminor on I
|P1
|P1
|~2
|~2
Line 1,318: Line 1,379:
| rowspan="3" | +2, -4, -5
| rowspan="3" | +2, -4, -5
|-
|-
!twin downminor on IV
!down-7 downminor on IV
|P1
|P1
|~2
|~2
Line 1,332: Line 1,393:
|544-45-44<u>43</u>4
|544-45-44<u>43</u>4
|-
|-
!twin downminor on V
!down-7 downminor on V
|P1
|P1
|m2
|m2
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=== Nineteen-tone - The chromatic scale ===
=== Nineteen-tone - The chromatic scale ===
Let's continue the analysis that starts 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, and there's not much reason to name them individually. Using the full 19 note scale is somewhat overkill, unless your song is about bumblebees.  
There's not much to say about these scales. All the modes sound fairly similar, and there's not much reason to name them individually. Using the full 19 note scale is somewhat overkill, unless your song is about bumblebees.  


The one-fret step implies several different ratios, and doesn't imply any particular prime subgroup. The step count is 3L 16s and the L/s ratio is 1.5. 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.
The one-fret step implies several different ratios, and doesn't imply any particular prime subgroup. The step count is 3L 16s and the L/s ratio is 1.5. 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.